Theoretical and numerical aspects of nonlinear reflection–transmission phenomena in acoustics

Equations of nonlinear acoustic wave motion in a non-classical lossy medium are used to derive generalised formulas describing the phenomena of reflection and transmission. Integral, non-local operators that are caused by the nonlinear effects in wave propagation and occur in reflection and transmission formulas are given in a form in which classical linear reflection and transmission coefficients are explicitly separated. Numerical calculations are performed for a simplified, one-dimensional wave travelling in a lossless medium. These simplifications reveal the pure effect of the impact of nonlinearities on the reflection and transmission phenomena. We consider adjacent media with different properties to illustrate various aspects of the problem. In particular, even if two media have the same linear impedance and the same material modules of the third order, we observe an explicit effect of the nonlinearity on the reflection phenomenon. The theoretical predictions are confirmed qualitatively by numerical calculations based on the finite difference time domain method.     

Theoretical and numerical aspects of nonlinear reflection–transmission phenomena in acoustics

Equations of nonlinear acoustic wave motion in a non-classical lossy medium are used to derive generalised formulas describing the phenomena of reflection and transmission. Integral, non-local operators that are caused by the nonlinear effects in wave propagation and occur in reflection and transmission formulas are given in a form in which classical linear reflection and transmission coefficients are explicitly separated. Numerical calculations are performed for a simplified, one-dimensional wave travelling in a lossless medium. These simplifications reveal the pure effect of the impact of nonlinearities on the reflection and transmission phenomena. We consider adjacent media with different properties to illustrate various aspects of the problem. In particular, even if two media have the same linear impedance and the same material modules of the third order, we observe an explicit effect of the nonlinearity on the reflection phenomenon. The theoretical predictions are confirmed qualitatively by numerical calculations based on the finite difference time domain method.     

Selective Gram–Schmidt orthonormalization for conic cutting surface algorithms

It is not straightforward to find a new feasible solution when several conic constraints are added to a conic optimization problem. Examples of conic constraints include semidefinite constraints and second order cone constraints. In this paper, a method to slightly modify the constraints is proposed. Because of this modification, a simple procedure to generate strictly feasible points in both the primal and dual spaces can be defined. A second benefit of the modification is an improvement in the complexity analysis of conic cutting surface algorithms. Complexity results for conic cutting surface algorithms proved to date have depended on a condition number of the added constraints. The proposed modification of the constraints leads to a stronger result, with the convergence of the resulting algorithm not dependent on the condition number. Research supported in part by NSF grant number DMS-0317323. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.     

On a product formula for the Conley–Zehnder index of symplectic paths and its applications

Using invariance by fixed-endpoints homotopies and a generalized notion of symplectic Cayley transform, we prove a product formula for the Conley–Zehnder index of continuous paths with arbitrary endpoints in the symplectic group. We discuss two applications of the formula, to the metaplectic group and to periodic solutions of Hamiltonian systems.      

On certain finite dimensional numerical ranges and numerical radii †

Let A be an invertible n×n matrix defined over a field k, and let A′denote the transpose of A. The object of this paper is to prove the following result.     

On Λ2-strong convergence of numerical sequences and Fourier series

We introduce the Λ²-strong convergence of numerical sequences and with it we generalize the concept of Λ-strong convergence of the results published by F. Móricz [2].     

Generalized symplectic Cayley transforms and a higher order formula for the Conley–Zehnder index of symplectic paths

Using a generalized notion of symplectic Cayley transform in the symplectic group, we introduce a sequence of integer valued invariants (higher order signatures) associated with a degeneracy instant of a smooth path of symplectomorphisms. In the real analytic case, we give a formula for the Conley–Zehnder index in terms of the higher order signatures.     

On the André motive of certain irreducible symplectic varieties

We show that if Y is an algebraic deformation of the Hilbert scheme of points on a K3 surface, then the André motive of Y is an object of the category generated be the motive of Y truncated in degree 2.     

On numerical methods and error estimates for degenerate fractional convection–diffusion equations

First we introduce and analyze a convergent numerical method for a large class of nonlinear nonlocal possibly degenerate convection diffusion equations. Secondly we develop a new Kuznetsov type theory and obtain general and possibly optimal error estimates for our numerical methods—even when the principal derivatives have any fractional order between 1 and 2! The class of equations we consider includes equations with nonlinear and possibly degenerate fractional or general Levy diffusion. Special cases are conservation laws, fractional conservation laws, certain fractional porous medium equations, and new strongly degenerate equations.     

On monodromies of a degeneration of irreducible symplectic Kähler manifolds

We study the monodromy operators on the Betti cohomologies associated to a good degeneration of irreducible symplectic manifold and we show that the unipotency of the monodromy operator on the middle cohomology is at least the half of the dimension. This implies that the “mildest” singular fiber of a good degeneration with non-trivial monodromy of irreducible symplectic manifolds is quite different from the generic degeneration of abelian varieties or Calabi–Yau manifolds.     

On the numerical approximations of stiff convection–diffusion equations in a circle

Our aim in this article is to study the numerical solutions of singularly perturbed convection–diffusion problems in a circular domain and provide as well approximation schemes, error estimates and numerical simulations. To resolve the oscillations of classical numerical solutions due to the stiffness of our problem, we construct, via boundary layer analysis, the so-called boundary layer elements which absorb the boundary layer singularities. Using a $P_1$ classical finite element space enriched with the boundary layer elements, we obtain an accurate numerical scheme in a quasi-uniform mesh.     

On the decomposable numerical range of λI-N

Let N be an nxn normal matrix. For 1≤m≤n we characterize the convexity of the mth decomposable numerical range of λI_n -N which is defined to be

{det(X?(λI_n ?N)X) [sdot] X?C _n×m X ^? X=I_m }.

A related problem on mixed decomposable numerical range of λI_n -N is also discussed.     

On the numerical solution of diffusion–reaction equations with singular source terms

A numerical study is presented of reaction–diffusion problems having singular reaction source terms, singular in the sense that within the spatial domain the source is defined by a Dirac delta function expression on a lower dimensional surface. A consequence is that solutions will be continuous, but not continuously differentiable. This lack of smoothness and the lower dimensional surface form an obstacle for numerical discretization, including amongst others order reduction. In this paper the standard finite volume approach is studied for which reduction from order two to order one occurs. A local grid refinement technique is discussed which overcomes the reduction.     

On improving the accuracy of Horner’s and Goertzel’s algorithms

It is known that Goertzels algorithm is much less numerically accurate than the Fast Fourier Transform (FFT) (cf. [2]). In order to improve accuracy we propose modifications of both Goertzels and Horners algorithms based on the divide-and-conquer techniques. The proof of the numerical stability of these two modified algorithms is given. The numerical tests in Matlab demonstrate the computational advantages of the proposed modifications. The appendix contains the proof of numerical stability of Goertzels algorithm of polynomial evaluation. AMS subject classification 65F35, 65G50     

Classification of four-dimensional real Lie bialgebras of symplectic type and their Poisson–Lie groups

We classify all four-dimensional real Lie bialgebras of symplectic type and obtain the classical r-matrices for these Lie bialgebras and Poisson structures on all the associated four-dimensional Poisson–Lie groups. We obtain some new integrable models where a Poisson–Lie group plays the role of the phase space and its dual Lie group plays the role of the symmetry group of the system.     

On some invariants in numerical semigroups and estimations of the order bound

Let S={s _i } _i∈??? be a numerical semigroup. For s _i ∈S, let ν(s _i ) denote the number of pairs (s _i ?s _j ,s _j )∈S ². When S is the Weierstrass semigroup of a family $\{\mathcal{C}_{i}\}_{i\in\mathbb{N}}Let S={s _i } _i∈ℕ⊆ℕ be a numerical semigroup. For s _i ∈S, let ν(s _i ) denote the number of pairs (s _i −s _j ,s _j )∈S ². When S is the Weierstrass semigroup of a family {C_i}_{i ? \mathbbN}\{\mathcal{C}_{i}\}_{i\in\mathbb{N}} of one-point algebraic-geometric codes, a good bound for the minimum distance of the code C_i\mathcal{C}_{i} is the Feng and Rao order bound d _ORD (C _i ). It is well-known that there exists an integer m such that d _ORD (C _i )=ν(s _i+1) for each i≥m. By way of some suitable parameters related to the semigroup S, we find upper bounds for m and we evaluate m exactly in many cases. Further we conjecture a lower bound for m and we prove it in several classes of semigroups.     

On convergence of numerical methods for variational–hemivariational inequalities under minimal solution regularity

Hemivariational inequalities have been successfully employed for mathematical and numerical studies of application problems involving nonsmooth, nonmonotone and multivalued relations. In recent years, error estimates have been derived for numerical solutions of hemivariational inequalities under additional solution regularity assumptions. Since the solution regularity properties have not been rigorously proved for hemivariational inequalities, it is important to explore the convergence of numerical solutions of hemivariational inequalities without assuming additional solution regularity. In this paper, we present a general convergence result enhancing existing results in the literature.