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Constructions of (q, K, λ, t, Q) almost difference families
Lu QIU,Dianhua WU
Front. Math. China. 2014, 9 (2): 377-386.
https://doi.org/10.1007/s11464-014-0332-3
The concept of a (q, k, λ, t) almost difference family (ADF) has been introduced and studied by C. Ding and J. Yin as a useful generalization of the concept of an almost difference set. In this paper, we consider, more generally, (q, K, λ, t, Q)-ADFs, where K = {k1, k2, ..., kr} is a set of positive integers and Q = (q1, q2,..., qr) is a given block-size distribution sequence. A necessary condition for the existence of a (q, K, λ, t, Q)-ADF is given, and several infinite classes of (q, K, λ, t, Q)-ADFs are constructed.
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Approximation by semigroup of spherical operators
Yuguang WANG,Feilong CAO
Front. Math. China. 2014, 9 (2): 387-416.
https://doi.org/10.1007/s11464-014-0361-y
This paper concerns about the approximation by a class of positive exponential type multiplier operators on the unit sphere Sn of the (n + 1)-dimensional Euclidean space for n≥2.We prove that such operators form a strongly continuous contraction semigroup of class (C0) and show the equivalence between the approximation errors of these operators and the K-functionals. We also give the saturation order and the saturation class of these operators. As examples, the rth Boolean of the generalized spherical Abel-Poisson operator ⊕rVtγ and the rth Boolean of the generalized spherical Weierstrass operator ⊕rWtk for integer r≥1 and reals γ, κ∈ (0, 1] have errors ∥⊕rVtγf-f∥X≈ωrγ(f,t1/γ)X and ∥⊕rWtkf-f∥X≈ω2rk(f,t1/(2k))X for all f∈X and 0≤t≤2π, where Xis the Banach space of all continuous functions or all Lpintegrable functions, 1≤p<+∞, on Sn with norm ∥⋅∥X, and ωs(f,t)Xis the modulus of smoothness of degree s>0 for f∈X. Moreover, ⊕rVtγ and ⊕rWtk have the same saturation class if γ=2κ.
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Weak Galerkin finite element method for valuation of American options
Ran ZHANG,Haiming SONG,Nana LUAN
Front. Math. China. 2014, 9 (2): 455-476.
https://doi.org/10.1007/s11464-014-0358-6
We introduce a weak Galerkin finite element method for the valuation of American options governed by the Black-Scholes equation. In order to implement, we need to solve the optimal exercise boundary and then introduce an artificial boundary to make the computational domain bounded. For the optimal exercise boundary, which satisfies a nonlinear Volterra integral equation, it is resolved by a higher-order collocation method based on graded meshes. With the computed optimal exercise boundary, the front-fixing technique is employed to transform the free boundary problem to a one-dimensional parabolic problem in a half infinite area. For the other spatial domain boundary, a perfectly matched layer is used to truncate the unbounded domain and carry out the computation. Finally, the resulting initial-boundary value problems are solved by weak Galerkin finite element method, and numerical examples are provided to illustrate the efficiency of the method.
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