Harmonic equiangular tight frames comprised of regular simplices

An equiangular tight frame (ETF) is a sequence of unit-norm vectors in a Euclidean space whose coherence achieves equality in the Welch bound, and thus yields an optimal packing in a projective space. A regular simplex is a simple type of ETF in which the number of vectors is one more than the dimension of the underlying space. More sophisticated examples include harmonic ETFs which equate to difference sets in finite abelian groups. Recently, it was shown that some harmonic ETFs are comprised of regular simplices. In this paper, we continue the investigation into these special harmonic ETFs. We begin by characterizing when the subspaces that are spanned by the ETF's regular simplices form an equi-isoclinic tight fusion frame (EITFF), which is a type of optimal packing in a Grassmannian space. We shall see that every difference set that produces an EITFF in this way also yields a complex circulant conference matrix. Next, we consider a subclass of these difference sets that can be factored in terms of a smaller difference set and a relative difference set. It turns out that these relative difference sets lend themselves to a second, related and yet distinct, construction of complex circulant conference matrices. Finally, we provide explicit infinite families of ETFs to which this theory applies

Mergers with Quality Differentiated Products

We consider the impact of merger on the equilibrium price and quality of products. Consumer demand for both products depends not only on own price and quality, but also on the price and quality of the other product. We consider both the case in which the merging firms produce gross complements, and the case in which the firms produce gross substitutes. In both cases, merger may lower or increase both product price and quality. In the case in which firms produce complementary products, it may happen that firms both lower price and increase product quality when merged. This happens when the cross quality elasticities of demand and the cross price elasticities of demand are equal in magnitude. Surprisingly, we also find that there are situations under which merger between firms producing substitutes increases welfare. For example, it is possible that merger between firms producing gross complements may result in higher product quality but lower social welfare, and merger between firms producing substitute products may result in lower product quality but higher social welfare.

Risk Aversion and Double Marginalization

In vertical markets, eliminating double marginalization with a two-part tariff may not be possible due to downstream firms\u27 risk aversion. When demand is uncertain, contracts with large fixed fees expose the downstream rm to more risk than contracts that are more reliant on variable fees. In equilibrium, contracts may thus rely on variable fees, giving rise to double marginalization. Counterintuitively, we show that increased demand risk or risk aversion can actually mitigate double marginalization. We also characterize several sufficient conditions under which increased risk or risk aversion does exacerbate double marginalization. We conclude with an application to merger analysis

Harmonic Equiangular Tight Frames Comprised of Regular Simplices

An equiangular tight frame (ETF) is a sequence of unit-norm vectors in a Euclidean space whose coherence achieves equality in the Welch bound, and thus yields an optimal packing in a projective space. A regular simplex is a simple type of ETF in which the number of vectors is one more than the dimension of the underlying space. More sophisticated examples include harmonic ETFs which equate to difference sets in finite abelian groups. Recently, it was shown that some harmonic ETFs are comprised of regular simplices. In this paper, we continue the investigation into these special harmonic ETFs. We begin by characterizing when the subspaces that are spanned by the ETF\u27s regular simplices form an equi-isoclinic tight fusion frame (EITFF), which is a type of optimal packing in a Grassmannian space. We shall see that every difference set that produces an EITFF in this way also yields a complex circulant conference matrix. Next, we consider a subclass of these difference sets that can be factored in terms of a smaller difference set and a relative difference set. It turns out that these relative difference sets lend themselves to a second, related and yet distinct, construction of complex circulant conference matrices. Finally, we provide explicit infinite families of ETFs to which this theory applies

Harmonic Equiangular Tight Frames Comprised of Regular Simplices

An equiangular tight frame (ETF) is a sequence of unit-norm vectors in a Euclidean space whose coherence achieves equality in the Welch bound, and thus yields an optimal packing in a projective space. A regular simplex is a simple type of ETF in which the number of vectors is one more than the dimension of the underlying space. More sophisticated examples include harmonic ETFs which equate to difference sets in finite abelian groups. Recently, it was shown that some harmonic ETFs are comprised of regular simplices. In this paper, we continue the investigation into these special harmonic ETFs. We begin by characterizing when the subspaces that are spanned by the ETF\u27s regular simplices form an equi-isoclinic tight fusion frame (EITFF), which is a type of optimal packing in a Grassmannian space. We shall see that every difference set that produces an EITFF in this way also yields a complex circulant conference matrix. Next, we consider a subclass of these difference sets that can be factored in terms of a smaller difference set and a relative difference set. It turns out that these relative difference sets lend themselves to a second, related and yet distinct, construction of complex circulant conference matrices. Finally, we provide explicit infinite families of ETFs to which this theory applies

Harmonic Equiangular Tight Frames Comprised of Regular Simplices

An equiangular tight frame (ETF) is a sequence of unit-norm vectors in a Euclidean space whose coherence achieves equality in the Welch bound, and thus yields an optimal packing in a projective space. A regular simplex is a simple type of ETF in which the number of vectors is one more than the dimension of the underlying space. More sophisticated examples include harmonic ETFs which equate to difference sets in finite abelian groups. Recently, it was shown that some harmonic ETFs are comprised of regular simplices. In this paper, we continue the investigation into these special harmonic ETFs. We begin by characterizing when the subspaces that are spanned by the ETF\u27s regular simplices form an equi-isoclinic tight fusion frame (EITFF), which is a type of optimal packing in a Grassmannian space. We shall see that every difference set that produces an EITFF in this way also yields a complex circulant conference matrix. Next, we consider a subclass of these difference sets that can be factored in terms of a smaller difference set and a relative difference set. It turns out that these relative difference sets lend themselves to a second, related and yet distinct, construction of complex circulant conference matrices. Finally, we provide explicit infinite families of ETFs to which this theory applies

The role of cerebellar circuitry alterations in the pathophysiology of autism spectrum disorders

The cerebellum has been repeatedly implicated in gene expression, rodent model and post-mortem studies of autism spectrum disorder (ASD). How cellular and molecular anomalies of the cerebellum relate to clinical manifestations of ASD remains unclear. Separate circuits of the cerebellum control different sensorimotor behaviors, such as maintaining balance, walking, making eye movements, reaching, and grasping. Each of these behaviors has been found to be impaired in ASD, suggesting that multiple distinct circuits of the cerebellum may be involved in the pathogenesis of patients' sensorimotor impairments. We will review evidence that the development of these circuits is disrupted in individuals with ASD and that their study may help elucidate the pathophysiology of sensorimotor deficits and core symptoms of the disorder. Preclinical studies of monogenetic conditions associated with ASD also have identified selective defects of the cerebellum and documented behavioral rescues when the cerebellum is targeted. Based on these findings, we propose that cerebellar circuits may prove to be promising targets for therapeutic development aimed at rescuing sensorimotor and other clinical symptoms of different forms of ASD

High-resolution soft x-ray spectrometry using the electron-multiplying charge-coupled device (EM-CCD)

The Electron-Multiplying Charge-Coupled Device (EM-CCD) shares a similar structure to the CCD except for the inclusion of a gain register that multiplies signal before the addition of read-noise, offering sub-electron effective readnoise at high frame-rates. EM-CCDs were proposed for the dispersive spectrometer on the International X-ray Observatory (IXO) to bring sub-300 eV X-rays above the noise, increasing the science yield. The high-speed, low-noise performance of the EMCCD brought added advantages of reduced dark current and stray-light per frame, reducing cooling and filtering requirements. To increase grating efficiency, several diffracted spectral orders were co-located so the inherent energy resolution of the detector was required for order separation. Although the spectral resolution of the EM-CCD is degraded by the gain process, it was shown that the EM-CCD could achieve the required separation. The RIXS spectrometer at the Advanced Resonant Spectroscopy beamline (ADRESS) of the Swiss Light Source (SLS) at the Paul Scherrer Institute currently uses a CCD, with charge spreading between pixels limiting the spatial resolution to 24 μm (FWHM). Through improving the spatial resolution below 5 μm alongside upgrading the grating, a factor of two energy resolution improvement could theoretically be made. With the high-speed, low-noise performance of the EM-CCD, photon-counting modes could allow the use of centroiding techniques to improve the resolution. Using various centroiding techniques, a spatial resolution of 2 μm (FWHM) has been achieved experimentally, demonstrating the benefits of this detector technology for soft X-ray spectrometry. This paper summarises the use of EM-CCDs from our first investigations for IXO through to our latest developments in ground-based testing for synchrotron-research and looks beyond to future possibilities

Inhibitory Control Processes and the Strategies That Support Them during Hand and Eye Movements

Background and Aims: Adaptive behavior depends on the ability to voluntarily suppress context-inappropriate behaviors, a process referred to as response inhibition. Stop Signal tests (SSTs) are the most frequently studied paradigm used to assess response inhibition. Previous studies of SSTs have indicated that inhibitory control behavior can be explained using a common model in which GO and STOP processes are initiated independent from one and another, and the process that is completed first determines whether the behavior is elicited (GO process) or terminated (STOP process). Consistent with this model, studies have indicated that individuals strategically delay their behaviors during SSTs in order to increase their stopping abilities. Despite being controlled by distinct neural systems, prior studies have largely documented similar inhibitory control performance across eye and hand movements. Though, no existing studies have compared the extent to which individuals strategically delay behavior across different effectors is not yet clear. Here, we compared the extent to which inhibitory control processes and the cognitive strategies that support them during oculomotor and manual motor behaviors. Methods: We examined 29 healthy individuals who performed parallel oculomotor and manual motor SSTs. Participants also completed a separate block of GO trials administered prior to the Stop Signal tests to assess baseline reaction times for each effector and reaction time increases during interleaved GO trials of the SST. Results: Our results showed that stopping errors increased for both effectors as the interval between GO and STOP cues was increased (i.e., stop signal delay), but performance deteriorated more rapidly for eye compared to hand movements with increases in stop signal delay. During GO trials, participants delayed the initiation of their responses for each effector, and greater slowing of reaction times on GO trials was associated with increased accuracy on STOP trials for both effectors. However, participants delayed their eye movements to a lesser degree than their hand movements, and strategic reaction time slowing was a stronger determinant of stopping accuracy for hand compared to eye movements. Overall, stopping accuracies for eye and hand movements were only modestly correlated, and the time it took individuals to cancel a response was not related for eye and hand movements. Discussion and Conclusion: Our findings that GO and STOP processes are independent and that individuals strategically delay their behavioral responses to increase stopping accuracy regardless of effector indicate that inhibitory control of Frontiers in Psychology | www.frontiersin.org 1 December 2016 | Volume 7 | Article 1927 Schmitt et al. Inhibitory Control of Hand and Eye Movements oculomotor and manual motor behaviors both follow common guiding principles. Yet, our findings document that eye movements are more difficult to inhibit than hand movements, and the timing, magnitude, and impact of cognitive control strategies used to support voluntary response inhibition are less robust for eye compared to hand movements. This suggests that inhibitory control systems also show unique characteristics that are behavior-dependent. This conclusion is consistent with neurophysiological evidence showing important differences in the architecture and functional properties of the neural systems involved in inhibitory control of eye and hand movements. It also suggests that characterizing inhibitory control processes in health and disease requires effector-specific analysis.NIH Autism Center of Excellence P50HD055751; MH092696, and the Kansas Center for Autism Research and Training Research Investment Council Strategic Initiative Gran

Saccadic eye movement abnormalities in autism spectrum disorder indicate dysfunctions in cerebellum and brainstem

BACKGROUND: Individuals with autism spectrum disorder (ASD) show atypical scan paths during social interaction and when viewing faces, and recent evidence suggests that they also show abnormal saccadic eye movement dynamics and accuracy when viewing less complex and non-social stimuli. Eye movements are a uniquely promising target for studies of ASD as their spatial and temporal characteristics can be measured precisely and the brain circuits supporting them are well-defined. Control of saccade metrics is supported by discrete circuits within the cerebellum and brainstem - two brain regions implicated in magnetic resonance (MR) morphometry and histopathological studies of ASD. The functional integrity of these distinct brain systems can be examined by evaluating different parameters of visually-guided saccades. METHODS: A total of 65 participants with ASD and 43 healthy controls, matched on age (between 6 and 44-years-old), gender and nonverbal IQ made saccades to peripheral targets. To examine the influence of attentional processes, blocked gap and overlap trials were presented. We examined saccade latency, accuracy and dynamics, as well as the trial-to-trial variability of participants’ performance. RESULTS: Saccades of individuals with ASD were characterized by reduced accuracy, elevated variability in accuracy across trials, and reduced peak velocity and prolonged duration. In addition, their saccades took longer to accelerate to peak velocity, with no alteration in the duration of saccade deceleration. Gap/overlap effects on saccade latencies were similar across groups, suggesting that visual orienting and attention systems are relatively spared in ASD. Age-related changes did not differ across groups. CONCLUSIONS: Deficits precisely and consistently directing eye movements suggest impairment in the error-reducing function of the cerebellum in ASD. Atypical increases in the duration of movement acceleration combined with lower peak saccade velocities implicate pontine nuclei, specifically suggesting reduced excitatory activity in burst cells that drive saccades relative to inhibitory activity in omnipause cells that maintain stable fixation. Thus, our findings suggest that both cerebellar and brainstem abnormalities contribute to altered sensorimotor control in ASD. ELECTRONIC SUPPLEMENTARY MATERIAL: The online version of this article (doi:10.1186/2040-2392-5-47) contains supplementary material, which is available to authorized users