Search

This paper determined the components of the generalized curvature tensor for the class of Kenmotsu type and established the mentioned class is {\eta}-Einstein manifold when the generalized curvature tensor is flat; the converse holds true under suitable conditions. It also introduced the notion of generalized {\Phi}-holomorphic sectional (G{\Phi}SH-) curvature tensor and thus found the necessary and sufficient conditions for the class of Kenmotsu type to be of constant G{\Phi}SH-curvature. In addition, the notion of {\Phi}-generalized semi-symmetric was introduced and its relationship with the class of Kenmotsu type and {\eta}-Einstein manifold established. Furthermore, this paper generalized the notion of the manifold of constant curvature and deduced its relationship with the aforementioned ideas. It finally showed that the class of Kenmotsu type exists as a hypersurface of the Hermitian manifold and derived a relation between the components of the Riemannian curvature tensors of the almost Hermitian manifold and its hypersurfaces.

In this paper, we analyze the effects of expansion on large scale structure formation in our Universe. We do that by incorporating a cosmological constant term in the gravitational partition function. This gravitational partition function with a cosmological constant is used for analyzing the thermodynamics of this system. We analyze the virial expansion for this system, and obtain its equation of state. It is observed that the generalization of this equation of state is like the Van der Waals equation. We also analyze a gravitational phase transition in this system using the mean field theory. We construct the cosmic energy equation for this system of galaxies, and discuss its consequences. We obtain and analyze the distribution function for this system, using the gravitational partition function. We also compare the results obtained in this paper with the observational data., Comment: 14 pages, 12 figures

We consider a structural model where the survival/default state is observed together with a noisy version of the firm value process. This assumption makes the model more realistic than most of the existing alternatives, but triggers important challenges related to the computation of conditional default probabilities. In order to deal with general diffusions as firm value process, we derive a numerical procedure based on the recursive quantization method to approximate it. Then, we investigate the error approximation induced by our procedure. Eventually, numerical tests are performed to evaluate the performance of the method, and an application is proposed to the pricing of CDS options.

Suppressing the reflection from material interfaces or entire layer stacks across an extended spectral range is important for various applications. While many approaches have been described for this purpose, we look at the problem here from the perspective of helicity preservation. Specifically, the anti-reflection performance of nano-particle arrays on top of solar cell stacks is related to two conditions: a high enough degree of discrete rotational symmetry and the ability of the system to suppress cross-talk between the two handednesses (helicities) of the electromagnetic field upon light-matter interaction. For particle-lattice systems with high enough degree of discrete rotational symmetry ($2\pi/n$ for $n>2$), our numerical studies link the suppression of backscattering to the ability of the system to avoid the mixing between the two helicity components of the incident field. In an exemplary design, we use an array of TiO$_2$ disks placed on top of a flat heterojunction solar cell stack, and analyze the helicity preservation properties of the system. We show numerically that a hexagonal array lattice, featuring a higher degree of discrete rotational symmetry, can improve over the anti-reflection performance of a square lattice. Importantly, the disks are introduced in an electrically decoupled manner such that the passivation and electric properties of the device are not disturbed., Comment: 23 pages, 4 figures

We present a Bloch modes' based Green's function scattering formalism for cost efficient forward modelling of disordered binary surface textures. The usage of Bloch modes of an unperturbed reference ordered system as ansatz allows our formalism to address surface scattering beyond the shallow amplitude regime. The main advantage of our formalism is the possibility to utilize a small amount of plane waves to represent the assumed Bloch modes thereby reducing computational costs, while still allowing one to estimate the scattering response to all channels accessible by the disordered system. Benchmarking calculations discussed in the paper demonstrate how the usage of our Bloch modes ansatz provides an excellent estimate of the scattering response over an important regime of disorder. As an example of our method's strength, we examine an electrically decoupled binary light trapping texture and demonstrate how introducing disorder may improve light incoupling into the considered solar cell structure., Comment: 12 pages, 9 figures, submitted to PRB

Observing that the recent developments of the recursive (product) quantization method induces a family of Markov chains which includes all standard discretization schemes of diffusions processes , we propose to compute a general error bound induced by the recursive quantization schemes using this generic markovian structure. Furthermore, we compute a marginal weak error for the recursive quantization. We also extend the recursive quantization method to the Euler scheme associated to diffusion processes with jumps, which still have this markovian structure, and we say how to compute the recursive quantization and the associated weights and transition weights.

The establishment of a rigorous theory on thermodynamics of light management in photovoltaics that accommodates various loss mechanisms as well as wave-optical effects in the absorption and reemission of light is at stake in this contribution. To this end, we propose a theoretical framework to calculate the open-circuit voltage enhancement resulting from photon recycling ($\Delta V^{\mathrm{PR}}_{\mathrm{oc}}$) with rigorous wave-optical treatment. It can be applied to both planar thin-film and nanostructured single-junction solar cells. We derive an explicit expression for $\Delta V^{\mathrm{PR}}_{\mathrm{oc}}$, which reveals its dependence on internal quantum luminescence efficiency, parasitic reabsorption, and on photon escape probabilities of reemmited photons. While the internal quantum luminescence efficiency is an intrinsic material property, both latter quantities can be determined rigorously for an arbitrary solar cell architecture by three-dimensional electrodynamic dipole emission calculations. We demonstrate the strengths and validity of our framework by determining the impact of photon recycling on the $V_{\mathrm{oc}}$ of a conventional planar organo-metal halide perovskite thin-film solar cell and compare it to established reference cases with perfect antireflection and Lambertian light scattering. Our calculations reveal $\Delta V^{\mathrm{PR}}_{\mathrm{oc}}$ values of up to 80 mV for the considered device stack in the absence of angular restriction and up to 240 mV when the escape cone above the cell is restricted to $\theta_{\mathrm{out}}=2.5^\circ$ around the cell normal. These improvements impose severe constraints on the parasitic absorption as a parasitic reabsorption probability of only 2\% reduces the $\Delta V^{\mathrm{PR}}_{\mathrm{oc}}$ to 100 mV for the same angular restriction. Our work here can be used to provide design guidelines., Comment: 12 pages, 10 figures, Accepted in PRB

We introduce a new approach to quantize the Euler scheme of an $\mathbb{R}^d$-valued diffusion process. This method is based on a Markovian and componentwise product quantization and allows us, from a numerical point of view, to speak of {\em fast online quantization} in dimension greater than one since the product quantization of the Euler scheme of the diffusion process and its companion weights and transition probabilities may be computed quite instantaneously. We show that the resulting quantization process is a Markov chain, then, we compute the associated companion weights and transition probabilities from (semi-) closed formulas. From the analytical point of view, we show that the induced quantization errors at the $k$-th discretization step $t_k$ is a cumulative of the marginal quantization error up to time $t_k$. Numerical experiments are performed for the pricing of a Basket call option, for the pricing of a European call option in a Heston model and for the approximation of the solution of backward stochastic differential equations to show the performances of the method.

We propose a new approach to quantize the marginals of the discrete Euler diffusion process. The method is built recursively and involves the conditional distribution of the marginals of the discrete Euler process. Analytically, the method raises several questions like the analysis of the induced quadratic quantization error between the marginals of the Euler process and the proposed quantizations. We show in particular that at every discretization step $t\_k$ of the Euler scheme, this error is bounded by the cumulative quantization errors induced by the Euler operator, from times $t\_0=0$ to time $t\_k$. For numerics, we restrict our analysis to the one dimensional setting and show how to compute the optimal grids using a Newton-Raphson algorithm. We then propose a closed formula for the companion weights and the transition probabilities associated to the proposed quantizations. This allows us to quantize in particular diffusion processes in local volatility models by reducing dramatically the computational complexity of the search of optimal quantizers while increasing their computational precision with respect to the algorithms commonly proposed in this framework. Numerical tests are carried out for the Brownian motion and for the pricing of European options in a local volatility model. A comparison with the Monte Carlo simulations shows that the proposed method may sometimes be more efficient (w.r.t. both computational precision and time complexity) than the Monte Carlo method., Comment: 29 pages

The model consists of a signal process $X$ which is a general Brownian diffusion process and an observation process $Y$, also a diffusion process, which is supposed to be correlated to the signal process. We suppose that the process $Y$ is observed from time 0 to $s>0$ at discrete times and aim to estimate, conditionally on these observations, the probability that the non-observed process $X$ crosses a fixed barrier after a given time $t>s$. We formulate this problem as a usual nonlinear filtering problem and use optimal quantization and Monte Carlo simulations techniques to estimate the involved quantities.

By tuning the radiative coupling of localized surface plasmons to diffracted orders, we demonstrate how stop-gaps in plasmonic crystals of nanorods may be opened and tuned. The stop-gap arises from the mutual coupling of surface lattice resonances (SLRs), which are collective resonances associated with counter-propagating surface polaritons. We present experimental results for three different nanorod arrays, where we show how the dispersion of SLRs can be controlled by modifying the size of the rods. Combining experiments with numerical simulations, we show how the properties of the stop-gap can be tailored by tuning a single structural factor. We find that the central frequency of the stop-gap falls quadratically, the frequency width of the stop-gap rises linearly, and the in-plane momentum width of the standing waves rises quadratically, as the width of the nanorods increases. These relationships hold for a broad range of nanorod widths, including duty cycles of the array between 20% to 80%. We discuss the physics in terms of a coupled oscillator analog, which relates the tunability of the stop-gaps to the coupling strength of plasmonic modes., Comment: 8 pages, 8 figures

We investigate in this paper an alternative method to simulation based recursive importance sampling procedure to estimate the optimal change of measure for Monte Carlo simulations. We propose an algorithm which combines (vector and functional) optimal quantization with Newton-Raphson zero search procedure. Our approach can be seen as a robust and automatic deterministic counterpart of recursive importance sampling by means of stochastic approximation algorithm which, in practice, may require tuning and a good knowledge of the payoff function in practice. Moreover, unlike recursive importance sampling procedures, the proposed methodology does not rely on simulations so it is quite generic and can come along on the top of Monte Carlo simulations. We first emphasize on the consistency of quantization for designing an importance sampling algorithm for both multi-dimensional distributions and diffusion processes. We show that the induced error on the optimal change of measure is controlled by the mean quantization error. We illustrate the effectiveness of our algorithm by pricing several options in a multi-dimensional and infinite dimensional framework.

We demonstrate the coupling of bright and dark Surface Lattice Resonances (SLRs), which are collective Fano resonances in 2D plasmonic crystals. As a result of this coupling, a frequency stop-gap in the dispersion relation of SLRs is observed. The different field symmetries of the low and high frequency SLR bands lead to pronounced differences in their coupling to free space radiation. Standing waves of very narrow spectral width compared to localized surface plasmon resonances are formed at the high frequency band edge, while subradiant damping onsets at the low frequency band edge leading the resonance into darkness. We introduce a coupled oscillator analog to the plasmonic crystal, which serves to elucidate the physics of the coupled plasmonic resonances and to estimate very high quality factors (Q>700) for SLRs, which are the highest known for any 2D plasmonic crystal., Comment: 13 pages (in preprint), 4 figures, 1 table

This paper is devoted to the pricing of Barrier options by optimal quadratic quantization method. From a known useful representation of the premium of barrier options one deduces an algorithm similar to one used to estimate nonlinear filter using quadratic optimal functional quantization. Some numerical tests are fulfilled in the Black-Scholes model and in a local volatility model and a comparison to the so called Brownian Bridge method is also done.

In this paper we use a hybrid Monte Carlo-Optimal quantization method to approximate the conditional survival probabilities of a firm, given a structural model for its credit defaul, under partial information. We consider the case when the firm's value is a non-observable stochastic process $(V_t)_{t \geq 0}$ and inverstors in the market have access to a process $(S_t)_{t \geq 0}$, whose value at each time t is related to $(V_s, s \leq t)$. We are interested in the computation of the conditional survival probabilities of the firm given the "investor information". As a application, we analyse the shape of the credit spread curve for zero coupon bonds in two examples., Comment: 22 pages

Let $P$ be a probability distribution on $\mathbb{R}^d$ (equipped with an Euclidean norm $|\cdot|$). Let $ r> 0 $ and let $(\alpha_n)_{n \geq1}$ be an (asymptotically) $L^r(P)$-optimal sequence of $n$-quantizers. We investigate the asymptotic behavior of the maximal radius sequence induced by the sequence $(\alpha_n)_{n \geq1}$ defined for every $n \geq1$ by $\rho(\alpha_n) = \max{|a|, a \in\alpha_n}$. When $\card(\supp(P))$ is infinite, the maximal radius sequence goes to $\sup{|x|, x \in\operatorname{supp}(P)}$ as $n$ goes to infinity. We then give the exact rate of convergence for two classes of distributions with unbounded support: distributions with hyper-exponential tails and distributions with polynomial tails. In the one-dimensional setting, a sharp rate and constant are provided for distributions with hyper-exponential tails., Comment: 31 pages

Let $ r, s>0 $. For a given probability measure $P$ on $\mathbb{R}^d$, let $(\alpha_n)_{n \geq 1}$ be a sequence of (asymptotically) $L^r(P)$- optimal quantizers. For all $\mu \in \mathbb{R}^d $ and for every $\theta >0$, one defines the sequence $(\alpha_n^{\theta, \mu})_{n \geq 1}$ by : $\forall n \geq 1, \alpha_n^{\theta, \mu} = \mu + \theta(\alpha_n - \mu) = \{\mu + \theta(a- \mu), a \in \alpha_n \} $. In this paper, we are interested in the asymptotics of the $L^s$-quantization error induced by the sequence $(\alpha_n^{\theta, \mu})_{n \geq 1}$. We show that for a wide family of distributions, the sequence $(\alpha_n^{\theta, \mu})_{n \geq 1}$ is $L^s$-rate-optimal. For the Gaussian and the exponential distributions, one shows how to choose the parameter $\theta$ such that $(\alpha_n^{\theta, \mu})_{n \geq 1}$ satisfies the empirical measure theorem and probably be asymptotically $L^s$-optimal., Comment: 26 pages