On the Convergence Rate of a Space Decomposition Method

Domain decomposition method and multigrid method can be unified in the framework of the space decomposition method. This paper has obtained a new result on the convergence rate of the space decomposition method, which can be applied to some nonuniformly elliptic problems.     

Schmidt Decomposition of Quaternion Matrix and the Orthonormalization of Vectors in a Generalized Unitary Space

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On the Convergence of a Space Decomposition Method for Nonlinear Pròblems

SupportedbyNSFCproject.ContractNo.19771034TheSchwarzalternatingmethodofpartialdifferentialequationappearedonehundredyearsagoandhasbeenappliedtonumericalcomputationsince1960s.NowithasdevelopedintoSchwarzalgorithmconvenientforparallelcomputation.P.L.Lions[1]firststudiedtheSchwarzalgoritlunbyusingviewpointofspacedecomposition(seealso[2]).ThenJ.C.Xu[3]hasproposedspacedecompositionmethodtounifydomaindecompositionmethod(includingSchwarzalgorithms)andmultigridmethod.Since1990sX.C.Taihasstudie…     

On some properties of the positive m-Hessian operators in C 2(Ω)

We consider the p-Hessian operator T _p [u] := tr _p u _xx on the set ${\{u \in C^2(\bar{\Omega}), T_p[u] \geq \nu > 0, 1 \leq p \geq 0\}}$ and extend the common approach to the uniqueness and existence theorems. For instance, we prove the uniqueness of solutions to the Dirichlet problem for m-Hessian equations in ${C^2(\bar{\Omega})}$ for constant boundary data. We also give nonexistence theorems and, in addition, we establish the extremal properties of the functionals I _p [u; f] in ${C^2(\bar{\Omega})}$ and we derive some applications.     

On the error term in Weyl’s law for the Heisenberg manifolds (II)

In this paper we study the mean square of the error term in the Weyl’s law of an irrational (2l + 1)-dimensional Heisenberg manifold. An asymptotic formula is established. This work was supported by National Natural Science Foundation of China (Grant No. 10771127)     

A weighted estimate of best approximations in L2(Ω)

The best approximation [in the space L₂(Ω)] of a function f, satisfying a Lipschitz condition with exponent α, 0?α?1, with the aid of certain spaces of local functions, dependent on a parameter h, is discussed. We obtain the estimate $$\left\| {f - \tilde f} \right\|_\beta \leqslant \tilde C\left( f \right)h\min \left\{ {\alpha , \beta } \right\}$$ , where $$\left\| u \right\|_\beta = \mathop {\max }\limits_{x \in \bar \Omega } \left| {r^\beta u\left( x \right)} \right|,\beta \geqslant 0, u \in C\left( {\bar \Omega } \right)$$ and r = r(x) is the distance of the point x from the boundary of the domain Ω.     

On the error term in Weyl’s law for the Heisenberg manifolds (Ⅱ)

In this paper we study the mean square of the error term in the Weyl’s law of an irrational (2l + 1)-dimensional Heisenberg manifold. An asymptotic formula is established.     

On the L2-Discrepancy of the Sobol–Hammersley Net in Dimension 3

With the help of Walsh series analysis we show that the symmetrized Sobol–Hammersley net in base 2 and dimension 3 has almost best possible order of L₂ -discrepancy.     

On the Uniqueness Theorems for the Closed Convex Hypersurfaces in a Space of Constant Curvature

In this paper we generalized the uniqueness theorem of. Alexadroff-Fenchel-Jessen, the cohn-Vossen theorem and the Hilbert-Liebmann-Hsiung theorem to hypersurfaces in a sphere S~(n+1) or a hyperbolic space H~(n+1)     

On the Existence of Suprema of Infinite Many Elements in a Riesz Space

<正> Let E be a Riesz space with a disjoint system {u_i :i∈I}. We denote by E_i the principal band generated by u_i(i∈I). A necessary and sufficient condition is found in this paper under which sup{X_i:i∈I} exists in E for every x_i∈E (i∈I).     

Additional Note on Partial Regularity of Weak Solutions of the Navier-Stokes Equations in the Class L ∞(0, T, L 3(Ω)3)

We present a simplified proof of a theorem proved recently concerning the number of singular points of weak solutions to the Navier-Stokes equations. If a weak solution u belongs to L (0, T, L ³()³), then the set of all possible singular points of u in is at most finite at every time t₀ (0,T).     

On the limit theorems for random variables with values in the spaces L p (2≦p<∞)

Summary We prove that whenever B is an infinite dimensional Banach space, there exists a B-valued random variable X failing the Central Limit Theorem (in short the CLT) and such that IEX²= but yet satisfying the Law of the Iterated Logarithm (in short the LIL). We obtain a new sufficient condition for the LIL in Hilbert space and we characterize the random variables with values in l _p or L _p with 2<p< which satisfy the CLT. As an application we show that in l _p (2<p<) the stochastic boundedness of the weighed partial sums does not imply the CLT.Research partially supported by NSF Grant MCS 75-07605 A01     

On the torsion in K_2 of a field

For a field F,let Gn(F) = {{a,Φn(a)} ∈ K2(F) | a,Φn(a) ∈ F*},where Φn(x) is the n-th cyclotomic polynomial.At first,by using Faltings' theorem on Mordell conjecture it is proved that if F is a number field and if n = 4,8,12 is a positive integer having a square factor then Gn(F) is not a subgroup of K2(F),and then by using the results of Manin,Grauert,Samuel and Li on Mordell conjecture theorem for function fields,a similar result is established for function fields over an algebraically closed field.     

On the Size of a Cap in PG(n,q) with q Even and n ≥ 3

Let m2(n,q), m2(n,q) be, respectively, the maximum value, the second largest value of k for which there exists a complete k-cap in PG(n,q). In this paper, the known upper bound on m2(3,q), q even, q 8, is improved. This new upper bound on m2(3,q) is then used to improve the upper bounds on m2(n,q), q even, q 8 and n 4.     

On the description of the spacesL_p^{\overrightarrow \alpha } (Ω)

We consider the problem of describing the spaces \(L_p^{\overrightarrow \alpha } \) (Ω), Ω ≠E _n of Bessel potentials in terms of the “modified” fractional derivatives of Marchaud.     

On 2-Factors with Prescribed Properties in a Bipartite Graph

Liu and Yan gave the degree condition for a balanced bipartite graph G = （V1, V2; E） to have k vertex-disjoint quadrilaterals containing any given k independent edges e1,……, ek of G, respectively. They also conjectured that for any k independent edges e1,……, ek of G, G has a 2-factor with k cycles C1, C2, ……, Ck with respect to {e1, e2,……, ek} such that k - 1 of them are quadrilaterals. In this paper, we prove this conjecture.