Skew convolution semigroups and affine Markov processes

A general affine Markov semigroup is formulated as the convolution of a homogeneous one with a skew convolution semigroup. We provide some sufficient conditions for the regularities of the homogeneous affine semigroup and the skew convolution semigroup. The corresponding affine Markov process is constructed as the strong solution of a system of stochastic equations with non-Lipschitz coefficients and Poisson-type integrals over some random sets. Based on this characterization, it is proved that the affine process arises naturally in a limit theorem for the difference of a pair of reactant processes in a catalytic branching system with immigration.Comment: Published at http://dx.doi.org/10.1214/009117905000000747 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Conservation relations and anisotropic transmission resonances in one-dimensional PT-symmetric photonic heterostructures

We analyze the optical properties of one-dimensional (1D) PT-symmetric structures of arbitrary complexity. These structures violate normal unitarity (photon flux conservation) but are shown to satisfy generalized unitarity relations, which relate the elements of the scattering matrix and lead to a conservation relation in terms of the transmittance and (left and right) reflectances. One implication of this relation is that there exist anisotropic transmission resonances in PT-symmetric systems, frequencies at which there is unit transmission and zero reflection, but only for waves incident from a single side. The spatial profile of these transmission resonances is symmetric, and they can occur even at PT-symmetry breaking points. The general conservation relations can be utilized as an experimental signature of the presence of PT-symmetry and of PT-symmetry breaking transitions. The uniqueness of PT-symmetry breaking transitions of the scattering matrix is briefly discussed by comparing to the corresponding non-Hermitian Hamiltonians.Comment: 10 pages, 10 figure

Extracting Energy from a Black Hole through Its Disk

When some magnetic field lines connect a Kerr black hole with a disk rotating around it, energy and angular momentum are transferred between them. If the black hole rotates faster than the disk, $ca/GM_H>0.36$ for a thin Keplerian disk, then energy and angular momentum are extracted from the black hole and transferred to the disk ($M_H$ is the mass and $a M_H$ is the angular momentum of the black hole). This way the energy originating in the black hole may be radiated away by the disk. The total amount of energy that can be extracted from the black hole spun down from $ca/GM_H = 0.998$ to $ca/GM_H = 0.36$ by a thin Keplerian disk is $\approx 0.15 M_Hc^2$. This is larger than $\approx 0.09 M_Hc^2$ which can be extracted by the Blandford-Znajek mechanism.Comment: 8 pages, 2 figure

A knowledge-based geometry repair system for robust parametric CAD models

In modern multi-objective design optimization (MDO) an effective geometry engine is becoming an essential tool and its performance has a significant impact on the entire MDO process. Building a parametric geometry requires difficult compromises between the conflicting goals of robustness and flexibility. This article presents a method of improving the robustness of parametric geometry models by capturing and modeling engineering knowledge with a support vector regression surrogate, and deploying it automatically for the search of a more robust design alternative while trying to maintain the original design intent. Design engineers are given the opportunity to choose from a range of optimized designs that balance the ‘health’ of the repaired geometry and the original design intent. The prototype system is tested on a 2D intake design repair example and shows the potential to reduce the reliance on human design experts in the conceptual design phase and improve the stability of the optimization cycle. It also helps speed up the design process by reducing the time and computational power that could be wasted on flawed geometries or frequent human intervention

The Hahn Quantum System

Using a formulation of quantum mechanics based on the theory of orthogonal polynomials, we introduce a four-parameter system associated with the Hahn and continuous Hahn polynomials. The continuum energy scattering states are written in terms of the continuous Hahn polynomial whose asymptotics give the scattering amplitude and phase shift. On the other hand, the finite number of discrete bound states are associated with the Hahn polynomial.Comment: 18 pages, 7 figure

Method for classifying multiqubit states via the rank of the coefficient matrix and its application to four-qubit states

We construct coefficient matrices of size 2^l by 2^{n-l} associated with pure n-qubit states and prove the invariance of the ranks of the coefficient matrices under stochastic local operations and classical communication (SLOCC). The ranks give rise to a simple way of partitioning pure n-qubit states into inequivalent families and distinguishing degenerate families from one another under SLOCC. Moreover, the classification scheme via the ranks of coefficient matrices can be combined with other schemes to build a more refined classification scheme. To exemplify we classify the nine families of four qubits introduced by Verstraete et al. [Phys. Rev. A 65, 052112 (2002)] further into inequivalent subfamilies via the ranks of coefficient matrices, and as a result, we find 28 genuinely entangled families and all the degenerate classes can be distinguished up to permutations of the four qubits. We also discuss the completeness of the classification of four qubits into nine families

Totems

In modern multi-objective design optimization (MDO) an effective geometry engine is becoming an essential tool and its performance has a significant impact on the entire MDO process. Building a parametric geometry requires difficult compromises between the conflicting goals of robustness and flexibility. This article presents a method of improving the robustness of parametric geometry models by capturing and modeling engineering knowledge with a support vector regression surrogate, and deploying it automatically for the search of a more robust design alternative while trying to maintain the original design intent. Design engineers are given the opportunity to choose from a range of optimized designs that balance the ‘health’ of the repaired geometry and the original design intent. The prototype system is tested on a 2D intake design repair example and shows the potential to reduce the reliance on human design experts in the conceptual design phase and improve the stability of the optimization cycle. It also helps speed up the design process by reducing the time and computational power that could be wasted on flawed geometries or frequent human intervention

Self-learning Multiscale Simulation for Achieving High Accuracy and High Efficiency Simultaneously

We propose a new multi-scale molecular dynamics simulation method which can achieve high accuracy and high sampling efficiency simultaneously without aforehand knowledge of the coarse grained (CG) potential and test it for a biomolecular system. Based on the resolution exchange simulations between atomistic and CG replicas, a self-learning strategy is introduced to progressively improve the CG potential by an iterative way. Two tests show that, the new method can rapidly improve the CG potential and achieve efficient sampling even starting from an unrealistic CG potential. The resulting free energy agreed well with exact result and the convergence by the method was much faster than that by the replica exchange method. The method is generic and can be applied to many biological as well as non-biological problems.Comment: 14 pages, 6 figure