A C finite element method for an inverse problem

A C⁰ finite element method is presented for an inverse problem in which the coefficient in the differential operator is to be determined from the measurement of the solution of a boundary value problem. The unknown in the inverse problem is approximated by a minimizer of a cost function that includes both the output error and equation error. Error estimates in a weighted H⁻¹ norm and L² are given. Numerical examples are presented to show features of the method.     

Asymptotically optimal algorithm for finding one and two edge-disjoint traveling salesman routes of maximal weight in Euclidean space

The paper presents a polynomial approximation algorithm A solving the problem of finding one and two edge-disjoint Hamiltonian cycles (traveling salesman routes) of maximal weight in a complete weighted undirected graph in multidimensional Euclidean space. The asymptotic optimality of the algorithm is established.     

Phase Retrieval of Real-valued Functions in Sobolev Space

The Sobolev space H^s(R^d) with s > d/2 contains many important functions such as the bandlimited or rational ones. In this paper we propose a sequence of measurement functions {ϕ_j,k^γ} ⊆ H^-s(R^d) to the phase retrieval problem for the real-valued functions in H^s(R^d). We prove that any real-valued function f ∈ H^s(R^d) can be determined, up to a global sign, by the phaseless measurements {|<f, ϕ_j,k^γ>|}. It is known that phase retrieval is unstable in infinite dimensional spaces with respect to perturbations of the measurement functions. We examine a special type of perturbations that ensures the stability for the phase-retrieval problem for all the real-valued functions in H^s(R^d) ∩ C¹(R^d), and prove that our iterated reconstruction procedure guarantees uniform convergence for any function f ∈ H^s(R^d) ∩ C¹(R^d) whose Fourier transform f is L¹-integrable. Moreover, numerical simulations are conducted to test the efficiency of the reconstruction algorithm.     

Asymptotic and Partial Asymptotic Hankel Operators on $$H^2(\mathbb{D}^n)$$ H 2 ( D n ) )

In this paper, we generalize the concept of asymptotic Hankel operators on H²(D) to the Hardy space H²(Dⁿ) (over polydisk) in terms of asymptotic Hankel and partial asymptotic Hankel operators and investigate some properties in case of its weak and strong convergence. Meanwhile, we introduce i^th-partial Hankel operators on H²(Dⁿ) and obtain a characterization of its compactness for n > 1. Our main results include the containment of Toeplitz algebra in the collection of all strong partial asymptotic Hankel operators on H²(Dⁿ). It is also shown that a Toeplitz operator with symbol φ is asymptotic Hankel if and only if φ is holomorphic function in L^∞(Tⁿ).     

二阶双曲方程的全离散格式下的混合元超收敛分析

通过在空间方向上使用双线性元和最低阶的Nedelec元(即Q₁₁+Q₀₁×Q₁₀)以及在时间方向上使用二阶精度的数值逼近格式,得到了在矩形网格上二阶双曲方程全离散混合元格式下的对原始变量的L^∞(H¹)和流量的L^∞((L²)²)的超逼近和超收敛的误差结果.在分析过程中,巧妙地使用了上述混合单元对在矩形网格上的特有的高精度积分恒等式和精确解的投影和插值之间的在H¹范数意义下的超逼近的估计.最后,给出一些数值结果来验证理论分析的正确性.     

Least-squares Galerkin procedures for parabolic integro-differential equations

Two least-squares Galerkin finite element schemes are formulated to solve parabolic integro-differential equations. The advantage of this method is that it is not subject to the LBB condition. The convergence analysis shows that the least-squares mixed element schemes yield the approximate solution with optimal accuracy in H(div;Ω)×H¹(Ω) and (L²(Ω))²×L²(Ω), respectively.     

Beurling Type Theorem on the Hilbert Space Generated by a Positive Sequence

Let H²(γ) be the Hilbert space over the bidisk D² generated by a positive sequence γ={γ_nm}_n,m ≥ 0. In this paper, we prove that the Beurling type theorem holds for the shift operator on H²(γ) with γ={γ_nm}_n,m ≥ 0 satisfying certain series of inequalities. As a corollary, we give several applications to a class of classical analytic reproducing kernel Hilbert spaces over the bidisk D².     

Crank–Nicolson least-squares Galerkin procedures for parabolic integro-differential equations

Two Crank–Nicolson least-squares Galerkin finite element schemes are formulated to solve parabolic integro-differential equations. The advantage of this method is that it is not subject to the LBB condition. The convergence analysis shows that the methods yield the approximate solutions with optimal accuracy in H(div; Ω) × H¹(Ω) and (L²(Ω))² × L²(Ω), respectively. Moreover, the two methods both get the approximate solutions with second-order accuracy in time increment.     

New constructions of divisible designs

A construction is given for a (p^2a(p+1),p²,p^2a+1(p+1),p^2a+1,p^2a(p+1)) (p a prime) divisible difference set in the group H×Z²_pa+1 where H is any abelian group of order p+1. This can be used to generate a symmetric semi-regular divisible design; this is a new set of parameters for λ₁≠0, and those are fairly rare. We also give a construction for a (p^a−1+p^a−2+…+p+2,p^a+2, p^a(p^a+p^a−1+…+p+1), p^a(p^a−1+…+p+1), p^a−1(p^a+…+p²+2)) divisible difference set in the group H×Z_p2×Z^a_p. This is another new set of parameters, and it corresponds to a symmetric regular divisible design. For p=2, these parameters have λ₁=λ₂, and this corresponds to the parameters for the ordinary Menon difference sets.     

A stochastic maximum principle for processes driven by fractional Brownian motion

We prove a stochastic maximum principle for controlled processes X(t)=X^(u)(t) of the form

dX(t)=b(t,X(t),u(t)) dt+σ(t,X(t),u(t)) dB^(H)(t),

where B^(H)(t) is m-dimensional fractional Brownian motion with Hurst parameter . As an application we solve a problem about minimal variance hedging in an incomplete market driven by fractional Brownian motion.     

Large 2P3-free graphs with bounded degree

Let ex^* (D;H) be the maximum number of edges in a connected graph with maximum degree D and no induced subgraph H; this is finite if and only if H is a disjoint union of paths. If the largest component of such an H has order m, then ex^*(D; H) = O(D²ex^*(D; P_m)). Constructively, ex^*(D;qP_m) = Θ(gD²ex^*(D;P_m)) if q>1 and m> 2(Θ(gD²) if m = 2). For H = 2P₃ (and D 8), the maximum number of edges is if D is even and

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if D is odd, achieved by a unique extremal graph.     

Three-dimensional stable nonconforming parallelepiped elements for the Stokes problem

Stability result is obtained for the approximation of the stationary Stokes problem with nonconforming elements proposed by Jim Douglas Jr. et al. [Math. Model. Numer. Anal. 33 (4) (1999) 747] for the velocity with conforming bubble functions and discontinuous piecewise linear for the pressure on parallelepiped elements. Optimal order H¹ and L² error estimates are derived.     

Efficient Energy-preserving Methods for General Nonlinear Oscillatory Hamiltonian System

In this paper, we propose and analyze two kinds of novel and symmetric energy-preserving formulae for the nonlinear oscillatory Hamiltonian system of second-order differential equations Aq"(t)+ Bq(t)=f(q(t)), where A ∈ R^m×m is a symmetric positive definite matrix, B ∈ R^m×m is a symmetric positive semi-definite matrix that implicitly contains the main frequencies of the problem and f(q)=-▽_qV(q) for a real-valued function V(q). The energy-preserving formulae can exactly preserve the Hamiltonian H(q', q)=(1)/2q'^τ Aq' + (1)/2q^τ Bq + V(q). We analyze the properties of energy-preserving and convergence of the derived energy-preserving formula and obtain new efficient energy-preserving integrators for practical computation. Numerical experiments are carried out to show the efficiency of the new methods by the nonlinear Hamiltonian systems.     

Wandering subspaces and quasi-wandering subspaces in the Hardy–Sobolev spaces

In this paper, we prove that for-1/2 ≤β≤0.suppose M is an invariant subspaces of the Hardy Sobolev spaces H_β~2(D) for T_z~β, then M() zM is a generating wandering subspace of M, that is,M=[MzM]_T_z~β Moreover, any non-trivial invariant subspace M of H_β~2(D) is also generated by the quasi-wandering subspace P_MT_z~βM~⊥ that is,M=[P_MT_z~βM~⊥]_(T_z~β).     

A branch and bound method for the job-shop problem with sequence-dependent setup times

This paper deals with the job-shop scheduling problem with sequence-dependent setup times. We propose a new method to solve the makespan minimization problem to optimality. The method is based on iterative solving via branch and bound decisional versions of the problem. At each node of the branch and bound tree, constraint propagation algorithms adapted to setup times are performed for domain filtering and feasibility check. Relaxations based on the traveling salesman problem with time windows are also solved to perform additional pruning. The traveling salesman problem is formulated as an elementary shortest path problem with resource constraints and solved through dynamic programming. This method allows to close previously unsolved benchmark instances of the literature and also provides new lower and upper bounds.     

Pairs of Adjacent Hamiltonian Circuits with Small Intersection

Consider the question: When can the edges in a pair of Hamiltonian circuits be redistributed to form another pair of circuits with the same union and intersection? A class of pairs is exhibited which intersect in two edges and cannot be rearranged in this way. A connection to algorithms for the traveling salesman problem is explained using the convex polytope of Hamiltonian circuits in . The exhibited pair is shown to be an edge of that polytope.     

Choosability conjectures and multicircuits

This paper starts with a discussion of several old and new conjectures about choosability in graphs. In particular, the list-colouring conjecture, that ch′=χ′ for every multigraph, is shown to imply that if a line graph is (a : b)-choosable, then it is (ta : tb)-choosable for every positive integer t. It is proved that ch(H²)=χ(H²) for many “small” graphs H, including inflations of all circuits (connected 2-regular graphs) with length at most 11 except possibly length 9; and that ch″(C)=χ″(C) (the total chromatic number) for various multicircuits C, mainly of even order, where a multicircuit is a multigraph whose underlying simple graph is a circuit. In consequence, it is shown that if any of the corresponding graphs H² or T(C) is (a : b)-choosable, then it is (ta : tb)-choosable for every positive integer t.     

Regularity results of solution uniform in time for complex Ginzburg-Landau equation

We provide the H²-regularity result of the solution ψ and its first- order time derivative ψ_t and the second-order time derivative ψ_tt for the complex Ginzburg-Landau equation with the Dirichlet or Neumann boundary conditions. The analysis shows that these regularity results are uniform when t tends to ∞ and 0 and are dependent of the powers of ε⁻¹.     

On the relationship of approximation algorithms for the minimum and the maximum traveling salesman problem

We consider the relationship between the minimum and the maximum traveling salesman problem. The paper is based on the idea of applying heuristics for the maximum traveling salesman problem to the minimum traveling salesman problem. Numerical results confirm the efficiency of the proposed method.