Several eigenvalue properties of the third-order boundary value problems with distributional potentials are investigated.Firstly,we prove that the operators associated with the problems are self-adjoint and the corresponding eigenvalues are real.Next,the continuity and differential properties of the eigenvalues of the problems are given,especially we find the differential expressions for the boundary conditions,the coefficient functions and the endpoints.Finally,we show a brief application to a kind of transmission boundary value problems of the problems studied here.
The emergence of meal replacement(MR)originates from physical exercise or fitness as a substitute for one or all meals and later expands to the field of weight loss.Indeed,the main application of current meal replacement is to lose body weight,whether patients with obesity,diabetes,fatty liver,infertile or pregnant women can benefit from weight loss.In addition,MRs still exhibit more biomedical potential in preventing and treating diseases,like anti-diabetes,improving fatty liver and kidney disease,preventing cancer,conceiving and reducing pregnancy complications,and improving life quality.Indeed,there are also disadvantages to meal replacement,including causing adverse effects,although most are acceptable and tolerated.To date,various commercially-developed MRs are walking from dining table to sickbed.Therefore,a scientific understanding of the advantages and disadvantages of meal replacements is crucial for their extensive application beyond biomedical potentials.
Hao QiangTianshu XuPeng MaSiyuan ZhangGuanhua DuGuifen QiangTengfei Ji
For s∈[0,1],b∈R and p∈[1,∞),let˙B_ (p,∞)^(s,b)(R^(n))be the logarithmic-Gagliardo–Lipschitz space,which arises as a limiting interpolation space and coincides to the classical Besov space when b=0 and s∈(0,1).In this paper,the authors study restricting principles of the Riesz potential space I_(α)(˙B _(p,∞)^(s,b)(R^(n)))into certain Radon–Campanato space.
Molecular dynamics(MD)is a powerful method widely used in materials science and solid-state physics.The accuracy of MD simulations depends on the quality of the interatomic potentials.In this work,a special class of exact solutions to the equations of motion of atoms in a body-centered cubic(bcc)lattice is analyzed.These solutions take the form of delocalized nonlinear vibrational modes(DNVMs)and can serve as an excellent test of the accuracy of the interatomic potentials used in MD modeling for bcc crystals.The accuracy of the potentials can be checked by comparing the frequency response of DNVMs calculated using this or that interatomic potential with that calculated using the more accurate ab initio approach.DNVMs can also be used to train new,more accurate machine learning potentials for bcc metals.To address the above issues,it is important to analyze the properties of DNVMs,which is the main goal of this work.Considering only the point symmetry groups of the bcc lattice,34 DNVMs are found.Since interatomic potentials are not used in finding DNVMs,they are exact solutions for any type of potential.Here,the simplest interatomic potentials with cubic anharmonicity are used to simplify the analysis and to obtain some analytical results.For example,the dispersion relations for small-amplitude phonon modes are derived,taking into account interactions between up to the fourth nearest neighbor.The frequency response of the DNVMs is calculated numerically,and for some DNVMs examples of analytical analysis are given.The energy stored by the interatomic bonds of different lengths is calculated,which is important for testing interatomic potentials.The pros and cons of using DNVMs to test and improve interatomic potentials for metals are discussed.Since DNVMs are the natural vibrational modes of bcc crystals,any reliable interatomic potential must reproduce their properties with reasonable accuracy.
