Rigorous Mathematical Derivation Research Articles

Near-field three-dimensional coherent imaging: Theory and simulations

This work presents a rigorous mathematical derivation of an effective approximate solution to the three-dimensional inverse scattering/imaging problem that is applicable for all imaging zones ranging from the near to the far field. Simulation results for the point spread function illustrate the range and cross-range resolution as a function of the optical f number. The model system operates in a synthetic aperture type mode, where the coherent signals are transmitted, and the scattered signals are subsequently received at individual transmitters and receivers. Potential applications of this technology include: Medical ultrasound, foliage penetrating synthetic aperture radar, ground penetrating radar for land mine detection, and electromagnetic millimeter-wave scanning for concealed weapon detection.

Efficient MLFMA, RPFMA, and FAFFA Algorithms for EM Scattering by Very Large Structures

Based on the addition theorem, the principle of a multilevel ray-propagation fast multipole algorithm (RPFMA) and fast far-field approximation (FAFFA) has been demonstrated for three-dimensional (3-D) electromagnetic scattering problems. From a rigorous mathematical derivation, the relation among RPFMA, FAFFA, and a conventional multilevel fast multipole algorithm (MLFMA) has been clearly stated. For very large-scale problems, the translation between groups in the conventional MLFMA is expensive because the translator is defined on an Ewald sphere with many sampling k/spl circ/ directions. When two groups are well separated, the translation can be simplified using RPFMA, where only a few sampling k/spl circ/ directions are required within a cone zone on the Ewald sphere. When two groups are in the far-field region, the translation can be further simplified by using FAFFA where only a single k/spl circ/ is involved in the translator along the ray-propagation direction. Combining RPFMA and FAFFA with MLFMA, three algorithms RPFMA-MLFMA, FAFFA-MLFMA, and RPFMA-FAFFA-MLFMA have been developed, which are more efficient than the conventional MLFMA in 3-D electromagnetic scattering and radiation for very large structures. Numerical results are given to verify the efficiency of the algorithms.

Multiparameter time-domain sensitivity computation

In this paper, a novel approach for the computation of multiparameter time-domain sensitivity during circuit simulation is presented. The computation of the sensitivity values of one or more circuit variables (e.g., node voltages or branch currents) with respect to both design parameters (e.g., R, C, MOS length, MOS width...) and process parameters (e.g., tox, nsub, u0...) is addressed by introducing a rigorous mathematical derivation of the proposed direct differentiation modified algorithm (DDMA). First, the circuit interpretation of the analytical formulas as well as their theoretical derivation from the Tellegen's theorem is discussed. Then, DDMA is compared with the direct method and the adjoint technique to emphasize the cost difference in terms of CPU operations. The accuracy of the new algorithm is also evaluated, by estimating the numerical error involved in the solution of discretized sensitivity equations. Finally, a set of circuit examples is presented to demonstrate the effectiveness of DDMA together with the implementation details needed to include the present sensitivity analysis in a circuit simulator such as Spice3F5.

Relationships between antiviral treatment effects and biphasic viral decay rates in modeling HIV dynamics

Recently, potent combination antiviral therapies consisting of reverse transcriptase inhibitor (RTI) drugs and protease inhibitor (PI) drugs, have been developed which can rapidly suppress HIV below the limit of detection. Two phases of plasma viral decay after initiation of treatment have been observed in clinical studies. Some researchers have suggested that the viral decay rates may reflect the potency (efficacy) of antiviral therapies. In this paper we model the effect of RTI drugs and PI drugs as inhibition rates of cell infection and infectious virus production, respectively, based on the biological mechanisms of these two different types of drugs. Through rigorous mathematical derivation, we show that the two viral decay rates are monotone functions of the treatment effects of these antiviral therapies. We derive approximation formulas for the relationships between viral decay rates and treatment effects. Computer simulations show that the approximation formulas approximate the true values very well. These formulas may be used to study what factors really affect the viral decay rates. The results in this paper provide a theoretical justification for using both viral decay rates for evaluation of the treatment efficacy of antiviral therapies.

SOME PROPERTIES OF RANDOMDEC SIGNATURES

Using complex modal superposition, the Randomdec signatures are investigated for the random responses of a linear time-invariant multiple-degree-of-freedom system with non-proportional damping. The input is assumed to be Gaussian white noise with zero mean. Through rigorous mathematical derivation and numerical simulations, this paper presents three properties of the Randomdec signatures to supplement the mathematical basis for the random decrement technique so that the Randomdec signatures will not be misused in the further system identification. First, the estimations of auto- and cross-Randomdec signatures are proved to be unbiased, and the variances of the estimated auto- and cross-Randomdec signatures are proved to be proportional to the variances of the raw data and inversely proportional to the number of records used. Second, under certain circumstances, the Randomdec displacement and velocity signatures are equivalent to the corresponding free decay responses of the system while the Randomdec acceleration signature is never equivalent to the free decay signals. Third, the singular behaviour of Randomdec acceleration signatures depends on the singularities of input correlation functions as well as the mass matrix of the system under consideration.

An analysis of corona currents between two concentric cylindrical electrodes

The electric field and space charge in a corona device are governed by nonlinear differential equations. With the symmetry present in the configuration involving concentric cylindrical electrodes, exact mathematical solution becomes tractable. However, some commonly used formulas in the literature have lacked rigorous mathematical derivation and analysis. In the present work, a general exact solution is derived for the problem of the unipolar charge current between two concentric cylindrical electrodes. In addition to rigorous mathematical analysis, an approximate expression based on the empirical form of well-established Warburg's formula for current–voltage relationship is derived systematically. For typical wire–cylinder corona devices, the presently derived coefficient in Warburg's formula turns out to be numerically very close to that in the commonly used Townsend's formula, despite their rather different mathematical forms. Moreover, the validity of the commonly used, quadratic form of Townsend's formula is shown to lack concrete mathematical reasoning.

Iterative optimization approach for the design of diffractive phase elements simultaneously implementing several optical functions

A new iterative optimization method based on the Yang–Gu (Y-G) algorithm for the design of diffractive phase elements (DPE’s) that incorporate several optical functions is presented. The Y-G algorithm has been previously used to deal with various amplitude-phase retrieval problems in linear optical transform systems involving a monochromatic wave. We extend the general theory to the case of linear imaging systems with the illumination consisting of a number of components at different wavelengths. Based on a rigorous mathematical derivation, an iterative algorithm is developed to integrate several functions into one DPE. In the first set of examples numerical simulations are carried out in the design of several one-dimensional DPE’s capable of both demultiplexing different wavelength components and focusing each component wave. The second example illustrates the design of a two-dimensional DPE that performs as a demultiplexer in one direction and an array generator in the other direction.

THEORY OF THE AMPLITUDE-PHASE RETRIEVAL IN ANY LINEAR TRANSFORM SYSTEM AND ITS APPLICATIONS

This paper is based on our previous works in the past decade and is a summary of the theory of the amplitude-phase retrieval problem in any linear transform system and its applications. We describe the general statement on the amplitude-phase retrieval problem in an imaging system and derive a set of equations governing the amplitude-phase distributions in terms of the rigorous mathematical derivation. By using these equations and an iterative algorithm, a variety of amplitude-phase problems can be successfully handled. We carry out systematical investigations and comprehensive numerical calculations to demonstrate the utilization of this new algorithm in various transform systems. We also indicated the relationship and distinction between our algorithm and the original Gerchberg-Saxton algorithm. This new algorithm possesses considerable capability to deal with the various phase-retrieval problems in the arbitrary linear transform system and some inverse problems in solid state physics.

Formulation of the Boltzmann equation in terms of scattering matrices

We present a rigorous mathematical derivation of the Scattering Matrix Approach from the path integral formulation of the Boltzmann equation. We also show that in the limiting cases, the Scattering Matrix Approach correctly reproduces McKelvey's earlier results. This proof puts the Scattering Matrix Approach in a more general perspective and relates it to other techniques for solving the Boltzmann equation.

Simple theory of the charge transfer rate in ion/molecule reactions. 3. Refinement in mathematical treatment

A more rigorous mathematical derivation of the parameter ω that determines the rate of the unimolecular steps in the simple theory of the charge transfer rate is offered. The refinement leads to better agreement with experiment.

An Extension to Newton's Method in Device Simulators--On An Efficient Algorithm to Evaluate Small-Signal Parameters and to Predict Initial Guess

The Newton iteration method used in conventional numerical device simulations is extended to include bias conditions as variable parameters in the system of semiconductor equations. This extension leads to a very simple algorithm to evaluate low-frequency, small-signal parameters based on dc solutions. It also provides an elegant way to project an initial guess for subsequent bias conditions. Rigorous mathematical derivation and applications to the evaluation of junction capacitance and breakdown voltage are given. The initial guess projection is also illustrated through examples.

Formation of prominences by condensation modes in magnetized cylindrical plasmas

Condensation modes in a magnetized cylindrical plasma are studied to shed light on the formation and stability of solar prominences. A rigorous mathematical derivation of the perturbation equation is developed, and the effect of field twist on the stability is studied for an equilibrium with uniform field twist, in which temperature increases, but density does not, as pressure increases. The results imply that prominences may form in globally magnetohydrodynamic-stable magnetic loops with very low field twist. Also, prominences are more likely to form in a region of weaker area-averaged magnetic field.

Gantmakher-Kaner oscillations at 79 GHz in single-crystal copper plates

Thin plates of high-purity, single-crystal copper have been investigated with the microwave transmission technique at 79 GHz with the utilization of a magnetic field perpendicular to the sample surfaces and parallel to the $〈100〉$, $〈110〉$, and $〈111〉$ crystal axes. The observed Gantmakher-Kaner oscillations are compared with a theoretical model first proposed by Phillips, Baraff, and Schmidt to explain their data in copper at 35 GHz for a $〈100〉$ orientation. With the aid of their model we find two classes of signals: (a) those arising from a sharp cutoff of the topological effectiveness, and (b) those due to an extremal helical trajectory. The generally excellent agreement between observation and the predictions of the model for all three crystal orientations confirms the physical significance of the model, although it is lacking a rigorous mathematical derivation.

ON THE AMPLITUDE-PHASE RETRIEVAL PROBLEM IN OPTICAL SYSTEMS

In this paper we propose a general description of the amplitude-phase retrieval problem. A set of equations for the determination of amplitude-phase based on rigorous mathematical derivation is established. Using these equations and the iteration computing method, a variety of reconstruction problems can be solved.

Introduction to radio and radio antennas

Electromagnetic wave propagation is a simple and beautiful phenomenon, but its rigorous mathematical derivation from Maxwell's equations is formidable. This article makes many of the concepts visible and plausible using only mathematics available to college freshmen and taught today in most high schools. Those who are familiar with the subject will appreciate the simple and direct way in which Dr. Friis' transmission formula is derived. This derivation is new and is published here for the first time.

Subdynamics of constrained many-body systems and the theory of transport coefficients

The concept of subdynamics previously introduced for homogeneous systems by our group is extended to the case of constrained systems. The case of a plasma in an external electric field is studied in detail. For this purpose, we introduce projection operators II (o) E and II (∨) E depending on the external field. The interest of this generalization appears when we show that the steady current for weak stationary external fields depends only on the II (o) E component of the N-body distribution function. Another very important consequence of our calculations is that we give a rigorous mathematical derivation for the standard formula which gives the steady current in a weak stationary and homogeneous field. The Chapman-Enskog derivation is recovered and very naturally generalized in the II (o) E framework.

A General Stability Index for Flow Near Plane Boundaries

SINCE THE TIME of Osborne Reynolds, the stability problem in fluid mechanics has consistently withstood every effort toward a general solution. The mathematical method of superposing a periodic disturbance upon a parallel flow has, to be sure, yielded qualitative results in specific cases, and experimental studies have supplied quantitative information in terms of characteristic Reynolds Numbers wherever required. However, a general criterion for stability is not yet at hand (as evidenced by the wide numerical range of critical Reynolds Numbers even for related types of flow), nor have existing analyses provided a physical picture of the controlling factors which is within the grasp of the average engineer. The stability index proposed in this paper does not have the justification of rigorous mathematical derivation, nor has it evolved from extensive laboratory study. Yet it does stem from logical and almost self-evident dimensional and physical concepts, and its general significance is indicated by experimental data already at hand for several basically different types of fluid motion.

On the KP I transonic limit of two-dimensional Gross-Pitaevskii travelling waves

We provide a rigorous mathematical derivation of the convergence in the long-wave transonic limit of the minimizing travelling waves for the two-dimensional Gross-Pitaevskii equation towards ground states for the Kadomtsev-Petviashvili equation (KP I).