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This study delves into the thermomechanical vibration behavior of functionally graded porous nanoplates under extreme thermal temperature and humidity conditions. The equation of motion of the nanoplate was derived using advanced theories in elasticity and deformation. The nanoplate consists of metal (SUS304) on the bottom surface and ceramic (Ni3S4) on the top surface, with the material distribution changing according to the power law across the plate thickness. The nanoplate was modeled with uniform and symmetric distributions of porosity reaching as high as 60%. Upon incorporating the thermal and moisture loads from the humid surroundings into the equations of motion derived from Hamilton's principle, the equations were solved using Navier's method and simplified to the eigenvalue equation. Analyzed within a broad framework are the thermomechanical vibration behavior of the nanoplate, temperature impact, humidity influence, porosity and its distribution, material grading parameter effects, and nonlocal integral elasticity effects. Observations indicate that variations in thermal temperature, humidity, and nonlocal parameters can lower the thermomechanical vibration frequency of the nanoplate, whereas porosity has the opposite effect. The effects mentioned are influenced by factors, such as the porosity ratio, porosity distribution, material ratios, and the size of the nonlocal parameter in the plate. The primary objective of this work is to uncover the nonlinear frequency response of nanoplates with high porosity in conditions characterized by high temperature and humidity. [ABSTRACT FROM AUTHOR]

We study the transport properties of Dirac fermions in ABC trilayer graphene (ABC-TLG) superlattices. More specifically, we analyze the impact of varying the physical parameters-the number of cells, barrier/well width, and barrier heights-on electron tunneling in the ABC-TLG. In the initial stage, we solved the eigenvalue equation to determine the energy spectrum solutions for the ABC-TLG superlattices. Subsequently, we applied boundary conditions to the eigenspinors and employed the transfer matrix method to calculate transmission probabilities and conductance. For the two-band model, we identified the presence of Klein tunneling, with a notable decrease as the number of cells increased. The introduction of interlayer bias opened a gap as the number of cells increased, accompanied by an asymmetry in scattered transmission. Increasing the barrier/well width and the number of cells resulted in an amplified number of gaps and oscillations in both two-band and six-band cases. We observed a corresponding decrease in conductance as the number of cells increased, coinciding with the occurrence of a gap region. Our study demonstrates that manipulating parameters such as the number of cells, the width of the barrier/well, and the barrier heights provides a means of controlling electron tunneling and the occurrence of gaps in ABC-TLG. Specifically, the interplay between interlayer bias and the number of cells is identified as a crucial factor influencing gap formation and transmission asymmetry. [ABSTRACT FROM AUTHOR]

The SL (2 , Z) -symmetry of Cherednik's spherical double affine Hecke algebras in Macdonald theory includes a distinguished generator which acts as a discrete time evolution of Macdonald operators, which can also be interpreted as a torus Dehn twist in type A. We prove for all twisted and untwisted affine algebras of type ABCD that the time-evolved q-difference Macdonald operators, in the t → ∞ q-Whittaker limit, form a representation of the associated discrete integrable quantum Q-systems, which are obtained, in all but one case, via the canonical quantization of suitable cluster algebras. The proof relies strongly on the duality property of Macdonald and Koornwinder polynomials, which allows, in the q-Whittaker limit, for a unified description of the quantum Q-system variables and the conserved quantities as limits of the time-evolved Macdonald operators and the Pieri operators, respectively. The latter are identified with relativistic q-difference Toda Hamiltonians. A crucial ingredient in the proof is the use of the "Fourier transformed" picture, in which we compute time-translation operators and prove that they commute with the Pieri operators or Hamiltonians. We also discuss the universal solutions of Koornwinder-Macdonald eigenvalue and Pieri equations, for which we prove a duality relation, which simplifies the proofs further. [ABSTRACT FROM AUTHOR]

We develop the Euclidean time method of the variational quantum eigensolver for solving the generalized eigenvalue equation A | ϕ n ⟩ = λ n B | ϕ n ⟩ , where A and B are hermitian operators, and | ϕ n ⟩ and λ n are called the eigenvector and the corresponding eigenvalue of this equation, respectively. For the purpose, we modify the usual Euclidean time formalism, which was developed for solving the time-independent Schrödinger equation. We apply our formalism to three numerical examples for test. It is shown that our formalism works very well in all numerical examples. We also apply our formalism to the hydrogen atom and compute the electric polarizability. It turns out that our result is slightly less than that of the perturbation method. [ABSTRACT FROM AUTHOR]

The 3D problem became the basis for modeling the critical states and shapes of bifurcation buckling of longitudinal force and torque-stressed nanotubes. The homogeneous system of ordinary differential equations built on the theory of rectilinear elastic rods was formulated. Its nontrivial solutions result in critical torque or longitudinal force levels at preset values of one of those entities. The closed-form solutions under chosen boundary conditions indicate that only spatial curves as variable diameter spirals with the left or right orientation consistent with that of the torque can be stability loss modes. The spiral is single-wound without longitudinal force, regardless of the tube length. If the tube is also prestressed with axial compression or tension, the bifurcation spiral is multi-wound; its number is determined by the eigenvalue of the equations, which increases with growing forces and tube length. [ABSTRACT FROM AUTHOR]

In this work, perforated and restrained nanobeams with deformable boundary conditions are modeled based on the non-local strain gradient elasticity theory with the elastic medium effect. In this approach in which the Fourier infinite series and Stokes' transformation are used together, the nanobeam is detached into parts from the two boundary points with the main part. Then using elastic force boundary conditions, a system of linear equations in terms of nonlinearity, elastic medium parameter, spring coefficients and small size effects are derived and the eigenvalue solution of these equations is also presented. The nonlinear stability of restrained nanobeam is examined and the effects of size parameters, perforation and elastic spring coefficients are studied. To reveal the accuracy and effectiveness of the offered model, several numerical applications are solved for the nonlinear stability response of restrained nanobeams with elastic medium effects. The outcomes of this method validated that the presented approach is appropriate for the stability behavior of rigid and restrained nanobeams with perforated cross section. [ABSTRACT FROM AUTHOR]

The conveyance of fluid through pipes is a common occurrence in practical applications, and the dynamic stability of such systems has been a subject of interest for many decades. In this paper, we investigate the dynamic stability of a cantilevered pipe conveying fluid, considering the lateral distributed load as a critical factor. To achieve this, we employ the differential quadrature method to solve the eigenvalue equations for various load patterns. Remarkably, we observe excellent agreement with exact solutions and the Argand diagram. By analyzing the relationship between circular frequency and dimensionless velocity, we demonstrate that the descending distributed load pattern exhibits the highest stability. For certain conditions, it is possible to enhance stability. The results obtained in this study provide valuable insights for improving the stability of cantilevered pipe conveying fluid through a different approach. [ABSTRACT FROM AUTHOR]

This paper investigates the size-dependent vibration and buckling behavior of nanoplates described by the nonlocal Kirchhoff model analytically. The nonlocal and local constitutive relations, governing equations and boundary conditions are comprehensively discussed. A shifted governing equation in terms of local stress resultants with associated local boundary conditions is chosen in the current work. The direct separation of variables method is formulated based on the Hamiltonian dual form of the governing differential equation to address the eigenvalue problems of nonlocal rectangular plates with each edge either simply supported or clamped. The closed-form eigen-solutions are acquired for the first time for the cases where two adjacent clamped edges exist. The validity and accuracy of the present approach are verified by comparison with published analytical and numerical results. Parametric studies are performed to investigate the influences of nonlocal parameter and boundary conditions on the size-dependency of natural frequencies and critical buckling loads of nanoplates. [ABSTRACT FROM AUTHOR]

We introduce the unitary almost-Mathieu operator, which is obtained from a two-dimensional quantum walk in a uniform magnetic field. We exhibit a version of Aubry–André duality for this model, which partitions the parameter space into three regions: a supercritical region and a subcritical region that are dual to one another, and a critical regime that is self-dual. In each parameter region, we characterize the cocycle dynamics of the transfer matrix cocycle generated by the associated generalized eigenvalue equation. In particular, we show that supercritical, critical, and subcritical behavior all occur in this model. Using Avila's global theory of one-frequency cocycles, we exactly compute the Lyapunov exponent on the spectrum in terms of the given parameters. We also characterize the spectral type for each value of the coupling constant, almost every frequency, and almost every phase. Namely, we show that for almost every frequency and every phase the spectral type is purely absolutely continuous in the subcritical region, pure point in the supercritical region, and purely singular continuous in the critical region. In some parameter regions, we refine the almost-sure results. In the critical case for instance, we show that the spectrum is a Cantor set of zero Lebesgue measure for arbitrary irrational frequency and that the spectrum is purely singular continuous for all but countably many phases. [ABSTRACT FROM AUTHOR]

Several numerical methods allow the prediction of guided dispersion curves. Due to some drawbacks of such methods, e.g., the challenge of divergence, which heavily depends on partial waves, the Legendre polynomial (LP) approach is applied to the magneto-electro-elastic (MEE) alloy. This approach is based on polynomial approximation, which replaces the problem of solving a transcendental dispersion equation with an eigenvalue problem. To apply the LP method for MEE alloy, this paper presents a numerical solution to calculate the waveguide dispersion curves. The roots prediction overlaps with the convergence character of this polynomial approach, leading to accurate phase velocities of guided waves. The results indicate the existence of the obvious physical phenomenon: non-continuous furcation in guided dispersion curves. It is thus now possible to confirm this physical phenomenon, differently reported in available data. However, with an incremental change of structure from bilayer plate to alloy, non-continuous furcation for SH modes appears more obviously, which was not discussed before. With such a phenomenon, the uniaxial residual stresses can affect these new trajectories, where these stress states could play a useful role in controlling the dispersion curves. Thereby, a comparison of the guided dispersion curves in stressed and unstressed 40%BaTiO3-60%CoFe2O4 alloy answers this need and sets forth to explore the effects of initial stress. Results show that the effects of initial stresses on phase velocity change are significant and give useful information, although their influence on the non-continuous furcation phenomenon is neglected. [ABSTRACT FROM AUTHOR]

We consider the periodic solutions of the quasi-linear Euler–Bernoulli equation arising from the forced vibrations of an I-beam with hard-wired ends. The presence of inhomogeneous (forced) term leads to the destruction of the natural symmetry ( Z 2 symmetry) in the corresponding homogeneous problem, and the effect of destroying the symmetry has been an open question for the multiplicity of periodic solutions [(see Rabinowitz (Trans Am Math Soc 272:753-769, 1982)]. The aim of this paper is to analyze the effect of inhomogeneous term and shows the existence of multiple periodic solutions for the forced Euler–Bernoulli beam equation. In the situation with fixed ends considered here, this problem involves a bulky transcendental equation for the related eigenvalue problem. By a sophisticated analysis, we first obtain the asymptotic estimates for the eigenvalues and eigenfunctions. On this basis, we may then investigate the fundamental properties of the linear differential operator and the corresponding functional. Finally, by constructing a modified functional and using the minimax method, we establish a theorem on the existence of infinitely many periodic solutions for the nonlinear term having power-law growth but not requiring its homogeneity and monotonicity. [ABSTRACT FROM AUTHOR]

Let (M , g (t) , e - ϕ d ν) be a measure space and (g (t) , ϕ (t)) evolve by the rescaled List's extended Ricci flow. In this paper, we derive the evolution equations for first eigenvalue of the geometric operators - Δ ϕ + c S under the rescaled List's extended Ricci flow, where Δ ϕ is the Witten Laplacian, ϕ ∈ C ∞ (M) , S = R - α | ∇ ϕ | 2 , and R is the scalar curvature with respect to the metric g(t). As an application, we obtain several monotonic quantities along the rescaled List's extended Ricci flow. Our results are natural extensions of some known results for Witten Laplace operator under various geometric flows. [ABSTRACT FROM AUTHOR]

Hartree's self-consistent determination of the ground state of the Helium atom marks the beginning of computational many-electron theory. The nonlinear nature of the problem seems to suggest that an iterative self-consistent field (SCF) type of approach is the only numerical approach to solve this problem. We will show that such an assumption is wrong. To this end we will re-interpret the Hartree equation as a tensor eigenvalue problem, and then we systematically guide the reader through an algebraic solution of the Hartree equation for the trivial example of a Helium atom with a single Gaussian as a basis set. In a second step we will study the Helium atom based on a basis set with two Gaussians, where we will compare numerical results obtained from an SCF calculation with results obtained from an algebraic solution of the corresponding Hartree equation. We will see that both solutions turn out to be identical. Our results suggest that the use of algebraic rather than iterative numerical techniques could serve as an alternative numerical approach to study Hartree-type systems, as well as a whole variety of similar problems throughout the physical and chemical literature. [ABSTRACT FROM AUTHOR]

Purpose: Functionally Graded Material (FGM) are materials that have different properties in different regions due to the gradual change of chemical composition, distribution and orientation, or the size of the amplifier phase in one or more dimensions. Methods: In this paper, the vibrations of an FGM plate under the effect of axial and lateral loading are investigated using the Variational Iteration Method (VIM). For this purpose, first the mathematical modeling of FGM material properties is performed. The Hamilton principle is used for dynamic modeling of the problem. Results: Using the VIM, the vibrational motion equation of the plane is obtained. The Ritz numerical method is used to solve the obtained equation. By simplification, the vibration equation of this plane is extracted in matrix form. Conclution: By solving the equation of the eigenvalue corresponding to the dynamic state, the natural frequencies are obtained. In the works done in the past, the boundary conditions of the simply supported have been mainly examined. In the present research, the natural frequencies of an FGM plate obtained in different boundary conditions. [ABSTRACT FROM AUTHOR]

We investigate the properties of entanglement between two quantum dots (QDs) with an azimuthal angle difference in two different plasmonic waveguide systems where a cavity coupled to the QDs is included or not. The real space formalism and the concurrence are used in solving the eigenvalue equation and calculating the entanglement, respectively. We analyze the influence of azimuthal angle difference on the entanglement and propose several effective ways to achieve high entanglement by adjusting the detuning, the QD-cavity coupling strength, and so on. Moreover, comparing the entanglement in the two models, we demonstrate that the addition of cavity can improve the entanglement of two QDs. [ABSTRACT FROM AUTHOR]

A full-vectorial meshfree finite cloud mode solver is presented, for the first time, for the analysis of optical waveguides with full anisotropic permittivity tensors, where the interface can be regular or irregular. The computational window are divided along the dielectric interfaces into appropriate number of clouds with homogeneous refractive index profiles and then patched by using the continuity conditions of the longitudinal field components among clouds. Present formulations yield a nonlinear eigenvalue matrix equation, and it is solved using an efficient iterative scheme after transforming the matrix equation to a standard linear eigenvalue equation, in which the convergent solutions can be achieved after performing only four iterations. The proposed method employs an unconstrained interior node distribution scheme as well as a novel scheme for dealing with dielectric interfaces, in which two extra rows of nodes are added on either side of the interfaces, significantly improving computational accuracy and efficiency. Comparing results for a nematic liquid crystal core channel waveguide and an E7-nematic-liquid crystal waveguide in silicon V-groove with the finite-difference frequency-domain method and pseudospectral-based eigenvalue method highlight the proposed method versatility, efficiency and accuracy. [ABSTRACT FROM AUTHOR]

We start to discuss some aspects of the scattering theory for the Sturm–Liouville operator L : 1 y - D 2 + q . In particular, we pose and solve the problem of reconstructing the function q when y is fixed and when a set S of scattering data is given. In the meanwhile, several relations concerning the spectral properties of L and the solutions of the related eigenvalue equation are established. [ABSTRACT FROM AUTHOR]

We investigate the isogeometric analysis approach based on the extended Catmull–Clark subdivision for solving the PDEs on surfaces. As a compatible technique of NURBS, subdivision surfaces are capable of the refinability of B-spline techniques, and overcome the major difficulties of the interior parameterization encountered by the isogeometric analysis. The surface is accurately represented as the limit form of the extended Catmull–Clark subdivision, and remains unchanged throughout the h-refinement process. The solving of the PDEs on surfaces is processed on the space spanned by the Catmull–Clark subdivision basis functions. In this work, we establish the interpolation error estimates for the limit form of the extended Catmull–Clark subdivision function space on surfaces. We apply the results to develop the approximation estimates for solving multiple second-order PDEs on surfaces, which are the Laplace–Beltrami equation, the Laplace–Beltrami eigenvalue equation and the time-dependent Cahn–Allen equation. Numerical experiments confirm the theoretical results and are compared with the classical linear finite element method to demonstrate the performance of the proposed method. [ABSTRACT FROM AUTHOR]

The construction of conformal blocks for the analysis of multipoint correlation functions with N > 4 local field insertions is an important open problem in higher dimensional conformal field theory. This is the first in a series of papers in which we address this challenge, following and extending our short announcement in [1]. According to Dolan and Osborn, conformal blocks can be determined from the set of differential eigenvalue equations that they satisfy. We construct a complete set of commuting differential operators that characterize multipoint conformal blocks for any number N of points in any dimension and for any choice of OPE channel through the relation with Gaudin integrable models we uncovered in [1]. For 5-point conformal blocks, there exist five such operators which are worked out smoothly in the dimension d. [ABSTRACT FROM AUTHOR]

The possibility of using quantum computers for electronic structure calculations has opened up a promising avenue for computational chemistry. Towards this direction, numerous algorithmic advances have been made in the last five years. The potential of quantum annealers, which are the prototypes of adiabatic quantum computers, is yet to be fully explored. In this work, we demonstrate the use of a D-Wave quantum annealer for the calculation of excited electronic states of molecular systems. These simulations play an important role in a number of areas, such as photovoltaics, semiconductor technology and nanoscience. The excited states are treated using two methods, time-dependent Hartree–Fock (TDHF) and time-dependent density-functional theory (TDDFT), both within a commonly used Tamm–Dancoff approximation (TDA). The resulting TDA eigenvalue equations are solved on a D-Wave quantum annealer using the Quantum Annealer Eigensolver (QAE), developed previously. The method is shown to reproduce a typical basis set convergence on the example H 2 molecule and is also applied to several other molecular species. Characteristic properties such as transition dipole moments and oscillator strengths are computed as well. Three potential energy profiles for excited states are computed for NH 3 as a function of the molecular geometry. Similar to previous studies, the accuracy of the method is dependent on the accuracy of the intermediate meta-heuristic software called qbsolv. [ABSTRACT FROM AUTHOR]

The grating has a significant role in sensing applications. Similarly, the grating-assisted coupler has excellent potential in chemical sensing applications. The power coupling between two closely coupled waveguide couplers can be significantly tuned by incorporating grating between them. The grating has been taken of silica material with sinusoidal shape in variation. The grating layer is assumed to be embedded within the sensing layer while considering a changeable effective refractive index depending on the sensing layer substances. In the present paper, grating assisted directional coupler has been numerically analysed using its own developed MATLAB-based algorithm of finite difference method (FDM) scheme. FDM method has been applied to solve the Eigenvalue equation to obtain allowed Eigenvalues and corresponding Eigen vectors (TE and TM cases). In FDM, the analysis domain has been fine discretized into the mesh of 1-D equal spacing for reasonable accurate computation results. In experimental validation, Fibre Bragg grating (FBG) has been suspended between two high refractive index coupler regions, which act as a power coupling zone. Also, the coupling length has been changed from 5 to 20 μ m for tuning purposes and then optimized for grating parameters viz. length, period, etc. The whole structure is 2-Dimensional (x and y directions) with invariant in the y-direction. [ABSTRACT FROM AUTHOR]

We study modular invariants arising in the four-point functions of the stress tensor multiplet operators of the N = 4 SU(N) super-Yang-Mills theory, in the limit where N is taken to be large while the complexified Yang-Mills coupling τ is held fixed. The specific four-point functions we consider are integrated correlators obtained by taking various combinations of four derivatives of the squashed sphere partition function of the N = 2∗ theory with respect to the squashing parameter b and mass parameter m, evaluated at the values b = 1 and m = 0 that correspond to the N = 4 theory on a round sphere. At each order in the 1/N expansion, these fourth derivatives are modular invariant functions of (τ, τ ¯ ). We present evidence that at half-integer orders in 1/N , these modular invariants are linear combinations of non-holomorphic Eisenstein series, while at integer orders in 1/N, they are certain "generalized Eisenstein series" which satisfy inhomogeneous Laplace eigenvalue equations on the hyperbolic plane. These results reproduce known features of the low-energy expansion of the four-graviton amplitude in type IIB superstring theory in ten-dimensional flat space and have interesting implications for the structure of the analogous expansion in AdS5× S5. [ABSTRACT FROM AUTHOR]

We propose to reinterpret Einstein's field equations as a nonlinear eigenvalue problem, where the cosmological constant Λ plays the role of the (smallest) eigenvalue. This interpretation is fully worked out for a simple model of scalar gravity. The essential ingredient for the feasibility of this approach is that the classical field equations be nonlinear, i.e., that the gravitational field is itself a source of gravity. The cosmological consequences and implications of this approach are developed and discussed. [ABSTRACT FROM AUTHOR]

For Hermitian and non-Hermitian Hamiltonian matrices H, we present the Schr¨odinger equation for qudit (spin-j system, N-level atom) with the state vector |ψ〉 in a new form of the linear eigenvalue equation for the matrix ℋ = (H ⊗ 1N) and the probability eigenvector |p〉 identified with quantum states in the probability representation of quantum mechanics. We discuss the possibility to experimentally detect the difference between the system states described by the solutions, corresponding to the Schrödinger equation with Hermitian and non-Hermitian Hamiltonians, by measuring the probabilities of artificial spin-1/2 projections m = ± 1/2, sets of which are identified with qudit states. We show that different symmetries of systems, including 𝒫𝒯 -symmetry and broken 𝒫𝒯 -symmetry, are determined by a set of N complex eigenvalues of the Hamiltonian matrix H. [ABSTRACT FROM AUTHOR]

The Hartree–Fock (HF) equation for atoms with closed (sub)shells is transformed with the pseudospectral (PS) method into a discrete eigenvalue equation for scaled orbitals on a finite radial grid. The Fock exchange operator and the Hartree potential are obtained from the respective Poisson equations also discretized using the PS representation. The numerical solution of the discrete HF equation for closed-(sub)shell atoms from He to No is robust, fast and gives extremely accurate results, with the accuracy superior to that of the previous HF calculations. A very moderate number of 33 to 71 radial grid points is sufficient to obtain total energies with 14 significant digits and occupied orbital energies with 12 to 14 digits in numerical calculations using the double precision (64-bit) of the floating-point format.The electron density at the nucleus is then determined with 13 significant digits and the Kato condition for the density and s orbitals is satisfied with the accuracy of 11 to 13 digits. The node structure of the exact HF orbitals is obtained and their asymptotic dependence, including the common exponential decay, is reproduced very accurately. The accuracy of the investigated quantities is further improved by performing the PS calculations in the quadruple precision (128-bit) floating-point arithmetic which provides the total energies with 25 significant digits while using only 80 to 130 grid points. [ABSTRACT FROM AUTHOR]

We present a classical and a quantum constant of motion used to study the behavior of an electron moving in the vicinity of a tridimensional charged nanostructure with electric dipole. It is the generalization of the bidimensional version studied in a previous work. The classical case is found by using two approaches: the Hamilton Jacobi equation and a Newtonian treatment. The quantum case is obtained by separating the Schrödinger equation in spherical coordinates. We use three quantum numbers to classify the states. The angular part is solved by using an expansion of spherical harmonics to get an eigenvalue equation for the new constant of motion. The solution shows that the probability density of the electron shifts toward the values where 𝜃 < π 2 . The eigenvalue λ is near the value l(l + 1) of the angular momentum, but their difference grows if the electric dipole increases, so that the effect is small for a cluster like (GaAs)3 but very important for fullerenes like RbC60 and LiC60. The radial part is solved using the shooting method and we find the ground state, and first and second excited states. [ABSTRACT FROM AUTHOR]

The present work focuses on the effect of rotational restraints on the shear buckling of symmetrically laminated curved composite panels. The Sanders–Koiter shell theory and a first-order shear deformation scheme were used for the mathematical representation of the deformation kinematics of cylindrical shells. The eigenvalue buckling equations were obtained using the principle of minimum total potential energy and by employing the Ritz method. The solution for the deformed shape was approximated as a series of trigonometric functions compatible with the essential boundary conditions of the problem. The effect of the rotational and torsional springs was incorporated by adding their corresponding potential energy to the total potential energy of the panel-loading system. Using the developed formulation, the effect of the influential parameters such as aspect ratio, panel curvature and restraint stiffness on the buckling strength of a specific class of laminates was studied extensively. To present the results in a more insightful manner, non-dimensional parameters were used in parametric studies. To normalize the effect of torsional elements, a new non-dimensional parameter was introduced. [ABSTRACT FROM AUTHOR]

The investigation concerns local buckling of compressed flanges of axially compressed composite channel columns. Cooperation of the member flange and web is taken into account here. The buckling mode of the member flange is defined by rotation angle a flange about the line of its connection with the web. The channel column under investigation is made of unidirectional fibre-reinforced laminate. Two approaches to member orthotropic material modelling are performed: the homogenization with the aid of theory of mixture and periodicity cell or homogenization upon the Voigt–Reuss hypothesis. The fundamental differential equation of local buckling is derived with the aid of the stationary total potential energy principle. The critical buckling stress corresponding to a number of buckling half-waves is assumed to be a minimum eigenvalue of the equation. Some numerical examples dealing with columns are given here. The analytical results are compared with the finite element stability analysis carried out by means of ABAQUS software. The paper is focused on a close analytical solution of the critical buckling stress and the associated buckling mode while the web–flange cooperation is assumed. [ABSTRACT FROM AUTHOR]

The 1/8 BPS D6ℛ4 coupling in type II string theory compactified on T2 receives contributions from worldsheet instantons and anti-instantons wrapping the T2, up to genus three in string perturbation theory. These involve contributions separately from bound states of instantons and anti-instantons, which are qualitatively similar to such contributions to the 1/2 and 1/4 BPS couplings. At genus two, the D6ℛ4 coupling also receives contributions from instanton/anti-instanton bound states unlike the 1/2 and 1/4 BPS couplings, which is a consequence of a T-duality invariant eigenvalue equation a term in the coupling satisfies. We solve this eigenvalue equation to obtain the complete structure of the worldsheet (anti)instanton contributions. In the type IIB theory, strong weak coupling duality leads to certain contributions involving bound states of D string (anti)instantons wrapping the T2. [ABSTRACT FROM AUTHOR]

The free vibration of axially functionally graded (FG) tapered Timoshenko curved beams is studied with the numerical approach. By using the non-uniform rational B-spline (NURBS) basis functions, the exact geometry and the generalized displacement field are formulated. Variable geometric parameters and material properties, including the curvature, cross-sectional area, area moment of inertia, mass density, and Young's modulus, are expanded as functions of the coordinate in a parametric domain. Based on Hamilton's principle, the weak formulation is derived by applying a refined constitutive relation which considers the thickness effect. Natural frequencies and mode shapes are obtained from the eigenvalue equation. Circular, elliptic, and parabolic curved beams are considered in numerical examples. The obtained results are in good agreement with those in the existing studies and those calculated by the finite element software ANSYS. Moreover, the effects of the material gradient, taper ratio, slenderness ratio, and height-span ratio on vibration behaviors are discussed. [ABSTRACT FROM AUTHOR]

We consider solutions of the eigenvalue equation at zero energy for a class of non-local Schrödinger operators with potentials decreasing to zero at infinity. Using a path integral approach, we obtain detailed results on the spatial decay at infinity of both L 2 and resonance solutions. We highlight the interplay of the kinetic term and the potential in these decay behaviours, and identify the decay mechanisms resulting from specific balances of global lifetimes with or without the potential. [ABSTRACT FROM AUTHOR]

While variation of the energy functional yields the Schrödinger equation in the usual, linear case, no such statement can be formulated in the general nonlinear situation when the Hamiltonian depends on its eigenvector. In this latter case, as we illustrate by sample numerical calculations, the points of the energy expectation value hypersurface where the eigenvalue equation is satisfied separate from those where the energy is stationary. We show that the variation of the energy at the eigensolution is determined by a generalized Hellmann–Feynman theorem. Functionals, other than the energy, can, however be constructed, that result the nonlinear Schrödinger equation upon setting their variation zero. The second centralized moment of the Hamiltonian is one example. [ABSTRACT FROM AUTHOR]

In a recent paper, Castellanos-Jaramillo and Castellanos-Moreno proposed a simple quantum-mechanical model for an electron in the vicinity of an ionized nanostructure with a permanent electric dipole. They chose the interaction of the electron with the charge and the dipole in such a way that the resulting Schrödinger equation is separable into radial and angular parts. In this comment, we show that those authors did not solve the angular eigenvalue equation with proper periodic boundary conditions and that they also made a mistake in the elimination of the first derivative in the radial equation. Such errors invalidate their results of the Einstein coefficients for the G a A s 3 system considered. [ABSTRACT FROM AUTHOR]

The onset of convection in a porous layer saturated by a power-law fluid is here investigated. The walls are considered to be isothermal, isobaric and permeable in such a way that a vertical throughflow is described. The threshold for a buoyancy-driven cellular flow is investigated by means of a linear stability analysis. This study consists in introducing disturbances with small amplitude. The disturbances are plane waves, i.e. a normal modes stability analysis of the basic stationary solution is performed. The resulting problem is an ordinary differential equation eigenvalue problem which is solved numerically by coupling the Runge–Kutta method with the shooting method. Results are presented in the form of marginal stability curves and their critical points representing the values of the control parameters such that the growth rate of the disturbances is zero. It is found that, among roll disturbances, the most unstable modes are stationary and uniform with infinite wavelength. For this reason, an asymptotic analysis for vanishing wave numbers is carried out. The results of this asymptotic analysis are obtained analytically displaying a very good agreement with the numerical solution. It is found that vertical throughflow plays a destabilising role for pseudoplastic fluids and a stabilising role for dilatant fluids. [ABSTRACT FROM AUTHOR]

We study some resonant equations related to the classical orthogonal polynomials on infinite intervals, i.e., the Hermite and the Laguerre orthogonal polynomials, and propose an algorithm for finding their particular and general solutions in the closed form. This algorithm is especially suitable for the computer-algebra tools, such as Maple. The resonant equations form an essential part of various applications, e.g., of the efficient functional-discrete method for the solution of operator equations and eigenvalue problems. These equations also appear in the context of supersymmetric Casimir operators for the di-spin algebra and in the solution of square operator equations, such as A2u = f (e.g., of the biharmonic equation). [ABSTRACT FROM AUTHOR]

We tackle with the Dirac equation in the presence of the general molecular potential (GMP). The spin symmetric solution of the relativistic wave equation is obtained by considering the Pekeris-type approximation scheme to deal with the centrifugal term, and in order to solve second-order differential equation, the asymptotic iteration method (AIM) is used. The closed form of the energy eigenvalue equation is found out for any values of the angular momentum quantum number. We calculate the relativistic vibrational bound state energies of 51Δg state of Na2 molecule and compare them with the Rydberg–Klein–Rees (RKR) data. We show that relativistic vibrational energies of this molecule, which are found in the spin symmetric case, are more convenient with experimental RKR data than non-relativistic vibrational energies. We also obtain normalization constant by considering inductive approach and investigate the radial eigenfunctions and probability density functions corresponding to different eigenvalues graphically for the Na2(51Δg) molecule. [ABSTRACT FROM AUTHOR]

We propose a unifying approach to the problem of measuring the inconsistency of judgments. More precisely, we define a general framework to allow several well-known inconsistency indices to be expressed as special cases of this new formulation. We consider inconsistency indices as aggregations of 'local', i.e. triple-based, inconsistencies. We show that few reasonable assumptions guarantee a set of good properties for the obtained general inconsistency index. Under this representation, we prove a property of Pareto efficiency and show that OWA functions and t-conorms are suitable aggregation functions of local inconsistencies. We argue that the flexibility of this proposal allows tuning of the index. For example, by using different types of OWA functions, the analyst can obtain the desired balance between an averaging behavior and a 'largest inconsistency-focused' behavior. [ABSTRACT FROM AUTHOR]

In this paper we study the spherical convexity of quadratic functions on spherically convex sets. In particular, conditions characterizing the spherical convexity of quadratic functions on spherical convex sets associated to the positive orthants and Lorentz cone are given. [ABSTRACT FROM AUTHOR]

We address the reduction to compact band forms, via unitary similarity transformations, for the solution of symmetric eigenvalue problems and the computation of the singular value decomposition (SVD). Concretely, in the first case, we revisit the reduction to symmetric band form, while, for the second case, we propose a similar alternative, which transforms the original matrix to (unsymmetric) band form, replacing the conventional reduction method that produces a triangular-band output. In both cases, we describe algorithmic variants of the standard Level 3 Basic Linear Algebra Subroutines (BLAS)-based procedures, enhanced with look-ahead, to overcome the performance bottleneck imposed by the panel factorization. Furthermore, our solutions employ an algorithmic block size that differs from the target bandwidth, illustrating the important performance benefits of this decision. Finally, we show that our alternative compact band form for the SVD is key to introduce an effective look-ahead strategy into the corresponding reduction procedure. [ABSTRACT FROM AUTHOR]

We discuss a fractional eigenvalue problem with the fractional Sturm-Liouville operator mixing the left and right derivatives of order in the range (1/2, 1], subject to a variant of Robin boundary conditions. The considered differential fractional Sturm-Liouville problem (FSLP) is equivalent to an integral eigenvalue problem on the respective subspace of continuous functions. By applying the properties of the explicitly calculated integral Hilbert-Schmidt operator, we prove the existence of a purely atomic real spectrum for both eigenvalue problems. The orthogonal eigenfunctions' systems coincide and constitute a basis in the corresponding weighted Hilbert space. An analogous result is obtained for the reflected fractional Sturm-Liouville problem. [ABSTRACT FROM AUTHOR]

In this article, we introduce a Laplacian for the reciprocal distance matrix of a connected graph, called the reciprocal distance Laplacian. Let δ1≥δ2≥⋯≥δn-1≥δn be the reciprocal distance Laplacian spectrum. In this short note, we show that δ1≤n with equality if and only if the complement graph G¯ of G is disconnected. [ABSTRACT FROM AUTHOR]

The suggested approach to maximizing the difference between the first and second eigenvalues of the Laplace operator is based on the introduction of nonlocal boundary conditions of a special form. It is shown that the difference can be arbitrarily large. [ABSTRACT FROM AUTHOR]

In this paper we study a Dirichlet-to-Neumann operator with respect to a second order elliptic operator with measurable coefficients, including first order terms, namely, the operator on L2(∂Ω) given by φ↦∂νu where u is a weak solution of -div(a∇u)+b·∇u-div(cu)+du=λuonΩ,u|∂Ω=φ.Under suitable assumptions on the matrix-valued function a, on the vector fields b and c, and on the function d, we investigate positivity, sub-Markovianity, irreducibility and domination properties of the associated Dirichlet-to-Neumann semigroups. [ABSTRACT FROM AUTHOR]

We write expressions connected with numerical differentiation formulas of order 2 in the form of Stieltjes integral, then we use Ohlin lemma and Levin-Stechkin theorem to study inequalities connected with these expressions. In particular, we present a new proof of the inequality fx+y2≤1(y-x)2∫xy∫xyfs+t2dsdt≤1y-x∫xyf(t)dtsatisfied by every convex function f:R→R and we obtain extensions of this inequality. Then we deal with non-symmetric inequalities of a similar form. [ABSTRACT FROM AUTHOR]

A path P in an edge-colored graph G is called a proper path if no two adjacent edges of P are colored the same, and G is proper connected if every two vertices of G are connected by a proper path in G. The proper connection number of a connected graph G, denoted by pc(G), is the minimum number of colors that are needed to make G proper connected. In this paper, we study the proper connection number on the lexicographic, strong, Cartesian, and direct products and present exact values or upper bounds for these products of graphs. [ABSTRACT FROM AUTHOR]

It has been proven that all Moufang loops of order p4q3 where p and q are odd primes with 3 are associative if and only if q≢1(modp). Also all Moufang loops of order p13p23⋯pn3 where each pi are a prime with 3≤p1 were proven to be associative if and only if pi≢1(modpj). In this paper, we extend these results to Moufang loops of order p4q13q23⋯qn3 where p and each qi is a prime with 3 and qi≢1(modqj). [ABSTRACT FROM AUTHOR]

This paper presents a numerical method for computing the conformal mappings onto the parabolic slit domain, the elliptic slit domain and the hyperbolic slit domain. The method relies on a boundary integral equation with the generalized Neumann kernel. For a given multiply connected domain of connectivity m+1, the proposed method requires O((m+1)nlogn) operations where n is the number of nodes in the discretization of each boundary component of the given domain. Several numerical examples are presented to illustrate the performance of the proposed method. [ABSTRACT FROM AUTHOR]

Using the restricted and associated Stirling numbers of the second kind, we define the incomplete multi-poly-Bernoulli numbers which generalize poly-Bernoulli numbers. We study their analytic and combinatorial properties. As an application, we present a new infinite series representation of the multiple zeta values via the Lambert W-function. We also give similar results for incomplete Bernoulli numbers of Hurwitz type and incomplete multi-poly-Bernoulli-star numbers. [ABSTRACT FROM AUTHOR]

One of the aims of this article is to provide a class of polynomial mappings for which the Jacobian conjecture is true. Also, we state and prove several global univalence theorems and present a couple of applications of them. [ABSTRACT FROM AUTHOR]

The generalized k-connectivity κk(G) of a graph G, introduced by Hager (J Comb Theory 38:179-189, 1985) is a generalization of the classical connectivity κ(G) with κ2(G)=κ(G). In this paper, we construct graphs to show that for every pair of integers m and n(1 there is a graph with the generalized 3-connectivity n whose line graph has the generalized 3-connectivity m. [ABSTRACT FROM AUTHOR]