Method of Moments

Nanoparticle inception remains one of the least constrained steps in predictive combustion particle modeling. In soot models, detailed gas-phase chemistry can describe fuel decomposition, aromatic growth, PAH chemistry, radical pathways, oxidation, surface growth, and coagulation, yet the conversion of molecular precursors into the first persistent particles is often represented by empirical nucleation rates, selected dimerization reactions, or source terms into a first particle bin. Similar uncertainty occurs in inorganic flame-aerosol models, where the earliest molecular-to-particle transition is frequently reduced to classical or semi-empirical nucleation expressions. This paper formulates the transient nano-dense molecular state (NDMS) hypothesis as a practical persistence-stabilization closure for this inception gap. NDMS is defined as a transient, non-equilibrium, locally dense ensemble of associated molecular or sub-molecular precursor units that remains reversible on the dissociation timescale and contributes to particle inception only when chemical or structural stabilization competes successfully with dissociation and other losses. The framework separates precursor association, cluster dissociation, competing non-stabilizing loss, and stabilization into distinct model components. It introduces an operational density-enhancement criterion, association-and stabilization-weighted precursor descriptors, bounded stabilization probabilities, and a minimal zero-dimensional demonstration. Implementation routes are outlined for detailed chemistry, sectional population balances, moment methods, and reduced CFD closures. For soot, the framework provides a structured way to couple PAH clustering with radical-driven stabilization and chemical aging. For inorganic systems, illustrated using TiO₂ formation, NDMS is treated cautiously as a system-specific modeling architecture for reactive oxide-cluster formation rather than as evidence of a universal microscopic pathway. The hypothesis is presented as falsifiable: it should be retained only where constrained NDMS closures improve inception-specific observables such as onset location, pressure dependence, early particle number density, precursor sensitivity, and cluster-sensitive diagnostics.

This research examines the presence of authentic leadership among managers of foreign owned companies operating in Turkey. It evaluates the effectiveness of the authentic leadership model in addressing cultural aspects within these organizations. It also examines whether there are distinct authentic leadership elements that are company-and leader-specific, whether authentic leadership characteristics are adapted to Turkish culture and business practice, and whether elements from different leadership styles are incorporated. The research aims to supply to the writings by uncovering the authentic leadership profile in foreign owned companies and presenting an alternative leadership profile that integrates globalization and localization. The investigation is designed as a qualitative study and content analysis was employed. Semi-structured interview forms were utilized to conduct in-depth interviews with 10 foreign owned company managers. The result obtained in the research, foreign owned company managers tend to exhibit authentic leadership model which is consistent with local culture. The analysis of the word trees shows that these leaders are consistently open, direct, and clear. They also show respect, commitment to their values and to universal ethics and the communication skills in a friendly way with their staff.

In attempting to describe the energy levels of heavy nuclei ), researchers were confronted with daunting calculations for a many bodied system with extremely complicated interaction forces. Unable to explicitly calculate the energy levels, physicists developed Random Matrix Theory to predict general properties of the system. In this chapter we give a brief introduction to classical Random Matrix Theory, Random Graphs, and L-Functions. The goal is to show how diverse systems exhibit similar universal behaviors, and introduce the techniques used in the proofs. In some sense, this is a continuation of the Poissonian behavior investigations of Chapter ??. The survey below is meant to only show the broad brush strokes of this rich landscape -detailed proofs will follow in later chapters. We assume familiarity with the basic concepts of probability theory (Chapter ??) and linear algebra (a quick review of the needed background is provided in Appendix ??). While we assume the reader has some familiarity with the basic concepts in physics for the historical introduction in §1.1, no knowledge of physics is required for the detailed expositions. For those interested in learning more (as well as a review of recent developments), we conclude this chapter with a brief summary of the literature. A central question in mathematical physics is the following: given some system with observables t 1 ≤ t 2 ≤ t 3 ≤ . . . , describe how the t i are spaced. For example, we could take the t i to be the energy levels of a heavy nuclei, or the prime numbers, or zeros of L-functions, or eigenvalues of real symmetric or complex Hermitian matrices (or as in Chapter ?? the fractional parts {n k α} arranged in increasing order). If we completely understood the system, we would know exactly where all the t i are; in practice we try and go from knowledge of how the t i are spaced to knowledge of the underlying system.

Given an ensemble of N × N random matrices, a natural question to ask is whether or not the empirical spectral measures of typical matrices converge to a limiting spectral measure as N → ∞. While this has been proved for many thin patterned ensembles sitting inside all real symmetric matrices, frequently there is no nice closed form expression for the limiting measure. Further, current theorems provide few pictures of transitions between ensembles. We consider the ensemble of symmetric m-block circulant matrices with entries i.i.d.r.v. These matrices have toroidal diagonals periodic of period m. We view m as a "dial" we can "turn" from the thin ensemble of symmetric circulant matrices, whose limiting eigenvalue density is a Gaussian, to all real symmetric matrices, whose limiting eigenvalue density is a semi-circle. The limiting eigenvalue densities f m show a visually stunning convergence to the semi-circle as m → ∞, which we prove. In contrast to most studies of patterned matrix ensembles, our paper gives explicit closed form expressions for the densities. We prove that f m is the product of a Gaussian and a certain even polynomial of degree 2m -2; the formula is the same as that for the m × m Gaussian Unitary Ensemble (GUE). The proof is by derivation of the moments from the eigenvalue trace formula. The new feature, which allows us to obtain closed form expressions, is converting the central combinatorial problem in the moment calculation into an equivalent counting problem in algebraic topology. We end with a generalization of the m-block circulant pattern, dropping the assumption that the m random variables be distinct. We prove that the limiting spectral distribution exists and is determined by the pattern of the independent elements within an m-period, depending on not only the frequency at which each element appears, but also the way the elements are arranged.

Given an ensemble of N × N random matrices, a natural question to ask is whether or not the empirical spectral measures of typical matrices converge to a limiting spectral measure as N → ∞. While this has been proved for many thin patterned ensembles sitting inside all real symmetric matrices, frequently there is no nice closed form expression for the limiting measure. Further, current theorems provide few pictures of transitions between ensembles. We consider the ensemble of symmetric m-block circulant matrices with entries i.i.d.r.v. These matrices have toroidal diagonals periodic of period m. We view m as a "dial" we can "turn" from the thin ensemble of symmetric circulant matrices, whose limiting eigenvalue density is a Gaussian, to all real symmetric matrices, whose limiting eigenvalue density is a semi-circle. The limiting eigenvalue densities f m show a visually stunning convergence to the semi-circle as m → ∞, which we prove. In contrast to most studies of patterned matrix ensembles, our paper gives explicit closed form expressions for the densities. We prove that f m is the product of a Gaussian and a certain even polynomial of degree 2m -2; the formula is the same as that for the m × m Gaussian Unitary Ensemble (GUE). The proof is by derivation of the moments from the eigenvalue trace formula. The new feature, which allows us to obtain closed form expressions, is converting the central combinatorial problem in the moment calculation into an equivalent counting problem in algebraic topology. We end with a generalization of the m-block circulant pattern, dropping the assumption that the m random variables be distinct. We prove that the limiting spectral distribution exists and is determined by the pattern of the independent elements within an m-period, depending on not only the frequency at which each element appears, but also the way the elements are arranged. MURAT KOLO GLU, GENE S. KOPP, AND STEVEN J. MILLER 3.1. The Limiting Spectral Measure of the Symmetric m-Block Circulant Ensemble 15 3.2. The m → ∞ Limit and the Semi-Circle 18 4. Convergence of the Limiting Spectral Measures 20 5. Future Research 21 5.1. Representation Theory 21 5.2. Spacings 22 Appendix A. Pointwise Convergence as m → ∞ 23 Appendix B. Generalized m-Block Circulant Matrices 26 B.1. Zone-wise Locations and Pairing Conditions 27 B.2. The Fourth Moment 28 B.3. Existence and Convergence of High Moments 32 References 34

Given an ensemble of N × N random matrices, a natural question to ask is whether or not the empirical spectral measures of typical matrices converge to a limiting spectral measure as N → ∞. While this has been proved for many thin patterned ensembles sitting inside all real symmetric matrices, frequently there is no nice closed form expression for the limiting measure. Further, current theorems provide few pictures of transitions between ensembles. We consider the ensemble of symmetric m-block circulant matrices with entries i.i.d.r.v. These matrices have toroidal diagonals periodic of period m. We view m as a "dial" we can "turn" from the thin ensemble of symmetric circulant matrices, whose limiting eigenvalue density is a Gaussian, to all real symmetric matrices, whose limiting eigenvalue density is a semi-circle. The limiting eigenvalue densities f m show a visually stunning convergence to the semi-circle as m → ∞, which we prove. In contrast to most studies of patterned matrix ensembles, our paper gives explicit closed form expressions for the densities. We prove that f m is the product of a Gaussian and a certain even polynomial of degree 2m -2; the formula is the same as that for the m × m Gaussian Unitary Ensemble (GUE). The proof is by derivation of the moments from the eigenvalue trace formula. The new feature, which allows us to obtain closed form expressions, is converting the central combinatorial problem in the moment calculation into an equivalent counting problem in algebraic topology. We end with a generalization of the m-block circulant pattern, dropping the assumption that the m random variables be distinct. We prove that the limiting spectral distribution exists and is determined by the pattern of the independent elements within an m-period, depending on not only the frequency at which each element appears, but also the way the elements are arranged.

Purpose: To develop and demonstrate a method to calculate the temperature rise that is induced by the radio frequency (RF) field in MRI at the electrode of an implanted medical lead. Materials and Methods: The electric field near the electrode is calculated by integrating the product of the tangential electric field and a transfer function along the length of the lead. The transfer function is numerically calculated with the method of moments. Transfer functions were calculated at 64 MHz for different lengths of model implants in the form of bare wires and insulated wires with 1 cm of wire exposed at one or both ends. Results: Heating at the electrode depends on the magnitude and the phase distribution of the transfer function and the incident electric field along the length of the lead. For a uniform electric field, the electrode heating is maximized for a lead length of approximately one-half a wavelength when the lead is terminated open. The heating can be greater for a worst-cas...

PurposeTo develop and demonstrate a method to calculate the temperature rise that is induced by the radio frequency (RF) field in MRI at the electrode of an implanted medical lead.Materials and MethodsThe electric field near the electrode is calculated by integrating the product of the tangential electric field and a transfer function along the length of the lead. The transfer function is numerically calculated with the method of moments. Transfer functions were calculated at 64 MHz for different lengths of model implants in the form of bare wires and insulated wires with 1 cm of wire exposed at one or both ends.ResultsHeating at the electrode depends on the magnitude and the phase distribution of the transfer function and the incident electric field along the length of the lead. For a uniform electric field, the electrode heating is maximized for a lead length of approximately one‐half a wavelength when the lead is terminated open. The heating can be greater for a worst‐case phase ...

To improve effectively sea surface wind speed retrieval accuracy when it is raining, it is very interesting to find out a more accurate normal relationship between Ku band backscatter coefficient  0 ku and C band backscatter coefficient  0 c for a dual frequency altimeter (13.58GHz/5.25GHz). In this paper an electromagnetic scattering model with a rough surface profiles is established. In terms of the established rough sea surface model  0 ku and  0 c are calculated by electromagnetic numerical methods of method of moment (MOM) and Kirchhoff approximatation (KA) when the rms of the rough surface is various. The simulation results are in a good agreement by these two different methods: MOM and KA. Those calculated results show that the relationship between  0 ku and  0 c obtained by calculating electromagnetic scattering from Gaussian random rough sea surface profiles exists positive differences from the normal relationships while those obtained by calculating electromagnetic scattering from Exponential rough sea surface profiles exist negative differences from thoses. It is obvious that the calculated relationship can be in a good agreement with an empirical normal relationship by adjusting compositions of those ocean spectrums. It is a good way to find out the normal relationship between  0 ku and  0 c by theoretical calculation and analysis, and also is an effective way to study the ocean spectrum and sea states.

Advances in fabrication techniques have recently allowed the synthesis of Carbon Nanotubes (CNTs) with controlled complex shapes. For example, CNTs with one or more branches sprouting off the main CNT axis, forming unique Y-shaped and K-shaped carbon nanostructures, have recently been fabricated. The angle at each junction and the properties of each branch can be varied by the careful control of the fabrication conditions. In addition to their complex shapes, the conductivity of these nanostructures is not uniform especially at the locations of defects and at junctions where the different CNT branches interface. In this work, we calculate the electromagnetic scattering characteristics of carbon nanotubes with complex shapes. Moreover, we quantify the electromagnetic response of these complex carbon nanostructures using Characteristic Mode Analysis (CMA). CMA decomposes the current of a scatterer in terms of a set of fundamental modes, and it evaluates the relative significance of ea...

CHAPTER SYNOPSIS 1.INTRODUCTION 1.1. Dielectric Grating Filters .ithms 1.2. Motivation for Use of Genetic Algor 1.3. Application of GAS to Design Problems in Optics 11. PGAPACK GENETIC ALGORITHM LIBRARY 11.1. The Merits of Different Types of Encoding 11.2. Data Structure Neutrality and Custom Data Types 11.3. Hybrid Genetic Algorithms 11.4. Parallel Implementation of PGAPACK 111. DIELECTRIC GRATING FILTER DESIGN 111.1. Forward Scattering via Rigorous Coupled Wave Technique 111.2. Forward Scattering via E-field Integral Equation IV. NUMERICAL RESULTS

The emergence of by-production technology as an alternative foundation for pollutiongenerating technology marks a significant turning point in environmental literature due
to its compatibility with the material balance principle. This approach considers two
independent sub-technologies: a primary production technology and a residualgenerating technology. The classical by-production technology can be estimated using
both parametric and nonparametric techniques. This study aims to identify the impact
of different shape constraint assumptions on by-production technology. We evaluate
four distinct model specifications: (i) two concerning the error terms, which may be
either composite or deterministic, and (ii) two concerning the shape of the technology,
which may be imposing either convexity or nonconvexity assumptions. In addition, we
evaluate the out-of-sample predictive performance of these alternative approaches
using machine learning techniques. To validate our estimation approach, we conduct
an empirical case study encompassing 47 Chinese cities from 2011 to 2019. Our
findings reveal that both StoNED by-production models exhibit higher consistency
compared to their deterministic counterparts. Furthermore, we observe that relaxing
convexity/concavity assumptions generates a lower bound for both sub-technologies.
Exploring the predictive power of nonconvex estimators on unseen data yields more
precise out-of-sample predictions in both stochastic and deterministic settings.

We analyze earlier applications of perturbation theory by the moment method (also called inner product method) to anharmonic oscillators. For concreteness we focus on two-dimensional models with symmetry C 4v and C 2v and reveal the reason why some of those earlier treatments proved unsuitable for the calculation of the perturbation corrections for some excited states. Point-group symmetry enables one to predict which states require special treatment. Many years ago there was great interest in perturbation theory without wavefunction. Fernández and Castro [1] developed an approach for multidimensional nonseparable problems that was based on the recurrence relations of the moments of the wavefunction instead of the wavefunction itself. They applied it to the Zeeman effect in Hydrogen and to the Hydrogen molecular ion. Austin [2] resorted to this approach in order to obtain the coefficients of the renormalized perturbation series for the lowest states of the Zeeman effect in Hydrogen and Arteca et al. [3] carried out a similar calculation by means of an order-dependent mapping. Fernández and Castro [4] also applied this moment perturbation theory (MPT) to the Hydrogen in parallel magnetic and electric fields and outlined its application to multidimensional anharmonic oscillators. Sometime later Killingbeck et al. [5] rediscovered the approach and baptized it inner-product method. They applied it to one-dimensional anharmonic oscillators and later Killingbeck and Jones [6] to two dimensional ones. Fernández et al. [7] developed F. M. Fernández (B) División Química Teórica, INIFTA (UNLP, CCT La Plata-CONICET), Blvd.

A thorough numerical assessment of Finite Difference Time Domain and Complex-Envelope Alternating-Direction-Implicit Finite-Difference-Time-Domain Methods has been carried out based on a basic single mode Plane Optical Waveguide structure. Simulation parameters for both methods were varied and the impact on the performance of both numerical methods is investigated.

AbstracC: Tactical antenna installations contain a multitude of collocated antennae, installed on a common mast. It is not uncommon to have installations containing dozens of VHF and UHF antenna operating simultaneously, with the need for minimal coupling and the consequent interference between them. The objective of this paper is to address the issue of the prefered configuration for the installation of collocated Tactical VHF (30 to 88 MHz) dipole antenna installed on a common mast. Method of Moments analysis is used for determining the decoupling between antennae in the different configurations, and practical recommendations for tactical antennae installations are thus derived.

Leadership remains a critical determinant of organisational effectiveness, social cohesion, and sustainable development in an increasingly complex and dynamic global environment. The Heart of Leadership: Enriching the Leadership in You presents a comprehensive and integrative exploration of leadership as both a professional practice and a deeply personal responsibility. The book examines leadership beyond positional authority, emphasising values, character, emotional intelligence, ethics, and self-awareness as the true foundations of effective leadership. Drawing from classical and contemporary leadership theories, the text systematically analyses leadership styles, forms, traits, skills, and competencies required to navigate modern organisational and societal challenges. Particular attention is given to contextual realities in developing economies, with practical illustrations and reflections relevant to Africa and Tanzania in particular. The book bridges theory and practice by translating scholarly concepts into actionable insights that can be applied in public institutions, private organisations, and community leadership settings. Designed for undergraduate and postgraduate students, academics, and practitioners, the book adopts a clear, accessible, and pedagogically sound approach that encourages critical thinking, reflection, and ethical decision-making. Ultimately, The Heart of Leadership argues that sustainable leadership excellence begins from within the individual leader and is sustained through integrity, service, and continuous personal development.

We establish general theorems quantifying the notion of recurrencethrough an estimation of the moments of passage times -for irreducible continuous-time Markov chains on countably infinite state spaces. Sharp conditions of occurrence of the phenomenon of explosion are also obtained. A new phenomenon of implosion is introduced and sharp conditions for its occurrence are proven. The general results are illustrated by treating models having a difficult behaviour even in discrete time.

In this paper a novel technique for cascading generalized scattering matrices based on a Krylov's iterative solver is presented. This new technique is fully general, since it can be applied to solve the connection of an arbitrary number of networks, each one with an arbitrary number of ports, and it is easy to implement. This technique is able to compute, not only the modal spectra at the free ports of the global network, but also the modal spectra at the connected ports, so that the field inside the full network can be computed for an arbitrary incidence. In addition, this technique can also be used to evaluate the scattering parameters of the global network.

We establish general theorems quantifying the notion of recurrencethrough an estimation of the moments of passage times -for irreducible continuous-time Markov chains on countably infinite state spaces. Sharp conditions of occurrence of the phenomenon of explosion are also obtained. A new phenomenon of implosion is introduced and sharp conditions for its occurrence are proven. The general results are illustrated by treating models having a difficult behaviour even in discrete time.

A closed form expression for the cavity field within a slim, cubical, metallic enclosure with three closed edges and one open edge is presented. The radiated far fields from the open slot are described by analytical formulations for the electric field and the radiated power. The formulations allow to efficiently analyze resonance frequencies, positions of maximum field density, and placement criteria for critical electronic components within the enclosure.

This work presents an extension of the formulation for the synthesis of dual-reflector antennas with omnidirectional coverage and arbitrary radiation pattern in elevation plane. The body-of-revolution subreflector is generated by a single conic section while the shaped main reflector is generated by a series of local conic sections sequentially consecutively concatenated. As case studies, four representative axis-displaced ellipse (ADE) configurations are synthesized to provide a sectoral coverage in the elevation plane. Two different ray structures are considered: with a real or a virtual main reflector caustic. The GO shaping results are validated using method-of-moments analysis.

This work investigates the synthesis of dual-reflector antennas for omnidirectional coverage. Both reflectors are bodies-of-revolution: the subreflector is generated by a single conic section while the shaped main reflector is generated by a series of local conic sections which are consecutively concatenated. The main-reflector is shaped to radiate, under GO principles, a prescribed power density in the elevation plane. To