Functional Boundary Conditions Research Articles

Solvability of functional third-order problems of Ambrosetti–Prodi-type

This work presents an Ambrosetti–Prodi alternative for functional problems composed of a fully third-order differential equation with two types of functional boundary conditions. The discussion of existence and non-existence of solution is obtained in a more general case, and the multiplicity of solution is done with restrictive boundary conditions-The main arguments are based on the lower and upper solutions method, together with the Leray–Schauder topological degree theory. We stress that the multiplicity situation requires different speed growths on the variables.An example illustrates the results’ applicability and shows a technique to estimate the bifurcation values of the parameter.

Emerging flat bands and spin polarization in nanodiamond island superlattices with varying carrier effective masses

For the first time, a set of regular 2D sp2/sp3 hybrid superlattices of nanodiamond islands confined between two graphene sheets (NDI-c2G) were proposed and explored using Density Functional Theory Periodic Boundary Conditions simulations. The fusion involves coronene and ovalene molecules with top and bottom graphene lattices, which can be treated with electron beam irradiation and atomic hydrogen at the nanodiamond site in three different configurations: (111)CD, (0001)HD, and (2¯110)HD. This creates distorted sp3 nanodiamond sublattices within the sp2 graphene fragments, serving as regions with confined electrons, making the charge carriers’ effective masses heavier. The presence of spin-splitting valence spin-up and conduction spin-down flat bands near the Fermi level in cubic ((111)CD-I and (111)CD-II) and hexagonal ((2¯110)HD)-II) superlattices resulted in magnetic moments of 3.99μB, 3.99μB, and 1.99μB per unit cell, respectively. Various configurations significantly impacted their electronic properties, resulting in spin-polarized and non-spin-polarized systems with flat bands near the Fermi level, exhibiting a semiconducting nature. The averaged spin moments per carbon atom were 0.16μB, 0.14μB, and 0.08μB, localized at the sp2/sp3 interfaces of the nanodiamond sites. Analysis of the electronic states, band structure, charge carrier effective masses, and spin density distribution suggests NDI-c2G shares similar electronic nature to strongly correlated twisted bilayer graphenes at small magic angles.

Solving Fractional Random Differential Equations by Using Fixed Point Methodologies under Mild Boundary Conditions

This manuscript aims to study the existence and uniqueness of solutions to a new system of differential equations. This system is a mixture of fractional operators and stochastic variables. The study has been completed under nonlocal functional boundary conditions. In the study, we used the fixed-point method to examine the existence of a solution to the proposed system, mainly focusing on the theorems of Leray, Schauder, and Perov in generalized metric spaces. Finally, an example has been provided to support and underscore our results.

Some results on a nonlinear fractional equation with nonlocal boundary condition

The aim of this paper is to derive sufficient conditions for the existence, uniqueness, and Hyers–Ulam stability of solutions to a new nonlinear fractional integro‐differential equation with functional boundary conditions, using several fixed‐point theorems, the multivariate Mittag‐Leffler function and Babenko's approach. A few examples are also presented to illustrate the applications of our results based on approximate values of a couple of Mittag‐Leffler functions calculated by Python codes. Furthermore, the approaches used have a wide range of applications to various fractional differential equations with initial or boundary conditions or integral equations in complete spaces.

Nonperturbative running of the tensor operator for Nf=3 QCD from the chirally rotated Schrödinger functional

We study the renormalization group (RG) running of the nonsinglet tensor operator, for Nf=3 QCD with Wilson fermions in a mixed action setup, with standard Schrödinger functional (SF) boundary conditions for sea quarks and chirally rotated Schrödinger functional (χSF) boundary conditions for valence quarks. Based on a recursive finite-size scaling technique we compute nonperturbatively the tensor step-scaling function for an energy range between a hadronic scale and an electroweak scale, above which perturbation theory may be safely applied. Our result is expressed as the RG-running factor TRGI/[T(μhad)]R, where the numerator is the scale independent (renormalization group invariant—RGI) tensor operator and the denominator is its renormalized counterpart at a hadronic scale μhad=233(8) MeV in a given scheme. We determine the step-scaling function in four distinct renormalization schemes. We also compute the renormalization parameters of these schemes at μhad which, combined with the RG-running factor, gives the scheme-independent quantity ZTRGI(g02) in four schemes and for a range of bare gauge couplings in which large volume hadronic matrix element simulations are performed by the CLS consortium in Nf=2+1 QCD. All four results are compatible and also agree with a recent determination based on a unitary setup for Wilson quarks with Schrödinger functional boundary conditions [arXiv:2309.04314]. This provides a strong universality test. Published by the American Physical Society 2024

Uniqueness and existence for a fractional differential equation with functional boundary condition

Using Banach’s contractive principle and Leray–Schauder’s fixed point theorem, we study the uniqueness and existence for a nonlinear fractional differential equation with functional boundary condition based on the two-parameter Mittag-Leffler function and an implicit integral equation. Several examples are also presented to demonstrate applications of our key theorems. The methods used can also deal with other types of differential equations with various initial or boundary conditions.

On fourth order retarded equations with functional boundary conditions: A unified approach

By means of a recent Birkhoff-Kellogg type theorem, we discuss the solvability of a fairly general class of parameter-dependent fourth order retarded differential equations subject to functional boundary conditions. We seek solutions within a translate cone of nonnegative functions. We provide an example to illustrate our theoretical results.

Mechanistic study of the atomic layer deposition of cobalt: a combined mass spectrometric and computational approach.

Cobaltcarbonyl-tert-butylacetylene (CCTBA) is a conventional precursor for the selective atomic layer deposition of Co onto silicon surfaces. However, a limited understanding of the deposition mechanism of such cobalt precursors curbs rational improvements on their design for increased efficiency and tuneable selectivity. The impact of using a less reactive internal alkyne instead of a terminal alkyne was investigated using experimental and computational methods. Using electrospray-ionization mass spectrometry, the formation of CCTBA analogs and their gas phase decomposition pathways were studied. Decomposition experiments show very similar decomposition pathways between the two complexes. The internal alkyne dissociates from the Co complex at slightly lower energies than the terminal alkyne, suggesting that an internal alkynyl ligand may be more suited to low temperature ALD. In addition, transition state calculations using the nudged elastic band method confirm an increased reaction barrier between the internal alkyne and the Si-H surface bonds on Si(111). These results suggests that using a less reactive internal alkyne will result in fewer embedded carbon impurities during deposition onto Si wafers. DFT calculations using the PBE functional and periodic boundary conditions also predict increased surface binding with the metal centers of the internal alkynyl complex.

Nearly flat bands and ferromagnetism in the terminated Mn2C MXene

Using Density Functional Theory and Periodic Boundary Conditions it is shown that the hydroxylated/oxygenated/halogenated Mn2C monolayer is a 2D ferromagnetic material with a local Mn ions magnetic moment of 2.7μв per unit cell. Upon oxygenation the ferromagnetic coupling between Mn ions can be transformed into a superposition of magnetic states. In particular, the intrinsic magnetic moments in the hydroxylated/halogenated Mn2C monolayer can attain up to 6μB per unit cell. It is found that oxygen termination induces flat bands in the band structure, which evidence for the strong electron correlations and could lead to the implementation of exotic quantum phases in 2D crystals and high-temperature superconductivity. Along with the potential of the hydroxylated Mn2C monolayer characterized by the half-metallicity for application in spintronic devices as a perfect spin injector/detector, this material like other conventional MXenes is promising for the use in energy storage, electromagnetic interference shielding, and sensing.

Functional Impulsive Fractional Differential Equations Involving the Caputo-Hadamard Derivative and Integral Boundary Conditions

In this paper, we investigate the existence and uniqueness of solutions for functional impulsive fractional differential equations and integral boundary conditions. Our results are based on some fixed point theorems. Finally, we provide an example to illustrate the validity of our main results.

From university basic research to firm innovation: diffusion mechanism and boundary conditions under a U-shaped relationship

The paradox between China's dazzling S&T output and the technological dilemma of enterprises challenges the positive effect of basic research on technological innovation, and extant research has not solved this problem. To bridge this theoretical gap, we integrated resource dependence theory and institutional theory, and matched the data of universities and governments at the provincial level and the data of enterprises at the micro level to clarify whether and how university basic research promotes firm innovation. We found that: (1) there is a U-shaped relationship between university basic research and firm innovation. (2) University-enterprise cooperation, S&T services and paper achievements played a mediating role, but invention patent achievements did not. (3) Neither the supporting nor the intervening hand of the government can moderate the U-shaped relationship. (4) Only when the enterprise has no political connection, is located in the eastern region, and is not in the provincial capital city or special economic zone, is the U-shaped relationship established. Furthermore, the functional mechanism and boundary conditions of the U-shaped relationship are further clarified through a robustness test of variable substitution and sample subdivision.

Coupled systems with Ambrosetti-Prodi-type differential equations

<abstract><p>In this paper, we consider some boundary value problems composed by coupled systems of second-order differential equations with full nonlinearities and general functional boundary conditions verifying some monotone assumptions. The arguments apply the lower and upper solutions method, and defining an adequate auxiliary, homotopic, and truncated problem, it is possible to apply topological degree theory as the tool to prove the existence of solution. In short, it is proved that for the parameter values such that there are lower and upper solutions, then there is also, at least, a solution and this solution is localized in a strip bounded by lower and upper solutions. As far as we know, it is the first paper where Ambrosetti-Prodi differential equations are considered in couple systems with different parameters.</p></abstract>

Nonzero positive solutions of fractional Laplacian systems with functional terms

AbstractWe study the existence of nonzero positive solutions of a class of systems of differential equations driven by fractional powers of the Laplacian. Our approach is based on the notion of fixed point index, and allows us to deal with nonlocal functional weights and functional boundary conditions. We present two examples to shed light on the type of functionals and growth conditions that can be considered with our approach.

Solvability of mixed Hilfer fractional functional boundary value problems with p-Laplacian at resonance

This article investigates the existence of solutions of mixed Hilfer fractional differential equations with p-Laplacian under the functional boundary conditions at resonance. By defining Banach spaces with appropriate norms, constructing suitable operators, and using the extension of the continuity theorem, some of the current results are extended to the nonlinear situation, and some new existence results of the problem are obtained. Finally, an example is given to verify our main results.

An affine Birkhoff–Kellogg‐type result in cones with applications to functional differential equations

In this short note, we prove, by means of classical fixed point index, an affine version of a Birkhoff–Kellogg‐type theorem in cones. We apply our result to discuss the solvability of a class of boundary value problems for functional differential equations subject to functional boundary conditions. We illustrate our theoretical results in an example.

Existence and uniqueness of solutions of nonlinear fractional stochastic differential systems with nonlocal functional boundary conditions

The authors study the existence and uniqueness of solutions to nonlinear first-order fractional stochastic differential systems driven by Brownian motion and with nonlocal functional boundary conditions. The technique of proof involves Perov’s fixed point theorem with matrices that converge to zero and the Leray–Schauder theorem.

Numerical method for system of space-fractional equations of superdiffusion type with delay and Neumann boundary conditions

We consider a system of two space-fractional superdiffusion equations with functional general delay and Neumann boundary conditions. For this problem, an analogue of the Crank-Nicolson method is constructed, based on the shifted Grünwald-Letnikov formulas for approximating fractional Riesz derivatives with respect to a spatial variable and using piecewise linear interpolation of discrete prehistory with extrapolation by continuation to take into account the delay effect. With the help of the Gershgorin theorem, the solvability of the difference scheme and its stability are proved. The order of convergence of the method is obtained. The results of numerical experiments are presented.

Nonperturbative running of the quark mass for Nf=3 QCD from the chirally rotated Schrödinger functional

We study the renormalization group (RG) running of the quark mass, for ${N}_{f}=3$ QCD with Wilson fermions in a mixed action setup, with standard Schr\"odinger functional (SF) boundary conditions for sea quarks and chirally rotated Schr\"odinger functional ($\ensuremath{\chi}\mathrm{SF}$) boundary conditions for valence quarks. This necessitates the tuning of the boundary factor ${z}_{\mathrm{f}}({g}_{0}^{2})$ of the $\ensuremath{\chi}\mathrm{SF}$ valence action, in order to ensure that QCD symmetries are fully recovered in the continuum. The properties of this novel setup are monitored through the ratios ${Z}_{S}/{Z}_{P}$ and ${\mathrm{\ensuremath{\Sigma}}}_{S}/{\mathrm{\ensuremath{\Sigma}}}_{P}$ of the renormalization parameters and step scaling functions of the scalar and pseudoscalar densities. Where comparison is possible, our ${Z}_{S}/{Z}_{P}$ results are found to agree with previous determinations, based on a mass ratio method [G. M. de Divitiis et al. (ALPHA Collaboration), Eur. Phys. J. C 79, 797 (2019)] and Ward identities [ J. Heitger et al. (ALPHA Collaboration), Eur. Phys. J. C 80, 765 (2020); J. Heitger et al., Eur. Phys. J. C 81, 606 (2021)], with Schr\"odinger functional boundary conditions. The behavior of ${\mathrm{\ensuremath{\Sigma}}}_{S}/{\mathrm{\ensuremath{\Sigma}}}_{P}$ confirms the theoretical expectations of $\ensuremath{\chi}\mathrm{SF}$ QCD, related to the restoration of the theory's symmetries in the continuum limit. From the step-scaling function of the pseudoscalar density we obtain the quark mass RG-running function from hadronic to perturbative energy scales. This is fully compatible with the earlier result obtained in a similar setup for Wilson quarks with Schr\"odinger functional boundary conditions [I. Campos et al., Eur. Phys. J. C 78, 387 (2018)], and provides a strong universality test for the two lattice setups.

Thermal buckling and vibro-acoustic behaviour of functionally graded graphene polymer layered composites subjected to in-plane temperature variance

The current study reports the thermal buckling, vibration and acoustic characteristics of functionally graded graphene polymer layer composite plates subjected to the in-plane temperature variance. The macroscopic properties of the composites are evaluated using the modified rule of mixtures to compute the layer-wise properties of an functionally graded graphene polymer layer composite plate. The critical buckling temperature is computed and compared for various functional gradings, boundary conditions and in-plane temperature variances. The in-plane temperature variance showed a major impact on the critical thermal buckling temperature and respective mode shapes. The vibro-acoustic behaviour of the functionally graded graphene polymer layer composite plate is investigated and documented keeping critical buckling temperature as a function. With an increase in thermal load and the nature of in-plane temperature variance, the vibro-acoustic results showed significant difference in velocity and acoustic response. For functionally graded graphene polymer layer composite plate with one free edge, the difference was statistically significant as indicated by an octave band plot. We conclude that the portion of the functionally graded graphene polymer layer composite plate that is subjected to the higher temperature in an in-plane temperature variance, as well as the nature of the boundary conditions may exacerbate the effect of in-plane temperature variance and are crucial in predicting vibro-acoustic characteristics.

On differentiability of the membrane-mediated mechanical interaction energy of discrete–continuum membrane–particle models

We consider a discrete–continuum model of a biomembrane with embedded particles. While the membrane is represented by a continuous surface, embedded particles are described by rigid discrete objects which are free to move and rotate in lateral direction. For the membrane we consider a linearized Canham–Helfrich energy functional and height and slope boundary conditions imposed on the particle boundaries resulting in a coupled minimization problem for the membrane shape and particle positions. When considering the energetically optimal membrane shape for each particle position we obtain a reduced energy functional that models the implicitly given interaction potential for the membrane-mediated mechanical particle–particle interactions. We show that this interaction potential is differentiable with respect to the particle positions and orientations. Furthermore, we derive a fully practical representation of the derivative only in terms of well defined derivatives of the membrane. This opens the door for the application of minimization algorithms for the computation of minimizers of the coupled system and for further investigation of the interaction potential of membrane-mediated mechanical particle–particle interaction. The results are illustrated with numerical examples comparing the explicit derivative formula with difference quotient approximations. We furthermore demonstrate the application of the derived formula to implement a gradient flow for the approximation of optimal particle configurations.