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Stochastic partial differential equations (SPDEs) driven by Lévy noise are extensively employed across various domains such as physics, finance, and engineering to simulate systems experiencing random fluctuations. In this paper, we focus on a specific type of such SPDEs, namely the nonlinear beam epitaxy equation driven by Lévy noise. The Feynman graph formalism emerges as a potent tool for analyzing these SPDEs, particularly in computing their correlation functions, which are essential for understanding the moments of the solution. In this context, the solution to the SPDE and its truncated moments can be expressed as a sum over particular Feynman graphs. Each graph is evaluated according to a set of established rules, providing a systematic method to derive the properties of the solution. Moreover, the study delves into the behavior of the truncated moments for large times. Truncated moments, which capture the statistical properties of the system up to a certain order, are crucial for characterizing the long-term behavior and stability of the solution. The paper will conclude with a discussion on potential applications, highlighting the broader implications of this approach in various scientific and engineering contexts.

Stochastic partial differential equations (SPDEs) driven by Lévy noise are extensively employed across various domains such as physics, finance, and engineering to simulate systems experiencing random fluctuations. In this paper, we focus on a specific type of such SPDEs, namely the nonlinear beam epitaxy equation driven by Lévy noise. The Feynman graph formalism emerges as a potent tool for analyzing these SPDEs, particularly in computing their correlation functions, which are essential for understanding the moments of the solution. In this context, the solution to the SPDE and its truncated moments can be expressed as a sum over particular Feynman graphs. Each graph is evaluated according to a set of established rules, providing a systematic method to derive the properties of the solution. Moreover, the study delves into the behavior of the truncated moments for large times. Truncated moments, which capture the statistical properties of the system up to a certain order, are crucial for characterizing the long-term behavior and stability of the solution. The paper will conclude with a discussion on potential applications, highlighting the broader implications of this approach in various scientific and engineering contexts. [ABSTRACT FROM AUTHOR]

We consider some non-linear non-homogeneous partial differential equations (PDEs) and derive their exact Green function solution as a functional Taylor expansion in powers of the source. The kind of PDEs we consider are dispersive ones where the exact solution of the corresponding homogeneous equations can have some known shape. The technique has a formal similarity with the Dyson–Schwinger set of equations to solve quantum field theories. However, there are no physical constraints. Indeed, we show that a complete coincidence with the statistical field model of a quartic scalar theory can be achieved in the Gaussian expansion of the cumulants of the partition function. [ABSTRACT FROM AUTHOR]

The oblique collision zone of Arabia-Eurasia is a seismically active region with complex crustal deformation patterns. While GPS measurements provide valuable data, their sparse distribution limits our understanding of the full extent of deformation. This study addresses this limitation by using a robust interpolation method for GPS velocity data in the collision zone. We utilized biharmonic splines to interpolate horizontal components of sparse GPS velocity data independently and in a coupled manner by altering Poisson ratio. This method is an effective means of interpolating sparse vector data in cases where deformation mechanics can be explained by elasticity principles. The interpolation process included fitting trends to the input data, calculating residuals, and analyzing them. The prediction process consisted of trend and spline fitting stages. We interpolate horizontal GPS velocities onto a standard geographic grid with a 30-minute interval, excluding data points with significant deviation. The data was partitioned into training and testing subsets, with the training set used for calibration and the testing set for evaluation of the interpolation method. Our analysis revealed an irregular spatial distribution of crustal movement. The northern component of the velocity field consistently points towards Eurasia and is greater than the eastern component. The amplitude of the northern component decreases from south to north and from west to east, indicating variations in deformation intensity. The eastern component exhibits a change in direction, moving westward in the western half of Iran and eastward in the eastern half, with a reversed trend in the north. This change in direction highlights the presence of solid blocks within the collision zone. Undeformed regions, major faults, convergence deformation, and compressing high-elevation regions are also observed in the collision zone. These findings provide a detailed picture of present-day crustal deformation in the Arabia-Eurasia collision zone, enhancing our understanding of the collision process. [ABSTRACT FROM AUTHOR]

In this work we present a formalism based on scalar Green's functions to deal with electromagnetic scattering problems. Although the formulations of the Mie theory and Born approximations in terms of electromagnetic scattering are well known and relevant, they have certain disadvantages: complexity, computational time, few symmetries, etc. Therefore, the study with scalar Green's functions allows dealing with these problems with greater simplicity and efficiency. However, the information provided by the vector formulation is sacrificed. Nevertheless, different cases of electromagnetic scattering of dielectric media with different dimensions, geometries and refractive indices will be presented. Thus, we will be able to verify the capacity of this scalar method in predicting light-scattering problems. [ABSTRACT FROM AUTHOR]

In the paper we use recently proposed concepts of granular derivative and horizontal membership function to introduce and study the equivalence relation between granular differential equations and a class of second-order linear fuzzy differential equations. It is well-known that in the study of boundary value problem (BVP) of differential equations, the equivalence of the analytical solution for BVP plays an extremely important role. The BVP of second-order linear fuzzy differential equation is solved by the inverse operator method of the second-order fuzzy differential operator and the Green function is defined. It is usually necessary to transform the BVP under consideration into its equivalent integral equations, and the key to the transformation is how to construct Green function for BVP of differential equation. The Green function method is mainly used to solve a class of second-order linear fuzzy differential equation semi-homogeneous BVP here. Moreover, the existence and uniqueness of Green function for semi-homogeneous BVP of self-conjugate equations with parameter λ is proposed. At the same time, several examples are also shown to illustrate the solution of BVP of fuzzy differential equations with Green function method. [ABSTRACT FROM AUTHOR]

Abstract In this study, Levinson-type inequalities for the class of n-convex ( n ≥ 3 $n \geq 3$ ) functions are generalized using new Green functions and the Hermite interpolating polynomial involving two types of data points. Some estimations for novel functionals are derived using f-divergence. Furthermore, different inequalities involving Shannon entropies are presented.

In this work, we use dyadic Green’s functions to calculate local density of states in a finite-size one-dimensional photonic crystal surrounded by air. The crystal composes alternating layers of polymer materials, with its translational periodicity disrupted by the introduction of a cavity. Its optical properties are affected by variations in the polymers’ dielectric constant under applied pressure, and the cavity is infiltrated with four types of cells: Normal, Jurkat, PC12, and MCF-7. Our findings reveal that, within the photonic band gaps, the local density of states reaches a maximum, a characteristic of localized modes. We show that when the dielectric constant of each cell type is increased, the localized mode shifts to shorter frequencies, exhibiting a specific value for the local density of states. Furthermore, we report that the localized mode shifts to higher frequencies when pressure increases while reducing the local density of states.

In this study, Levinson-type inequalities for the class of n-convex (n ≥ 3 ) functions are generalized using new Green functions and the Hermite interpolating polynomial involving two types of data points. Some estimations for novel functionals are derived using f-divergence. Furthermore, different inequalities involving Shannon entropies are presented. [ABSTRACT FROM AUTHOR]

We generalize Popoviciu's inequality utilizing the theory of 4-convex function under the effect of new Green functions. We formulate result for power means and quasi means. Also, we give applications in information theory by giving new estimations of generalized Csiszár divergence, Rényi divergence, Shannon entropy, Kullback-Leibler divergence, x² divergence, Zipf-Mandelbrot entropy, and Hybrid Zipf-Mandelbrot entropy. [ABSTRACT FROM AUTHOR]

Wang and Ye [Adv. Math. 230 (2012), pp. 294–320] prove a Hardy–Moser–Trudinger inequality in dimension two which improves both the classical Moser–Trudinger inequality and the classical Hardy inequality in the unit disc. In this paper, we generalize their result both to the higher dimensional unit ball and to the singular weighted cases as well. More precisely, we prove that \[ \sup _{\substack {u\in W^{1,n}_0(\mathbb {B}^n),\\ \int _{\mathbb {B}^n} |\nabla u|^n dx -\left (\frac {2(n-1)}n\right)^n \int _{\mathbb {B}^n} \frac {|u|^n}{(1-|x|^2)^n} dx \leq 1}}\int _{\mathbb {B}^n} e^{(1-\frac \beta n)\alpha _n |u|^{\frac n{n-1}}} |x|^{-\beta } dx \leq C \] for any \beta \in [0,n), n\geq 3 where \alpha _n = n \omega _{n-1}^{\frac 1{n-1}} and \omega _{n-1} is the surface area of the n-1 dimensional unit sphere. The proof of Wang and Ye is based on the blow-up analysis method which seems not work in the higher dimensions. In this paper, we propose a new approach based on the method of transplantation of Green's functions to prove our inequality. As a consequence, we obtain a singular Moser–Trudinger inequality in the hyperbolic spaces which confirms affirmatively a conjecture by Mancini, Sandeep and Tintarev [Adv. Nonlinear Anal., 2 (2013), pp. 309–324]. [ABSTRACT FROM AUTHOR]

Abstract In this paper, four new Green functions are used to generalize Levinson-type inequalities for the class of 3-convex functions. The f-divergence, Renyi entropy, Renyi divergence, Shannon entropy, and the Zipf–Mandelbrot law are also used to apply the main results to information theory.

The main aim of this paper is to prove several majorization type inequalities using Green and 4-convex functions. First of all, we drive generalized majorization inequality for arbitrary n-tuples and real weights. Further, we explore the inequality for majorized tuples, weighted majorization theorems given by Fuchs, Dragomir and Maligranda et al. For deriving another generalized majorization inequality, we use a simple form of Jensen's inequality, and by similar fashion we apply classical earlier majorization theorems for further elaborations of generalized inequality. Several applications of information theory are discussed at the end of the article. [ABSTRACT FROM AUTHOR]

We consider some elementary features of Newtonian gravity, or electrostatics, as defined on an N-sphere. In particular, we present and discuss "the shell theorem" for this system. [ABSTRACT FROM AUTHOR]

This paper develops the bitensorial formulation of the system of singularities associated with unbounded and bounded Stokes flows. The motivation for this extension is that Stokesian singularities and hydrodynamic fundamental solutions are multi-point functions, and bitensor calculus provides either the proper geometrical setting, in order to avoid inconsistencies and misunderstandings on the role of the different tensorial indices, or a way for compactly deriving hydrodynamic properties. A first relevant result is to provide a clear definition of the singularities (both bounded and unbounded) in Stokes flow, specifying the associated differential equations and boundary conditions. Using this formalism for bounded flows, we show the existence of an integro-differential operator providing the whole system of hydrodynamic singularities by acting on the unbounded Green function (Stokeslet) at its pole and we derive its explicit representation in terms of moments. In the case of an immersed body in a unbounded fluid, we show that, the operator furnishing the disturbance field of a purely $ n $-th order ambient flow, is a generalized $ n $-th order Faxén operator, i.e., it yields the $ n $-th moment on the body if applied to a generic ambient flow, and that a generic disturbance field can be expressed by a summation of the generalized $ n $-th order Faxén operators. Furthermore, we find that the operator providing the disturbance of an ambient flow coincides with the reflection operator for the Stokes solutions in the same flow geometry. We apply this result to the paradigmatic case of fundamental singularities for the Stokes flow bounded by a plane. In this way, we obtain in an alternative and easy way the image system for the Sourcelet and the Rotlet (already derived in the literature) and for the Source Doublet and the Strainlet (presented here for the first time).

In this work we present a formalism based on scalar Green’s functions to deal with electromagnetic scattering problems. Although the formulations of the Mie theory and Born approximations in terms of electromagnetic scattering are well known and relevant, they have certain disadvantages: complexity, computational time, few symmetries, etc. Therefore, the study with scalar Green’s functions allows dealing with these problems with greater simplicity and efficiency. However, the information provided by the vector formulation is sacrificed. Nevertheless, different cases of electromagnetic scattering of dielectric media with different dimensions, geometries and refractive indices will be presented. Thus, we will be able to verify the capacity of this scalar method in predicting light-scattering problems.

We generalize cyclic Jensen's inequality utilizing the theory of 4-convex function under the effect of introduced Green functions. We formulate result for power means and quasi means. Also we give applications in information theory by giving new estimations of generalized Csiszár divergence, Rényi-divergence, Shannon-entropy, Kullback-Leibler divergence and χ2-divergence. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to derive a new perturbation theory (PT) that has converging series. Such series arise in the nonlocal scalar quantum field theory (QFT) with fractional power potential. We construct a PT for the generating functional (GF) of complete Green functions (including disconnected parts of functions) Z j as well as for the GF of connected Green functions G j = ln Z j in powers of coupling constant g. It has infrared (IR)-finite terms. We prove that the obtained series, which has the form of a grand canonical partition function (GCPF), is dominated by a convergent series—in other words, has majorant, which allows for expansion beyond the weak coupling g limit. Vacuum energy density in second order in g is calculated and researched for different types of Gaussian part S 0 [ ϕ ] of the action S [ ϕ ] . Further in the paper, using the polynomial expansion, the general calculable series for G j is derived. We provide, compare and research simplifications in cases of second-degree polynomial and hard-sphere gas (HSG) approximations. The developed formalism allows us to research the physical properties of the considered system across the entire range of coupling constant g, in particular, the vacuum energy density. [ABSTRACT FROM AUTHOR]

In this paper, four new Green functions are used to generalize Levinson-type inequalities for the class of 3-convex functions. The f-divergence, Renyi entropy, Renyi divergence, Shannon entropy, and the Zipf–Mandelbrot law are also used to apply the main results to information theory. [ABSTRACT FROM AUTHOR]

This paper investigates the Einstein relation; the connection between the volume growth, the resistance growth and the expected time a random walk needs to leave a ball on a weighted graph. The Einstein relation is proved under different set of conditions. In the simplest case it is shown under the volume doubling and time comparison principles. This and the other set of conditions provide the basic framework for the study of (sub-) diffusive behavior of the random walks on weighted graphs. [ABSTRACT FROM AUTHOR]

We consider discrete (time and space) random walks confined to the quarter plane, with jumps only in directions (i , j) with i + j ≥ 0 and small negative jumps, i.e., i , j ≥ − 1. These walks are called singular, and were recently intensively studied from a combinatorial point of view. In this paper, we show how the compensation approach introduced in the 90ies by Adan, Wessels and Zijm may be applied to compute positive harmonic functions with Dirichlet boundary conditions. In particular, in case the random walks have a drift with positive coordinates, we derive an explicit formula for the escape probability, which is the probability to tend to infinity without reaching the boundary axes. These formulas typically involve famous recurrent sequences, such as the Fibonacci numbers. As a second step, we propose a probabilistic interpretation of the previously constructed harmonic functions and prove that they allow us to compute all positive harmonic functions of these singular walks. To that purpose, we derive the asymptotics of the Green functions in all directions of the quarter plane and use Martin boundary theory. [ABSTRACT FROM AUTHOR]

We propose a method for the determination of static thermoelastic states of multilayer bodies of canonical shapes with regard for the thermal radiation, convective heat exchange, and arbitrary temperature dependences of the physical and mechanical characteristics of the material under the action of surface and volume heat sources. For the solution of the corresponding heat-conduction and thermoelasticity problems with piecewise constant characteristics, we use the Kirchhoff transformation, Newton iterative method, generalized functions, and Green functions. The results of numerical investigations are presented. [ABSTRACT FROM AUTHOR]

Under the action of a stationary heat source, we construct the Boussinesq functions of the problem of thermoelasticity under the conditions of plane deformation of a semiinfinite body with free, rigidly, smoothly, or flexibly fastened boundary either kept at zero temperature or thermally insulated. The problem of construction of Boussinesq functions is reduced to solving boundary-value problems for harmonic functions in the half space. Relations for displacements and stresses are obtained. These relations are the corresponding Green functions and can be used to determine the thermoelastic state of the half space under the conditions of heat release in a ribbon-like domain. [ABSTRACT FROM AUTHOR]

Some problems of vibrational spectroscopy of particles in condensed systems are considered in this work. One of the aspects of theoretical research is the study of the vibrational properties of individual particles in view of the nano-dimension of the molecules of the condensed system surrounding the particle. Using the apparatus of temperature, Green functions of the operators of polarization of condensed systems, two main mechanisms of influence on impurity particles from the medium, solvation and fluctuation, are distinguished. Theoretical results are obtained within the framework of these two mechanisms for calculating changes in the vibrational spectrum of individual particles. The theoretical results are used to analyze the experimental data on the vibrational spectra of the N2O molecule in polar solvents: methanol, ethyl alcohol, acetone, and 1,2-dichloroethane. [ABSTRACT FROM AUTHOR]

Fault slip rate distribution plays an important role in earthquake studies. Faults are loaded at very slow rates in continental interiors. So, interaction among faults and resulting slip distribution can give rise to earthquakes on other faults after a long period of quiescence and seismicity that can migrate from one fault to the onother one. NW Iran-Eastern Turkey is a region of active deformation as a result of oblique collision of Arabia-Eurasia tectonic plates. In northwestof Iran, deformation between the Central Iranian block and the Caucasus domain is accommodated by a fault system and mainly by right lateral strike-slip on the North Tabriz fault. In the current study, we did slip rate partitioning in the fault system of northwest Iranian plateau using the concepts of dislocation theory. Modelling approach is described by Gomberg and Ellis (1994), Flerit et al., (2003) and Armijo et al., (2004) and it differs from rigid block models (Reilinger et al., 2006; Djamour et al., 2011) in which dislocation conditions at the boundaries of blocks are often incompatible with geological evidences. In the alternative method of Flerit et al., (2003), slip everywhere has a direction of motion consistent with geological constraints. The dislocations do not divide the region into closed rigid blocks and slip can vary along strike as observed geologically. Finally we obtain a tectonic model for NW Iran-Eastern Turkey that is more realistic than rigid block model (Reilinger et al., 2006; Djamour et al., 2011) or models based on seismic or geologic strain rates (Haines, 1982; Haines and Holt, 1993; Jackson et al., 1995; Masson et al., 2005). For this purpose we use a three dimensional boundary elements method. First, we consider an elastic and homogeneous half-space for the study area. Then geometric data of fault system are collected from geological and geophysical sources including fault length, width, dip, and locking depth. For Lame coefficients, we use global average values. Both mentioned geometrical and physical data are kept fixed in the modeling process. Then, strain tensor that best fits the GPS data is estimated for the study area using least squares method. Then, stress rate tensor is estimated using generalized Hook's law. Geomerical chracteristics of faults, physical characteristics of crust and stress rate tensor act as boundary conditions in the model. Faults are locked in normal direction but they are allowed to slip freely in strike and dip directions under the influence of boundary conditions. Regarding the strike changes of faults, the fault surfaces are divided by different segments in strike direction with constant strikes and dips. Then fault segment surfaces are divided into 1km elements. Finally, we have free slipping elements in strike and dip directions as inputs for modeling. Our model is fitted to the fault traces data set of NW Iran-eastern Turkey. The results indicate the dependency of the partitioned slip rate on the boundary conditions and confirm the existence of interaction among faults. Also, partitioned slip rates show that the Chalderan, Guilato-Siahcheshmeh-Khoy, Nakhchivan, North Tabriz and Pambak-Sevan-Sunik faults are right-lateral strike slip in all cases. Also, the slip rate in these faults is almost symmetric and reaches its maximum value around the center of the faults. We show that the maximum value of slip rate in the fault plane is reduced by partitioning, which it will be definitely closer to reality. According to the gridding for slip rate partitioning in the fault system, the highest value of slip rate is always related to the North Tabriz Fault. Previous studies show that the geological slip rate estimates are lower than the present-day GPS-derived slip-rates along the North Tabriz fault. We show that slip rate partitioning solves this discrepancy by considering the mechanical interaction among faults. Our partitioned slip rates for North Tabriz Fault are lower than geodetic rates and are more consistent with geological rates. Finally, we present a model that fits best with the geological constraints. The proximity of the partitioned slip rate to the paleo-seismic values indicates the closeness of the partitioning results to reality with the Boundary Elements Method, compared to other analytical and numerical methods. This research may open new research direction to handle the differene between geologic and geodetic slip rates values in the Iranian Plateau. The boundary elements method is both faster and more accurate for modeling compared to the finite element method used by Khodaverdian et al. (2015). Considering the effect of topography and sphericity of Earth, using the Galerkin boundary element method developed by Thompson (2019) is proposed to get more realistic results. The coefficients matrix in the of Boundary Elements Method is fully populated and in high dimensions it takes a lot of time to solve the resulting system of equations. Sparsing of the coefficient matrix using wavelet transforms is suggested (Ebrahimnejad et al., 2010) in this study. The use of iterative computational methods along with parallel processing will also reduce the computational time (Thompson and Meade, 2019). [ABSTRACT FROM AUTHOR]

The design of devices with coupled hybrid structures offers an approach to creating synthetic topological materials. This work discusses the topological and transport properties of low-dimensional binary heterostructures of topological and trivial materials. By adjusting the parameters of each component, we control the global topological properties to enhance tunneling and optimize the transmission of the topological edge states (TES). Considering a one-dimensional tight-binding model, we build heterostructures of coupled chains employing Green's functions (GF) formalism. We determine the topological characteristics of chains and couple them together, applying Dyson's equation to generate the heterostructure. The intensity and decay length of the TES vary depending on the coupling parameters and the size of each chain. We investigate the topological diagrams phase using the energy bands of the periodic system and calculating the invariant from the Zak phase. Using cross-band condition, we derive analytical functions of the parameter space to get the phase topological diagram, which can be compared with the LDOS maps at zero energy. Finally, we calculate the differential conductance with the Keldysh GF technique to demonstrate the tunneling of the TES at the zero bias voltage and discuss potential design and experimental applications. [ABSTRACT FROM AUTHOR]

We study the Cauchy-Dirichlet Problem (CDP) for a nonlinear and nonlocal diffusion equation of singular type of the form $ \partial_t u = - \mathcal{L} u^m $ posed on a bounded Euclidean domain $ \Omega\subset \mathbb{R}^N $ with smooth boundary and $ N\ge 1 $. The linear diffusion operator $ \mathcal{L} $ is a sub-Markovian operator, allowed to be of nonlocal type, while the nonlinearity is of singular type, namely $ u^m = |u|^{m-1}u $ with $ 0

We study semilinear problems in bounded $ C^{1, 1} $ domains for non-local operators with a boundary condition. The operators cover and extend the case of the spectral fractional Laplacian. We also study harmonic functions with respect to the non-local operator and boundary behaviour of Green and Poisson potentials. [ABSTRACT FROM AUTHOR]

In this paper, a relationship between the spectral zeta series of a class of higher order self-adjoint differential operators on the unit circle and the integral of Green functions is established by Mercer's Theorem. Furthermore, the explicit expression and the transcendental nature of the spectral series are obtained by the integral representation. Finally, several applications in physics about differential operators' spectral theory, yellow some further works, and the transcendental nature of some zeta series are listed. [ABSTRACT FROM AUTHOR]

For k-Riemann–Liouville fractional integral operators, the Hermite–Hadamard inequality is already well-known in the literature. In this regard, this paper presents the Hermite–Hadamard inequalities for k-Riemann–Liouville fractional integral operators by using a novel method based on Green's function. Additionally, applying these identities to the convex and monotone functions, new Hermite–Hadamard type inequalities are established. Furthermore, a different form of the Hermite–Hadamard inequality is also obtained by using this novel approach. In conclusion, we believe that the approach presented in this paper will inspire more research in this area. [ABSTRACT FROM AUTHOR]

The aim of this paper is to investigate the existence and uniqueness of solutions of the following boundary value problem concerning generalized ODEs dx d τ = D [ A (t) x + F (t) ] , ∫ a b d [ K (s) ] x (s) = r , for operators taking values in general Banach spaces. Up to now, this problem was only tackled in the case of finite dimensional space-valued functions. We establish necessary and sufficient conditions not only for the existence of at least one solution, but also for the uniqueness of a solution. Another important result describes the solution in terms of a Green function and the fundamental operator of the corresponding homogeneous problem. We also explore the problem dx d τ = D [ A (t) x + F (t) ] and we give necessary and sufficient conditions for the existence of a (θ , T) -periodic solution, where θ ∈ R , with θ ≠ 0 , and T > 0 , recalling that the notion of (θ , T) -periodic solution generalizes the notions of periodic, anti-periodic, almost periodic, and quasi-periodic solution. Examples are given to illustrate the main results. In addition, we apply the main theorems to abstract ODEs, as well as to a Volterra-Stieltjes-type integral equation, and we include examples of these as well. [ABSTRACT FROM AUTHOR]

In this paper, we consider a class of delay differential systems with two-point boundary value conditions. First, we give a criterion to guarantee the existence and uniqueness of solutions for the system. Then, by making use of lower order Newton–Cotes rules, we present a computer-oriented iterative scheme to approximate the exact solution. The efficiency of algorithm is verified by two examples. [ABSTRACT FROM AUTHOR]

We report the one-electron spectrum and eigenstates of infinite achiral and chiral (2m,m) carbon nanotubes found by using the analytic solution to the Schrödinger equation for the tight-binding Huckel-type Hamiltonian. With the help of matching the wave functions on the interfaces between the regions, where electrons have different site energies, we find and compare the transmission coefficients for zigzag, armchair and chiral nanotubes subjected to the action of an applied step-like potential. The correspondence between the nanotube band structure and the energy dependence of the transmission coefficient is demonstrated. It is shown that the (2m,m) nanotubes with a medium chiral angle reveal intermediate transport properties as compared with the achiral armchair, and zigzag nanotubes. [ABSTRACT FROM AUTHOR]

We give an elementary and self-contained proof of the uniformization theorem for noncompact simply connected Riemann surfaces. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to give a direct proof of an eigenfunction expansion formula for one-dimensional two-state quantum walks, which is an analog of that for Sturm–Liouville operators due to Weyl, Stone, Titchmarsh, and Kodaira. In the context of the theory of CMV matrices, it had been already established by Gesztesy–Zinchenko. Our approach is restricted to the class of quantum walks mentioned above, whereas it is direct and it gives some important properties of Green functions. The properties given here enable us to give a concrete formula for a positive-matrix-valued measure, which gives directly the spectral measure, in a simplest case of the so-called two-phase model. [ABSTRACT FROM AUTHOR]

Recent developments in nanoscale experimental techniques made it possible toutilize single molecule junctions as devices for electronics and energytransfer with quantum coherence playing an important role in theirthermoelectric characteristics. Theoretical studies on the efficiency ofnanoscale devices usually employ rate (Pauli) equations, which do not accountfor quantum coherence. Therefore, the question whether quantum coherence couldimprove the efficiency of a molecular device cannot be fully addressed withinsuch considerations. Here, we employ a nonequilibrium Green function approachto study the effects of quantum coherence and dephasing on the thermoelectricperformance of molecular heat engines. Within a generic bichromophoricdonor-bridge-acceptor junction model, we show that quantum coherence mayincrease efficiency compared to quasi-classical (rate equation) predictions andthat pure dephasing and dissipation destroy this effect.

The goal of this study is to calculate the eigenvalues that provide the eigenfrequencies and the critical loads for two heterogeneous beams with three supports: the (first) [second] beam is (fixed)[pinned] at the left end, the intermediate support is a roller while the right end of the beams can move vertically but the rotation is prevented there. The beams are referred to as FrsRp and PrsRp beams. Determination of the (eigenfrequencies) [critical loads] leads to three point eigenvalue problems associated with homogeneous boundary conditions. With the Green functions that belong to these eigenvalue problems we can transform them into eigenvalue problems governed by homogeneous Fredholm integral equations. The eigenvalue problems can then be reduced to algebraic eigenvalue problems that are solvable numerically by utilizing effective solution algorithms. [ABSTRACT FROM AUTHOR]

This study considers the core issue of evaluating flows created by general distributions of sources and dipoles over panels (typically flat/curved triangles/quadrilaterals) used to approximate the surface of a body (ship or offshore structure) in usual implementations of the Green function and boundary integral method in marine hydrodynamics. This crucial basic issue is considered – within the framework of the Fourier–Kochin (FK) method – for four classes of flows in deep-water ship and offshore hydrodynamics: diffraction-radiation of regular waves by an offshore structure, and flow around a ship that advances at a constant speed V in calm water or in regular waves of (encounter) frequency ω in the regimes τ ≡ ω V / g < 1 / 4 or 0. 3 ≤ τ where g is the acceleration of gravity; diffraction-radiation of regular waves by an offshore structure in water of uniform finite depth is also considered. Two notable features of the Green functions G used in the study of these five classes of flows are that (i) they are based on optimal decompositions into Rankine and Fourier components G R and G F , and that (ii) these Green functions are consistent. In particular, a new representation of the Green function for flows around ships advancing in regular waves at τ < 1 / 4 that is consistent with the Green functions for the special cases V = 0 or ω = 0 is given. The study also provides a complete and largely self-contained account of the FK method for evaluating the Fourier component ϕ F in the Rankine–Fourier decomposition ϕ R + ϕ F of the velocity potential ϕ (and the related velocity) of the flow created by a general distribution of singularities. An essential element of this FK theory is an optimal waves/local-effects (WL) decomposition ϕ F = ϕ W + ϕ L , given for a general dispersion relation associated with a broad class of dispersive waves, in which the component ϕ W represents the (far-field and near-field) waves contained in ϕ F and the component ϕ L corresponds to a non-oscillatory local disturbance that is mostly significant in a small near-field region. Applications of this general WL decomposition to the five classes of flows in marine hydrodynamics of primary interest in the study yield simple analytical flow representations – for general compact distributions of singularities – that offer two notable advantages over the classical direct Green function method: (i) Integration over hull-surface panels within the FK method only involves smooth ordinary functions , which are incomparably simpler than the highly singular functions G F and ∇ G F , and (ii) The computations related to G F and ∇ G F in the FK theory – implemented in the manner expounded in the study – are proportional to the number N of panels that approximate the body surface, whereas common panel methods require O (N 2) computations. Thus, the Fourier–Kochin method and the optimal Rankine–Fourier decompositions and optimal waves/local-flow decompositions given in the study lay the foundation of a new type of computational methods, which offers two compelling advantages over the classical 'direct Green function method' steadfastly applied in the past fifty years. [ABSTRACT FROM AUTHOR]

In this paper, we present the q-Lidstone polynomials which are q-Bernoulli polynomials generated by the third Jackson q-Bessel function, based on the Green's function of a certain q-difference equation. Also, we provide the q-Fourier series expansions of these polynomials and derive some results related to them. [ABSTRACT FROM AUTHOR]

In the work, the process of unsteady contact interaction of rigid stamp and elastic half-space having a recessed cavity of arbitrary geometry and location with a smooth boundary was investigated. Three variants of contact conditions are considered: free slip, rigid coupling, and bonded contact. The method for solving the problem is constructed using boundary integral equations. To obtain boundary integral equations, the dynamic reciprocal work theorem is used. The kernels of integral operators are bulk Green functions for the elastic plane. Because of straight-line approximations of the domain boundaries with respect to the spatial variable and straight-line approximations of the boundary values of the desired functions with respect to time, the problem is reduced to solving a system of algebraic equations with respect to the pivotal values of the desired displacements and stresses at each time interval. One of the axes is directed along the regular boundary of half-space, the second - deep into half-space.