Parity-Time Symmetry Breaking beyond One Dimension: The Role of Degeneracy

We consider the role of degeneracy in Parity-Time (PT) symmetry breaking for non-hermitian wave equations beyond one dimension. We show that if the spectrum is degenerate in the absence of T-breaking, and T is broken in a generic manner (without preserving other discrete symmetries), then the standard PT-symmetry breaking transition does not occur, meaning that the spectrum is complex even for infinitesimal strength of gain and loss. However the realness of the entire spectrum can be preserved over a finite interval if additional discrete symmetries X are imposed when T is broken, if X decouple all degenerate modes. When this is true only for a subset of the degenerate spectrum, there can be a partial PT transition in which this subset remains real over a finite interval of T-breaking. If the spectrum has odd-degeneracy, a fraction of the degenerate spectrum can remain in the symmetric phase even without imposing additional discrete symmetries, and they are analogous to dark states in atomic physics. These results are illustrated by the example of different T-breaking perturbations of a uniform dielectric disk and sphere, and a group theoretical analysis is given in the disk case. Finally, we show that multimode coupling is capable of restoring the T-symmetric phase at finite T-breaking. We also analyze these questions when the parity operator is replaced by another spatial symmetry operator and find that the behavior can be qualitatively different.Comment: 8 pages, 6 figure

Forecast Rationality and Monetary Policy Frameworks: Evidence from UK Interest Rate Forecasts

This paper explores the heterogeneity and rationality of professional forecasts at both short and long forecast horizons. We employ disaggregated survey data for forecasts of three-month inter-bank rates and ten-year gilt yields for the period 1989-2006. We find evidence of heterogeneity among forecasters. Moreover, forecasts violate both the unbiasedness and orthogonality conditions of the rational expectations hypothesis. The majority of biased forecasts underestimate the future spot rate. The rationality of forecasts varies across maturities and forecast horizons with short horizon and short maturity forecasts exhibiting more rationality. It also varies across sub-periods corresponding to different monetary policy frameworks. We produce evidence indicating that both monetary policy actions and elements of communication policy have information content regarding the rationality of forecasts. Changes in official bank rates and disagreement, as recorded in the minutes of the Monetary Policy Committee, influence the rationality of forecasts. The publication of inflation reports has no effect

A double bounded key identity for Goellnitz's (big) partition theorem

Given integers i,j,k,L,M, we establish a new double bounded q-series identity from which the three parameter (i,j,k) key identity of Alladi-Andrews-Gordon for Goellnitz's (big) theorem follows if L, M tend to infinity. When L = M, the identity yields a strong refinement of Goellnitz's theorem with a bound on the parts given by L. This is the first time a bounded version of Goellnitz's (big) theorem has been proved. This leads to new bounded versions of Jacobi's triple product identity for theta functions and other fundamental identities.Comment: 17 pages, to appear in Proceedings of Gainesville 1999 Conference on Symbolic Computation

Explaining Growth in Dutch Agriculture: Prices, Public R&D, and Technological Change

This paper analyzes the sources of growth of Dutch agriculture (arable, meat, and dairy sectors). Because the time series data (1950-1997) are non-stationary and not cointegrated, it is argued that a model estimated in first differences should be used. Estimated price elasticities turn out to be very inelastic, both in the short-run and the long-run. The direct distortionary effect of price support has therefore been rather limited. However, price support has an important indirect effect by improving the sectors investment possibilities and therewith the capital stock. Public R&D expenditure mainly affected agriculture by contributing to yield improvement therewith favoring intensification of production.growth, technology, cointegration, non-stationarity, agricultural policy, Agribusiness, Q18, O13,

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

PT-symmetry breaking and laser-absorber modes in optical scattering systems

Using a scattering matrix formalism, we derive the general scattering properties of optical structures that are symmetric under a combination of parity and time-reversal (PT). We demonstrate the existence of a transition beween PT-symmetric scattering eigenstates, which are norm-preserving, and symmetry-broken pairs of eigenstates exhibiting net amplification and loss. The system proposed by Longhi, which can act simultaneously as a laser and coherent perfect absorber, occurs at discrete points in the broken symmetry phase, when a pole and zero of the S-matrix coincide.Comment: 4 pages, 4 figure

Steady-State Ab Initio Laser Theory for N-level Lasers

We show that Steady-state Ab initio Laser Theory (SALT) can be applied to find the stationary multimode lasing properties of an N-level laser. This is achieved by mapping the N-level rate equations to an effective two-level model of the type solved by the SALT algorithm. This mapping yields excellent agreement with more computationally demanding N-level time domain solutions for the steady state

Fast Spectral Clustering Using Autoencoders and Landmarks

In this paper, we introduce an algorithm for performing spectral clustering efficiently. Spectral clustering is a powerful clustering algorithm that suffers from high computational complexity, due to eigen decomposition. In this work, we first build the adjacency matrix of the corresponding graph of the dataset. To build this matrix, we only consider a limited number of points, called landmarks, and compute the similarity of all data points with the landmarks. Then, we present a definition of the Laplacian matrix of the graph that enable us to perform eigen decomposition efficiently, using a deep autoencoder. The overall complexity of the algorithm for eigen decomposition is $O(np)$, where $n$ is the number of data points and $p$ is the number of landmarks. At last, we evaluate the performance of the algorithm in different experiments.Comment: 8 Pages- Accepted in 14th International Conference on Image Analysis and Recognitio