A long standing problem in lattice QCD has been the discrepancy between the experimental and calculated values for the axial charge of the nucleon, gA≡GA(Q2=0). Though finite volume effects have been ...shown to be large, it has also been suggested that excited state effects may also play a significant role in suppressing the value of gA. In this work, we apply a variational method to generate operators that couple predominantly to the ground state, thus systematically removing excited state contamination from the extraction of gA. The utility and success of this approach is manifest in the early onset of ground state saturation and the early onset of a clear plateau in the correlation function ratio proportional to gA. Through a comparison with results obtained via traditional methods, we show how excited state effects can suppress gA by as much as 8% if sources are not properly tuned or source–sink separations are insufficiently large.
A new table of the nuclear equation of state (EOS) based on realistic nuclear potentials is constructed for core-collapse supernova numerical simulations. Adopting the EOS of uniform nuclear matter ...constructed by two of the present authors with the cluster variational method starting from the Argonne v18 and Urbana IX nuclear potentials, the Thomas–Fermi calculation is performed to obtain the minimized free energy of a Wigner–Seitz cell in non-uniform nuclear matter. As a preparation for the Thomas–Fermi calculation, the EOS of uniform nuclear matter is modified so as to remove the effects of deuteron cluster formation in uniform matter at low densities. Mixing of alpha particles is also taken into account following the procedure used by Shen et al. (1998, 2011). The critical densities with respect to the phase transition from non-uniform to uniform phase with the present EOS are slightly higher than those with the Shen EOS at small proton fractions. The critical temperature with respect to the liquid–gas phase transition decreases with the proton fraction in a more gradual manner than in the Shen EOS. Furthermore, the mass and proton numbers of nuclides appearing in non-uniform nuclear matter with small proton fractions are larger than those of the Shen EOS. These results are consequences of the fact that the density derivative coefficient of the symmetry energy of our EOS is smaller than that of the Shen EOS.
In present paper, we study the following nonlinear Schrödinger equation with combined power nonlinearities −Δu+V(x)u+λu=|u|2∗−2u+μ|u|q−2uinRN,N≥3having prescribed mass ∫RNu2dx=a2,where μ,a>0, ...q∈(2,2∗), 2∗=2NN−2 is the critical Sobolev exponent, V is an external potential vanishing at infinity, and the parameter λ∈R appears as a Lagrange multiplier. Under some mild assumptions on V, combining the Pohožaev manifold, constrained minimization arguments and some analytical skills, we get the existence of normalized solutions for the problem with q∈(2,2∗). At the same time, the exponential decay property of the solutions is established, which is important for the instability analysis of the standing waves. Furthermore, we give a description of the ground state set and obtain the strong instability of the standing waves for q∈2+4N,2∗).
We study the following Kirchhoff type elliptic problem,(P){−(a+b∫Ω|∇u|2dx)Δu=λuq+μu3,u>0in Ω,u=0on ∂Ω, where Ω⊂R4 is a bounded domain with smooth boundary ∂Ω. Moreover, we assume a,λ,μ>0, b≥0 and ...1≤q<3. In this paper, we prove the existence of solutions of (P). Our tools are the variational method and the concentration compactness argument for PS sequences.
We consider the Schrödinger-Poisson system{−Δu+V(x)u+K(x)ϕu=a(x)|u|p−2u+|u|4u,x∈R3,−Δϕ=K(x)u2,x∈R3, where 4<p<6 and the potentials V, a are allowed to change their signs. Under some reasonable ...assumptions on V, K and a, we apply the constraint minimization argument to establish the existence of positive ground state solutions and ground state nodal solutions.
Using the Mountain-Pass Theorem of Ambrosetti and Rabinowitz we prove that the following fractional p-Laplacian equation with double critical nonlinearities ...(−Δp)su=|u|ps∗−2u+|u|ps∗(α)−2u|x|αinRN,admits a positive solution in the class Ẇs,p(RN). In the above, (−Δp)s is the fractional p-Laplacian, s∈(0,1), p>1, 0<α<ps<N, ps∗=NpN−ps is the critical fractional Sobolev exponent and ps∗(α)=p(N−α)N−ps is the critical Hardy–Sobolev exponent, Ẇs,p(RN) denotes the completion of C0∞(RN) with respect to Gagliardo norm us,pp≔∫RN∫RN|u(x)−u(y)|p|x−y|N+psdxdy.Our method is based on the existence of extremals of some fractional Hardy–Sobolev type inequalities, and coupled with some intricate estimates for the nonlocal (s,p)-gradient. Moreover, we also establish the existence of a nontrivial solution to an elliptic system which involves fractional p-Laplacian and critical Hardy–Sobolev exponents in RN.
In this paper we concern with the multiplicity and concentration of positive solutions for the semilinear Kirchhoff type equation{−(ε2a+bε∫R3|∇u|2)Δu+M(x)u=λf(u)+|u|4u,x∈R3,u∈H1(R3),u>0,x∈R3, where ...ε>0 is a small parameter, a, b are positive constants and λ>0 is a parameter, and f is a continuous superlinear and subcritical nonlinearity. Suppose that M(x) has at least one minimum. We first prove that the system has a positive ground state solution uε for λ>0 sufficiently large and ε>0 sufficiently small. Then we show that uε converges to the positive ground state solution of the associated limit problem and concentrates to a minimum point of M(x) in certain sense as ε→0. Moreover, some further properties of the ground state solutions are also studied. Finally, we investigate the relation between the number of positive solutions and the topology of the set of the global minima of the potentials by minimax theorems and the Ljusternik–Schnirelmann theory.
•The variational principle of the solution is established.•The new abundant solutions are constructed.•The absolute, real and imaginary parts of the solutions are illustrated.
In this paper, we aim ...to study the (1+2)-dimensional chiral nonlinear Schrödinger equation. A complex transform is adopted to convert the equation into the real and imaginary parts. The variational principle is developed by the Semi-inverse method. Then we, for the first time ever, extend He's variational method to construct the new abundant solutions, which include the bright soliton, bright-dark soliton, bright-like soliton, kinky bright soliton and the periodic solution. By using extended He's variational method, we can reduce the order of the studied equation through the variational principle, make the equation more simple and then obtain the optimal solutions by the stationary conditions. Finally, we use one example to verify the effectiveness and reliability of the extended He's variational method through the 3-D graphs. The obtained results in this work are helpful to be of significance to the study of traveling wave theory in physics.
Positron emission tomography/computed tomography (PET/CT) imaging can simultaneously acquire functional metabolic information and anatomical information of the human body. How to rationally fuse the ...complementary information in PET/CT for accurate tumor segmentation is challenging. In this study, a novel deep learning based variational method was proposed to automatically fuse multimodality information for tumor segmentation in PET/CT. A 3D fully convolutional network (FCN) was first designed and trained to produce a probability map from the CT image. The learnt probability map describes the probability of each CT voxel belonging to the tumor or the background, and roughly distinguishes the tumor from its surrounding soft tissues. A fuzzy variational model was then proposed to incorporate the probability map and the PET intensity image for an accurate multimodality tumor segmentation, where the probability map acted as a membership degree prior. A split Bregman algorithm was used to minimize the variational model. The proposed method was validated on a non-small cell lung cancer dataset with 84 PET/CT images. Experimental results demonstrated that: (1) Only a few training samples were needed for training the designed network to produce the probability map; (2) The proposed method can be applied to small datasets, normally seen in clinic research; (3) The proposed method successfully fused the complementary information in PET/CT, and outperformed two existing deep learning-based multimodality segmentation methods and other multimodality segmentation methods using traditional fusion strategies (without deep learning); (4) The proposed method had a good performance for tumor segmentation, even for those with Fluorodeoxyglucose (FDG) uptake inhomogeneity and blurred tumor edges (two major challenges in PET single modality segmentation) and complex surrounding soft tissues (one major challenge in CT single modality segmentation), and achieved an average dice similarity indexes (DSI) of 0.86 ± 0.05, sensitivity (SE) of 0.86 ± 0.07, positive predictive value (PPV) of 0.87 ± 0.10, volume error (VE) of 0.16 ± 0.12, and classification error (CE) of 0.30 ± 0.12.
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