Scattering poles for asymptotically hyperbolic manifolds

For a class of manifolds that includes quotients of real hyperbolic -dimensional space by a convex co-compact discrete group, we show that the resonances of the meromorphically continued resolvent kernel for the Laplacian on coincide, with multiplicities, with the poles of the meromorphically continued scattering operator for . In order to carry out the proof, we use Shmuel Agmon's perturbation theory of resonances to show that both resolvent resonances and scattering poles are simple for generic potential perturbations.

     

Ordered group invariants for nonorientable one-dimensional generalized solenoids

Let be an edge-wrapping rule which presents a one-dimensional generalized solenoid , and let be the adjacency matrix of . When is a wedge of circles and leaves the unique branch point fixed, we show that the stationary dimension group of is an invariant of homeomorphism of even if is not orientable.

     

Computation of symbolic dynamics for one-dimensional maps

In this paper we design and implement rigorous algorithms for computing symbolic dynamics for piecewise-monotone-continuous maps of the interval. The algorithms are based on computing forwards and backwards approximations of the boundary, discontinuity and critical points. We explain how to handle the discontinuities in the symbolic dynamics which occur when the computed partition element boundaries are not disjoint. The method is applied to compute the symbolic dynamics and entropy bounds for the return map of the singular limit of a switching system with hysteresis and the forced Van der Pol equation.     

Essential spectrum and -solutions of one-dimensional Schrödinger operators

In 1949, Hartman and Wintner showed that if the eigenvalue equations of a one-dimensional Schrödinger operator possess square integrable solutions, then the essential spectrum is nowhere dense. Furthermore, they conjectured that this statement could be improved and that under this condition the essential spectrum might always be void. This is shown to be false. It is proved that, on the contrary, every closed, nowhere dense set does occur as the essential spectrum of Schrödinger operators which satisfy the condition of existence of -solutions. The proof of this theorem is based on inverse spectral theory.

     

A heuristic for the one-dimensional cutting stock problem with usable leftover

A heuristic algorithm for the one-dimensional cutting stock problem with usable leftover (residual length) is presented. The algorithm consists of two procedures. The first is a linear programming procedure that fulfills the major portion of the item demand. The second is a sequential heuristic procedure that fulfills the remaining portion of the item demand. The algorithm can balance the cost of the consumed bars, the profit from leftovers and the profit from shorter stocks reduction. The computational results show that the algorithm performs better than a recently published algorithm.     

Domain of attraction of quasi-stationary distribution for one-dimensional diffusions

We study quasi-stationarity for one-dimensional diffusions killed at 0, when 0 is a regular boundary and +∞ is an entrance boundary. We give a necessary and sufficient condition for the existence of exactly one quasistationary distribution, and we also show that this distribution attracts all initial distributions.     

The Inverse Resonance Problem for Hermite Operators

In this paper the inverse resonance problem for the Hermite operator is investigated. The Hermite operator with the creation operator , the annihilation operator , and a finitely supported multiplication operator b, is an unbounded operator on ℓ ²(ℕ₀) having finitely many eigenvalues and infinitely many resonances (except for b=0, when there are no eigenvalues or resonances). It is shown that knowing the location of eigenvalues and resonances determines the potential b uniquely.      

Homogenization of the layered medium equation

A new partial differential equation to be called the layered medium equation is introduced, and it is proved that certain relevant initial or periodic boundary conditions give well-posed problems. Then, the homogenized limit of the layered medium equation is studied. It is shown to be preserved in limit in the limit in the physical problem in which the coefficients that arise from the dielectric layer are both proportional to thickness. Otherwise, a non-local problem is obtained as the limiting form     

A note on invariant regions in one-dimensional nonlinear viscoelasticity

We consider a nonlinear parabolic system arising as a viscosity regularization of the hyperbolic functional equation [001]-For such a system, the existence of invariant regions is proved on condition that the kernel K is dissipative in a certain sense and б satisfies some growth conditions     

The arctangent law for a certain random time related to one-dimensional diffusions

For a time-homogenous one-dimensional diffusion process X(t), we investigate the distribution of the first instant, after a given time r, at which X(t) exceeds its maximum in the interval [0, r], generalizing a result of Papanicolaou, holding for Brownian motion.     

Existence of infinitely many solutions for the one-dimensional Perona-Malik model

We establish the existence of infinitely many weak solutions for the the one-dimensional version of the well-known and widely used Perona-Malik anisotropic diffusion equation model in image processing. We establish the existence result under the homogeneous Neumann condition with smooth non-constant initial values. Our method is to convert the problem into a partial differential inclusion problem.     

Homology and AH conjecture for groupoids on one-dimensional solenoids

We show that Matui’s AH conjecture holds for groupoids of the Bratteli–Vershik systems embedded in the unstable equivalence relation on one-dimensional solenoids.     

On some geometric optimization problems in layered manufacturing

Efficient geometric algorithms are given for optimization problems arising in layered manufacturing, where a 3D object is built by slicing its CAD model into layers and manufacturing the layers successively. The problems considered include minimizing the stair-step error on the surfaces of the manufactured object under various formulations, minimizing the volume of the so-called support structures used, and minimizing the contact area between the supports and the manufactured object—all of which are factors that affect the speed and accuracy of the process. The stair-step minimization algorithm is valid for any polyhedron, while the support minimization algorithms are applicable only to convex polyhedra. The techniques used to obtain these results include construction and searching of certain arrangements on the sphere, 3D convex hulls, halfplane range searching, and constrained optimization.     

Conditioning a reflected one-dimensional diffusion via its canonical decomposition

This paper is concerned with a class of nonnegative reflected diffusionsX which have a canonical decompositionX=N-B, whereB is a Brownian motion andN a locally zero energy additive functional that decreases whenX>0. We condition the diffusion by the event N stays nonpositive. This conditioned diffusion naturally appears in various circumstances. In particular, we present a simple pathwise construction of the conditioned process from the original diffusion.     

A numerical solution of a one-dimensional blood flow model—moving grid approach

We consider a one-dimensional blood flow model suitable for larger arteries. It consists of a hyperbolic system of two coupled nonlinear equations. The model has already been successfully used in practice. Its numerical solution is usually achieved by means of an explicit Taylor–Galerkin scheme. We have proposed a different approach. The system can be transformed to characteristic directions emphasizing the physical nature of the problem. We solved this system by using an operator splitting on a moving grid.     

Existence of multiple positive solutions for one-dimensional p-Laplacian

In this paper we consider the multipoint boundary value problem for one-dimensional p-Laplacian

     

The Fredholm alternative for the one-dimensional p-Laplacian

In this paper we consider the solvability of the boundary value problem

     

Large gaps in one-dimensional cutting stock problems

Its linear relaxation is often solved instead of the one-dimensional cutting stock problem (1CSP). This causes a difference between the optimal objective function values of the original problem and its relaxation, called a gap. The size of this gap is considered in this paper with the aim to formulate principles for the construction of instances of the 1CSP with large gaps. These principles are complemented by examples for such instances.