Sliding invariants and classification of singular holomorphic foliations in the plane

By introducing a new invariant called the set of slidings, we give a complete strict classification of the class of germs of non-dicritical holomorphic foliations in the plan whose Camacho-Sad indices are not rational. Moreover, we will show that, in this class, the new invariant is finitely determined. Consequently, the finite determination of the class of isoholonomic non-dicritical foliations and absolutely dicritical foliations that have the same Dulac maps are proved.Comment: 21 pages, 2 figur

Lorentz Transformation in Flat 5D Complex-Hyperbolic Space

The Lorentz transfomation is derived in 5D flat pseudo-complex affine space or TT Space. The TT space or pseudo-Complex space accomodates one uncompactified time-like extra dimension. It is shown that the maximum allowable speed for particles living in TT space exceeds the speed of light, c, the absolute speed of the Minkowski space.Comment: Removal of non-alpha numeric characters from the title and abstrac

Extra-Dimensional Approach to Option Pricing and Stochastic Volatility

The generalized 5D Black-Scholes differential equation with stochastic volatility is derived. The projections of the stochastic evolutions associated with the random variables from an enlarged space or superspace onto an ordinary space can be achieved via higher-dimensional operators. The stochastic nature of the securities and volatility associated with the 3D Merton-Garman equation can then be interpreted as the effects of the extra dimensions. We showed that the Merton-Garman equation is the first excited state, i.e. n=m=1, within a family which contain an infinite numbers of Merton-Garman-like equations.Comment: Ease the time-independent restriction on the extra dimensional coordinates. Fixed typos and expand the conclusio

Commuting foliations

The aim of this paper is to extend the notion of commutativity of vector fields to the category of singular foliations, using Nambu structures, i.e. integrable multi-vector fields. We will classify the relationship between singular foliations and Nambu structures, and show some basic results about commuting Nambu structures.Comment: New version, with a completely new section which clarifies the relationship between singular foliations and Nambu structures. The size of the paper has doubled from 10 to 20 page

Non-classical properties and generation schemes of superposition of multiple-photon-added two-mode squeezed vacuum state

In this paper, we study some non-classical properties and propose the generation schemes of the superposition of multiple-photon-added two-mode squeezed vacuum state (SMPA-TMSVS). Based on the   Wigner function, we clarify that this state is a non-Gaussian state, while the original two-mode squeezed vacuum state (TMSVS) is a Gaussian state. Besides, the SMPA-TMSVS is sum squeezing, as well as difference squeezing. In particular, the manifestation of the sum squeezing and the difference squeezing in the SMPA-TMSVS becomes more pronounced when increasing parameters r and e. In addition, by exploiting the schemes of photon-added superposition in the usual order, we give some schemes that the SMPA-TMSVS can be generated with the higher-order photon-added superposition by using some optical devices

Hexa-aryl/alkylsubstituted Cyclopropanes

A series of penta-aryl/alkyl-1-(toluene-4-sulfonyl)-4,5-dihydro-1H-pyrazole 5a-c was synthesized by addition of methyllithium or phenylllithium followed by trapping the nitrogen anion intermediate with tosyl-fluoride to cyclic azines 2a,b. Addition of methyllithium or phenyllithium to 5a-c generated a series of hexa-aryl/alkylsubstituted-4,5-dihydro-3H-pyrazoles 6a-c. Neat thermolysis of hexa-aryl/alkylsubstituted-4,5-dihydro-3H-pyrazoles 6a-c at 200◦C produced hexa-aryl/alkylsubstituted cyclopropanes 7a-c in high yield

Theory and numerical modeling of photonic resonances: Quasinormal Modal Expansion -- Applications in Electromagnetics

The idea of the modal expansion in electromagnetics is derived from the research on electromagnetic resonators, which play an essential role in developments in nanophotonics. All of the electromagnetic resonators share a common property: they possess a discrete set of special frequencies that show up as peaks in scattering spectra and are called resonant modes. These resonant modes are soon recognized to dictate the interaction between electromagnetic resonators and light. This leads to a hypothesis that the optical response of resonators is the synthesis of the excitation of each physical-resonance-state in the system: Under the excitation of external pulses, these resonant modes are initially loaded, then release their energy which contributes to the total optical responses of the resonators. These resonant modes with complex frequencies are known in the literature as the Quasi-Normal Mode (QNM). Mathematically, these QNMs correspond to solutions of the eigenvalue problem of source-free Maxwell's equations. In the case where the optical structure of resonators is unbounded and the media are dispersive (and possibly anisotropic and non-reciprocal) this requires solving non-linear (in frequency) and non-Hermitian eigenvalue problems. Thus, the whole problem boils down to the study of the spectral theory for electromagnetic Maxwell operators. As a result, modal expansion formalisms have recently received a lot of attention in photonics because of their capabilities to model the physical properties in the natural resonance-state basis of the considered system, leading to a transparent interpretation of the numerical results. This manuscript is intended to extend the study of QNM expansion formalism, in particular, and nonlinear spectral theory, in general. At the same time, several numerical modelings are provided as examples for the application of modal expansion in computations.Comment: PhD thesi