Adaptive estimation of the density matrix in quantum homodyne tomography with noisy data

In the framework of noisy quantum homodyne tomography with efficiency parameter $1/2 < \eta \leq 1$, we propose a novel estimator of a quantum state whose density matrix elements $\rho_{m,n}$ decrease like $Ce^{-B(m+n)^{r/ 2}}$, for fixed $C\geq 1$, $B>0$ and $0<r\leq 2$. On the contrary to previous works, we focus on the case where $r$, $C$ and $B$ are unknown. The procedure estimates the matrix coefficients by a projection method on the pattern functions, and then by soft-thresholding the estimated coefficients. We prove that under the $\mathbb{L}_2$ -loss our procedure is adaptive rate-optimal, in the sense that it achieves the same rate of conversgence as the best possible procedure relying on the knowledge of $(r,B,C)$. Finite sample behaviour of our adaptive procedure are explored through numerical experiments

B-Splines Based Finite Difference Schemes For Fractional Partial Differential Equations

Fractional partial differential equations (FPDEs) are considered to be the extended formulation of classical partial differential equations (PDEs). Several physical models in certain fields of sciences and engineering are more appropriately formulated in the form of FPDEs. FPDEs in general, do not have exact analytical solutions. Thus, the need to develop new numerical methods for the solutions of space and time FPDEs. This research focuses on the development of new numerical methods. Two methods based on B-splines are developed to solve linear and non-linear FPDEs. The methods are extended cubic B-spline approximation (ExCuBS) and new extended cubic B-spline approximation (NExCuBS). Both methods have the same basis functions but for the NExCuBS, a new approximation is used for the second order space derivative

Optimal Reconstruction of Inviscid Vortices

We address the question of constructing simple inviscid vortex models which optimally approximate realistic flows as solutions of an inverse problem. Assuming the model to be incompressible, inviscid and stationary in the frame of reference moving with the vortex, the "structure" of the vortex is uniquely characterized by the functional relation between the streamfunction and vorticity. It is demonstrated how the inverse problem of reconstructing this functional relation from data can be framed as an optimization problem which can be efficiently solved using variational techniques. In contrast to earlier studies, the vorticity function defining the streamfunction-vorticity relation is reconstructed in the continuous setting subject to a minimum number of assumptions. To focus attention, we consider flows in 3D axisymmetric geometry with vortex rings. To validate our approach, a test case involving Hill's vortex is presented in which a very good reconstruction is obtained. In the second example we construct an optimal inviscid vortex model for a realistic flow in which a more accurate vorticity function is obtained than produced through an empirical fit. When compared to available theoretical vortex-ring models, our approach has the advantage of offering a good representation of both the vortex structure and its integral characteristics.Comment: 33 pages, 10 figure

An overview on deep learning-based approximation methods for partial differential equations

It is one of the most challenging problems in applied mathematics to approximatively solve high-dimensional partial differential equations (PDEs). Recently, several deep learning-based approximation algorithms for attacking this problem have been proposed and tested numerically on a number of examples of high-dimensional PDEs. This has given rise to a lively field of research in which deep learning-based methods and related Monte Carlo methods are applied to the approximation of high-dimensional PDEs. In this article we offer an introduction to this field of research, we review some of the main ideas of deep learning-based approximation methods for PDEs, we revisit one of the central mathematical results for deep neural network approximations for PDEs, and we provide an overview of the recent literature in this area of research.Comment: 23 page

Mid-Infrared Self-Similar Compression of Picosecond Pulse in an Inversely Tapered Silicon Ridge Waveguide

On chip high quality and high degree pulse compression is desirable in the realization of integrated ultrashort pulse sources, which are important for nonlinear photonics and spectroscopy. In this paper, we design a simple inversely tapered silicon ridge waveguide with exponentially decreasing dispersion profile along the propagation direction, and numerically investigate self-similar pulse compression of the fundamental soliton within the mid-infrared spectral region. When higher-order dispersion (HOD), higher-order nonlinearity (HON), losses (α), and variation of the Kerr nonlinear coefficient γ(z) are considered in the extended nonlinear Schrödinger equation, a 1 ps input pulse at the wavelength of 2490 nm is successfully compressed to 57.29 fs in only 5.1-cm of propagation, along with a compression factor Fc of 17.46. We demonstrated that the impacts of HOD and HON are minor on the pulse compression process, compared with that of α and variation of γ(z). Our research results provide a promising solution to realize integrated mid-infrared ultrashort pulse sources

MHD free convective flow past a vertical plate

The free convective flow in incompressible viscous fluid past a vertical plate is studied under the presence of magnetic field. The flow is considered along the vertical plate at x-axis in upward direction and y-axis is taken normal to it. The governing equations are written in vector form. Afterwards, the equations are solved numerically using finite element method with automated solution techniques. Later, the effects of magnetic field strength to the velocity and temperature of the fluid are obtained. It is found that for heated plate, the velocity and the temperature of the fluid decreases when the magnetic field strength increases. Meanwhile for cooled plate, the velocity decreases but the temperature increases when the magnetic field strength increases

Generation and Characterization of Isolated Attosecond Pulse in the Soft X-ray Regime

The observation of any atomic and molecular dynamics requires a probe that has a timescale comparable to the dynamics itself. Ever since the invention of laser, the temporal duration of the laser pulse has been incrementally reduced from several nanoseconds to just attoseconds. Picosecond and femtosecond laser pulses have been widely used to study molecular rotation and vibration. In 2001, the first single isolated attosecond pulse (1 attosecond = 10^-18 seconds.) was demonstrated. Since this breakthrough, attoscience has become a hot topic in ultrafast physics. Attosecond pulses typically have span between EUV to X-ray photon energies and sub-femtosecond pulse duration. It becomes an ideal tool for experimentalists to study ultrafast electron dynamics in atoms, molecules and condensed matter. The conventional scheme for generating attosecond pulses is focusing an intense femtosecond laser pulse into inert gases. The bound electrons are ionized into continuum through tunneling ionization under the strong electrical field. After ionization, the free electron will be accelerated by the laser field away from the parent ion and then recombined with its parent ion and releases its kinetic energy as a photon burst that lasts for a few hundred attoseconds. According to the classical three-step model , high order harmonic will have a higher cutoff photon energy when driven by a longer wavelength laser field. Compared to Ti:sapphire lasers center at a wavelength of 800 nm, an optical parametric amplifier could offer a broad bandwidth at infrared range, which could support few cycle pulses for driving high harmonic generation in the X-ray spectrum range. In this work, an optical parametric chirped-pulse amplification system was developed to deliver CEP-stable 3-mJ, 12-fs pulses centered at 1.7 micron. We implement a chirped-pump technique to phase match the board parametric amplification bandwidth with high conversion efficiency. Using such a laser source, isolated attosecond pulses with photon exceeding 300 eV are achieved by applying the polarization gating technique at 1.7 micron. The intrinsic positive chirp of the attosecond pulses is measured by the attosecond streak camera and retrieved with our PROOF technique. Sn metal filters with negative dispersion were chosen to compensate the intrinsic attochirp. As a result, isolated 53-attosecond soft x-ray pulses are achieved. Such water window attosecond source will be a powerful tool for studying charge distribution/migration in bio-molecules and will bring opportunities to study high field physics or attochemistry

Mathematical Modeling and Simulation in Mechanics and Dynamic Systems

The present book contains the 16 papers accepted and published in the Special Issue “Mathematical Modeling and Simulation in Mechanics and Dynamic Systems” of the MDPI “Mathematics” journal, which cover a wide range of topics connected to the theory and applications of Modeling and Simulation of Dynamic Systems in different field. These topics include, among others, methods to model and simulate mechanical system in real engineering. It is hopped that the book will find interest and be useful for those working in the area of Modeling and Simulation of the Dynamic Systems, as well as for those with the proper mathematical background and willing to become familiar with recent advances in Dynamic Systems, which has nowadays entered almost all sectors of human life and activity

New Challenges Arising in Engineering Problems with Fractional and Integer Order

Mathematical models have been frequently studied in recent decades, in order to obtain the deeper properties of real-world problems. In particular, if these problems, such as finance, soliton theory and health problems, as well as problems arising in applied science and so on, affect humans from all over the world, studying such problems is inevitable. In this sense, the first step in understanding such problems is the mathematical forms. This comes from modeling events observed in various fields of science, such as physics, chemistry, mechanics, electricity, biology, economy, mathematical applications, and control theory. Moreover, research done involving fractional ordinary or partial differential equations and other relevant topics relating to integer order have attracted the attention of experts from all over the world. Various methods have been presented and developed to solve such models numerically and analytically. Extracted results are generally in the form of numerical solutions, analytical solutions, approximate solutions and periodic properties. With the help of newly developed computational systems, experts have investigated and modeled such problems. Moreover, their graphical simulations have also been presented in the literature. Their graphical simulations, such as 2D, 3D and contour figures, have also been investigated to obtain more and deeper properties of the real world problem

Modelling transmission dynamics of covid-19 during Pre-vaccination period in Malaysia: a predictive guiseird model using streamlit

Coronavirus disease (COVID-19) is a major health threat worldwide pandemic, first identified in Malaysia on 25 January 2020. This outbreak can be represented in the mathematical expressions of a non-linear system of ordinary differential equations (ODEs). With the lack of a predictive SEIRD model in terms of Graphical Users Interface (GUI) in Malaysia, this paper aims to model the COVID-19 outbreak in Malaysia during the pre-vaccination period using the Susceptible-Exposed-Infected-Recovered-Death (SEIRD) model with time-varying parameters, then develop a GUI-SEIRD predictive model using Streamlit Python library. This GUI-SEIRD predictive model considers different values of the proportion of the quarantine-abiding population (r) and three different decisions of MCO lifted date to forecast the number of active cases (I) on 15 October 2020 that gives insightful information to government agencies. The mathematical model is solved using Scipy odeint function, which uses Livermore Solver for Ordinary Differential Equations with an Automatic method switching (LSODA) algorithm. The time-varying coefficients of SEIRD model that best fit the real data of COVID-19 cases are obtained using the Nelder-Mead optimization algorithm. This an extended SIRD model with exposed (E) compartment becoming SEIRD, leads to a robust model. It adequately fitted two datasets of Malaysian COVID-19 indicated by the slightest average values of root mean square error (RMSE) as compared to other existing models. The results highlight that the larger the values of the proportion of the quarantine-abiding population (r) and the later the date of the lifted MCO, the faster Malaysia reaches disease free equilibrium