THE EMBEDDING THEORY OF NATIVE SPACES

In the theory of radial basis functions, mathematicians use linear combinations of the translates of the radial basis functions as interpolants. The set of these linear combinations is a normed vector space. This space can be completed and become a Hilbert space, called native space, which is of great importance in the last decade. The native space then contains some abstract elements which are not linear combinations of radial basis functions. The meaning of these abstract elements is not fully known. This paper presents some interpretations for the these elements. The native spaces are embedded into some well-known spaces. For example, the Sobolev-space is shown to be a native space. Since many differential equations have solutions in the Sobolev-space, we can therefore approximate the solutions by linear combinations of radial basis functions. Moreover, the famous question of the embedding of the native space into L2(Ω) is also solved by the author.

An optimized cluster density matrix embedding theory

We propose an optimized cluster density matrix embedding theory(CDMET).It reduces the computational cost of CDMET with simpler bath states.And the result is as accurate as the original one.As a demonstration,we study the distant correlations of the Heisenberg J_(1)-J_(2)model on the square lattice.We find that the intermediate phase(0.43≤sssim J_(2)≤sssim 0.62)is divided into two parts.One part is a near-critical region(0.43≤J_(2)≤0.50).The other part is the plaquette valence bond solid(PVB)state(0.51≤J_(2)≤0.62).The spin correlations decay exponentially as a function of distance in the PVB.

Efficient Virtual Network Embedding Algorithm Based on Restrictive Selection and Optimization Theory Approach

Network virtualization(NV) is widely considered as a key component of the future network and promises to allow multiple virtual networks(VNs) with different protocols to coexist on a shared substrate network(SN). One main challenge in NV is virtual network embedding(VNE). VNE is a NPhard problem. Previous VNE algorithms in the literature are mostly heuristic, while the remaining algorithms are exact. Heuristic algorithms aim to find a feasible embedding of each VN, not optimal or sub-optimal, in polynomial time. Though presenting the optimal or sub-optimal embedding per VN, exact algorithms are too time-consuming in smallscaled networks, not to mention moderately sized networks. To make a trade-off between the heuristic and the exact, this paper presents an effective algorithm, labeled as VNE-RSOT(Restrictive Selection and Optimization Theory), to solve the VNE problem. The VNERSOT can embed virtual nodes and links per VN simultaneously. The restrictive selection contributes to selecting candidate substrate nodes and paths and largely cuts down on the number of integer variables, used in the following optimization theory approach. The VNE-RSOT fights to minimize substrate resource consumption and accommodates more VNs. To highlight the efficiency of VNERSOT, a simulation against typical and stateof-art heuristic algorithms and a pure exact algorithm is made. Numerical results reveal that virtual network request(VNR) acceptance ratio of VNE-RSOT is, at least, 10% higher than the best-behaved heuristic. Other metrics, such as the execution time, are also plotted to emphasize and highlight the efficiency of VNE-RSOT.

Phase diagram,correlations,and quantum critical point in the periodic Anderson model

Periodic Anderson model is one of the most important models in the field of strongly correlated electrons. With the recent developed numerical method density matriX embedding theory, we study the ground state properties of the periodic Anderson model on a two-dimensional square lattice. We systematically investigate the phase diagram away from half filling. We find three different phases in this region, which are distinguished by the local moment and the spin-spin correlation functions. The phase transition between the two antiferromagnetic phases is of first order. It is the so-called Lifshitz transition accompanied by a reconstruction of the Fermi surface. As the filling is close to half filling, there is no difference between the two antiferromagnetic phases. From the results of the spin-spin correlation, we find that the Kondo singlet is formed even in the antiferromagnetic phase.

Continuity of Some Classes of Functions Related to Degenerate Parabolic Equations and Its Applications

We study some classes of functions satisfying the assumptions similar to but weaker than those for the classical B2 function classes used in the research of quasi-linear parabolic equations as well as the ones used in the research of degenerate parabolic equations including porous medium equations. Consequently, we prove that a function in such a class is continuous. As an application, we obtain the estimate for the continuous modulus of the solutions of a few degenerate parabolic equations in divergence form, including the anisotropic porous equations.