Analysis and numerics for a parabolic equation with impulsive forcing

The paper considers a one-dimensional particle-continuum model, with impulsive interaction between the fluid and a number of pointwise particles. A simplification results in a system of ODEs coupled with a parabolic PDE forced by a nonlinear term involving a sum of Dirac delta functions. The existence of a mild solution is proved using a combination of energy estimates and semigroup theory. However, the regularity of these solutions is shown to be limited to C^0,1 by the impulsive terms. The convergence of a Galerkin method is established simultaneously with a proof of continuous dependence, and thus uniqueness, of solutions for the underlying system. The peculiarities of the system imply this analysis must be performed in L_∞. The C^0,1 regularity of the solution determines a suboptimal rate of convergence for the Galerkin method. The theoretical results are verified by MATLAB computations.     

Bounded rank of C-algebras

We introduce a concept of the bounded rank (with respect to a positive constant) for unital C^*-algebras as a modification of the usual real rank and present a series of conditions insuring that bounded and real ranks coincide. These observations are then used to prove that for a given n and K>0 there exists a separable unital C^*-algebra Z_n^K such that every other separable unital C^*-algebra of bounded rank with respect to K at most n is a quotient of Z_n^K.     

Boundary Behavior of Large Solutions to the Monge-Ampère Equation in a Borderline Case

This paper is concerned with the boundary behavior of strictly convex large solutions to the Monge-Ampère equation detD²u(x)=b(x)f(u(x)), u > 0, x ∈ Ω, where Ω is a strictly convex and bounded smooth domain in R^N with N ≥ 2, f is normalized regularly varying at infinity with the critical index N and has a lower term, and b ∈ C^∞(Ω) is positive in Ω, but may be appropriate singular on the boundary.     

Enumeration of irreducible binary words

Overlap free words over two letters are called irreducible binary words. Let d(n) denote the number of irreducible binary words of length n. In this paper we show that there are positive constants C₁ and C₂ such that C₁n^1.155<d(n)<C₂n^1.587 holds for all n>0.     

Analysis and geometry of Floer theory of Landau-Ginzburg model on ℂn

This article studies the Floer theory of Landau-Ginzburg (LG) model on ℂⁿ: We perturb the Kähler form within a xed Kähler class to guarantee the transversal intersection of Lefschetz thimbles. The C⁰ estimate for solutions of the LG Floer equation can be derived then by our analysis tools. The Fredholm property is guaranteed by all these results.     

Harnack Differential Inequalities for the Parabolic Equation ut=LF(u) on Riemannian Manifolds and Applications

In this paper, let(M~n, g) be an n-dimensional complete Riemannian manifold with the mdimensional Bakry–mery Ricci curvature bounded below. By using the maximum principle, we first prove a Li–Yau type Harnack differential inequality for positive solutions to the parabolic equation u_t= LF(u)=ΔF(u)-f·F(u),on compact Riemannian manifolds Mn, where F∈C~2(0, ∞), F0 and f is a C~2-smooth function defined on M~n. As application, the Harnack differential inequalities for fast diffusion type equation and porous media type equation are derived. On the other hand, we derive a local Hamilton type gradient estimate for positive solutions of the degenerate parabolic equation on complete Riemannian manifolds. As application, related local Hamilton type gradient estimate and Harnack inequality for fast dfiffusion type equation are established. Our results generalize some known results.     

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.     

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.     

Mixed finite element models for free vibrations of thin-walled beams

Simple mixed finite element models are developed for the free vibration analysis of curved thin-walled beams with arbitrary open cross section. The analytical formulation is based on a Vlasov's type thin-walled beam theory which includes the effects of flexural-torsional coupling, and the additional effects of transverse shear deformation and rotary inertia. The fundamental unknowns consist of seven internal forces and seven generalized displacements of the beam. The element characteristic arrays are obtained by using a perturbed Lagrangian-mixed variational principle. Only C⁰ continuity is required for the generalized displacements. The internal forces and the Lagrange multiplier are allowed to be discontinuous at interelement boundaries.

Numerical results are presented to demonstrate the high accuracy and effectiveness of the elements developed. The standard of comparison is taken to be the solutions obtained by using two-dimensional plate/shell models for the beams.     

Diffeomorphisms with the ℳ0-shadowing Property

This paper studies the M₀-shadowing property for the dynamics of diffeomorphisms defined on closed manifolds. The C¹ interior of the set of all two dimensional diffeomorphisms with the M₀-shadowing property is described by the set of all Anosov diffeomorphisms. The C¹-stably M₀-shadowing property on a non-trivial transitive set implies the diffeomorphism has a dominated splitting.     

A semilinear beam equation with nonconstant loads

We investigate multiplicity of solutions u(x, t) for a piecewise linear perturbation −(bu⁺−au⁻) of the one-dimensional beam operator u_tt + u_xxx under Dirichlet boundary condition on the interval (fr|Sol|π/2,π/2) π/2) and periodic codition on the vasible t. Our concern is to investigate multiplicity of solutions of the equation when the nonlinearity crosses finite eigenvalues and the source term is generated by two eigenfunctions.     

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.     

Spectrum of Normal Weighted Composition Operators on the Fock Space Over ℂN

By using commutative C^*-algebra techniques, spectrum and essential spectrum of normal weighted composition operators on the Fock space over C^N are completely characterized. As an application, spectrum of self-adjoint weighted composition operators on the Fock space are obtained also.     

Consequences of an algorithm for bridged graphs

Chepoi showed that every breadth first search of a bridged graph produces a cop-win ordering of the graph. We note here that Chepoi's proof gives a simple proof of the theorem that G is bridged if and only if G is cop-win and has no induced cycle of length four or five, and that this characterization together with Chepoi's proof reduces the time complexity of bridged graph recognition. Specifically, we show that bridged graph recognition is equivalent to (C₄,C₅)-free graph recognition, and reduce the best known time complexity from O(n⁴) to O(n^3.376).     

Homogenization of Elliptic Problems with Neumann Boundary Conditions in Non-smooth Domains

We consider a family of second-order elliptic operators {L_ε} in divergence form with rapidly oscillating and periodic coefficients in Lipschitz and convex domains in R~n. We are able to show that the uniform W~(1,p) estimate of second order elliptic systems holds for 2n/(n+1)-δ p 2n/(n-1)+ δ where δ 0 is independent of ε and the ranges are sharp for n = 2, 3. And for elliptic equations in Lipschitz domains, the W~(1,p) estimate is true for 3/2-δ p 3 + δ if n ≥ 4, similar estimate was extended to convex domains for 1 p ∞.     

A new finite element of C cubic splines

In this paper, we give a new finite element with dimension 16 of C¹ cubic splines which have interpolation schemes on the Morgan-Scott construction ₀ of a triangle.     

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.     

Higher-order theories for symmetric and unsymmetric fiber reinforced composite beams with C finite elements

A simple C⁰ isoparametric finite element formulation based on a set of higher-order displacement models for the analysis of symmetric and asymmetric multilayered composite and sandwich beams subjected to sinusoidal loading is presented. These theories do not require the usual shear correction coefficients which are generally associated with the Timoshenko theory. The four-noded Lagrangian cubic element with kinematic models having four, five and six degrees of freedom per node is used. A computer algorithm is developed which incorporates realistic prediction of transverse interlaminar stresses from equilibrium equations. By comparing the results obtained with the elasticity solution and the CPT (classical laminated plate theory) it is shown that the present higher-order theories give a much better approximation to the behaviour of laminated composite beams, both thick and thin. In addition numerical results for unsymmetric sandwich beams are presented which may serve as benchmark for future investigations.     

Quasi-orders on the positive cone of a C

Motivated by an inequality of Choi for unital positive linear maps we introduce a quasi-order on the positive cone of a C^*-algebra and present several intrinsic characterizations for it. We consider also another quasi-order based on an inequality of Kadison, and study interrelationship between those two quasi-orders.