Minkowski Metric Research Articles

Weighted consensus clustering and its application to Big data

The aim of this study is the development of a weighted consensus clustering that assigns weights to single clustering methods using the purity utility function. In the case of Big data that does not contain labels, the utility function based on the Davies-Bouldin index is proposed in this paper. The Banknote authentication, Phishing, Diabetic, Magic04, Credit card clients, Covertype, Phone accelerometer, and NSL-KDD datasets are used to assess the efficiency of the proposed consensus approach. The proposed approach is evaluated using the Euclidean, Minkowski, squared Euclidean, cosine, and Chebychev distance metrics. It is compared with single clustering algorithms (DBSCAN, OPTICS, CLARANS, k-means, and shared nearby neighbor clustering). The experimental results show the effectiveness of the proposed approach to the Big data clustering in comparison to single clustering methods. The proposed weighted consensus clustering using the squared Euclidean distance metric achieves the highest accuracy, which is a very promising result for Big data clustering. It can be applied to expert systems to help experts make group decisions based on several alternatives. The paper also provides directions for future research on consensus clustering in this area.

Determining the Rating of Ukrainian Banks on the Risk of Legalization of Illegally Obtained Income

The article examines a scientific and methodological approach to the rating of banks on the risk of money laundering. A sample of 18 indicators of 65 Ukrainian banks in 2019 was selected. The relative indicators that characterize the risk of using the bank's operations to legalize illegal income are considered. Logically, the indicators are divided into three parts. The first group of indicators characterizes the quantity and quality of the bank's compliance with the current legislation of Ukraine in the field of financial monitoring. The second group of indicators reflects the size of cash turnover in the bank, which is a characteristic of the bank's participation as a conversion center. The third group of indicators characterizes the bank's involvement in international income laundering cycles, considering transactions in countries - offshore zones and dubious transactions without explicit confirmation by a foreign trade contract. The study of input data on multicollinearity was carried out, based on which 5 indicators that are collinear with others were excluded. Normalization of the input data set based on nonlinear normalization is carried out. The weights of each indicator are calculated based on the principal component’s method. The optimal number of factors was determined based on the percentage of the variance explained by each factor and the graph of the scree plot. Minkowski metric was used to construct the integral index. Based on the integrated indicator, the rating of banks on the risk of money laundering was formed. The verbal-numerical Harrington scale provided a qualitative characterization of the risk of using bank operations to legalize illicit income. MS Office Excel software and Correlations of the statistical package STATISTICA 10 were used for calculations. Keywords: Anti-money laundering, the rating of banks, on the risk of legalization, integral indicator, Minkowski metric.

ОЦІНЮВАННЯ РИЗИКУ ВИКОРИСТАННЯ БАНКІВ З МЕТОЮ ЛЕГАЛІЗАЦІЇ КРИМІНАЛЬНИХ ДОХОДІВ НА ОСНОВІ ГРАВІТАЦІЙНОГО МОДЕЛЮВАННЯ

The article is stressed on a method for assessing the risk of using banks for money laundering based on gravity modeling. Stimulatory factors are reduced to a comparable form by applying relative normalization. The priority of factors is determined using the principal components method. It is determined an integration indicator of a quantitative assessment of a country's rating concerning the characteristics of determining the level of money laundering risk by using Minkowski's metrics. It is built an economic-mathematical model for estimating the risk of money laundering based on the equation of the law of gravitational gravity and gravitational force in social phenomena. It is proved the expediency of application of the developed methodology in the decision of actual questions connected with reduction of risks for the country from the side of money laundering which serves as a basis for perfection of economic policy standards of the state concerning the strengthening of national economic security.

A Mathematical Comparison of the Schwarzschild and Kerr Metrics

A few physicists have recently constructed the generating compatibility conditions (CC) of the Killing operator for the Minkowski (M) , Schwarzschild (S) and Kerr (K) metrics. They discovered second order CC, well known for M, but also third order CC for S and K. In a recent paper, we have studied the cases of M and S, without using specific technical tools such as Teukolski scalars or Killing-Yano tensors. However, even if S($m$) and K($m,a$) are depending on constant parameters in such a way that S $\rightarrow $ M when $m \rightarrow 0$ and K $\rightarrow$ S when $ a \rightarrow 0$, the CC of S do not provide the CC of M when $m \rightarrow 0$ while the CC of K do not provide the CC of S when $a\rightarrow 0$. In this paper, using tricky motivating examples of operators with constant or variable parameters, we explain why the CC are depending on the choice of the parameters. In particular, the only purely intrinsic objects that can be defined, namely the extension modules, may change drastically. As the algebroid bracket is compatible with the {\it prolongation/projection} (PP) procedure, we provide for the first time all the CC for K in an intrinsic way, showing that they only depend on the underlying Killing algebras and that the role played by the Spencer operator is crucial. We get K$<$S$<$M with $2 < 4 < 10$ for the Killing algebras and explain why the formal search of the CC for M, S or K are strikingly different, even though each Spencer sequence is isomorphic to the tensor product of the Poincar\'{e} sequence for the exterior derivative by the corresponding Lie algebra.

Magic identities for the conformal four-point integrals; the Minkowski metric case

The original “magic identities” are due to J. M. Drummond, J. Henn, V. A. Smirnov and E. Sokatchev; they assert that all n-loop box integrals for four scalar massless particles are equal to each other [3]. The authors give a proof of the magic identities for the Euclidean metric case only and claim that the result is also true in the Minkowski metric. However, the Minkowski case is much more subtle and requires specification of the relative positions of cycles of integration to make these identities correct. In this article we prove the magic identities in the Minkowski metric case and, in particular, specify the cycles of integration.Our proof of magic identities relies on previous results from [7,8], where we give a mathematical interpretation of the n-loop box integrals in the context of representations of a Lie group U(2,2) and quaternionic analysis. The main result of [7,8] is a (weaker) operator version of the “magic identities”.No prior knowledge of physics or Feynman diagrams is assumed from the reader. We provide a summary of all relevant results from quaternionic analysis to make the article self-contained.

Isoparametric hypersurfaces of a class of Finsler manifolds induced by navigation problem in Minkowski spaces

In this paper, we mainly discuss the isoparametric hypersurfaces of a class of Finsler manifolds (M˜,F˜) produced by navigation datum (F,v), where F is a Minkowski metric. By studying the relationship between the principal curvatures with respect to F˜ and F, we get a general Cartan-type identity and a complete classification of isoparametric hypersurfaces in (M˜,F˜(F,v)). Moreover, for the case v=2cx, when (M˜,F˜) is a Funk-type space with negative constant flag curvature −c2, we also get a complete classification of dμBH-isoparametric hypersurfaces.

Mimetic f(T) teleparallel gravity and cosmology

We formulate mimetic theory in f(T) teleparallel gravity where T is torsion scalar. It is shown that the construction of the mimetic theory in the teleparallel gravity requires the vierbeins to be left unchanged and the confomal transformation performed on the Minkowski metric of the tangent space. It is further argued that the conformal degree of freedom in this teleparallel mimetic theory becomes dynamical and mimics the behavior of cold dark matter. We also show that it is possible to employ the Lagrange multipliers method to formulate the mimetic theory in an f(T) theory without any auxiliary metric. The mimetic f(T) theory is examined by the method of dynamical system and it is found that there are five fixed points representing inflation, radiation, matter, mimetic dark matter and dark energy dominated eras in the theory if some conditions are satisfied. We examined the power-law model and its phase trajectories with these conditions.

A theory of special relativity based on four-displacement of particles instead of Minkowski four-position

All Lorentz four-vectors of Special Relativity (SR) are derived from a basic Lorentz four-position in a Minkowski space. This article explores use of a Lorentz four-displacement, describing translatory motion of particles in a 4-dimensional space irrespective of position, as a basic four-vector. Derivations based on this assumption have striking outcomes and impact: All four-vectors of SR associated with frame invariants are intact and valid. All postulates and predictions of SR are intact and valid, with two exceptions - Length contraction in the boost direction and relativity of simultaneity are both obviated. Notably, these two aspects of SR have never been definitively proven in experiments and can therefore be challenged. We propose an adapted SR theory, based on four-displacement of particles, predicting the following revolutionary and far-reaching consequences: 1. 4D-space is absolute and Euclidean relative to which all particles, mass and massless, are never at rest and perform an equal distance of translatory displacement at a universal displacement rate equivalent to the speed of light. Inertial frames of 3D-space+time remain relative. 2. What appear as Lorentz four-vectors in a Minkowski metric of a relative spacetime are actually Lorentz four-vectors in an absolute Euclidian 4D-space. 3. Special Relativity actually emerges from describing properties of particles in 4 momentum-space, irrespective of position and localization in space. In this form, SR achieves inherent compatibility with Quantum theories, reflecting the particle-wave duality of particle fields. 4. Time is a property of mass particles associated with their displacement in 4D-space, as opposed to a Minkowski-space property. 5. Frames of 3D-space+time represent mixed domain coordinates: 3 spatial coordinates of position-space and a temporal coordinate of momentum-space, unlike the position-space status of 4 spacetime coordinates in conventional SR.

Acoustic black holes in curved spacetime and the emergence of analogue Minkowski spacetime

Gravity is not only able to be mimicked in flat spacetimes, but also in curved spacetimes. We study analogue gravity models in curved spacetime by considering the relativistic Gross-Pitaevskii theory and Yang-Mills theory in the fixed background spacetime geometry. The results show that acoustic metrics can be emergent from curved spacetimes yielding a Hadamard product of a real metric-tensor and an analogue metric-tensor. Taking \emph{quantum vortices} as \emph{test particles}, we evaluate their released energy ratio during the "gravitational binding". The $2+1$-dimensional flat Minkowski metric is derived from the $3+1$-dimensional Anti-de Sitter space by considering perturbations of the Yang-Mills field, which implies that Minkowski spacetime can be also simulated and the derivations presented here have some deep connections with the holographic principle.

On conformally flat fourth root (α,β)-metrics

The theory of 4-th root metrics plays an important role in Biology, General Relativity and Seismic Ray Theory. Using the Shen's sy-separating trick, we prove that every conformally flat weakly Einstein 4-th root (α,β)-metric on a manifold M of dimension n≥3 is either a Riemannian metric or a locally Minkowski metric. Then, we show that every conformally flat 4-th root (α,β)-metric of almost vanishing Ξ-curvature on a manifold M of dimension n≥3 reduces to a Riemannian metric or a locally Minkowski metric.

The Innovative Approach to Increasing Cybersecurity of Transactions Through Counteraction to Money Laundering

A current task is to provide the economic security of any country in the context of creating effective and reliable measures of the banking cybersecurity system against money laundering. First of all, it relates to the fact that the money laundering processes and financing of terrorism negatively influence the economy of any country and reduce the economic security level. Secondly, the high level of money laundering in the country promotes the emergence of such negative processes as corruption, extortion, drug production, people smuggling, gangsterism, terrorism, which leads to an increase of the crime situation in the country and endangers the lives of the population. Thirdly, the existing cybersecurity measures of banks do not affect the security of transactions in a timely manner to identify funds obtained illegally. Therefore, exactly this aspect is required changes and modernization in order to accomplish the task. The aim of the article is to develop the innovative scientific and methodic approach to the country’s attractiveness modelling for proceeds laundering by other countries. This technique is one of the tools of the bank’s cybersecurity system for making further decisions regarding the risk of legalization. In order to solve this problem authors suggest the approach, which is based on gravity modelling. Eight factors: Gross Domestic Product per capita, Claims on the central government, Internally displaced persons, associated with conflict and violence; Automatic Exchange of Information; Corruption Perceptions Index; Global Terrorism Index; Legatum Prosperity Index; Happy Planet Index are proposed to be evaluated using the expert approach to implement the above approach. Then the integral indicator is calculated using the Minkowski metric and taking into account the factors normalization. Using the gravity model, the country’s attractiveness degree considering is defined for proceeds laundering on the part of another country. Data for 105 countries are used for calculation and results for Ukraine, Poland and Germany are represented. As a result, we can see that developed countries with high welfare level are attractive for developing countries for money laundering, countries with low welfare level, low economic development and unstable political situation are attractive for the developed countries. The proposed methodology is recommended to be introduced in the activity of banks' cybersecurity systems. It will allow identifying transactions of those countries for which the risk of legalization is high and introduce additional monitoring to regard the legitimacy of their financial sources. In addition, it is expedient to use the model in the activities of the country’s regulatory authorities, which will promote the introduction of cybersecurity standards and increase the ethics of financial relations between countries. Keywords: cybersecurity, money laundering, gravity modelling, country attractiveness, risk, expert approach, normalization, Minkowski metric.

Generating Compatibility Conditions and General Relativity

The search for the generating compatibility conditions (CC) of a given operator is a very recent problem met in general relativity in order to study the Killing operator for various standard useful metrics. Accordingly, this paper can be considered as a natural continuation of a previous paper recently published in JMP under the title Minkowski, Schwarschild and Kerr metrics revisited. In particular, we prove that the intrinsic link existing between the lack of formal exactness of an operator sequence on the jet level, the lack of formal exactness of its corresponding symbol sequence and the lack of formal integrability (FI) of the initial operator is of a purely homological nature as it is based on the long exact connecting sequence provided by the so-called snake lemma in homological algebra. It is therefore quite difficult to grasp it in general and even more difficult to use it on explicit examples. It does not seem that any one of the results presented in this paper is known as most of the other authors who studied the above problem of computing the total number of generating CC are confusing this number with the degree of generality introduced by A. Einstein in his 1930 letters to E. Cartan. One of the motivating examples that we provide is so striking that it is even difficult to imagine that such an example could exist. We hope this paper could be used as a source of testing examples for future applications of computer algebra in general relativity and, more generally, in mathematical physics.

Sector models—a toolkit for teaching general relativity: III. Spacetime geodesics

Sector models permit a model-based approach to the general theory of relativity. The approach has its focus on the geometric concepts and uses no more than elementary mathematics. This contribution shows how to construct the paths of light and free particles on a spacetime sector model. Radial paths close to a black hole are used by way of example. We outline two workshops on gravitational redshift and on vertical free fall, respectively, that we teach for undergraduate students. The workshop on redshift does not require knowledge of special relativity; the workshop on particles in free fall presumes familiarity with the Lorentz transformation. The contribution also describes a simplified calculation of the spacetime sector model that students can carry out on their own if they are familiar with the Minkowski metric. The teaching materials presented in this paper are available online for teaching purposes at https://www.spacetimetravel.org.

Chirality in gravitational and electromagnetic interactions with matter

It has been suggested that single and double jets observed emanating from certain astrophysical objects may have a purely gravitational origin. We discuss new classes of pulsed gravitational wave solutions to the equation for perturbations of Ricci-flat spacetimes around Minkowski metrics, as models for the genesis of such phenomena. We discuss how these solutions are motivated by the analytic structure of spatially compact finite energy pulse solutions of the source-free Maxwell equations generated from complex chiral eigen-modes of a chirality operator. Complex gravitational pulse solutions to the linearized source-free Einstein equations are classified in terms of their chirality and generate a family of non-stationary real spacetime metrics. Particular members of these families are used as backgrounds in analyzing time-like solutions to the geodesic equation for test particles. They are found numerically to exhibit both single and double jet-like features with dimensionless aspect ratios suggesting that it may be profitable to include such backgrounds in simulations of astrophysical jet dynamics from rotating accretion discs involving electromagnetic fields.

Deep-learned Top Tagging with a Lorentz Layer

We introduce a new and highly efficient tagger for hadronically decaying top quarks, based on a deep neural network working with Lorentz vectors and the Minkowski metric. With its novel machine learning setup and architecture it allows us to identify boosted top quarks not only from calorimeter towers, but also including tracking information. We show how the performance of our tagger compares with QCD-inspired and image-recognition approaches and find that it significantly increases the performance for strongly boosted top quarks.

On strongly convex projectively flat and dually flat complex Finsler metrics

In this paper, we prove that a strongly convex complex Finsler metric F on a domain $$D\subset \mathbb {C}^n$$ is projectively flat (resp. dually flat) if and only if F comes from a strongly convex complex Minkowski metric.

Representation of the Minkowski metric as a fuzzy set

This paper proposes a representation of the family of Minkowski distances using fuzzy sets. The proposed method helps to represent human-like perceptions about distances, which can help decision making in presence of non-probabilistic uncertainties such as imprecision and ambiguity. This way we propose to define a fuzzy set regarding the concept of closeness of two elements/sets measured by a Minkowski metric. Two application examples are presented, solved, and compared to some classical approaches. Finally some concluding remarks are provided and some interpretation issues are explained.

Symplectic Wick rotations between moduli spaces of 3-manifolds

Given a closed hyperbolic surface $S$, let $\cQF$ denote the space of quasifuchsian hyperbolic metrics on $S\times\R$ and $\cGH_{-1}$ the space of maximal globally hyperbolic anti-de Sitter metrics on $S\times\R$. We describe natural maps between (parts of) $\cQF$ and $\cGH_{-1}$, called "Wick rotations", defined in terms of special surfaces (e.g. minimal/maximal surfaces, CMC surfaces, pleated surfaces) and prove that these maps are at least $C^1$ smooth and symplectic with respect to the canonical symplectic structures on both $\cQF$ and $\cGH_{-1}$. Similar results involving the spaces of globally hyperbolic de Sitter and Minkowski metrics are also described. These 3-dimensional results are shown to be equivalent to purely 2-dimensional ones. Namely, consider the double harmonic map $\cH:T^*\cT\to\cTT$, sending a conformal structure $c$ and a holomorphic quadratic differential $q$ on $S$ to the pair of hyperbolic metrics $(m_L,m_R)$ such that the harmonic maps isotopic to the identity from $(S,c)$ to $(S,m_L)$ and to $(S,m_R)$ have, respectively, Hopf differentials equal to $i q$ and $-i q$, and the double earthquake map $\cE:\cT\times\cML\to\cTT$, sending a hyperbolic metric $m$ and a measured lamination $l$ on $S$ to the pair $(E_L(m,l), E_R(m,l))$, where $E_L$ and $E_R$ denote the left and right earthquakes. We describe how such 2-dimensional double maps are related to 3-dimensional Wick rotations and prove that they are also $C^1$ smooth and symplectic.

Boundedness of the total energy of relativistic membranes evolving in a curved spacetime

We establish a global existence theory for the equation governing the evolution of a relativistic membrane in a (curved) background spacetime, when the spacetime metric is a perturbation of the Minkowski metric. Relying on a hyperboloidal foliation and generalizing an observation made by LeFloch and Ma, we revisit a theorem established by Lindblad (who treated membranes in the flat Minkowski spacetime) and provide a simpler proof of existence. Our result is valid in a curved spacetime and, most importantly, establishes the important property that the total energy of the membrane is globally bounded in time.

The zero mass limit of Kerr and Kerr-(anti-)de-Sitter space-times: exact solutions and wormholes

Heun-type exact solutions emerge for both the radial and the angular equations for the case of a scalar particle coupled to the zero mass limit of both the Kerr and Kerr-(anti)de-Sitter spacetime. Since any type D metric has Heun-type solutions, it is interesting that this property is retained in the zero mass case. This work further refutes the claims that $M$ going to zero limit of the Kerr metric is both locally and globally the same as the Minkowski metric.