Borehole-GPR numerical simulation of full wave field based on convolutional perfect matched layer boundary

The absorbing boundary is the key in numerical simulation of borehole radar.Perfect match layer(PML) was chosen as the absorbing boundary in numerical simulation of GPR.But CPML(convolutional perfect match layer) approach that we have chosen has the advantage of being media independent.Beginning with the Maxwell equations in a two-dimensional structure,numerical formulas of finite-difference time-domain(FDTD) method with CPML boundary condition for transverse electric(TE) or transverse magnetic(TM) wave are presented in details.Also,there are three models for borehole-GPR simulation.By analyzing the simulation results,the features of targets in GPR are obtained,which can provide a better interpretation of real radar data.The results show that CPML is well suited for the simulation of borehole-GPR.

Results on energies for trees with a given diameter having perfect matching

Let Γd2nbe the set of trees with a given diameter d having a perfect matching,where 2n is the number of vertex.For a tree T in Γd2n,let Pd＋1be a diameter of T and q = d m,where m is the number of the edges of perfect matching inPd＋1.It can be found that the trees with minimal energy in Γd2nfor four cases q = d 2,d 3,d 4,[d2],and two remarks aregiven about the trees with minimal energy in Γd2nfor2d 33q d 5 and [d2] ＋ 1 q2d 33 1.

Perfect matchings on a type of lattices with toroidal boundary

Enumeration of perfect matchings on graphs has a longstanding interest in combinatorial mathematics. In this paper, we obtain some explicit expressions of the number of perfect matchings for a type of Archimedean lattices with toroidal boundary by applying Tesler's crossing orientations to obtain some Pfaffan orientations and enumerating their Pfaffans.

Ordering of Unicyclic Graphs with Perfect Matchings by Minimal Matching Energies

In 2012, Gutman and Wagner proposed the concept of the matching energy of a graph and pointed out that its chemical applications can go back to the 1970s. The matching energy of a graph is defined as the sum of the absolute values of the zeros of its matching polynomial. Let u and v be the non-isolated vertices of the graphs G and H with the same order, respectively. Let wi?be a non-isolated vertex of graph Gi?where i=1, 2, …, k. We use Gu(k)?(respectively, Hv(k)) to denote the graph which is the coalescence of G (respectively, H) and G1, G2,…, Gk?by identifying the vertices u (respectively, v) and w1, w2,…, wk. In this paper, we first present a new technique of directly comparing the matching energies of Gu(k)?and Hv(k), which can tackle some quasi-order incomparable problems. As the applications of the technique, then we can determine the unicyclic graphs with perfect matchings of order 2n with the first to the ninth smallest matching energies for all n≥211.

Perfect 1-k Matchings of Bipartite Graphs

Let k be a positive integer and G a bipartite graph with bipartition (X,Y). A perfect 1-k matching is an edge subset M of G such that each vertex in Y is incident with exactly one edge in M and each vertex in X is incident with exactly k edges in M. A perfect 1-k matching is an optimal semi-matching related to the load-balancing problem, where a semi-matching is an edge subset M such that each vertex in Y is incident with exactly one edge in M, and a vertex in X can be incident with an arbitrary number of edges in M. In this paper, we give three sufficient and necessary conditions for the existence of perfect 1-k matchings and for the existence of 1-k matchings covering | X |−dvertices in X, respectively, and characterize k-elementary bipartite graph which is a graph such that the subgraph induced by all k-allowed edges is connected, where an edge is k-allowed if it is contained in a perfect 1-k matching.

The Number of Perfect Matchings in (3,6)-Fullerene

A(3,6)-fullerene is a connected cubic plane graph whose faces are only triangles and hexagons,and has the connectivity 2 or 3.The(3,6)-fullerenes with connectivity 2 are the tubes consisting of l concentric hexagonal layers such that each layer consists of two hexangons,capped on each end by two adjacent triangles,denoted by T_(l)(l≥1).A(3,6)-fullerene Tl with n vertices has exactly 2n/4+1 perfect matchings.The structure of a(3,6)-fullerene G with connectivity 3 can be determined by only three parameters r,s and t,thus we denote it by G=(r,s,t),where r is the radius(number of rings),s is the size(number of spokes in each layer,s(≥4,s is even),and t is the torsion(0≤t<s,t≡r mod 2).In this paper,the counting formula of the perfect matchings in G=n+1,4,t)is given,and the number of perfect matchpings is obtained.Therefore,the correctness of the conclusion that every bridgeless cubic graph with p vertices has at least 2p/3656perfect matchings proposed by Esperet et al is verified for(3,6)-fullerene G=(n+1,4,t).

Maximizing the Minimum and Maximum Forcing Numbers of Perfect Matchings of Graphs

Let G be a simple graph with 2n vertices and a perfect matching.The forcing number f(G,M) of a perfect matching M of G is the smallest cardinality of a subset of M that is contained in no other perfect matching of G.Among all perfect matchings M of G,the minimum and maximum values of f(G,M) are called the minimum and maximum forcing numbers of G,denoted by f(G) and F(G),respectively.Then f(G)≤F(G) ≤n-1.Che and Chen(2011) proposed an open problem:how to characterize the graphs G with f(G)=n-1.Later they showed that for a bipartite graph G,f(G)=n-1 if and only if G is complete bipartite graph K_(n,n).In this paper,we completely solve the problem of Che and Chen,and show that f(G)=n-1 if and only if G is a complete multipartite graph or a graph obtained from complete bipartite graph K_(n,n) by adding arbitrary edges in one partite set.For all graphs G with F(G)=n-1,we prove that the forcing spectrum of each such graph G forms an integer interval by matching 2-switches and the minimum forcing numbers of all such graphs G form an integer interval from [n/2] to n-1.

Finite-difference modeling of Maxwell viscoelastic media developed from perfectly matched layer

In numerical simulation of wave propagation,both viscoelastic materials and perfectly matched layers(PMLs)attenuate waves.The wave equations for both the viscoelastic model and the PML contain convolution operators.However,convolution operator is intractable in finite-difference time-domain(FDTD)method.A great deal of progress has been made in using time stepping instead of convolution in FDTD.To incorporate PML into viscoelastic media,more memory variables need to be introduced,which increases the code complexity and computation costs.By modifying the nonsplitting PML formulation,I propose a viscoelastic model,which can be used as a viscoelastic material and/or a PML just by adjusting the parameters.The proposed viscoelastic model is essentially equivalent to a Maxwell model.Compared with existing PML methods,the proposed method requires less memory and its implementation in existing finite-difference codes is much easier.The attenuation and phase velocity of P-and S-waves are frequency independent in the viscoelastic model if the related quality factors(Q)are greater than 10.The numerical examples show that the method is stable for materials with high absorption(Q=1),and for heterogeneous media with large contrast of acoustic impedance and large contrast of viscosity.

Stability of Perfectly Matched Layers for Time Fractional Schrödinger Equation

It is an important issue to numerically solve the time fractional Schrödinger equation on unbounded domains, which models the dynamics of optical solitons propagating via optical fibers. The perfectly matched layer approach is applied to truncate the unbounded physical domain, and obtain an initial boundary value problem on a bounded computational domain, which can be efficiently solved by the finite difference method. The stability of the reduced initial boundary value problem is rigorously analyzed. Some numerical results are presented to illustrate the accuracy and feasibility of the perfectly matched layer approach. According to these examples, the absorption parameters and the width of the absorption layer will affect the absorption effect. The larger the absorption width, the better the absorption effect. There is an optimal absorption parameter, the absorption effect is the best.

ON THE LAPLACIAN SPECTRAL RADII OF TREES WITH NEARLY PERFECT MATCHINGS

Let ：T2k＋1 be the set of trees on 2k＋ 1 vertices with nearly perfect matchings, and let S2k＋2 be the set of trees on 2k ＋ 2 vertices with perfect matchings. The largest Laplacian spectral radii of trees in ：T2k＋l and S2k＋2 and the corresponding trees were given by Guo （2003）. In this paper, the authors determine the second to the sixth largest Laplacian spectral radii among all trees in T2k＋1 and give the corresponding trees.

The Application of the Nonsplitting Perfectly Matched Layer in Numerical Modeling of Wave Propagation in Poroelastic Media

The nonsplitting perfectly matched layer （NPML） absorbing boundary condition （ABC） was first provided by Wang and Tang （2003） for the finite-difference simulation of elastic wave propagation in solids. In this paper, the method is developed to extend the NPML to simulating elastic wave propagation in poroelastic media. Biot＇s equations are discretized and approximated to a staggered-grid by applying a fourth-order accurate central difference in space and a second-order accurate central difference in time. A cylindrical twolayer seismic model and a borehole model are chosen to validate the effectiveness of the NPML. The results show that the numerical solutions agree well with the solutions of the discrete wavenumber （DW） method.

A study of perfectly matched layers for joint multicomponent reverse-time migration

Reverse-time migration in finite space requires effective boundary processing technology to eliminate the artificial truncation boundary effect in the migration result.On the basis of the elastic velocity-stress equations in vertical transversely isotropic media and the idea of the conventional split perfectly matched layer（PML）,the PML wave equations in reverse-time migration are derived in this paper and then the high order staggered grid discrete schemes are subsequently given.Aiming at the＂reflections＂from the boundary to the computational domain,as well as the effect of seismic event＇s abrupt changes at the two ends of the seismic array,the PML arrangement in reverse-time migration is given.The synthetic and real elastic,prestack,multi-component,reverse-time depth migration results demonstrate that this method has much better absorbing effects than other methods and the joint migration produces good imaging results.

Perfectly matched layer-absorbing boundary condition for finite-element time-domain modeling of elastic wave equations

The perfectly matched layer （PML） is a highly efficient absorbing boundary condition used for the numerical modeling of seismic wave equation. The article focuses on the application of this technique to finite-element time-domain numerical modeling of elastic wave equation. However, the finite-element time-domain scheme is based on the second- order wave equation in displacement formulation. Thus, the first-order PML in velocity-stress formulation cannot be directly applied to this scheme. In this article, we derive the finite- element matrix equations of second-order PML in displacement formulation, and accomplish the implementation of PML in finite-element time-domain modeling of elastic wave equation. The PML has an approximate zero reflection coefficients for bulk and surface waves in the finite-element modeling of P-SV and SH wave propagation in the 2D homogeneous elastic media. The numerical experiments using a two-layer model with irregular topography validate the efficiency of PML in the modeling of seismic wave propagation in geological models with complex structures and heterogeneous media.

Smallest Close to Regular Bipartite Graphs without an Almost Perfect Matching

A graph G is close to regular or more precisely a （d, d ＋ k）-graph, if the degree of each vertex of G is between d and d ＋ k. Let d ≥ 2 be an integer, and let G be a connected bipartite （d, d＋k）-graph with partite sets X and Y such that ｜X｜- ｜Y｜＋1. If G is of order n without an almost perfect matching, then we show in this paper that·n ≥ 6d ＋7 when k = 1,·n ≥ 4d＋ 5 when k = 2,·n ≥ 4d＋3 when k≥3.Examples will demonstrate that the given bounds on the order of G are the best possible.

On the Upper Bounds of the Numbers of Perfect Matchings in Graphs with Given Parameters

Let φ（G）, κ（G）, α（G）, χ（G）, cl（G）, diam（G） denote the number of perfect matchings, connectivity, independence number, chromatic number, clique number and diameter of a graph G, respectively. In this note, by constructing some extremal graphs, the following extremal problems are solved： 1. max {φ（G）： ｜V（G）｜ = 2n, κ（G）≤ k} = k[（2n - 3）!!], 2. max{φ（G）： ｜V（G）｜ = 2n,α（G） ≥ k} =[∏ i=0^k-1 （2n - k-i]（2n - 2k - 1）!!], 3. max{φ（G）： ｜V（G）｜=2n, χ（G） ≤ k} =φ（Tk,2n） Tk,2n is the Turán graph, that is a complete k-partitc graph on 2n vertices in which all parts are as equal in size as possible, 4. max{φ（G）： ｜V（G）｜ = 2n, cl（G） = 2} = n!, 5. max{φ（G）： ｜V（G）｜ = 2n, diam（G） ≥〉 2} = （2n - 2）（2n - 3）[（2n - 5）!!], max{φ（G）： ｜V（G）｜ = 2n, diam（G） ≥ 3} = （n - 1）^2[（2n - 5）!!].

Some Ore-type Results for Matching and Perfect Matching in k-uniform Hypergraphs

Let SI and S2 be two （k- 1）-subsets in a k-uniform hypergraph H. We call S1 and S2 strongly or middle or weakly independent if H does not contain an edge e ∈ E（H） such that S1 ∩ e≠ 0 and S2 ∩ e ≠0 or e S1 ∪ S2 or e S1 ∪ S2, respectively. In this paper, we obtain the following results concerning these three independence. （1） For any n ≥ 2k2 - k and k ≥ 3, there exists an n-vertex k-uniform hypergraph, which has degree sum of any two strongly independent （k - 1）-sets equal to 2n - 4（k - 1）, contains no perfect matching; （2） Let d ≥ 1 be an integer and H be a k-uniform hypergraph of order n ≥ kd＋ （k- 2）k. If the degree sum of any two middle independent （k- 1）-subsets is larger than 2（d- 1）, then H contains a d-matching; （3） For all k ≥ 3 and sufficiently large n divisible by k, we completely determine the minimum degree sum of two weakly independent （k - 1）-subsets that ensures a perfect matching in a k-uniform hypergraph H of order n.

Ordering Trees with Nearly Perfect Matchings by Algebraic Connectivity

Let T2k＋1 be the set of trees on 2k＋1 vertices with nearly perfect matchings and α（T） be the algebraic connectivity of a tree T. The authors determine the largest twelve values of the algebraic connectivity of the trees in T2k＋1. Specifically, 10 trees T2,T3,... ,T11 and two classes of trees T（1） and T（12） in T2k＋1 are introduced. It is shown in this paper that for each tree T^′1,T^″1∈T（1）and T^′12,T^″12∈T（12） and each i,j with 2≤i〈j≤11,α（T^′1）=α（T^″1）〉α（Tj）〉α（T^′12）=α（T^″12）.It is also shown that for each tree T with T∈T2k＋1／（T（1）∪{T2,T3,…,T11}∪T（12））,α（T^′12）〉α（T）.

Perfect Matching in k-partite k-graphs and 3-uniform HM-bipartite Hypergraphs

Let H=(V,E)be an n-balanced k-partite k-graph with partition classes V1,...,Vk.Suppose for every legal(k-1)-tuple f contained in V\V1 and for every legal(k-1)-tuple g contained in V\Vk such that f∪g■E(H),we have d(f)+d(g)≥n+1.In this paper,we prove that under this condition H must have a perfect matching.Another result of this paper is about the perfect matching in 3-uniform hm-bipartite hypergraphs.Let G be a 3-uniform hm-bipartite hypergraph with one of whose sides V1 has the size n,the another side V2 has size 2 n.If for all the legal 2-tuple f with|f∩V1|=1 and for all the legal 2-tuple g with|g∩V1|=0,we have d(f)≥n-2 and d(g)>n/2,then G has a perfect matching.

DISCRETE SINGULAR CONVOLUTION METHOD WITH PERFECTLY MATCHED ABSORBING LAYERS FOR THE WAVE SCATTERING BY PERIODIC STRUCTURES

A new computational algorithm is introduced for solving scattering problem in periodic structure. The PML technique is used to deal with the difficulty on truncating the unbounded domain while the DSC algorithm is utilized for the spatial discretization. The present study reveals that the method is efficient for solving the problem.

A THEOREM CONCERNING PERFECT MATCHINGS IN HEXAGONAL SYSTEMS

In the present paper a theorem is established which offers some necessary and sufficient eonditionsfor a hexagonal system to have perfect matchings.