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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]

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]

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 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 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.

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]

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.

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.

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).

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]

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]

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]

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.

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.

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.

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.

We discuss the non-equilibrium dynamics of condensed matter/quantum field systems in the framework of Keldysh technique. In order to deal with the inhomogeneous systems we use the Wigner-Weyl formalism. Unification of the mentioned two approaches is demonstrated on the example of Hall conductivity. We express Hall conductivity through the Wigner transformed two-point Green's functions. We demonstrate how this expression is reduced to the topological number in thermal equilibrium at zero temperature. At the same time both at finite temperature and out of equilibrium the topological invariance is lost. Moreover, Hall conductivity becomes sensitive to interaction corrections. [ABSTRACT FROM AUTHOR]

We consider a projection operator approach to the non-equilibrium Green function equation-of-motion (PO-NEGF EOM) method. The technique resolves problems of arbitrariness in truncation of an infinite chain of EOMs and prevents violation of symmetry relations resulting from the truncation (equivalence of left- and right-sided EOMs is shown and symmetry with respect to interchange of Fermi or Bose operators before truncation is preserved). The approach, originally developed by Tserkovnikov (1999 Theor. Math. Phys. 118 85) for equilibrium systems, is reformulated to be applicable to time-dependent non-equilibrium situations. We derive a canonical form of EOMs, thus explicitly demonstrating a proper result for the non-equilibrium atomic limit in junction problems. A simple practical scheme applicable to quantum transport simulations is formulated. We perform numerical simulations within simple models and compare results of the approach to other techniques and (where available) also to exact results.

Abstract In this paper, Levinson type inequalities are studied for the class of higher order convex functions by using Abel–Gontscharoff interpolation. Cebyšev, Grüss, and Ostrowski-type new bounds are also found for the functionals involving data points of two types.

One of the most accurate approaches to the lightning-related transient analysis of grounding systems in layered soil is based on the method of moments solution of the integral form of the Maxwell equations. The practical application of this rigorous model to large systems is hindered by a great amount of computer time required for the numerical integration of oscillatory and slowly convergent Sommerfeld integrals (SI). As a result, approximate methods that avoid the solution of SI, e.g., image theory-based methods, are often used for computation of the response of large grounding systems. In this article, we extend the application of the rigorous model for large grounding systems in multilayered soil by applying a procedure that minimizes the number of direct computations of the SI using interpolation over a grid of a small number of sample points. We implement a three-dimensional interpolation scheme that adapts to the problem to minimize the interpolation grid, maintaining the accuracy of the rigorous approach. The developed model is applied for the analysis of transient voltages that might couple to secondary cables and disrupt the operation of connected equipment in case of lightning. The analysis shows that there are cases when the image theory-based models might underestimate the possibly harmful transient voltages. [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.

Nutrient transport and productivity teleconnections with the Southern Ocean are diagnosed in a data-assimilated circulation model coupled to a jointly optimized simple phosphorus cycling model. The North Atlantic has the strongest extratropical teleconnections with the Southern Ocean: phosphate (PO4) last utilized in the Southern Ocean sustains 29 ± 6% of the subpolar and 14 ± 6% of the subtropical, new production in the North Atlantic. A PO4 path-density diagnostic shows that these teleconnections are mediated by thermocline paths and reveals that most paths to anywhere north of 40°S lie in the deep Pacific. Forcing nearly complete Southern-Ocean nutrient utilization increases the overall number of paths to anywhere north of 40°S, but reduces the number of paths from the Pacific to the North Atlantic by trapping nutrients in return paths to the Southern Ocean. At the same time, the mean export-to-uptake transit times to anywhere north of 40°S increase, while the mean transit times to the North Atlantic decrease. Correspondingly, the amount of North-Atlantic production sustained by Southern-Ocean export increases in spite of decreased total production in response to Southern-Ocean nutrient trapping. The distributions of export-to-uptake, export-to-surface, and surface-to-surface transit times are computed and summarized in terms of their mean transit times and their mean interior residence times. The combined particle and advective–diffusive transport of nutrients is characterized by broad, skewed transit-time distributions, which result in mean residence times much longer than the mean transit times, in turn much longer than the most probable transit times.

In this paper, we consider the definition of “useful” Csiszár divergence and “useful” Zipf-Mandelbrot law associated with the real utility distribution to give the results for majorizatioQn inequalities by using monotonic sequences. We obtain the equivalent statements between continuous convex functions and Green functions via majorization inequalities, “useful” Csiszár functional and “useful” Zipf-Mandelbrot law. By considering “useful” Csiszár divergence in the integral case, we give the results for integral majorization inequality. Towards the end, some applications are given.

A mathematical theory is developed for constructing integral transformations in a partially bounded region with a radial heat flow - a massive body bounded from the inside by a cylindrical cavity. Constructed: an integral transformation, the image of the operator on the right side of the equation of unsteady heat conduction, the inversion formula for the image of the desired function. The proposed approach favorably differs from the classical theory of differential equations of mathematical physics for the construction of generalized integral transformations based on the eigenfunctions of the corresponding singular Sturm-Liouville problems. The developed method is based on the operational solution of the initial boundary problems of unsteady heat conduction with an initial function of a general form L2(r0,∞) belonging to the r > r0 region and homogeneous boundary conditions and is associated with the calculation of the Riemann-Mellin contour integrals from images containing various combinations of modified Bessel functions. At the same time, for the above-mentioned region, the method of Green's functions was developed by constructing integral representations of analytical solutions of the first, second and third boundary value problems through inhomogeneities in the initial formulation of the problem (boundary conditions, source function in the initial equation). Mathematical models for finding the corresponding Green's functions are formulated, and functional relations of all three Green functions included in the presented integral formula are written out with the help of the developed theory of integral transformations. The functional relations constructed in the article can be used when considering numerous special cases of practical thermal physics. The specific possible applications of the presented results in many areas of science and technology are given.

This paper deals with structural static or stability analysis of straight beams, when the bending-torsion coupling is considered. A practical case is the lateral buckling analysis of high cross-section beams which involves the coupled bending deflection and twist. Simple rectangular cross-section or thin walled open profile symmetric cross-section beams can buckle laterally when they are subjected to bending with respect to the stiffer plane. Bending-torsion coupling occurs also for some composite beams. The presented approximate integral method uses flexibility influence functions (Green functions) and a matrix formulation, leading to an eigenvalue problem in the case of stability analysis. The formulation can be also used for static or dynamic analysis. Some numerical examples are presented in comparison with the results of other methods.

Let G (q) be a finite group of Lie type over a field with q elements, where q is a prime power. The Green functions of G (q) , as defined by Deligne and Lusztig, are known in almost all cases by work of Beynon–Spaltenstein, Lusztig und Shoji. Open cases exist for groups of exceptional type E 6 2 , E 7 , E 8 in small characteristics. We propose a general method for dealing with these cases, which proceeds by a reduction to the case where q is a prime and then uses computer algebra techniques. In this way, all open cases in type E 6 2 , E 7 are solved, as well as at least one particular open case in type E 8. [ABSTRACT FROM AUTHOR]

A new finite element method/boundary element method (FEM/BEM) scheme is proposed for the solution of the 2D magnetic static and quasi-static problems with unbounded domains. The novelty is an original approach in the treatment of the outer region. The related domain integral is eliminated at the discrete level by using the finite element approximation of the fundamental solutions (Green’s functions) at every node of the related mesh. This “FEM-Green” approach replaces the standard boundary element method. It is simpler to implement because no integration on the boundary of the domain is required. Then, the method leads to a substantially reduced computational burden. Moreover, the coupling with finite elements is more natural since it is based on the same Galerkin approximation. Some examples with open boundary and nonlinear materials are presented and compared with the standard finite element method.

This work presents a synthesis of the use of an integral approximate method based on structural influence functions (Green’s functions) concerning the behavior of beam-like structures. This integral method is used in the areas of static, dynamic, aeroelasticity and stability analysis. The method starts from the differential equations governing the bending or/and torsional behavior of a beam. These equations are put in integral form by using appropriate Green’s functions, according to the boundary conditions. Choosing a number of n collocation points on the beam axis, each integral are then computed by a summation using weighting numbers. This approach is suitable for conventional Euler-Bernoulli beams and also for the thin-walled open or closed cross-section beams which can have bending-torsion coupling. Generally, for a static analysis this approach leads to a linear system of equations (the case of the lift aeroelastic distribution analysis) or to an eigenvalues and eigenvectors problem in the case of dynamic, stability or divergence analysis.

This paper gives extensions and improvements of Sherman’s inequality for n-convex functions obtained by using new identities which involve Green’s functions and Fink’s identity. Moreover, extensions and improvements of Majorization inequality as well as Jensen’s inequality are obtained as direct consequences. New inequalities between geometric, logarithmic and arithmetic means are also established.

In this paper we introduce new effective resistances on a given network, associated with a positive value and two weights on the vertex set and investigate under which conditions they determine a distance. We prove that this property is closely related with superharmonicity. Moreover, we analyze the behavior of these distances under the usual network transformations, specially the so-called star-mesh transformation. We also compute the effective resistance for an extended network ; that is, the network obtained from the former one by joining a new vertex, and then study the effect of the contraction of this new network ; that is, we apply a star-mesh transformation with center in the joined vertex. [ABSTRACT FROM AUTHOR]

We aim here at determining the Green function for general Schrödinger operators on product networks. The first step consists in expressing Schrödinger operators on a product network as sum of appropriate Schrödinger operators on each factor network. Hence, we apply the philosophy of the separation of variables method in PDE, to express the Green function for the Schrödinger operator on a product network using Green functions on one of the factors and the eigenvalues and eigenfunctions of some Schrödinger operator on the other factor network. We emphasize that our method only needs the knowledge of eigenvalues and eigenfunctions of one of the factors, whereas other previous works need the spectral information of both factors. We apply our results to compute the Green function of P m × S h , where P m is a Path with m vertices and S h is a Star network with h + 1 vertices. [ABSTRACT FROM AUTHOR]

This paper deals with the study of the existence and non-existence of solutions of a three-parameter family of nonlinear fractional differential equation with mixed-integral boundary value conditions. We consider the α -Riemann-Liouville fractional derivative, with α ∈ ( 1 , 2 ] . To deduce the existence and non-existence results, we first study the linear equation, by deducing the main properties of the related Green functions. We obtain the optimal set of parameters where the Green function has constant sign. After that, by means of the index theory, the nonlinear boundary value problem is studied. Some examples, at the end of the paper, are showed to illustrate the applicability of the obtained results.

A non-stationary problem of determining the electromagnetic field components in a thick wall spherical shell according to the assumed axially symmetric displacement field is considered. A linearized model, which includes Maxwell’s equations and Ohm’s generalized law, is used. For solving the problem, the expansions in series are used, as well as Laplace’s transformation in the time domain and integral representations with nuclei in the form of Green’s functions. These functions are deduced in a quasi-static approximation. An analytical solution is described. A translational shell motion is provided as an example.

In this paper we consider the nonlinear third-order coupled system composed by the differential equations {0-u‷ (t) = f (t, u(t), u′(t), u‶(t), v(t), v′(t), v‶(t)), -v‷ (t) = h (t, u(t), u′(t), u‶(t), v(t), v′(t), v‶(t)), with f, h: [0, 1] × ⁕6 ℯ ⁕ continuous functions, and the boundary conditions{-u(0) = u′ (0) = u′ (1) = 0, v(0) = v′ (0) = v′ (1) = 0. We remark that the nonlinearities can depend on all derivatives of both unknown functions, which is new in the literature, as far as we know. This is due to an adequate auxiliary integral problem with a truncature, applying lower and upper solutions method with bounded perturbations. The main theorem is an existence and localization result, which provides some qualitative data on the system solution, such as, sign, variation, bounds, etc., as it can be seen in the example. [ABSTRACT FROM AUTHOR]

The identification of sources of advection–diffusion transport is based usually on solving complex ill-posed inverse models against the available state-variable data records. However, if there are several sources with different locations and strengths, the data records represent mixtures rather than the separate influences of the original sources. Importantly, the number of these original release sources is typically unknown, which hinders reliability of the classical inverse-model analyses. To address this challenge, we present here a novel hybrid method for identification of the unknown number of release sources. Our hybrid method, called HNMF, couples unsupervised learning based on Non-negative Matrix Factorization (NMF) and inverse-analysis Green’s functions method. HNMF synergistically performs decomposition of the recorded mixtures, finds the number of the unknown sources and uses the Green’s function of advection–diffusion equation to identify their characteristics. In the paper, we introduce the method and demonstrate that it is capable of identifying the advection velocity and dispersivity of the medium as well as the unknown number, locations, and properties of various sets of synthetic release sources with different space and time dependencies, based only on the recorded data. HNMF can be applied directly to any problem controlled by a partial-differential parabolic equation where mixtures of an unknown number of sources are measured at multiple locations. [ABSTRACT FROM AUTHOR]

This paper deals with structural static or stability analysis of straight beams, when the bending-torsion coupling is considered. A practical case is the lateral buckling analysis of high crosssection beams which involves the coupled bending deflection and twist. Simple rectangular crosssection or thin walled open profile symmetric cross-section beams can buckle laterally when they are subjected to bending with respect to the stiffer plane. Bending-torsion coupling occurs also for some composite beams. The presented approximate integral method uses flexibility influence functions (Green functions) and a matrix formulation, leading to an eigenvalue problem in the case of stability analysis. The formulation can be also used for static or dynamic analysis. Some numerical examples are presented in comparison with the results of other methods. [ABSTRACT FROM AUTHOR]

Yukawa scattering is pedagogically interpreted, by the Schwinger's quantum dynamical principle involving the generating function, which is replaced by a functional differential operation. As for the results, we get the asymptotically free Green function that explains the behavior of the Yukawa potential when the mass parameter is increasing and it can also lead to scattering amplitude and differential cross section respectively. [ABSTRACT FROM AUTHOR]

By using Fink’s Identity, Green functions, and Montgomery identities we prove some identities related to Steffensen’s inequality. Under the assumptions of n-convexity and n-concavity, we give new generalizations of Steffensen’s inequality and its reverse. Generalizations of some inequalities (and their reverse), which are related to Hardy-type inequality. New bounds of Gr u ¨ ss and Ostrowski-type inequalities have been proved. Moreover, we formulate generalized Steffensen’s-type linear functionals and prove their monotonicity for the generalized class of ( n + 1 ) -convex functions at a point. At the end, we present some applications of our study to the theory of exponentially convex functions.

The 2-D problem of interacting periodic set of edge dislocations and point forces with planar traction-free surface of semi-infinite elastic solid at the nanoscale is considered. Complex variable based technique and Gurtin-Murdoch model of surface elasticity, which leads to the hypersingular integral equation in surface stress, are used. The solution of this equation and explicit formulas for stress field (Green functions) are obtained in terms of Fourier series. The detailed numerical investigation of stress field induced by the dislocations at the nanometer distance from the surface and the force acting on each dislocation in classical and non-classical (with surface stress) solutions is presented. It is shown that formulas derived for the periodic set of dislocations can be applied to the analysis of the interaction of a single dislocation with the surface as well. The fundamental solutions obtained in the work can be used for applying the boundary integral equation method to an analysis of defects such as cracks and inhomogeneities, periodically distributed at the nanometer distance from the boundary. [ABSTRACT FROM AUTHOR]

This work presents a synthesis of the use of an integral approximate method based on structural influence functions (Green's functions) concerning the behavior of beam-like structures. This integral method is used in the areas of static, dynamic, aeroelasticity and stability analysis. The method starts from the differential equations governing the bending or/and torsional behavior of a beam. These equations are put in integral form by using appropriate Green's functions, according to the boundary conditions. Choosing a number of n collocation points on the beam axis, each integral are then computed by a summation using weighting numbers. This approach is suitable for conventional Euler-Bernoulli beams and also for the thin-walled open or closed cross-section beams which can have bending-torsion coupling. Generally, for a static analysis this approach leads to a linear system of equations (the case of the lift aeroelastic distribution analysis) or to an eigenvalues and eigenvectors problem in the case of dynamic, stability or divergence analysis. [ABSTRACT FROM AUTHOR]

This paper provides sufficient conditions to guarantee the existence, non-existence and multiplicity of solutions for a third order eigenvalue fully differential equation, coupled with three point boundary value conditions. Although the change of sign, some bounds for the second derivative of the Green's function are obtained, which allow to define a different kind of cone that, as far as we know, has not been previously used in the literature. The main arguments are based on the fixed point index theory for bounded and unbounded sets. Some examples are presented in order to show that the different existence theorems proved are not comparable. [ABSTRACT FROM AUTHOR]

Recent developments in nanoscale experimental techniques made it possible to utilize single molecule junctions as devices for electronics and energy transfer with quantum coherence playing an important role in their thermoelectric characteristics. Theoretical studies on the efficiency of nanoscale devices usually employ rate (Pauli) equations, which do not account for quantum coherence. Therefore, the question whether quantum coherence could improve the efficiency of a molecular device cannot be fully addressed within such considerations. Here, we employ a nonequilibrium Green function approach to study the effects of quantum coherence and dephasing on the thermoelectric performance of molecular heat engines. Within a generic bichromophoric donor-bridge-acceptor junction model, we show that quantum coherence may increase efficiency compared to quasi-classical (rate equation) predictions and that pure dephasing and dissipation destroy this effect. [ABSTRACT FROM AUTHOR]

This work studies the solvability of the nonlinear third order coupled system composed by the differential equations ... An adequate truncature together with Nagumo-type conditions allow the dependence of the nonlinearities on the second derivatives. By lower and upper solutions method we obtain strips where the unknown functions and their derivatives must lie, which provides some qualitative data on the solutions. [ABSTRACT FROM AUTHOR]

The inhomogeneous time-fractional telegraph equation with Caputo derivatives with constant coefficients is considered. For the considered equation, general representation of regular solution in rectangular domain is obtained and the fundamental solution is presented. Using this representation and the properties of the fundamental solution, the Cauchy problem and the main boundary value problems in half-strip and rectangular domains are studied. For the Cauchy problem theorems of existence and uniqueness of solution are proved, and the explicit form of the solution is constructed. The solutions of the investigated problems are constructed in terms of the appropriate Green functions, which are also constructed in explicit form. [ABSTRACT FROM AUTHOR]

We show that the anomalous diffusion equations with a fractional spatial derivative in the Caputo or Riesz sense are strictly related to the special convolution properties of the Lévy stable distributions which stem from the evolution properties of stretched or compressed exponential function. The formal solutions of these fractional differential equations are found by using the evolution operator method where the evolution is conceived as integral transform whose kernel is the Green function. Exact and explicit examples of the solutions are reported and studied for various fractional order of derivatives and for different initial conditions. [ABSTRACT FROM AUTHOR]