Spatial Spreads Research Articles

Multinational Corporations and Global Governance

Operating as they do in multiple national environments characterized by divergent levels of development, MNCs are uniquely positioned to extract arbitrage advantages from two simultaneous phenomena: the relative infancy and feebleness of institutions of global governance, and the inability of individual nation states to police an entity whose reach extends beyond national boundaries. However, theories of the MNC in organizational studies have not kept pace with their dynamic structures, spatial spreads, and their ability to exploit economic, cultural, governance‐based and geopolitical heterogeneities to their advantage. In this symposium, we will attempt to re‐ examine traditional ideas about MNC in light of current global developments (the financial crisis, offshoring, environmental shifts, migrations etc.), and offer ways in which we can theorize MNC governance more accurately and critically. In this symposium, we seek to contribute to a discussion of the MNC as a contested space of global governance. We re- examine some of the current theoretical resources available to management scholars to understand both the antecedents and consequences of MNC governance. By examining these resources with their possibilities and limitations, management scholars attending the symposium will help to determine the next steps that need to be taken to develop theory. Financialization and Global Corporate Governance: A Discussion of Cultural Tensions Presenter: Galit Ailon; Bar Ilan U.

APCV 2014, 19–22 July 2014, Takamatsu, Japan Abstracts

How are the image statistics of global image contrast computed? We answered this by using a contrast-matching task for checkerboard configurations of ‘battenberg’ micro-patterns where the contrasts and spatial spreads of interdigitated pairs of micro-patterns were adjusted independently. Test stimuli were 20 × 20 arrays with various sized cluster widths, matched to standard patterns of uniform contrast. When one of the test patterns contained a pattern with much higher contrast than the other, that determined global pattern contrast, as in a max() operation. Crucially, however, the full matching functions had a curious intermediate region where low contrast additions for one pattern to intermediate contrasts of the other caused a paradoxical reduction in perceived global contrast. None of the following models predicted this: RMS, energy, linear sum, max, Legge and Foley. However, a gain control model incorporating wide-field integration and suppression of nonlinear contrast responses predicted the results with no free parameters. This model was derived from experiments on summation of contrast at threshold, and masking and summation effects in dipper functions. Those experiments were also inconsistent with the failed models above. Thus, we conclude that our contrast gain control model (Meese & Summers, 2007) describes a fundamental operation in human contrast vision.

Inverse scattering analysis of an elastic half-space by means of the fast volume integral equation method

A method for inverse scattering analysis of an elastic half-space using the fast volume integral equation method is presented. For the formulation of the inverse scattering analysis, the volume integral equation for the forward scattering analysis is modified so that it describes the relationship between the observed scattered waves at the free surface and the fluctuation of the wavefield. The inversion equation is obtained for reconstructing spatial spreads and the amplitude of fluctuations of the wavefield from the observed scattered waves due to point sources at the free surface. Linearization by the Born approximation as well as the Tikhonov regularization method is applied to the inversion equation. The fast generalized Fourier transform is also applied in order to solve the inversion equation. Numerical calculations are performed in order to investigate the convergence properties of the solution with respect to different regularization parameters. In addition, the effects of the number of point sources on the accuracy of the reconstruction results are examined.

Textures shape the attentional focus: Evidence from exogenous and endogenous cueing

The spatial cueing paradigm (Posner Quarterly Journal of Experimental Psychology 32:3-25, 1980) has often been used to investigate the time course of the deployment of visual attention in space. In a series of eight experiments we investigated whether spatial cues would not only enhance processing of stimuli presented at cued locations, but also enhance processing of the entire texture in which the stimuli were presented. Results showed highest accuracy for responses to stimuli presented at cued locations, a replication of the traditional cueing effect (Posner 1980). Additionally, stimuli presented at uncued locations were responded to with higher accuracy when they were presented inside the same texture as the cued location, as compared with stimuli presented outside the texture with the cued location. To investigate this texture advantage for both automatic and voluntary attention deployment, exogenous and endogenous cues were used. The texture advantage was observed for short interstimulus intervals (ISIs) of 50 and 100 ms for exogenous cues and for a longer ISI of 200 ms for endogenous cues. These findings indicate that the arrangement of task-irrelevant visual stimuli also can have a large impact on the cueing effect. This suggests that visual spatial attention spreads texture-wise across the visual field. Control experiments revealed that the homogeneity within texture elements contributes most to the effect but that the texture advantage is a function of both orientation contrast at the texture border and homogeneity within texture elements.

SU-E-I-111: Evaluation of the Analytical Scattering Models of 1) Lynch-Dahl 2) Highland and 3) Rossi for Proton Beams and Comparison with GEANT4 Monte Carlo Simulations as a Prerequisite for Proton Radiography Applications for Patients.

To evaluate the approximate 1) Lynch-Dahl 2) Highland and 3) Rossi scattering models for proton beams with GEANT4 Monte Carlo. This is a prerequisite for proton radiography applications for patients. A Matlab program developed in-house at MGH was used to obtain a semianalytical generalized Fermi-Eyges theory estimation of the spatial and angular spreads of a 230 MeV zero-spread incident proton beam as a function of depth. The constants of 1) Lynch-Dahl 2) Highland and 3) Rossi were used respectively for each model. MC simulations will determine which approximation provides the best prediction for different media configurations. Further, the calculated spreads were used to inform proton radiography imaging by calculating two limiting angles, a positional Acut and a directional Ccut. Acut is defined as the viewing angle of a point of incidence observer at which they see a point displaced by one positional standard deviation. Ccut is defined as the direction cosine of one angular standard deviation momenta. Both the angular and spatial spreads as well as their respective model differences rose monotonically with depth in water. At 30 cm depth the angular spread reached values around 3 degrees with about 0.32 degrees model difference, translating to Ccut differences in the first or second significant digit. At the same depth the spatial spread reached values around 1.2 cm with about 0.7 mm model difference, translating to Acut differences in the first or second significant digit. Preliminary MC data (not shown) indicate that the signals obtained due to the influence of inhomogeneities are small and the model differences may be relevant. We observed non-negligible differences between the models using MC. Further analysis is required to understand, which model provides most accurate scattering predictions for protons penetrating different media configurations.

Priming processes in the Simon task: More evidence from the lexical decision task for a third route in the Simon effect.

Recently, it was proposed that the Simon effect would result not only from two interfering processes, as classical dual-route models assume, but from three processes. It was argued that priming from the spatial code to the nonspatial code might facilitate the identification of the nonspatial stimulus feature in congruent Simon trials. In the present study, the authors provide evidence that the identification of the nonspatial information can be facilitated by the activation of an associated spatial code. In three experiments, participants first associated centrally presented animal and fruit pictures with spatial responses. Subsequently, participants decided whether laterally presented letter strings were words (animal, fruit, or other words) or nonwords; stimulus position could be congruent or incongruent to the associated spatial code. As hypothesized, animal and fruit words were identified faster at congruent than at incongruent stimulus positions from the association phase. The authors conclude that the activation of the spatial code spreads to the nonspatial code, resulting in facilitated stimulus identification in congruent trials. These results speak to the assumption of a third process involved in the Simon task.

Measuring attention using flash-lag effect

We investigated the effect of attention on the flash-lag effect (FLE) in order to determine whether the FLE can be used to estimate the effect of visual attention. The FLE is the effect that a flash aligned with a moving object is perceived to lag the moving object, and several studies have shown that attention reduces its magnitude. We measured the FLE as a function of the number or speed of moving objects. The results showed that the effect of cueing, which we attributed the effect of attention, on the FLE increased monotonically with the number or the speed of the objects. This suggests that the amount of attention can be estimated by measuring the FLE, assuming that more amount of attention is required for a larger number or faster speed of objects to attend. On the basis of this presumption, we attempted to measure the spatial spread of visual attention by FLE measurements. The estimated spatial spreads were similar to those estimated by other experimental methods.

The effect of non-specific LTD on pattern recognition in cerebellar Purkinje cells

Long-term depression (LTD) of synapses between parallel fibres (PFs) and Purkinje cells in the cerebellar cortex has been suggested as mechanism for cerebellar learning and pattern recognition [1]. PF LTD can be induced by coincident PF and climbing fibre (CF) input to Purkinje cells, and existing theories of cerebellar learning assume that LTD is specific to the PF synapses that are activated when a CF signal is presented [2]. However, experimental evidence indicates that this view is too simplified. When PF LTD is induced by pairing PF input with Purkinje cell depolarization that mimics the effect of CF input, a non-specific form of PF LTD also develops that spreads to neighbouring synapses [3]. The amount of this non-specific LTD decreases with distance from the stimulated PF synapses in 3D. As many learning theories in neural networks assume that pattern recognition is implemented by specific weight changes at individual storage sites, non-specific LTD is expected to result in a decreased performance. A degradation of performance based on non-specific plasticity has recently been shown for an unsupervised learning task using Oja’ sr ule [4]. Here we study the effect of non-specific LTD in a simple artificial neural network (ANN) with a LTD based learning rule. We store a number of input patterns in the ANN and compare its performance in the absence and presence of non-specific LTD with varying spatial spreads. The performance of the ANN is evaluated by measuring its responses to the stored patterns and an equal number of novel patterns. The ability to discriminate between stored and novel patterns is quantified by calculating a signal-to-noise ratio [5]. Not surprisingly, non-specific LTD results in loss of performance. However, when the ANN is presented with stored patterns that have been degraded by adding different amounts of spatial noise, implemented by shifting active inputs to neighbouring locations, the effect of the nonspecific LTD is reversed. While the performance of an ANN without any non-specific LTD drops dramatically in the presence of spatial noise in the input patterns, the presence of non-specific LTD introduces robustness against spatial noise. This effect is particularly pronounced when the spatial spread of the LTD is matched with the spatial spread of the noise in the input patterns. Our results suggest a potential computational role for nonspecific LTD in Purkinje cells. We are currently studying non-specific LTD in simulations of a multi-compartmental conductance-based Purkinje cell model [6].

Underwater acoustic beam dynamics

Ray- and mode-based theoretical predictions of the spreads of directionally narrow beams are presented and compared to parabolic-equation-based simulations in deep-ocean environments. Both the spatial and temporal spreads of beams are considered. The environments considered consist of a range-independent deep-ocean background sound channel on which a highly structured sound-speed perturbation, associated with either internal waves or homogeneous isotropic single-scale turbulence, is superimposed. The simulation results are shown to be in good agreement with simple theoretical expressions which predict that beam spreading, in both the unperturbed and perturbed environments, is largely controlled by a property of the background sound channel-the ray-based stability parameter alpha or the asymptotically equivalent mode-based waveguide invariant beta. These results are consistent with earlier results showing that wavefield structure and stability are largely controlled by alpha (or beta).

Adaptive Transmission With Single-Carrier Multilevel BICM

In this paper, we introduce adaptive link control techniques for multiple-input multiple-output (MIMO) systems with broadband single-carrier signaling. Soft cancellation and minimum mean squared error turbo equalization is assumed, where matching between coding and equalization plays crucial roles to achieve high throughput. This paper uses multilevel coded bit-interleaved coded modulation (ML-BICM) with linear mapping as a core part of the transmission, on top of which this paper applies automatic repeat request (ARQ) with adaptive coding (AC) for link control. The reason behind the use of the linear mapping ML-BICM is the separability between the layers in the modulation format: This should bring us a significant benefit in designing the link control strategies because each layer has its own reliability, and the adaptive link control should be optimized based on the each layer's reliabilities. For ARQ with ML-BICM, we introduce layer-by-layer retransmission control, where turbo equalization and retransmission control can be performed independently over the layers because of the layer-separability. The layer-by-layer concept is then applied to AC with ML-BICM, where the code parameters such as code rate and generator polynomials are chosen so that after several iterations the mutual information between the transmitted and the soft-input soft-output decoder output information can reach a value very close to one while minimizing the rate loss due to the mismatch between the equalization and decoder. Extrinsic information transfer analysis is performed for the MIMO channel realization being given. The transmitter is notified of the selected codes for the each layer via the feedback channel, and uses the selected codes for the following transmission, assuming that the channel state information (CSI) stays the same at least over two consecutive frames. Finally, this paper evaluates the throughput performances of the ML-BICM ARQ and AC techniques using multidimensional field measurement data. The performance tendencies are correlated with propagation properties, obtained as results of the high-resolution channel analysis, such as spatial spreads at the transmitter and receiver sides.

Manuel Delgado's Urban Anthropology: From Multidimensional Space to Interdisciplinary Spatial Theory

Manuel Delgado's Urban Anthropology:From Multidimensional Space to Interdisciplinary Spatial Theory Benjamin R. Fraser (bio) "Una antropología urbana, en el sentido de lo urbano, sería, pues, una antropología de configuraciones sociales escasamente orgánicas, poco o nada solidificadas, sometidas a la oscilación constante y destinadas a desvanecerse enseguida […] una antropología de lo inestable, de lo no estructurado, no porque esté desestructurado, sino por estar estructurándose […]." —(Manuel Delgado, El animal público 12, original emphasis) "Though all the photographs of a city taken from all possible points of view indefinitely complete one another, they will never equal in value that dimensional object, the city along whose streets one walks." —(Henri Bergson, The Creative Mind 160-61) As spatial theory spreads from the social sciences into the humanities—from geography into literary circles—it becomes important to recognize spatial theory itself as an interdisciplinary area of investigation, one that shares significant methodological concerns with literary analysis and philosophical inquiry. Through the articulation of urban anthropologist Manuel Delgado's texts with key problems from the writings of Jane Jacobs, Susan Sontag, Juan Benet, Henri Bergson, and Henri Lefebvre, this essay argues that urban criticism depends on the humanities as much as the humanities need spatial theory. The result is stronger support for the Lefebvrian argument that to change the quality of urban life it is necessary to change the process through which space is perceived, conceived, and produced and, ultimately, to change the movement of thought itself. [End Page 57] Taken as a whole, Manuel Delgado's El animal público and his two conferences (edited for publication) "Memoria y lugar: El espacio público como crisis de significado" and "Tránsitos: Espacio público y masas corpóreas" are important contributions to the contemporary study of urban spaces.1 In these works, Delgado builds upon the fundamentally multidimensional view of space articulated by Henri Lefebvre (The Production of Space) while emphasizing space as a process over the reification of given spaces. Moreover, throughout Lefebvre's and Delgado's works there is a strong Bergsonian undercurrent that points to the differential unity of memory (representative of consciousness) and space (understood as matter) and consequently to the interconnection of philosophical, anthropological, and geographical concerns. This interconnection is just as crucial in approaching the complexity of 'public' and 'private' city-spaces as it is in restructuring the hard lines of disciplinary borders and in renovating the aims of traditional scholarship. Delgado's contribution to studies of space is in fact an enfolding of Lefebvre's triadic model of spatial productions and Bergson's non-dualistic spatial and temporal philosophy. This amalgam of paradigm shifts, digested as they are from the realms of geography and philosophy, suggests as much for the study of space as it does for the re-conceptualization of the very idea of 'space' itself. Incorporating the phenomenological questioning of perceiver and perceived into a superb materialist critique of capitalist spatial practices, Delgado draws together the imagined and physical aspects of our spatial production. The consequences of this type of scholarship cannot be overemphasized given the contemporary accelerated drive for capital accumulation through the production and reproduction of city-space (Harvey, Condition; Justice; Spaces). Urbanism suffers, in Delgado's (2001) words, from the need to "architecturalize everything"/ "arquitecturizarlo todo," [c]omo si la tarea en última instancia colonizadora del urbanista tuviera como su peor enemigo la tendencia que todo espacio socializado experimenta hacia la ambigüedad, hacia la indefinición, como consecuencia de la propia naturaleza indeterminada de los usos que registra. (Memoria 8) In what follows, I want to emphasize the above premise of Delgado, one that he shares with Lefebvre and Bergson, that process enfolds and includes the particular—that immaterial forces enfold and include material forces. I will first show how Delgado's urban anthropology, one that poses the question of public space as a central one, unites immaterial and material forces in a non-dualistic relation. I argue that this special theory of geographical processes is not far at all from the way innovative critics of art and literature, such as Susan Sontag and Juan Benet, have dissolved the clear distinction between...

Outage Probabilities of Wireless Systems With Imperfect Beamforming

This paper investigates the outage probability of a wireless system with conventional beamforming using a uniform linear array beamformer. The focus is to examine the impact of beamforming impairments, such as direction-of-arrival (DOA) estimation errors, signal spatial spreads, antenna array perturbation, and mutual coupling. Fading statistics of Rayleigh, Rician, and Nakagami are used to characterize the desired signal, whereas interferers are assumed to be subject to Rayleigh fading. A simplified beamforming model is used in deriving closed-form outage probability expressions. Numerical results illustrate the increase of outage probabilities in the presence of DOA estimation errors, signal spatial spreads, and antenna perturbation. However, negligible outage performance impact is observed due to mutual coupling

PKCα: a versatile key for decoding the cellular calcium toolkit

Conventional protein kinases C (cPKCs) play an essential role in signal transduction and are believed to integrate both global Ca2+ transients and diacylglycerol signals. We provide evidence that PKCα is a ubiquitous readout sensor for the cellular Ca2+ toolkit, including highly restricted elementary Ca2+ release.Threshold stimulations of cells with Ca2+-mobilizing agonists resulted in PKCα translocation events with limited spatial spreads (<4 μm) comprising two groups of lifetimes; brief events (400–1,500 ms) exclusively mediated by Ca2+–C2 domain membrane interactions and long-lasting events (>4 s) resulting from longer DAG-C1a domain–mediated membrane interactions.Although upon uncaging NP-EGTA, which is a caged Ca2+ compound, WT-PKCα displayed rapid membrane translocations within <250 ms, PKCα constructs with C2 domains mutated in their Ca2+-binding region lacked any Ca2+-dependent translocation. Flash photolysis of diazo-2, a photosensitive caged Ca2+ buffer, revealed a biphasic membrane dissociation (slow and fast period) of WT-PKCα. The slow phase was absent in cells expressing PKCα-constructs containing mutated C1a-domains with largely reduced DAG binding. Thus, two groups of PKCα membrane interactions coexist; C2- and C1a-mediated interactions with different lifetimes but rapid interconversion.We conclude that PKCα can readout very fast and, spatially and temporally, very complex cellular Ca2+ signals. Therefore, cPKCs are important transducers for the ubiquitous cellular Ca2+ signaling toolkit.

Performance Evaluation of CDMA Reverse Links With Imperfect Beamforming in a Multicell Environment Using a SimplifiedBeamforming Model

Reverse link capacity of a direct-sequence codedivision multiple-access (DS-CDMA) system in a multicell environment has been studied recently, and significant capacity improvements due to the use of beamforming have been observed. However, system performance with beamforming will be affected by several impairments, such as direction of arrival estimation errors, array perturbations, mutual coupling, and signal spatial spreads. In this paper, reverse link performance of CDMA systems with beamforming under these impairments (imperfect beamforming) is investigated. A simplified beamforming model is developed to evaluate the system performance in terms of user capacity, bit-error rates (BER), and outage probabilities. Both signal-to-interference-ratio-based power control and strengthbased power control are considered in this paper. The capacity and BER degradations due to different impairments are shown, and outage probabilities under different power control schemes are examined.

Space–Time Communication for OFDM With Implicit Channel Feedback

We consider wideband communication (e.g., using orthogonal frequency-division multiplexed (OFDM) systems) over a typical cellular "downlink," in which both the base station and the mobile may have multiple antennas, but the number of antennas at the mobile is assumed to be small. Implicit channel feedback can play a powerful role in such systems, especially for outdoor channels, which typically exhibit narrow spatial spreads. A summary of our findings is as follows: a) Implicit channel feedback regarding the covariance matrix for the downlink space-time channel can be obtained, without any power or bandwidth overhead, by suitably averaging uplink channel measurements across frequency. Since this approach relies on statistical reciprocity, it applies to both time-division duplex (TDD) and frequency-division duplex (FDD) systems. Using such covariance feedback yields significantly better performance at lower complexity than conventional space-time or space-frequency codes, which do not employ feedback; b) We provide guidelines for optimizing antenna spacing in systems with covariance feedback. Theoretical investigation of a hypothetical system with completely controllable channel eigenvalues shows that the optimal number of channel eigenmodes is roughly matched to the (small) number of receive antenna elements. Thus, while antenna elements in conventional systems without feedback should be spaced far apart in order to ensure uncorrelated responses, the optimal antenna spacing with covariance feedback is much smaller, thereby concentrating the channel energy into a small number of eigenmodes.

Nonstationary nonlinear dust-acoustic waves in unmagnetized plasma

The nonstationary and nonlinear dust-acoustic wave is considered in the framework of the Lagrangian fluid description. A single nonlinear equation and its solutions are obtained for dust-acoustic waves with nontrivial space and time dependence in both the large and small amplitude limit. In the absence of linear dispersion solutions in the large amplitude limit, it is demonstrated that under well-defined initial and boundary conditions the amplitude of the solutions is decreasing and the spatial profile spreads. This is a new class of nonlinear solutions leading to short-lived nonlinear structures. In the small amplitude limit, the soliton-like solution indicates close association with Korteweg–de Vries solitons.

Comparison in spatial spreads of secondary electron information between scanning ion and scanning electron microscopy.

Monte Carlo simulations have been carried out to compare the spatial spreads of secondary electron (SE) information in scanning ion microscopy (SIM) with scanning electron microscopy (SEM). Under Ga ion impacts, the SEs are excited by three kinds of collision-partners, that is, projectile ion, recoiled target atom, and target electron. The latter two partners dominantly contribute to the total SE yield gamma for the materials of low atomic number Z2. For the materials of high Z2, on the other hand, the projectile ions dominantly contribute to gamma. These Z2 dependencies generally cause the gamma yield to decrease with an increasing Z2, in contrast with the SE yield delta under electron impacts. Most of the SEs are produced in the surface layer of about 5lambda in depth (lambda: the mean free path of SEs), as they are independent of the incident probe. Under 30 keV Ga ion impacts, the spatial spread of SE information is roughly as small as 10 nm, decreasing with an increasing Z2. Under 10 keV electron impacts, the SEI excited by the primary electrons has a small spatial spread of about 5lambda, but the SEII excited by the backscattered electrons has a large one of several 10 to several 100 nanometers, decreasing with an increasing Z2. The main cause of a small spread of SE information at ion impact is the short ranges of the projectile ions returning to the surface to escape as backscattered ions, the recoiled target atoms, and the target electrons in collision cascade. The 30 keV Ga-SIM imaging is better than the 10 keV SEM imaging in spatial resolution for the structure/material measurements. Here, zero-size probes are assumed.

Calcium puffs are generic InsP3-activated elementary calcium signals and are downregulated by prolonged hormonal stimulation to inhibit cellular calcium responses

Elementary Ca(2+) signals, such as "Ca(2+) puffs", which arise from the activation of inositol 1,4,5-trisphosphate receptors, are building blocks for local and global Ca(2+) signalling. We characterized Ca(2+) puffs in six cell types that expressed differing ratios of the three inositol 1,4,5-trisphosphate receptor isoforms. The amplitudes, spatial spreads and kinetics of the events were similar in each of the cell types. The resemblance of Ca(2+) puffs in these cell types suggests that they are a generic elementary Ca(2+) signal and, furthermore, that the different inositol 1,4,5-trisphosphate isoforms are functionally redundant at the level of subcellular Ca(2+) signalling. Hormonal stimulation of SH-SY5Y neuroblastoma cells and HeLa cells for several hours downregulated inositol 1,4,5-trisphosphate expression and concomitantly altered the properties of the Ca(2+) puffs. The amplitude and duration of Ca(2+) puffs were substantially reduced. In addition, the number of Ca(2+) puff sites active during the onset of a Ca(2+) wave declined. The consequence of the changes in Ca(2+) puff properties was that cells displayed a lower propensity to trigger regenerative Ca(2+) waves. Therefore, Ca(2+) puffs underlie inositol 1,4,5-trisphosphate signalling in diverse cell types and are focal points for regulation of cellular responses.

Energy model for contrast detection: spatial-frequency and orientation selectivity in grating summation

Models of spatial vision usually assume a ‘front-end’ of spatial-frequency and orientation selective channels. Subthreshold-summation studies have provided some of the strongest support for this notion. We applied a single-channel energy model and a multiple-channels probability-summation model to explore subthreshold-summation phenomena. We measured the contrast thresholds for detection of two superimposed Gabor patches as a function of the spatial-frequency and orientation difference between the components. The stimuli were centred 7.5 deg above the fixation point and were windowed by a Gaussian function with one of two different spatial spreads. We have shown that the spatial-frequency and orientation selectivity in subthreshold summation of Gabor patches is determined by the similarity (cross-correlation) between the stimulus components. A single-channel energy model as well as a multiple-channels probability-summation model could explain the summation data.

Uncertainty products for nonparaxial wave fields.

Although maximal localization is a basic notion in the consideration of phase-space representations of fields, it has not yet been pursued for general wave fields. We develop measures of spatial and directional spreads for nonparaxial waves in free space. These measures are invariant under translation and rotation and are shown to reduce to the conventional ones when applied to paraxial fields. The associated uncertainty relation sets limits to joint localization in coordinate and frequency space. This relation provides a basis for the definition of a joint localization measure that is analogous to the beam propagation factor (i.e., M2) of paraxial optics. The results are first developed for two-dimensional fields and then generalized to three dimensions.