Several studies showed evidence supporting the possibility of hand trajectory decoding from low-frequency electroencephalography. However, the decoding in the source space via source localization is scarcely investigated. In this study, we tried to tackle the problem of collinearity due to the higher number of signals in the source space by two folds: first, we selected signals in predefined regions of interest ; second, we applied dimensionality reduction techniques to each ROI. The dimensionality reduction techniques were computing the mean, principal (...) component analysis, and locality preserving projections. We also investigated the effect of decoding between utilizing a template head model and a subject-specific head model during the source localization. The results indicated that applying source-space decoding with PCA yielded slightly higher correlations and signal-to-noise ratios than the sensor-space approach. We also observed slightly higher correlations and SNRs when applying the subject-specific head model than the template head model. However, the statistical tests revealed no significant differences between the source-space and sensor-space approaches and no significant differences between subject-specific and template head models. The decoder with Mean and PCA utilizes information mainly from precuneus and cuneus to decode the velocity kinematics similarly in the subject-specific and template head models. (shrink)

Car sickness is a major concern for car passengers, and with the development of autonomous vehicles, increasing numbers of car occupants are likely to be affected. Previous laboratory studies have used EEG measurements to better understand the cerebral changes linked to symptoms. However, the dynamics of motion in labs/simulators differ from those of a real car. This study sought to identify specific cerebral changes associated with the level of car sickness experienced in real driving conditions. Nine healthy volunteers participated as (...) front passengers in a slalom session inducing lateral movements at very low frequency. They were continuously monitored via EEG recordings and subjectively rated their level of symptoms after each slalom, using a 5-point likert scale. Car-sickness symptoms evolved concomitantly with changes in theta and alpha power in the occipital and parietal areas. These changes may reflect altered sensory integration, as well as a possible influence of sleepiness mitigating symptoms. (shrink)

The principal goal of the brain-computer interface is to translate brain signals into meaningful commands to control external devices or neuroprostheses to restore lost functions of patients with severe motor disabilities. The invasive recording of brain signals involves numerous health issues. Therefore, BCIs based on non-invasive recording modalities such as electroencephalography are safer and more comfortable for the patients. The BCI requires reconstructing continuous movement parameters such as position or velocity for practical application of neuroprostheses. The BCI studies (...) in continuous decoding have extensively relied on extracting features from the amplitude of brain signals, whereas the brain connectivity features have rarely been explored. This study aims to investigate the feasibility of using phase-based connectivity features in decoding continuous hand movements from EEG signals. To this end, the EEG data were collected from seven healthy subjects performing a 2D center-out hand movement task in four orthogonal directions. The phase-locking value and magnitude-squared coherence are exploited as connectivity features along with multiple linear regression for decoding hand positions. A brute-force search approach is employed to find the best channel pairs for extracting features related to hand movements. The results reveal that the regression models based on PLV and MSC features achieve the average Pearson correlations of 0.43 ± 0.03 and 0.42 ± 0.06, respectively, between predicted and actual trajectories over all subjects. The delta and alpha band features have the most contribution in regression analysis. The results also demonstrate that both PLV and MSC decoding models lead to superior results on our data compared to two recently proposed feature extraction methods solely based on the amplitude or phase of recording signals. This study verifies the ability of PLV and MSC features in the continuous decoding of hand movements with linear regression. Thus, our findings suggest that extracting features based on brain connectivity can improve the accuracy of trajectory decoder BCIs. (shrink)

Continuous sedation at the end of life is a practice that has attracted a great deal of attention. An increasing number of guidelines on the proposed correct performance of the practice have been drafted. All of the guidelines stress the importance of using sedation in proportion to the severity of the patient’s symptoms, thus to reduce the patient’s consciousness no more than is absolutely necessary. As different patients can have different experiences of suffering, the amount of suffering should, ideally, (...) be assessed subjectively; that is, via communication with the patient. Continuously sedated patients are often unable to communicate, however, making subjective methods of pain assessment unusable. For these patients, the degree of consciousness is the sole available measure. It therefore seems important to adequately measure how deeply the patient is sedated, thereby allowing sedation to be increased when it is too light and decreased when it is too heavy. This is in accordance with the idea that reducing consciousness is not an ethically neutral act.Although consciousness measuring techniques are a hot topic in anesthesiology, almost no research exists on the use of such techniques in the context of continuous sedation at the end of life. This article aims to review existing techniques to measure consciousness and to evaluate their applicability, efficiency, and invasiveness for patients who are continuously sedated until death.Techniques commonly used to assess the depth of sedation in continuously sedated patients are basic clinical assessment and sedation scales, as they are often considered reliable and noninvasive. These techniques might not be very reliable, however, since it is known that some patients are nonresponsive and yet aware. Moreover, sedation scales require stimulation of the patient (for example prodding, shaking, or providing painful stimuli), and can thus be considered invasive of one’s bodily integrity or dignity. Other techniques, such as EEG (electroencephalography) derivatives, may score better on reliability and invasiveness. Yet these have so far never been compared to sedation scale scores for patients receiving continuous sedation at the end of life.Therefore, we conclude that, for both clinical and ethical reasons, research into the efficiency and applicability of other techniques, such as derivatives of EEG, are urgently needed. (shrink)

Obstructive sleep apnea (OSA) severely impacts sleep and has long-term health consequences. Treating sleep apnea with continuous positive airway pressure (CPAP) not only relieves obstructed breathing, but also improves sleep. CPAP improves sleep by reducing apnea-induced awakenings. CPAP may also improve sleep by enhancing features of sleep architecture assessed with electroencephalography (EEG) that maximize sleep depth and neuronal homeostasis, such as the slow oscillation and spindle EEG activity, and by reducing neurophysiological arousal during sleep (i.e., beta EEG activity). (...) We examined cross-sectional differences in quantitative EEG characteristics of sleep, assessed with power spectral analysis, in 29 adults with type 2 diabetes treated with CPAP and 24 adults undergoing SHAM CPAP treatment (total n = 53). We then examined changes in spectral characteristics of sleep as the SHAM group crossed over to active CPAP treatment (n = 19). Polysomnography (PSG) from the CPAP titration night was used for the current analyses. Analyses focused on EEG frequencies associated with sleep maintenance and arousal. These included the slow oscillation (0.5–1 Hz), sigma activity (12–16 Hz, spindle activity), and beta activity (16–20 Hz) in F3, F4, C3, and C4 EEG channels. Whole night non-rapid eye movement (NREM) sleep and the first period of NREM spectral activity were examined. Age and sex were included as covariates. There were no group differences between CPAP and SHAM in spectral characteristics of sleep architecture. However, SHAM cross-over to active CPAP was associated with an increase in relative 12–16 Hz sigma activity across the whole night and a decrease in average beta activity across the whole night. Relative slow oscillation power within the first NREM period decreased with CPAP, particularly for frontal channels. Sigma and beta activity effects did not differ by channel. These findings suggest that CPAP may preferentially enhance spindle activity and mitigate neurophysiological arousal. These findings inform the neurophysiological mechanisms of improved sleep with CPAP and the utility of quantitative EEG measures of sleep as a treatment probe of improvements in neurological and physical health with CPAP. (shrink)

In the automatic detection of epileptic seizures, the monitoring of critically ill patients with time varying EEG signals is an essential procedure in intensive care units. There is an increasing interest in using EEG analysis to detect seizure, and in this study we aim to get a better understanding of how to visualize the information in the EEG time-frequency feature, and design and train a novel random forest algorithm for EEG decoding, especially for multiple-levels of illness. Here, we propose an (...) automatic detection framework for epileptic seizure based on multiple time-frequency analysis approaches; it involves a novel random forest model combined with grid search optimization. The short-time Fourier transformation visualizes seizure features after normalization. The dimensionality of features is reduced through principal component analysis before feeding them into the classification model. The training parameters are optimized using grid search optimization to improve detection performance and diagnostic accuracy by in the recognition of three different levels epileptic of conditions (healthy subjects, seizure-free intervals, seizure activity). Our proposed model was used to classify 500 samples of raw EEG data, and multiple cross-validations were adopted to boost the modeling accuracy. Experimental results were evaluated by an accuracy, a confusion matrix, a receiver operating characteristic curve and an area under the curve. The evaluations indicated that our model achieved the more effective classification than some previous typical methods. Such a scheme for computer-assisted clinical diagnosis of seizures has a potential guiding significance, which not only relieves the suffering of patient with epilepsy to improve quality of life, but also helps neurologists reduce their workload. (shrink)

Humans are fundamentally social and tend to create emergent organizations when interacting with each other; from dyads to families, small groups, large groups, societies and civilizations. The study of the neuronal substrate of human social behavior is currently gaining momentum in the young field of social neuroscience. Hyperscanning is a neuroimaging technique by which we can study two or more brain simultaneously while participants interact with each other. The aim of this article is to discuss several factors that we deem (...) important in designing hyperscanning experiments. We first review hyperscanning studies performed by means of electroencephalography (EEG) that have been relying on a continuous interaction paradigm. Then, we provide arguments for favoring ecological paradigms, for studying the emotional component of social interactions and for performing longitudinal studies, the last two aspects being largely neglected so far in the hyperscanning literature despite their paramount importance in social sciences. Based on these premises we argue that music performance is a suitable experimental setting for hyperscanning and that for such studies EEG is an appropriate choice as neuroimaging modality. (shrink)

Neurotherapies for diagnostics and treatment—such as electroencephalography (EEG) neurofeedback, single-photon emission computerized tomography (SPECT) imaging for neuropsychiatric evaluation, and off-label/experimental uses of brain stimulation—are continuously being offered to the public outside mainstream healthcare settings. Because these neurotherapies share many key features of complementary and alternative medicine (CAM) techniques—and meet the definition of CAM as set out in Kaptchuk and Eisenberg—here we refer to them as “alternative neurotherapies.” By explicitly linking these alternative neurotherapy practices under a common conceptual framework, this (...) paper draws attention to, and critically considers, the cross-cutting ethical and legal issues related to the provision of these services. The first section of this paper provides an updated empirical overview of uses of SPECT neuropsychiatric evaluations, EEG neurofeedback, and experimental/off-label forms of brain stimulation. Next, drawing on CAM bioethics scholarship, we highlight the pertinent ethical issues in the alternative neurotherapy context, including the truthful representation of evidence base, marketing to vulnerable populations, potential harms, provider competency, and conflicts of interest. Finally, we consider the principal legal issues at stake for the provision of alternative neurotherapies in the U.S., namely those related to licensing and scope-of-practice considerations. We conclude with recommendations for future research in this domain. (shrink)

While previous studies have analyzed mean neurophysiological responses to musical stimuli, the current study aimed to identify specific time courses of electroencephalography (EEG) oscillations, which are associated with dynamic changes in the acoustic features of the musical stimulus. In addition, we were interested in whether these time courses change during a repeated presentation of the same musical piece. A total of 16 subjects repeatedly listened to the well-known aria “Nessun dorma,” sung by Paul Potts, while continuous 128-channel EEG (...) and heart rate, as well as electrodermal responses, were recorded. The time courses for the EEG oscillations were calculated using a time resolution of 1 second for several frequency bands, on the basis of individual alpha-peak frequencies (theta, low alpha-1, low alpha-2, upper alpha, and beta). For all frequency bands, we identified a more or less continuous increase in power relative to a baseline period, indicating strong event-related synchronization (ERS) during music listening. The ERS time courses, however, did not correlate strongly with the time courses of the acoustic features of the aria. In addition, we did not observe changes in EEG oscillations after repeated presentation of the same musical piece. Aside from this distinctive feature, we identified a remarkable variability in EEG oscillations, both within and between the repeated presentations of the aria. We interpret the continuous increase in ERS observed in all frequency bands during music listening as an indicator of a particular neurophysiological and psychological state evoked by music listening. We suggest that this state is characterized by increased internal attention (accompanied by reduced external attention), increased inhibition of brain networks not involved in the generation of this internal state, the maintenance of a particular level of general alertness, and a type of brain state that can be described as “mind wandering.” The overall state can be categorized as a psychological process that may be seen as a “drawing in” to the musical piece. However, this state is not stable and varies considerably throughout the music listening session and across subjects. Most important, however, is the finding that the neurophysiological activations occurring during music listening are dynamic and not stationary. (shrink)

Passive Brain-Computer interfaces (pBCIs) are a human-computer communication tool where the computer can detect from neurophysiological signals the current mental or emotional state of the user. The system can then adjust itself to guide the user towards a desired state. One challenge facing developers of pBCIs is that the system's parameters are generally set at the onset of the interaction and remain stable throughout, not adapting to potential changes over time such as fatigue. The goal of this paper is to (...) investigate the improvement of pBCIs with settings adjusted according to the information provided by a second neurophysiological signal. With the use of a second signal, making the system a hybrid pBCI, those parameters can be continuously adjusted with dynamic thresholding to respond to variations such as fatigue or learning. In this experiment, we hypothesize that the adaptive system with dynamic thresholding will improve perceived game experience and objective game performance compared to two other conditions: an adaptive system with single primary signal biocybernetic loop and a control non-adaptive game. A within-subject experiment was conducted with 16 participants using three versions of the game Tetris. Each participant plays 15 minutes of Tetris under three experimental conditions. The control condition is the traditional game of Tetris with a progressive increase in speed. The second condition is a cognitive load only biocybernetic loop with the parameters presented in Ewing et al. (2016). The third condition is our proposed biocybernetic loop using dynamic threshold selection. Electroencephalography was used as the primary signal and automatic facial expression analysis as the secondary signal. Our results show that, contrary to our expectations, the adaptive systems did not improve the participants' experience as participants had more negative affect from the BCI conditions than in the control condition. We endeavored to develop a system that improved upon the authentic version of the Tetris game, however, our proposed adaptive system neither improved players’ perceived experience, nor their objective performance. Nevertheless, this experience can inform developers of hybrid passive BCIs on a novel way to employ various neurophysiological features simultaneously. (shrink)

The potential benefits of priming intermittent theta burst stimulation with continuous theta burst stimulation have not been examined in regard to sensorimotor oscillatory activities recorded in electroencephalography. The objective of this study was to investigate the modulatory effect of priming iTBS delivered to the motor cortex on movement-related and mirror visual feedback -induced sensorimotor event-related desynchronization, compared with iTBS alone, on healthy adults. Twenty participants were randomly allocated into Group 1: priming iTBS—cTBS followed by iTBS, and Group 2: (...) non-priming iTBS—sham cTBS followed by iTBS. The stimulation was delivered to the right primary motor cortex daily for 4 consecutive days. EEG was measured before and after 4 sessions of stimulation. Movement-related ERD was evaluated during left-index finger tapping and MVF-induced sensorimotor ERD was evaluated by comparing the difference between right-index finger tapping with and without MVF. After stimulation, both protocols increased movement-related ERD and MVF-induced sensorimotor ERD in high mu and low beta bands, indicated by significant time effects. A significant interaction effect favoring Group 1 in enhancing movement-related ERD was observed in the high mu band [F = 4.47, p = 0.049], compared with Group 2. Our experiment suggests that among healthy adults priming iTBS with cTBS delivered to the motor cortex yields similar effects with iTBS alone on enhancing ERD induced by MVF-based observation, while movement-related ERD was more enhanced in the priming iTBS condition, specifically in the high mu band. (shrink)

The effects of mental fatigue on both cognitive and physical performance are well described in the literature, but the recovery aspects of mental fatigue have been less investigated. The present study aimed to fill this gap by examining the persistence of mental fatigue on behavior and electrophysiological mechanisms. Fifteen participants performed an arm-pointing task consisting of reaching a target as fast as possible, before carrying out a 32-min cognitively demanding task [Time Load Dual Back task], and immediately, 10 and 20 (...) min after completion of the TLDB task. During the experiment, electroencephalography was continuously recorded. The significant increase in mental fatigue feeling after the TLDB task was followed by a decrease during the 20 min of recovery without returning to premeasurement values. Brain oscillations recorded at rest during the recovery period showed an increase in both theta and alpha power over time, suggesting a persistence of mental fatigue. Arm-pointing movement duration increased gradually over time during the recovery period, indicating that behavioral performance remained impaired 20 min after the end of the cognitively demanding task. To conclude, subjective measurements indicated a partial recovery of mental fatigue following a cognitively demanding task, whereas electrophysiological and behavioral markers suggested that the effects of mental fatigue persisted for at least 20 min. While the subjective evaluation of mental fatigue is a very practical way to attest the presence of mental fatigue, electrophysiological and behavioral measures seem more relevant to evaluate the time course of mental fatigue effects. (shrink)

Cushman, Young, & Greene (2010) urge the consolidation of moral psychology around a dual-system consensus. On this view, a slow, often-overstretched rational system tends to produce consequentialist intuitions and action-tendencies, while a fast, affective system produces virtuous (or vicious) intuitions and action-tendencies that perform well in their habituated ecological niche but sometimes disastrously outside of it. This perspective suggests a habit-corrected-by-reason picture of moral behavior. Recent research, however, has raised questions about the adequacy of dual-process theories of cognition and behavior, (...) which tend to distinguish categorically between fast, affect-laden, automatic, sometimes-unconscious, nearly-incorrigible mental processes and slow, rational, effortful, conscious, corrigible mental processes. Processes of one alleged type often have properties associated with the contrary type. For instance, rational processes can be emotional, and fast thinking can perform logical operations. In addition, the fast-slow binary transforms a continuous temporal scale into a dichotomous one, losing potentially important information along the way. While these problems are beginning to be addressed in some subfields of psychology, moral psychology has yet to grapple with them. In this theoretical paper, we argue that moral psychology needs a framework that accounts for our knowledge of both temporal and spatial (anatomical) dimensions of the brain basis of moral cognition and virtuous decision-making. We argue that methodologies with very fine-grained temporal granularity, such as electroencephalography (EEG), should be used to complement more frequently used methodologies such as functional magnetic resonance imaging (fMRI). We outline how this might be accomplished using generosity, gratitude, and integrity as a case study. (shrink)

Using a longitudinal case study design, we have tracked the recovery of motor function following severe traumatic brain injury (TBI) through a multimodal neuroimaging approach. In 2006, Canadian Soldier Captain (retired) Trevor Greene (TG) was attacked with an axe to the head while on tour in Afghanistan. TG continues intensive daily rehabilitation, which recently included the integration of physical therapy (PT) with neuromodulation using translingual neurostimulation (TLNS) to facilitate neuroplasticity. Recent findings with PT+TLNS demonstrated that recovery of motor function occurred (...) beyond conventional time limits, currently extending past 14-years post-injury. To investigate whether PT+TLNS similarly resulted in associated cognitive function improvements, we examined event-related potentials (ERPs) with the brain vital signs framework. In parallel with motor function improvements, brain vital signs detected significant increases in basic attention (as measured by P300 response amplitude) and cognitive processing (as measured by contextual N400 response amplitude). These objective cognitive improvements corresponded with TG’s self-reported improvements, including a noteworthy and consistent reduction in ongoing symptoms of post-traumatic stress disorder (PTSD). The findings provide valuable insight into the potential importance of non-invasive neuromodulation in cognitive rehabilitation, in addition to initial indications for physical rehabilitation. (shrink)

This book presents a detailed analysis of three ancient models of spatial magnitude, time, and local motion. The Aristotelian model is presented as an application of the ancient, geometrically orthodox conception of extension to the physical world. The other two models, which represent departures from mathematical orthodoxy, are a "quantum" model of spatial magnitude, and a Stoic model, according to which limit entities such as points, edges, and surfaces do not exist in (physical) reality. The book is unique in its (...) discussion of these ancient models within the context of later philosophical, scientific, and mathematical developments. (shrink)

On the observational equivalence of continuous-time deterministic and indeterministic descriptions Content Type Journal Article Pages 193-225 DOI 10.1007/s13194-010-0011-5 Authors Charlotte Werndl, Department of Philosophy, Logic and Scientific Method, London School of Economics, Houghton Street, London, WC2A 2AE UK Journal European Journal for Philosophy of Science Online ISSN 1879-4920 Print ISSN 1879-4912 Journal Volume Volume 1 Journal Issue Volume 1, Number 2.

Value Co-Creation plays a major role in engaging knowledgeable individuals in a community via innovation, problem solving, and new service/product development. This study investigates the personal factors that influence individuals’ engagement in value co-creation in Higher Education Institutions through the use of online platforms. Some higher education institutions have successfully established or used appropriate online platforms, such as online forums, web applications, and mobile applications to engage their community in ideation or crowdsourcing as a part of the value co-creation process. (...) On the other hand, some HEIs have failed to engage their community in value co-creation activities, and even if they managed to engage some individuals in value co-creation once, they failed to sustain these individuals’ engagement in value co-creation using online platforms. Using the Stimulus Organism Response framework, this study examines the relationship between relevant personal factors and other motivational factors that provide perceived benefits with value co-creation engagement. Data was collected from 308 respondents at five Malaysian research universities. The software analysis tool Smart PLS is used for data analysis and validation. The results demonstrate that personal factors and perceived benefits as a motivational factor has a significant effect on individual engagement in value co-creation. However, the significance of these findings varies from one individual to another. The implications of these findings are discussed. (shrink)

A continuoxis t- norm is a continuous map * from [0, 1]² into [0,1] such that is a commutative totally ordered monoid. Since the natural ordering on [0,1] is a complete lattice ordering, each continuous t-norm induces naturally a residuation → and becomes a commutative naturally ordered residuated monoid, also called a hoop. The variety of basic hoops is precisely the variety generated by all algebras, where * is a continuous t-norm. In this paper we investigate the (...) structure of the variety of basic hoops and some of its subvarieties. In particular we provide a complete description of the finite subdirectly irreducible basic hoops, and we show that the variety of basic hoops is generated as a quasivariety by its finite algebras. We extend these results to Hájek's BL-algebras, and we give an alternative proof of the fact that the variety of BL-algebras is generated by all algebras arising from continuous t-norms on [ 0,1] and their residua. The last part of the paper is devoted to the investigation of the subreducts of BL- algebras, of Gödel algebras and of product algebras. (shrink)

Excerpt from the first paragraph: In response to the interesting comments of Kate Distin and Roger Harnden, let me focus on the relationship between the biological (and evolutionary) systems view and the socio-cultural perspective.

The controversy over the equivalence of continuous sedation until death (CSD) and physician-assisted suicide/euthanasia (PAS/E) provides an opportunity to focus on a significant extended use of CSD. This extension, suggested by the equivalence of PAS/E and CSD, is designed to promote additional patient autonomy at the end-of-life. Samuel LiPuma, in his article, “Continuous Sedation Until Death as Physician-Assisted Suicide/Euthanasia: A Conceptual Analysis” claims equivalence between CSD and death; his paper is seminal in the equivalency debate. Critics contend that (...) sedation follows proportionality protocols for which LiPuma’s thesis does not adequately account. Furthermore, sedation may not eliminate consciousness, and as such LiPuma’s contention that CSD is equivalent to neocortical death is suspect. We not only defend the equivalence thesis, but also expand it to include additional moral considerations. First, we explain the equivalence thesis. This is followed by a defense of the thesis against five criticisms. The third section critiques the current use of CSD. Finally, we offer two proposals that, if adopted, would broaden the use of PAS/E and CSD and thereby expand options at the end-of-life. (shrink)

Seven studies explored the empirical basis for claims that infants represent cardinal values of small sets of objects. Many studies investigating numerical ability did not properly control for continuous stimulus properties such as surface area, volume, contour length, or dimensions that correlate with these properties. Experiment 1 extended the standard habituation/dishabituation paradigm to a 1 vs 2 comparison with three-dimensional objects and confirmed that when number and total front surface area are confounded, infants discriminate the arrays. Experiment 2 revealed (...) that infants dishabituated to a change in front surface area but not to a change in number when the two variables were pitted against each other. Experiments 3 through 5 revealed no sensitivity to number when front surface area was controlled, and Experiments 6 and 7 extended this pattern of findings to the Wynn (1992) transformation task. Infants’ lack of a response to number, combined with their demonstrated sensitivity to one or more dimensions of continuous extent, supports the hypothesis that the representations subserving object-based attention, rather than those subserving enumeration, underlie performance in the above tasks. (shrink)

Representation is central to contemporary theorizing about the mind/brain. But the nature of representation--both in the mind/brain and more generally--is a source of ongoing controversy. One way of categorizing representational types is to distinguish between the analog and the digital: the received view is that analog representations vary smoothly, while digital representations vary in a step-wise manner. I argue that this characterization is inadequate to account for the ways in which representation is used in cognitive science; in its place, I (...) suggest an alternative taxonomy. I will defend and extend David Lewis's account of analog and digital representation, distinguishing analog from continuous representation, as well as digital from discrete representation. I will argue that the distinctions available in this four-fold account accord with representational features of theoretical interest in cognitive science more usefully than the received analog/digital dichotomy. (shrink)

A central claim of Deleuze's reading of Bergson is that Bergson's distinction between space as an extensive multiplicity and duration as an intensive multiplicity is inspired by the distinction between discrete and continuous manifolds found in Bernhard Riemann's 1854 thesis on the foundations of geometry. Yet there is no evidence from Bergson that Riemann influences his division, and the distinction between the discrete and continuous is hardly a Riemannian invention. Claiming Riemann's influence, however, allows Deleuze to argue that (...) quantity, in the form of ‘virtual number’, still pertains to continuous multiplicities. This not only supports Deleuze's attempt to redeem Bergson's argument against Einstein in Duration and Simultaneity, but also allows Deleuze to position Bergson against Hegelian dialectics. The use of Riemann is thereby an important element of the incorporation of Bergson into Deleuze's larger early project of developing an anti-Hegelian philosophy of difference. This article first reviews the role of discrete and continuous multiplicities or manifolds in Riemann's Habilitationsschrift, and how Riemann uses them to establish the foundations of an intrinsic geometry. It then outlines how Deleuze reinterprets Riemann's thesis to make it a credible resource for Deleuze's Bergsonism. Finally, it explores the limits of this move, and how Deleuze's later move away from Bergson turns on the rejection of an assumption of Riemann's thesis, that of ‘flatness in smallest parts’, which Deleuze challenges with the idea, taken from Riemann's contemporary, Richard Dedekind, of the irrational cut. (shrink)

Within the traditional Hilbert space formalism of quantum mechanics, it is not possible to describe a particle as possessing, simultaneously, a sharp position value and a sharp momentum value. Is it possible, though, to describe a particle as possessing just a sharp position value (or just a sharp momentum value)? Some, such as Teller, have thought that the answer to this question is No - that the status of individual continuous quantities is very different in quantum mechanics than in (...) classical mechanics. On the contrary, I shall show that the same subtle issues arise with respect to continuous quantities in classical and quantum mechanics; and that it is, after all, possible to describe a particle as possessing a sharp position value without altering the standard formalism of quantum mechanics. (shrink)

Two different kinds of stochastic processes in Hilbert space used to introduce spontaneous localization into the quantum evolution are investigated. In the processes of the first type, finite changes of the wave function take place instantaneously with a given mean frequency. The processes of the second type are continuous. It is shown that under a suitable infinite frequency limit the discontinuous process transforms itself into the continuous one.

A journey of a thousand miles begins, obviously, with a single step. But isn’t it equally obvious that a step of a single metre must begin with a single millimetre? And before you can begin the last micron of that millimetre, don’t you have to get through 999 other microns first? And so ad infinitum? That “ad infinitum” bit is what worried the philosopher Zeno of Elea. Can our every action really consist of sub actions each consisting of sub sub (...) actions ... so that before we can move at all, we have to perform a literally infinite number of distinct, consecutive actions? (shrink)

A general metatheorem is proved which reduces a wide class of statements about continuous time stochastic processes to statements about discrete time processes. We introduce a strong language for stochastic processes, and a concept of forcing for sequences of discrete time processes. The main theorem states that a sentence in the language is true if and only if it is forced. Although the stochastic process case is emphasized in order to motivate the results, they apply to a wider class (...) of random variables. At the end of the paper we illustrate how the theorem can be used with three applications involving submartingales. (shrink)

It is a classic result in modal logic, often referred to as Jónsson-Tarski duality, that the category of modal algebras is dually equivalent to the category of descriptive frames. The latter are Kripke frames equipped with a Stone topology such that the binary relation is continuous. This duality generalizes the celebrated Stone duality for boolean algebras. Our goal is to generalize descriptive frames so that the topology is an arbitrary compact Hausdorff topology. For this, instead of working with the (...) boolean algebra of clopen subsets of a Stone space, we work with the ring of continuous real-valued functions on a compact Hausdorff space. The main novelty is to define a modal operator on such a ring utilizing a continuous relation on a compact Hausdorff space.Our starting point is the well-known Gelfand duality between the category ${\sf KHaus}$ of compact Hausdorff spaces and the category $\boldsymbol {\mathit {uba}\ell }$ of uniformly complete bounded archimedean $\ell $ -algebras. We endow a bounded archimedean $\ell $ -algebra with a modal operator, which results in the category $\boldsymbol {\mathit {mba}\ell }$ of modal bounded archimedean $\ell $ -algebras. Our main result establishes a dual adjunction between $\boldsymbol {\mathit {mba}\ell }$ and the category ${\sf KHF}$ of what we call compact Hausdorff frames; that is, Kripke frames equipped with a compact Hausdorff topology such that the binary relation is continuous. This dual adjunction restricts to a dual equivalence between ${\sf KHF}$ and the reflective subcategory $\boldsymbol {\mathit {muba}\ell }$ of $\boldsymbol {\mathit {mba}\ell }$ consisting of uniformly complete objects of $\boldsymbol {\mathit {mba}\ell }$. This generalizes both Gelfand duality and Jónsson-Tarski duality. (shrink)

Polysemy has recently emerged as a popular topic in philosophy of language. While much existing research focuses on the relatedness among senses, this article introduces a novel perspective that emphasizes the continuity of sense individuation, sense regularity, and sense productivity. This new perspective has only recently gained traction, largely due to advancements in computational linguistics. It also poses a serious challenge to semantic minimalism, so I present three arguments against minimalism from the continuous perspective that touch on the minimal (...) concept, the distinction from homonymy, and the quasi‐rule‐like nature of polysemy. Last, I provide an account of polysemy that incorporates this continuous perspective. (shrink)

-/- Continuous first-order logic has found interest among model theorists who wish to extend the classical analysis of “algebraic” structures (such as fields, group, and graphs) to various natural classes of complete metric structures (such as probability algebras, Hilbert spaces, and Banach spaces). With research in continuous first-order logic preoccupied with studying the model theory of this framework, we find a natural question calls for attention. Is there an interesting set of axioms yielding a completeness result? -/- The (...) primary purpose of this article is to show that a certain, interesting set of axioms does indeed yield a completeness result for continuous first-order logic. In particular, we show that in continuous first-order logic a set of formulae is (completely) satisfiable if (and only if) it is consistent. From this result it follows that continuous first-order logic also satisfies an approximated form of strong completeness, whereby Σ⊧φ (if and) only if Σ⊢φ ∸2−n for all n < ω. This approximated form of strong completeness asserts that if Σ⊧φ, then proofs from Σ, being finite, can provide arbitrarily better approximations of the truth of φ. -/- Additionally, we consider a different kind of question traditionally arising in model theory—that of decidability. When is the set of all consequences of a theory (in a countable, recursive language) recursive? Say that a complete theory T is decidable if for every sentence φ, the value φ T is a recursive real, and moreover, uniformly computable from φ. If T is incomplete, we say it is decidable if for every sentence φ the real number φ T o is uniformly recursive from φ, where φ T o is the maximal value of φ consistent with T. As in classical first-order logic, it follows from the completeness theorem of continuous first-order logic that if a complete theory admits a recursive (or even recursively enumerable) axiomatization then it is decidable. (shrink)

Inspired by a framework of multi lingual sequent calculus, we introduce a formal logical system called locally continuous sequent calculus to represent _L_-domains. By considering the logic states defined on locally continuous sequent calculi, we show that the collection of all logic states of a locally continuous sequent calculus with respect to set inclusion forms an _L_-domain, and every _L_-domain can be obtained in this way. Moreover, we define conjunctive consequence relations as morphisms between our sequent calculi, (...) and prove that the category of locally continuous sequent calculi and conjunctive consequence relations is equivalent to that of _L_-domains and Scott-continuous functions. This result extends Abramsky’s “Domain theory in logical form” to a continuous setting. (shrink)