CALCULATIONS OF BIVARIATE SPLINES——Ⅰ. Truncated Powers and BM-Splines in S_2~1, (?)2 and S_4~2,(?)_2

Determining the lowest degree, dimensions and basis functions of btp (bivariate truncated power(s)) spaces with given smoothness requirements and establishing the calculation formulae of btp on a three-and four-direction mesh, we give the necessary and sufficient condition for the existence of the bivariate BM-spline in S_4~2, and the bivariate BM-spline in S_2~1, in terms of linear combinations of btp. The so-called "revolving around" teahnique is mentioned.     

A SERENDIPITY FAMILY 0F LOCALLY SUPPORTED SPLINES IN S_8~2(△)

For a given triangulation Δ of a region R of interest, let S_8~2(Δ): = {s∈C~2(R): s|_1∈P_8,l∈Δ}be the space of all splines of degree 8 and smoothness 2, where t denotes any triangle of Δ and P_8 thespace of polynomials of total degree≤8. Furthermore, let S_8~2(Δ):={s∈S_8~3(Δ):s∈C~3 at v, v∈Δ}be a super spline subspace of S_8~2(Δ). We construel a collection of locally supported splines in S_8~2(Δ)which can be used to achieve lhe full approximation order of S_8~2(Δ) and its cardinalily is less than the di-mension of super spline space space S_8~2(Δ).     

Cyclical Deformations of Quadrangles in S_κ~2 and Defining Properties of Alexandrov Spaces

This paper is mainly devoted to proving the four equivalent defining properties of a CBA(κ)space.The proof is based on an interesting tool we established which describes the cyclical five-step deformation procedure for quadrangles in the model 2-plane S_κ~2, including the limit shape of each step.As a byproduct we give a complete list of cyclical deformation procedures for all kinds of quadrangles on S_1~2.At last we make a contrast of geometric properties of CBA with CBB spaces, including a comparison between their defining properties and a discussion about Alexandrov's Lemma.     

The Algebra of Pseudo-Differential Operator on the Functional Space W_λ S_(p,σ)~m

For popularizing the functional space S_(p,δ)~m which is common in use, a Frechet functional space W_λS_(p,δ)~m is defined in this paper and an exploration is attempted on the algebraic characteristics of the pseudo-differential operators stipulated by the functional space W_λS_(p,δ)~m. Definition 1 We say P(x,)∈C~∞(R_x~n×R_(?)~n) is a symbol of the functional set W_λS_(p,δ)~m (-∞相似文献   

2-CONNECTED k-REGULAR GRAPHS ON AT MOST 3k+3 VERTICES TO BE HAMILTONIAN (CONTINUED)

Let X~* and Y~* be generated by S(?){v_0},where G(S)is Hamiltonian connected and|X~*|=x~*,|Y~*|=Y~* and S_1~*,S_2~*,…,S_(x*)~* be the sets of vertices contained in the opensegments of C between vertices of X~*.Let S_1~*,x_1,S_2~*,x_2,…,S_(x*)~*,x_(?)~* be the segmentsand vertices of X~* in order around C.S_i~* is said to be an X~*(3)-interval if one ofx_(i-1)and x_i belongs to X_j~*—X_2~*.Let S=S_1~*,and S={a_1,c_1,c_2,…,c_1,b_1}.It is easy to see that the statement inLemma 2 can be modified as(?)({a_1,b_1},S_i~*)≤(?)     

ON INTERPOLATION BY S_2~1(△_(m,n)~((2)))

Let △_(m,n)~(2)be a subdivision of D:[a,b]×[c,d](Fig.1),_1arelengths of[a,b]and[c,d]respectively,and h_1=_1/m,h_2=_2/n.     

A Comment on a Necessary and Sufficient Condition for Transitive Permutation Group by P.J.Cameron

P.J. Cameron had mentioned that "It can be shown that a permutation group is transitive if and only if its centralizer in the symmetric group is semiregular, and vice versa (Wielandt, page 9)."[1] The latter is true, i.e. G≤S_Ω is semiregular(?)Cs_Ω(G) is transitive. [cf. 2] But in the former statement fails, i.e. Cs_Ω(G) is semiregular(?)G≤S_Ω is transitive. In this note we give a series of the     

The Order of Hypersubstitutions of Type （2, 2）

Hypersubstitutions are mappings which map operation symbols to terms of the corresponding arities. They were introduced as a way of making precise the concept of a hyperidentity and generalizations to M-hyperidentities. A variety in which every identity is satisfied as a hyperidentity is called solid. If every identity is an M-hyperidentity for a subset M of the set of all hypersubstitutions, the variety is called M-solid. There is a Galois connection between monoids of hypersubstitutions and sublattices of the lattice of all varieties of algebras of a given type. Therefore, it is interesting and useful to know how semigroup or monoid properties of monoids of hypersubstitutions transfer under this Galois connection to properties of the corresponding lattices of M-solid varieties. In this paper, we study the order of each hypersubstitution of type （2, 2）, i.e., the order of the cyclic subsemigroup generated by that hypersubstitution of the monoid of all hypersubstitutions of type （2, 2）. The main result is that the order is 1, 2, 3, 4 or infinite.     

Low Dimensional Cohomology of Hom-Lie Algebras and q-deformed W（2, 2） Algebra

This paper aims to study low dimensional cohomology of Hom-Lie algebras and the qdeformed W(2, 2) algebra. We show that the q-deformed W(2, 2) algebra is a Hom-Lie algebra. Also,we establish a one-to-one correspondence between the equivalence classes of one-dimensional central extensions of a Hom-Lie algebra and its second cohomology group, leading us to determine the second cohomology group of the q-deformed W(2, 2) algebra. In addition, we generalize some results of derivations of finitely generated Lie algebras with values in graded modules to Hom-Lie algebras.As application, we compute all αk-derivations and in particular the first cohomology group of the q-deformed W(2, 2) algebra.     

THE WEIGHTED HERZ-TYPE HARDY SPACES hK_q~(α,p)(ω_1,ω_2)

Let (.the Muckenhoupt class). In this paper, the author introduce the weighted Herz-type Hardy spaces (w2) and present their atomic decomposition. Using the atomic decomposition, the author find out their dual spaces, establish the boundedness on these spaces of the pseudo-differential operators of order zero and show that , the class of C(Rn)-functions with compactly support, is dense in and there is a subsequence, which converges in distrbutional sense to some distribution of , of any bounded sequence in In addition, the author also set up the boundedness of some non-linear quantities in compensated compactness.     

Analytic Properties for Holomorphic Matrix-valued Maps in C~(2×2)

In this paper, we investigate some analytic properties for a class of holomorphic matrixvalued functions. In particular, we give a Picard type theorem which depicts the characterization of Picard omitting value in these functions. We also study the relation between asymptotic values and Picard omitting values, and the relation between periodic orbits of the canonical extension on C~(2×2) and Julia set of one dimensional complex dynamic system.     

GABOR ANALYSIS OF THE SPACES M(p,q,w)(R~d) AND S(p,q,r,w,ω)(R~d)

Let g be a non-zero rapidly decreasing function and w be a weight function. In this article in analog to modulation space, we define the space M(p, q, w)(Rd) to be the subspace of tempered distributions f ∈ S′(Rd) such that the Gabor transform Vg(f) of f is in the weighted Lorentz space L(p, q, wdμ) (R2d). We endow this space with a suitable norm and show that it becomes a Banach space and invariant under time frequence shifts for 1 ≤ p, q ≤ ∞. We also investigate the embeddings between these spaces and the...     

A necessary and sufficient condition for a surface sum of two handlebodies to be a handlebody

The main results of the paper are that we give a necessary and sufficient condition for a surface sum of two handlebodies along a connected surface to be a handlebody as follows:(1) The annulus sum H = H_1∪_AH_2 of two handlebodies H_1 and H_2 is a handlebody if and only if the core curve of A is a longitude for either H_1 or H_2;(2) Let H = H_1 ∪S_(g,b) H_2 be a surface sum of two handlebodies H_1 and H_2 along a connected surface S = S_(g,b), b 1, n_i = g(H_i) 2, i = 1, 2. Suppose that S is incompressible in both H_1 and H_2. Then H is a handlebody if and only if there exists a basis J = {J_1,..., J_m} with a partition(J_1, J_2) of J such that J_1 is primitive in H_1 and J_2 is primitive in H_2.     

Some Results on Minimum Aberration Mixed-level （2^r）2^n Factorial Designs

Mukerjee and Wu(2001) employed projective geometry theory to find the wordlength pattern of a regular mixed factorial design in terms of its complementary set, but only for the numbers of words of length 3 or 4.In this paper,by introducing a concept of consulting design and based on the connection between factorial design theory and cod- ing theory,we obtain some combinatorial identities that relate the wordlength pattern of a regular mixed-level (2~r)2~n factorial design to that of its consulting design.Consequently,a general rule for identifying minimum aberration (2~r)2~n factorial designs through their con- sulting designs is established.It is an improvement and generalization of the related result in Mukerjee and Wu(2001).     

THE NEAR DIFFERENCE SETS OF TYPE 2 WITH PARAMETERS (9λ+6, 3λ+1, λ)

It is proved in this paper that the number of the near difference sets of type 2 with parameters ( 9λ +6, 3λ+1, λ) is finite by studying the relationship between the cyclic sequence and its normai subsequences. It is determined accordingly that there exists a ( 9λ + 6, 3λ + 1, λ) -near difference set of type 2 if and only if λ. = 1 or 2. Together with the results of [1,2,3,4] we can conclude that there are only 6 groups of parameters corresponding to the near difference sets of type 2 if k ≤ 3λ 4+2 .     

ON THE DIRICHLET PROBLEMS OF DOUBLY-PERIODIC ANALYTIC FUNCTIONS

Let Im(ω_2/ω_1)>0 and S_0 be the parallelogram with vertices ±ω_1±ω_2 (the fundamental cell). Let L_0 be a closed smooth curve in So with the counterclockwise sense as its positive direction which bounds a region S_0~+ Denote     

ON THE CAUCHY PROBLEM OF A COHERENTLY COUPLED SCHR?DINGER SYSTEM

In this article, we consider the well-posedness of a coherently coupled Schrdinger system with four waves mixing in space dimension n ≤ 4. The Cauchy problem for the cubic system is studied in L~2 for n ≤ 2 and in H~1 for n ≤ 4. We obtain two sharp conditions between global existence and blow up.     

I.I.D. STATISTICAL CONTRACTION OPERATORS AND STATISTICALLY SELF－SIMILAR SETS

I.i.d. random sequence is the simplest but very basic one in stochastic processes, and statistically self-similar set is the simplest but very basic one in random recursive sets in the theory of random fractal. Is there any relation between i.i.d. random sequence and statistically self-similar set? This paper gives a basic theorem which tells us that the random recursive set generated by a collection of i.i.d. statistical contraction operators is always a statistically self-similar set.     

A note on Faber operator

For a rectifable Jordan curve Γ with complementary domainsD and D,Anderson conjectured that the Faber operator is a bounded isomorphism between the Besov spaces Bp(1 p ∞) of analytic functions in the unit disk and in the inner domain D,respectively.We point out that the conjecture is not true in the special case p=2,and show that in this case the Faber operator is a bounded isomorphism if and only if Γ is a quasi-circle.     

Piecewise linear approximation for the dynamical Φ_3~4 model

We construct a piecewise linear approximation for the dynamicalΦ3~4 model on T^3.The approximation is based on the theory of regularity structures developed by Hairer(2014).They proved that renormalization in a dynamicalΦ3~4 model is necessary for defining the nonlinear term.In contrast to Hairer(2014),we apply piecewise linear approximations to space-time white noise,and prove that the solutions of the approximating equations converge to the solution of the dynamicalΦ_3~4 model.In this case,the renormalization corresponds to multiplying the solution by a t-dependent function,and adding it to the approximating equation.