Beyond the Spectral Theorem: Spectrally Decomposing Arbitrary Functions of Nondiagonalizable Operators

Nonlinearities in finite dimensions can be linearized by projecting them into infinite dimensions. Unfortunately, often the linear operator techniques that one would then use simply fail since the operators cannot be diagonalized. This curse is well known. It also occurs for finite-dimensional linear operators. We circumvent it by developing a meromorphic functional calculus that can decompose arbitrary functions of nondiagonalizable linear operators in terms of their eigenvalues and projection operators. It extends the spectral theorem of normal operators to a much wider class, including circumstances in which poles and zeros of the function coincide with the operator spectrum. By allowing the direct manipulation of individual eigenspaces of nonnormal and nondiagonalizable operators, the new theory avoids spurious divergences. As such, it yields novel insights and closed-form expressions across several areas of physics in which nondiagonalizable dynamics are relevant, including memoryful stochastic processes, open non unitary quantum systems, and far-from-equilibrium thermodynamics. The technical contributions include the first full treatment of arbitrary powers of an operator. In particular, we show that the Drazin inverse, previously only defined axiomatically, can be derived as the negative-one power of singular operators within the meromorphic functional calculus and we give a general method to construct it. We provide new formulae for constructing projection operators and delineate the relations between projection operators, eigenvectors, and generalized eigenvectors. By way of illustrating its application, we explore several, rather distinct examples.Comment: 29 pages, 4 figures, expanded historical citations; http://csc.ucdavis.edu/~cmg/compmech/pubs/bst.ht

Network dependence in multi-indexed data on international trade flows

Faced with the problem that conventional multidimensional fixed effects models only focus on unobserved heterogeneity, but ignore any potential cross-sectional dependence due to network interactions, we introduce a model of trade flows between countries over time that allows for network dependence in flows, based on sociocultural connectivity structures. We show that conventional multidimensional fixed effects model specifications exhibit cross-sectional dependence between countries that should be modeled to avoid simultaneity bias. Given that the source of network interaction is unknown, we propose a panel gravity model that examines multiplenetwork interaction structures, using Bayesian model probabilities to determine those most consistent with the sample data. This is accomplished with the use of computationally efficient Markov Chain Monte Carlo estimation methods that produce a Monte Carlo integration estimate of the log-marginal likelihood that can be used for model comparison. Application of the model to a panel of trade flows points to network spillover effects, suggesting the presence of network dependence and biased estimates from conventional trade flow specifications. The most important sources of network dependence were found to be membership in trade organizations, historical colonial ties, common currency, and spatial proximity of countries.Series: Working Papers in Regional Scienc

Dynamics of wind-driven upwelling and relaxation between Monterey Bay and Point Arena: Local-, regional-, and gyre-scale controls

In north and central California, equatorward winds drive equatorward flows and the upwelling of cold dense water over the shelf during the midspring and summer upwelling season. When the winds temporarily weaken, the upwelling flows between Point Reyes and Point Arena relax,\u27\u27 becoming strongly poleward over the shelf. Analytical and numerical models are used to describe the effect of alongshore variability of winds, bathymetry, and basin-scale pressure gradients on the strength of upwelling and its relaxation. Alongshore winds weaken to the south of Point Reyes, and the shelf becomes narrower from Point Reyes to Monterey Bay. Both of these lead to reduced upwelling at and to the north of Point Reyes, causing an alongshore gradient of temperature and density on the shelf. These alongshore gradients lead to an along-isobath pressure gradient over the shelf that drive the relaxation flows. A simple analytical model is used to explain the dynamics, magnitude, and structure of the relaxation flows. The modeling also suggests that the depth of origin of the upwelled waters, and thus their temperature, is controlled by the along-isobath pressure gradient that exists over the continental slope. This along-slope pressure gradient is also responsible for the California undercurrent in this region. This pressure gradient is not generated in a model of the Californian coast extending from 32 degrees N to 42 degrees N and integrated for several months, suggesting it is caused by dynamics whose spatial or temporal scales are larger than the Californian coast and/or longer than several months

The dynamics and optimal control of spinning spacecraft with movable telescoping appendages. Part C: Effect of flexibility during boom deployment

The dynamics of a spinning symmetrical spacecraft system during the deployment (or retraction) of flexible boom-type appendages were investigated. The effect of flexibility during boom deployment is treated by modelling the deployable members as compound spherical pendula of varying length (according to a control law). The orientation of the flexible booms with respect to the hub, is described by a sequence of two Euler angles. The boom members contain a flexural stiffness which can be related to an assumed effective restoring linear spring constant, and structural damping which effects the entire system. Linearized equations of motion for this system, when the boom length is constant, involve periodic coefficients with the frequency of the hub spin. A bounded transformation is found which converts this system into a kinematically equivalent one involving only constant coefficients

Conventional versus network dependence panel data gravity model specifications

Past focus in the panel gravity literature has been on multidimensional fixed effects specifications in an effort to accommodate heterogeneity. After introducing conventional multidimensional fixed effects, we find evidence of cross-sectional dependence in flows. We propose a simultaneous dependence gravity model that allows for network dependence in flows, along with computationally efficient Markov Chain Monte Carlo estimation methods that produce a Monte Carlo integration estimate of log-marginal likelihood useful for model comparison. Application of the model to a panel of trade flows points to network spillover effects, suggesting the presence of network dependence and biased estimates from conventional trade flow specifications. The most important sources of network dependence were found to be membership in trade organizations, historical colonial ties, common currency and spatial proximity of countries.Series: Working Papers in Regional Scienc

Ecosystem (dis)benefits arising from formal and informal land-use in Manchester (UK); a case study of urban soil characteristics associated with local green space management

Urban soils are subject to anthropogenic influences and, reciprocally, provide benefits and dis-benefits to human wellbeing; for example carbon storage, nutrient cycling and the regulation trace element and contaminant mobility. Collective stewardship of urban green commons provides contemporary examples of the diversity of uses and management of green space in cities and represents a growing movement in user participation in, and awareness of, the importance of urban ecological health. Exploring the range of social-ecological benefits exemplified in the urban environment has generally focused on above-ground processes, with few studies examining the potential for (dis)benefits arising from edaphic characteristics of collectively-managed spaces. An investigation into the influence of formal and informal green space management on carbon fluxes and heavy metal concentrations in urban soils was carried out in Manchester (UK) finding that carbon storage in soils of collectively managed urban green commons (7.15 ±1.42 kg C m⁻²) was significantly greater than at formally managed sites (for example city parks: 5.08 ±0.69 kg C m⁻²), though the latter exhibited reduced losses through CO2 emission. Variation in heavy metal concentrations and mobility were likewise observed, exemplified by the acidification of surface soils by leaf litter at orchard sites, and the resultant increase in the mobility of lead (Pb) and zinc (Zn). The results of this study indicate the importance of small-scale contemporary urban green space management on selected ecosystem services provided by the limited soil resource of cities. Thus, a greater consideration of the effects of horticultural and amenity activities with regards to soil quality/functionality is required to ensure available urban green commons retain or increase their ecological quality over time