Through Taylor's expansions and a thorough analysis of the necessary and sufficient conditions that will entail for fixed point and Newton's iterative methods to be of higher order convergence in ...Banach space, we are able to present a unified way to make these methods faster. Numerical examples illustrate the theoretical results.
Celotno besedilo
Dostopno za:
BFBNIB, DOBA, GIS, IJS, IZUM, KILJ, KISLJ, NUK, PILJ, PNG, SAZU, UILJ, UKNU, UL, UM, UPUK
It was known before Archimedes (287-212 BC) that the circumference of a circle was proportional to its diameter and that the area was proportional to the square of its radius. It was Archimedes who ...first supplied a rigorous proof that these two proportionality constants were the same, now called π 1. He started with inscribed and circumscribed hexagons and increased the number of sides from 6 up to 96 by successively doubling it. His result was not a single value. In fact he generated five intervals each of which contained π. He calculated a lower bound from the inscribed polygon and an upper bound from the circumscribed polygon of 96 sides. This gave him the interval () or (3.140845, 3.142857), which is less accurate than the interval bounded by half-perimeters of the inscribed and circumscribed 96-gons, which is (3.141031, 3.142714).
On Euler–Maclaurin formula Dubeau, François
Journal of computational and applied mathematics,
April 2016, 2016-04-00, 20160401, Letnik:
296
Journal Article
Recenzirano
Odprti dostop
We use a simple approach to show that an Euler–Maclaurin like formula can be associated to any interpolatory quadrature rule. This result is obtained by successively adding correcting terms that are ...exact for polynomials of increasing degree. A decomposition of the coefficients of the Euler–Maclaurin formula in terms of the integral to compute and representations of powers of the nodes is pointed out. Optimal truncation error bounds are also obtained.
Objective
Interictal epileptiform anomalies such as epileptiform discharges or high‐frequency oscillations show marked variations across the sleep‐wake cycle. This study investigates which state of ...vigilance is the best to localize the epileptogenic zone (EZ) in interictal intracranial electroencephalography (EEG).
Methods
Thirty patients with drug‐resistant epilepsy undergoing stereo‐EEG (SEEG)/sleep recording and subsequent open surgery were included; 13 patients (43.3%) had good surgical outcome (Engel class I). Sleep was scored following standard criteria. Multiple features based on the interictal EEG (interictal epileptiform discharges, high‐frequency oscillations, univariate and bivariate features) were used to train a support vector machine (SVM) model to classify SEEG contacts placed in the EZ. The performance of the algorithm was evaluated by the mean area under the receiver‐operating characteristic (ROC) curves (AUCs) and positive predictive values (PPVs) across 10‐minute sections of wake, non–rapid eye movement sleep (NREM) stages N2 and N3, REM sleep, and their combination.
Results
Highest AUCs were achieved in NREM sleep stages N2 and N3 compared to wakefulness and REM (P < .01). There was no improvement when using a combination of all four states (P > .05); the best performing features in the combined state were selected from NREM sleep. There were differences between good (Engel I) and poor (Engel II‐IV) outcomes in PPV (P < .05) and AUC (P < .01) across all states. The SVM multifeature approach outperformed spikes and high‐frequency oscillations (P < .01) and resulted in results similar to those of the seizure‐onset zone (SOZ; P > .05).
Significance
Sleep improves the localization of the EZ with best identification obtained in NREM sleep stages N2 and N3. Results based on the multifeature classification in 10 minutes of NREM sleep were not different from the results achieved by the SOZ based on 12.7 days of seizure monitoring. This finding might ultimately result in a more time‐efficient intracranial presurgical investigation of focal epilepsy.
Summary
Objective
Rapid eye movement (REM) sleep has a suppressing effect on epileptic activity. This effect might be directly related to neuronal desynchronization mediated by cholinergic ...neurotransmission. We investigated whether interictal epileptiform discharges (IEDs) and high frequency oscillations—a biomarker of the epileptogenic zone—are evenly distributed across phasic and tonic REM sleep. We hypothesized that IEDs are more suppressed during phasic REM sleep because of additional cholinergic drive.
Methods
Twelve patients underwent polysomnography during long‐term combined scalp‐intracerebral electroencephalography (EEG) recording. After sleep staging in the scalp EEG, we identified segments of REM sleep with rapid eye movements (phasic REM) and segments of REM sleep without rapid eye movements (tonic REM). In the intracerebral EEG, we computed the power in frequencies <30 Hz and from 30 to 500 Hz, and marked IEDs, ripples (>80 Hz) and fast ripples (>250 Hz). We grouped the intracerebral channels into channels in the seizure‐onset zone (SOZ), the exclusively irritative zone (EIZ), and the normal zone (NoZ).
Results
Power in frequencies <30 Hz was lower during phasic than tonic REM sleep (p < 0.001), most likely reflecting increased desynchronization. IEDs, ripples and fast ripples, were less frequent during phasic than tonic REM sleep (phasic REM sleep: 39% of spikes, 35% of ripples, 18% of fast ripples, tonic REM sleep: 61% of spikes, 65% of ripples, 82% of fast ripples; p < 0.001). In contrast to ripples in the epileptogenic zone, physiologic ripples were more abundant during phasic REM sleep (phasic REM sleep: 73% in NoZ, 30% in EIZ, 28% in SOZ, tonic REM sleep: 27% in NoZ, 70% in EIZ, 72% in SOZ; p < 0.001).
Significance
Phasic REM sleep has an enhanced suppressive effect on IEDs, corroborating the role of EEG desynchronization in the suppression of interictal epileptic activity. In contrast, physiologic ripples were increased during phasic REM sleep, possibly reflecting REM‐related memory consolidation and dreaming.
•Analysis of spike dynamics shows that epilepsy surgery outcome depends on strong, single and stable sources.•Spatio-temporal spike dynamics is an objective and easy-to-apply marker for epilepsy ...surgery outcome prediction.•The best results for outcome prediction were achieved by 18-h periods or longer.
We hypothesized that spatio-temporal dynamics of interictal spikes reflect the extent and stability of epileptic sources and determine surgical outcome.
We studied 30 consecutive patients (14 good outcome). Spikes were detected in prolonged stereo-electroencephalography recordings. We quantified the spatio-temporal dynamics of spikes using the variance of the spike rate, line length and skewness of the spike distribution, and related these features to outcome. We built a logistic regression model, and compared its performance to traditional markers.
Good outcome patients had more dominant and stable sources than poor outcome patients as expressed by a higher variance of spike rates, a lower variance of line length, and a lower variance of positive skewness (ps < 0.05). The outcome was correctly predicted in 80% of patients. This was better or non-inferior to predictions based on a focal lesion (p = 0.016), focal seizure-onset zone, or complete resection (ps > 0.05). In the five patients where traditional markers failed, spike distribution predicted the outcome correctly. The best results were achieved by 18-h periods or longer.
Analysis of spike dynamics shows that surgery outcome depends on strong, single and stable sources.
Our quantitative method has the potential to be a reliable predictor of surgical outcome.
This paper is a survey of topics related to Hermite interpolation. In the first part we present the standard analysis of the Hermite interpolation problem. Existence, uniqueness and error formula are ...included. Then some computational aspects are studied including Leibnitz' formula and devided differences for monomials. Moreover continuity and differentiation properties of divided differences are analyzed. Finally we represent Hermite polynomial with respect to different basis and give links between them.
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