Complex Numbers for Relativistic Operations

When presenting special relativity, it is customary to single out the so-called paradoxes. One of these paradoxes is the formal occurrence of speeds exceeding the speed of light. An essential part of such paradoxes arises from the incompleteness of the relativistic calculus of velocities. In special relativity, the additive group is used for velocities. However, the use of only group operations imposes artificial restrictions on possible computations. Naive expansion to vector space is usually done by using non-relativistic operations. We propose to consider arithmetic operations in the special theory of relativity in the framework of the Cayley–Klein model for projective spaces. We show that such paradoxes do not arise in the framework of the proposed relativistic extension of algebraic operations.

Pricing Vulnerable Options in the Bifractional Brownian Environment with Jumps

In this paper, we study the valuation of European vulnerable options where the underlying asset price and the firm value of the counterparty both follow the bifractional Brownian motion with jumps, respectively. We assume that default event occurs when the firm value of the counterparty is less than the default boundary. By using the actuarial approach, analytic formulae for pricing the European vulnerable options are derived. The proposed pricing model contains many existing models such as Black–Scholes model (1973), Merton jump-diffusion model (1976), Klein model (1996), and Tian et al. model (2014).

Temporal Klein Model for Particle-Pair Creation

This work considers the creation of electron-positron pairs from intense electric fields in vacuum, for arbitrary temporal field variations. These processes can be useful to study quantum vacuum effects with ultra-intense lasers. We use the quantized Dirac field to explore the temporal Klein model. This model is based on a vector potential discontinuity in time, in contrast with the traditional model based on a scalar potential discontinuity in space. We also extend the model by introducing a finite time-scale for potential variations. This allows us to study the transition from a singular electric field spike, with infinitesimal duration, to the opposite case of a static field where the Schwinger formula would apply. The present results are intrinsically non-perturbative. Explicit expressions for pair-creation as a function of the potential time-scales are derived. This work explores the spacetime symmetry associated with pair creation in vacuum: the space symmetry breaking of the old Klein paradox model, in contrast with the time symmetry breaking of the temporal Klein model.

Noncompact CPN as a phase space of superintegrable systems

We propose the description of superintegrable models with dynamical [Formula: see text] symmetry, and of the generic superintegrable deformations of oscillator and Coulomb systems in terms of higher-dimensional Klein model (the noncompact analog of complex projective space) playing the role of phase space. We present the expressions of the constants of motion of these systems via Killing potentials defining the [Formula: see text] isometries of the Kähler structure.

Hype-HAN: Hyperbolic Hierarchical Attention Network for Semantic Embedding

Hyperbolic space is a well-defined space with constant negative curvature. Recent research demonstrates its odds of capturing complex hierarchical structures with its exceptional high capacity and continuous tree-like properties. This paper bridges hyperbolic space's superiority to the power-law structure of documents by introducing a hyperbolic neural network architecture named Hyperbolic Hierarchical Attention Network (Hype-HAN). Hype-HAN defines three levels of embeddings (word/sentence/document) and two layers of hyperbolic attention mechanism (word-to-sentence/sentence-to-document) on Riemannian geometries of the Lorentz model, Klein model and Poincaré model. Situated on the evolving embedding spaces, we utilize both conventional GRUs (Gated Recurrent Units) and hyperbolic GRUs with Möbius operations. Hype-HAN is applied to large scale datasets. The empirical experiments show the effectiveness of our method.

Reflection-Like Maps in High-Dimensional Euclidean Space

In this paper, we introduce reflection-like maps in n-dimensional Euclidean spaces, which are affinely conjugated to θ : ( x 1 , x 2 , … , x n ) → 1 x 1 , x 2 x 1 , … , x n x 1 . We shall prove that reflection-like maps are line-to-line, cross ratios preserving on lines and quadrics preserving. The goal of this article was to consider the rigidity of line-to-line maps on the local domain of R n by using reflection-like maps. We mainly prove that a line-to-line map η on any convex domain satisfying η ∘ 2 = i d and fixing any points in a super-plane is a reflection or a reflection-like map. By considering the hyperbolic isometry in the Klein Model, we also prove that any line-to-line bijection f : D n ↦ D n is either an orthogonal transformation, or a composition of an orthogonal transformation and a reflection-like map, from which we can find that reflection-like maps are important elements and instruments to consider the rigidity of line-to-line maps.

Evaluating the Performance of Islamic Banks Using a Modified Monti-Klein Model

The development of Islamic banking continues to increase in many Muslim (majority) countries. Substituting interest with profit shares in the assets of a given Islamic bank as one of the bases of operation has many interesting implications, one of which is the need for more involved risk and return measures. In this paper, we take a balance sheet analysis-based approach to formulating profit in order to assess the performance of an Islamic bank. Then the implementation of this approach is demonstrated using data provided by Indonesia’s financial services authority, known as the OJK. We develop formulae for the calculation of profit share between funding and financing funds as well as the appropriate rates of return. The resulting figures are then used to construct statistical models for short-term forecasting of the volumes of funding fund from the depositors and financing fund for business people who need funds for their investment projects. The approach we develop is innovative for Islamic banks and would be a welcome addition to their performance assessment toolkit. One of the results of our model indicates an increasing pattern on the equivalent rates of returns for funding and financing funds every year, which is caused by the fact that the reported income from the financing fund seems to have been accumulated from the beginning until the end of year in the Islamic bank.

Nonlinear Non-Equilibrium Thermodynamics Based on the Ehrenfest–Klein Model

Nonlinear non-equilibrium thermodynamic relations have been constructed based on the generalized Ehrenfest–Klein model. Using these relations, the behavior of the entropy and its production in time at arbitrary deviations from equilibrium has been studied. It has been shown that the transient fluctuation theorem is valid for this model if a dissipation functional is treated as the thermodynamic entropy production.

Evaluation of Curriculum Localization Approach in Master's Degree of Curriculum Planning Filed Based on Klein Model

The results of evaluated elements showed that faculty members of Tehran state university ranked elements learning activities, teaching methods, time, materials and resources, grouping, place, and evaluation at a relatively desirable level and target and content elements as undesirable level.