Entire Interval Research Articles

Total negativity: Characterizations and single-vector tests

A matrix is called totally negative (totally non-positive) of order k, if all its minors of size at most k are negative (non-positive). The objective of this article is to provide several novel characterizations of total negativity via the (a) variation diminishing property, (b) sign non-reversal property, and (c) Linear Complementarity Problem. (The last of these provides a novel connection between total negativity and optimization/game theory.) More strongly, each of these three characterizations uses a single test vector whose coordinates alternate in sign. As an application of the sign non-reversal property, we study the interval hull of two rectangular matrices. In particular, we identify two matrices C±(A,B) in the interval hull of matrices A and B that test total negativity of order k, simultaneously for the entire interval hull. We also show analogous characterizations for totally non-positive matrices and provide a finite set of test matrices to detect the total non-positivity property of an interval hull. These novel characterizations may be considered similar in spirit to fundamental results characterizing totally positive matrices by Brown et al. (1981) [5] (see also Gantmacher–Krein, 1950), Choudhury et al. (2021) [9] and Choudhury (2022) [8]. Finally, we show that totally negative/non-positive matrices can not be detected by (single) test vectors from orthants other than the open bi-orthant that have coordinates with alternating signs, via the variation diminishing property or the sign non-reversal property.

An evaluation of the inter-rater reliability in a clinical skills objective structured clinical examination

Background. An objective structured clinical examination (OSCE) is a performance-based examination used to assess health sciences students and is awell-recognised tool to assess clinical skills with or without using real patients.Objectives. To determine the inter-rater reliability of experienced and novice assessors from different clinical backgrounds on the final mark allocationsduring assessment of third-year medical students’ final OSCE at the University of the Free State.Methods. This cross-sectional analytical study included 24 assessors and 145 students. After training and written instructions, two assessors per station(urology history taking, respiratory examination and gynaecology skills assessment) each independently assessed the same student for the same skill bycompleting their individual checklists. At each station, assessors could also give a global rating mark (from 1 to 5) as an overall impression.Results. The urology history-taking station had the lowest mean score (53.4%) and the gynaecology skills station the highest (71.1%). Seven (58.3%) ofthe 12 assessor pairs differed by >5% regarding the final mark, with differences ranging from 5.2% to 12.2%. For two pairs the entire confidence interval(CI) was within the 5% range, whereas for five pairs the entire CI was outside the 5% range. Only one pair achieved substantial agreement (weightedkappa statistic 0.74 ‒ urology history taking). There was no consistency within or across stations regarding whether the experienced or novice assessorgave higher marks. For the respiratory examination and gynaecology skills stations, all pairs differed for the majority of students regarding the globalrating mark. Weighted kappa statistics indicated that no pair achieved substantial agreement regarding this mark.Conclusion. Despite previous experience, written instructions and training in the use of the checklists, differences between assessors were found inmost cases.

Intrinsic optical absorption in Dirac metals

A Dirac metal is a doped (gated) Dirac material with the Fermi energy (EF) lying either in the conduction or valence bands. In the non-interacting picture, optical absorption in gapless Dirac metals occurs only if the frequency of incident photons (Ω) exceeds the direct (Pauli) frequency threshold, equal to 2EF. In this work, we study, both analytically and numerically, the role of electron–electron (ee) and electron–hole (eh) interactions in optical absorption of two-dimensional (2D) and three-dimensional (3D) Dirac metals in the entire interval of frequencies below 2EF. We show that, for Ω≪EF, the optical conductivity, ℜσ(Ω), arising from the combination of ee and certain eh scattering processes, scales as Ω2lnΩ in 2D and as Ω2 in 3D, respectively, both for short-range (Hubbard) and long-range (screened Coulomb) interactions. Another type of eh processes, similar to Auger–Meitner (AM) processes in atomic physics, starts to contribute for Ω above the direct threshold, equal to EF. Similar to the case of doped semiconductors with parabolic bands studied in prior literature, the AM contribution to ℜσ(Ω) in Dirac metals is manifested by a threshold singularity, ℜσ(Ω)∝(Ω−EF)d+2, where d is the spatial dimensionality and 0<Ω−EF≪EF. In contrast to doped semiconductors, however, the AM contribution in Dirac metals is completely overshadowed by the ee and other eh contributions. Numerically, ℜσ(Ω) happens to be small in almost the entire range of Ω<2EF. This finding may have important consequences for collective modes in Dirac metals lying below 2EF.

754 A Longitudinal Study of Compression Therapy in Pediatric Burn Patients

Abstract Introduction Compression Therapy is crucial to managing and preventing hypertrophic scarring. The pediatric population can be especially at risk for hypertrophic scarring due to rapid growth periods and/or discomfort which interferes with compliance. Our previous practice was for pediatric patients to be scheduled every 6 -12 months like the adult burn population. To address this issue, we hypothesized pediatric patients would benefit from more frequent follow up visits. We arbitrarily determined to follow our pediatric patients every 3-4 months for garment remeasuring to improve proper garment fit, efficacy and compliance. Methods A longitudinal study was conducted to determine whether 3–4 month interval check-ups would improve compliance of garment wearing and presumably efficacy of overall management of hypertrophic scarring. Data was collected between June 2020 – December 2021 (1 ½ years). The data was manually retrieved from our Electronic Medical Record. Results Prior to June 2021, children were routinely scheduled in the Burn Clinic at intervals between six and twelve months for scar checks (same as adults). Children were not able to participate in long term compression therapy for the entire interval for several reasons including growth, size changes, garment failure, expired insurance authorization, and possibly lack of knowledge. We decided to schedule consistent follow-up of pediatric patients every 3-4 months, to ensure consistent remeasuring of garments was completed. Pre intervention results noted that only 35% of patients had a 2nd garment remeasured as compared to our post intervention results which showed that 88% of patients successfully had a 2nd garment remeasured. Conclusions Our data suggests that short interval follow up with pediatric burn patients for scar management has led to improved rates of compression therapy compliance and presumably efficacy. Applicability of Research to Practice We have continued the practice of short term follow up in pediatric burn patients for scar management and our data has also provided opportunities to reinforce education with patients and their families. We have found that with better understanding, our patients and families are motivated to seek out and comply with longitudinal care.

Microscale Searching Algorithm for Coupling Matrix Optimization of Automated Microwave Filter Tuning.

Automated tuning can significantly improve productivity and save the costs of manual operation in the microwave filter manufacturing industry. This article proposes a mathematical model of scattering data optimization to find the accurate coupling matrix for multiple-version microwave filters, a core step of automated microwave filter tuning. For the large-scale problem of coupling coefficient combination, we propose a decision set decomposition strategy that evenly divides the entire frequency interval into several subintervals according to the correlation between scattering data. With this strategy, we design a microscale (small-size subsets of the decomposed decision set) searching algorithm, which solves each suboptimization problem by searching the decision subset instead of the entire decision set. To verify the validity of the proposed algorithm for multiple-version microwave filters, experiments are conducted on three versions of microwave filters from a real-world production line, including the two-port eighth-order, ninth-order, and tenth-order microwave filters. Experimental results show that the proposed model is feasible within the industrial error for the multiversion microwave filter tuning problem. Besides, the proposed algorithm outperforms the state-of-the-art optimization algorithms in the coupling matrix optimization problem.

Multi-target CFAR detector based on compressed sensing radar system

ABSTRACT In the multi-target scenarios, we consider the problem of constant false alarm rate (CFAR) target detection. For the conventional CFAR algorithms, the multi-target detection performance is mainly limited by the interference target tolerance within the reference window. In this paper, we propose a local ordered statistic CFAR (OS-CFAR) target detector based on compressed sensing (CS) radar system to address the degradation of detection performance in multi-target scenarios. With the analog-information-converter (AIC), the radar intermediate frequency (IF) signal is compressively sampled into discrete linear measurements. At each detection stage, multiple signal components are generated by correlation tests between the sensing matrix and linear measurements. By support set merge and proxy pruning, the signal components with the largest energy are retained. These components are the echoes reflected from the targets. Since target components are output in decreasing order of correlation, the proposed detector only requires a local OS-CFAR decision in the interval where the least correlated target is located, rather than traversing the entire interval. By continuously updating the detection stage, all targets that satisfy the false alarm rate requirement can be screened out. Finally, the performance of the proposed detector in the multi-target scenarios is confirmed by simulation results.

SYNTHESIS METHODS AND ALGORITHMS FOR EXPANDING THE DYNAMIC RANGE OF THE ADC

The modern development of new technologies, in particular, the intensive use of programmable logic integrated circuits in digital information processing tasks, allows for algorithmically and programmatically solving problems, opens up real opportunities to increase accuracy and ensure the required signal conversion speed, and also, it will be possible to take into account the effects of noise and their elimination over the entire time interval band. The article discusses the main methods of digital information processing, analyzes the ways and possibilities of expanding the dynamic range of analog-to-digital converters, as well as the synthesis of appropriate conversion devices, primarily digital filters.

Smoothness of solutions to the damping problem for nonstationary control system with delay of neutral type on the whole interval

We consider the damping problem for a nonstationary control system described by a system of differential-difference equations of neutral type with smooth matrix coe cients and several delays. This problem is equivalent to the boundary-value problem for a system of second-order differentialdifference equations, which has a unique generalized solution. It is proved that the smoothness of this solution can be violated on the considered interval and is preserved only on some subintervals. Su cient conditions for the initial function are obtained to ensure the smoothness of the generalized solution over the entire interval.

Field coupling-induced synchronization of a functional neuron with magnetic field perception

It is difficult to explore the induced mechanism of neuronal firing activity and the cooperation between neurons experimentally, so some neuronal circuits are established to train mode transformation and selection. Furthermore, some functional device, for example, piezoelectric ceramic, thermistor, phototube are inset in neuronal circuits and the functional neurons are developed to perceive some specific physical signal. In this paper, a linear resistance and an ideal Josephson junction are parallelly connected to FitzHugh-Nagumo (FHN) neuronal circuit, and then a functional neuron is established to percept the induction currents induced by the external magnetic field. Some basic dynamics are analyzed in terms of two-parameter bifurcation, one-parameter bifurcation and interspike interval(ISI for short) bifurcation, it is indicated that the functional neuron is sensitive to the electromagnetic stimulation and has rich multimodal transformations. The period-adding bifurcations accompanied by multi-period and chaos between two period cascades occupy the entire bifurcation interval when the external magnetic field is given in steady state, while the irregular modal transformations between period and chaos can be observed when the functional neuron encounters a periodic external magnetic field. Furthermore, two functional neurons are bridged with a capacitor, so field coupling is induced, then the effects of coupling strengths on complete synchronization are investigated by calculating the synchronization error function and the bifurcation diagram. In addition, the local dynamics of the network node play an important role in collective behavior and synchronous transition, so two capacitor-coupled functional neurons is presented as the network nodes, and a chain neural network is constructed to explore the effects of external magnetic field and coupling strength on network synchronous behavior. Obviously, the orderliness of the neural network can enhance or destroy under different modulation of external magnetic stimulation and coupling strength. It can give insights to investigate synchronization on neural networks with field coupling and useful guidance for implementing artificial synapse for signal processing.

Verbal Fluency in Prelingually Deaf, Early Implanted Children and Adolescents With Cochlear Implants.

Verbal fluency tasks assess the ability to quickly and efficiently retrieve words from the mental lexicon by requiring subjects to rapidly generate words within a phonological or semantic category. This study investigated differences between cochlear implant users and normal-hearing peers in the clustering and time course of word retrieval during phonological and semantic verbal fluency tasks. Twenty-eight children and adolescents (aged 9-17 years) with cochlear implants and 33 normal-hearing peers completed measures of verbal fluency, nonverbal intelligence, speech perception, and verbal short-term/working memory. Phonological and semantic verbal fluency tests were scored for total words generated, words generated in each 10-s interval of the 1-min task, latency to first word generated, number of word clusters, average cluster size, and number of word/cluster switches. Children and adolescents with cochlear implants generated fewer words than normal-hearing peers throughout the entire 60-s time interval of the phonological and semantic fluency tasks. Cochlear implant users also had slower start latency times and produced fewer clusters and switches than normal-hearing peers during the phonological fluency task. Speech perception and verbal working memory scores were more strongly associated with verbal fluency scores in children and adolescents with cochlear implants than in normal-hearing peers. Cochlear implant users show poorer phonological and semantic verbal fluency than normal-hearing peers, and their verbal fluency is significantly associated with speech perception and verbal working memory. These findings suggest deficits in fluent retrieval of phonological and semantic information from long-term lexical memory in cochlear implant users.

A case of impulsive singularity

The paper considers an impulsive system with singularities. Different types of problems with singular perturbations have been discussed in many books. In Bainov and Kovachev’s book [4]several articles cited therein consider impulse systems with small parameter involving only differential equations. The parameter is not in the impulsive equation of the systems. In our present the small parameter is inserted into the impulse equation. This is the principal novelty of our study. Furthermore, for the impulsive function, we found a condition that prevents the impulsive function to blow up as the parameter tends to zero. So we have significantly extended the singularity concept for discontinuous dynamics. The singularity of the impulsive part of the system can be treated in the manner of perturbation theory methods. This article is a continuation of [1] work. In our present research, we apply the method of the paper [1]. Our goal is to construct an approximation with higher accuracy and to obtain the complete asymptotic expansion. We construct a uniform asymptotic approximation of the solution that is valid in the entire close interval by using the method of boundary functions [22]. An illustrative example using numerical simulations is given to support the theoretical results.

Aperiodic Sampled-Data Control for Stabilization of Memristive Neural Networks With Actuator Saturation: A Dynamic Partitioning Method.

This article is concerned with the local stabilization of memristive neural networks subject to actuator saturation via aperiodic sampled-data control. A dynamic partitioning point is elegantly introduced, which is placed between the latest sampling instant and the present time to utilize more information of the inner sampling. To analyze the stability of the closed-loop system, a time-dependent two-side looped functional, which fully utilizes the state information on the entire sampling interval as well as at the dynamic partitioning point, is constructed. It relaxes the positive definiteness of traditional Lyapunov functional inside the sampling interval and therefore, provides the possibility to derive less conservative stability results. Besides, an auxiliary system is established to describe the dynamics at the partitioning point. On the basis of the constructed looped functional, the discrete-time Lyapunov theorem, and some estimation approaches, a linear matrix inequalities-based stability criterion is developed, and then, the sampled-data saturated controller is designed to ensure the local asymptotic stability of the closed-loop system. Thereafter, two optimization problems are developed to seek the desired feedback gain and to expand the estimation of the region of attraction or to enlarge the upper bound of the sampling interval. Eventually, a numerical example is given to demonstrate the effectiveness and the superiority of the proposed results.

Uniqueness of the potential in a time-fractional diffusion equation

Abstract This article concerns the uniqueness of an inverse coefficient problem of identifying a spatially varying potential in a one-dimensional time-fractional diffusion equation. The input sources are given by a complete system in L 2 ⁢ ( 0 , 1 ) {L^{2}(0,1)} , and measurements are observed at the end point of the spatial interval. Firstly, we provide the positive lower bound of the Green function for the differential operator with different boundary conditions. Then, based on the positive lower bound estimation of the Green function, the relationship between the Green function, the solution of the forward problem, and the potential, such measurements uniquely determine the potential on the entire interval under different boundary conditions.

New insights on the interpretation of the provenance and evolution of the Silurian units in the central Precordillera, Argentina

In the central region of the Precordillera, San Juan Province, Silurian silicoclastic sedimentites of the Los Espejos Formation crop out in the Jáchal River area. To the south of this region, an equivalent unit is recognized in the San Juan River area (Tambolar Formation). Both units present similar lithological characteristics, however, it has not yet been defined if they share source areas of detrital contributions. On the other hand, for the Jáchal River sector, it is proposed to establish if there were changes in the regions from where the sub-basin received sediment contributions during the Devonian, which can be seen reflected in the detrital zircon contribution patterns of the overlying unit (Talacasto Formation). The present work is part of a series of studies tending to determine the nature and provenance of the Silurian-Devonian sequences of the Central Precordillera. On this occasion, detrital zircon patterns of the detrital sources of the Los Espejos Formation are analyzed and compared with the information obtained for the Silurian Tambolar Formation (San Juan River area) and the overlying Devonian Talacasto and Punta Negra formations (Jáchal and San Juan rivers areas). To characterize and compare the studied units, different methodologies were applied, namely sedimentary petrography, heavy minerals studies, and morphological and isotopic analyses of detrital zircons. The analysis of thin sections allowed determining textural and compositional parameters. Through the predominance of detrital minerals, it was possible to establish that the studied units are composed of quartzite-type rocks coming from mature areas, with low percentages of lithic components and abundant opaque heavy minerals of the hematite group. The study of heavy minerals, especially morphological and typological parameters of detrital zircons, allowed to establish recycled and plutonic sources as main modes, as well as the changes that occurred during the basin filling dynamics for Silurian and Devonian times. On the other hand, U–Pb isotopic analysis in detrital zircons indicate that the Pampean-Brasiliano orogenic cycle composes the main source of detritus with ages between 511 and 816 Ma. In second place are the Mesoproterozoic ages, represented by the interval from 1000 to 1350 Ma. The youngest detrital ages show a maximum sedimentation age of 478.5 ± 4.4 Ma (Tremadocian), indicating that younger sources of contribution correspond to the Famatinian Orogen. The Kolmogorov-Smirnoff test revealed that the studied Silurian-Devonian units studied have similar patterns of sedimentary contributions, which suggests that the sources of provenance were common in both regions and remained active throughout the entire time interval, without significant changes.

Time-Frequency Range Estimation Method for Pulsed LiDAR

Light detection and ranging (LiDAR) is the key sensor behind the 3D perception of self-driving vehicles. The range, precision and accuracy of a LiDAR sensor mainly depend on how each received pulse is captured, sampled and processed. However, to the best of our knowledge, such a topic is yet to be investigated in depth. This work aims to fill such a gap by introducing a new range estimation method. The method takes only 3 samples from the rising edge of the received pulse and combines two simple estimation techniques: time-domain edge detection and frequency-domain phase estimation. The proposed method was demonstrated over a dynamic range of 45 dB, proving to be robust over the entire interval. Sub-centimeter precision and accuracy were achieved over a dynamic range of 24 dB. The obtained estimations are bounded by a derived theoretical limit that is identical to a Cramér-Rao lower bound of a phase estimator.

Efficacy of emicizumab is maintained throughout dosing intervals for bleed prophylaxis

BackgroundAcross the HAVEN clinical trial program, the efficacy of emicizumab has been demonstrated in children, adolescents, and adults with hemophilia A, with or without factor VIII inhibitors. After the 4-week loading dose period, emicizumab concentrations are expected to remain at levels that provide bleed protection throughout the entire dosing interval, regardless of the chosen maintenance dosing regimen, ie, weekly, every 2 weeks, or every 4 weeks. ObjectivesThe objective of this study was to examine the timing of treated bleeds within the dosing intervals for emicizumab administered during the HAVEN 1 to 4 studies. MethodsIn this post hoc analysis, we pooled data from all the participants of the HAVEN 1 to 4 studies and analyzed the timing of treated bleeds in relation to the emicizumab dose. ResultsA total of 392 participants were included in this analysis, with a median (range) age of 28.0 years (1.1-77.0 years). Target joints were identified in 237 of 392 (60.5%) participants before the study entry. Overall, 211 of 392 (53.8%) participants experienced 907 treated bleeding events. The total mean (SD) annualized bleeding rate across the 4 studies was 1.6 (5.9). There was no evidence that bleeding events clustered on any 1 particular day in any dosing schedule from HAVEN 1 to 4 (P > .05 for all 3 treatment regimens). ConclusionData from the HAVEN 1 to 4 trials show consistent bleed prevention within the dosing interval, regardless of the dosing regimen chosen. These findings provide further evidence of the sustained efficacy of emicizumab across all approved dosing regimens to reduce bleeding in people with hemophilia A.

Relationship of Maastrichtian–Thanetian benthic foraminiferal species diversity, palaeooxygenation, and palaeoproductivity in shallow waters of the Western Desert, Egypt

Benthic foraminiferal analysis (315 samples, 16,271 specimens) of the shallow water (< 100 m) Maastrichtian–Thanetian rocks from the Dakhla Oasis (Western Desert, Egypt) was studied to infer the inter–relationships between species diversity, palaeooxygenation, palaeoproductivity, and palaeodepth and changes at the Cretaceous–Paleogene (K/Pg) boundary. Positive and significant correlations are noted between these proxies, suggesting a well-oxygenated oligotrophic environment. However, a brief interval (mid–lower Maastrichtian) of increased palaeoproductivity with reduced diversity and oxygenation (ventilation) is noted (a characteristic of mesotrophic–eutrophic settings) that coincides with very shallow waters during a highstand system tract (HST) and dominated by the dysoxic agglutinated species Ammobaculites khargaensis. The diversity index, Fisher’s α (< 5), and paleodepth proxy (foraminiferal wall structure types) also suggest a shallow neritic (largely littoral) depth for the entire study interval. At the bottom of the study section (Planktic Foraminiferal Zones CF8b-CF7), species diversity, palaeooxygenation, and palaeoproductivity are high. From the K/Pg boundary to the post-K/Pg period, these variables are low and fluctuate with moderate species dominance. Data suggests an overall 40% benthic foraminiferal species (38% of agglutinated and 40% of calcareous extinct species) extinction rate after the K/Pg hiatus. The period immediately following the K/Pg boundary is characterized by increased basinal ventilation and decreased palaeoproductivity, which are attributed to changes in sea level and concurrent regional subsidence. However, as stable as the community structure was at or just after the K/Pg boundary, the changes in species composition (assemblage) were dramatic and marked by a change from a pre–K/Pg agglutinated–dominated fauna (Haplophragmoides–Ammobaculites) to a post–K/Pg calcareous one (Cibicodoides–Cibicides–Anomalinoides).

Associations between spontaneous electroencephalogram oscillations and oxygen saturation across normobaric and hypobaric hypoxia.

High-altitude indoctrination (HAI) trains individuals to recognize symptoms of hypoxia by simulating high-altitude conditions using normobaric (NH) or hypobaric (HH) hypoxia. Previous studies suggest that despite equivalent inspired oxygen levels, physiological differences could exist between these conditions. In particular, differences in neurophysiological responses to these conditions are not clear. Our study aimed to investigate correlations between oxygen saturation (SpO2 ) and neural responses in NH and HH. We recorded 5-min of resting-state eyes-open electroencephalogram (EEG) and SpO2 during control, NH, and HH conditions from 13 participants. We applied a multivariate framework to characterize correlations between SpO2 and EEG measures (spectral power and multiscale entropy [MSE]), within each participant and at the group level. Participants were desaturating during the first 150 s of NH versus steadily desaturated in HH. We considered the entire time interval, first and second half intervals, separately. All the conditions were characterized by statistically significant participant-specific patterns of EEG-SpO2 correlations. However, at the group level, the desaturation period expressed a robust pattern of these correlations across frequencies and brain locations. Specifically, the first 150 s of NH during desaturation differed significantly from the other conditions with negative absolute alpha power-SpO2 correlations and positive MSE-SpO2 correlations. Once steadily desaturated, NH and HH had no significant differences in EEG-SpO2 correlations. Our findings indicate that the desaturating phase of hypoxia is a critical period in HAI courses, which would require developing strategies for mitigating the hypoxic stimulus in a real-world situation.

A Practical Approach for Determining Multi-Dimensional Spatial Rainfall Scaling Relations Using High-Resolution Time–Height Doppler Data from a Single Mobile Vertical Pointing Radar

The rescaling of rainfall requires measurements of rainfall rates over many dimensions. This paper develops one approach using 10 m vertical spatial observations of the Doppler spectra of falling rain every 10 s over intervals varying from 15 up to 41 min in two different locations and in two different years using two different micro-rain radars (MRR). The transformation of the temporal domain into spatial observations uses the Taylor “frozen” turbulence hypothesis to estimate an average advection speed over an entire observation interval. Thus, when no other advection estimates are possible, this paper offers a new approach for estimating the appropriate frozen turbulence advection speed by minimizing power spectral differences between the ensemble of purely spatial radial power spectra observed at all times in the vertical and those using the ensemble of temporal spectra at all heights to yield statistically reliable scaling relations. Thus, it is likely that MRR and other vertically pointing Doppler radars may often help to obviate the need for expensive and immobile large networks of instruments in order to determine such scaling relations but not the need of those radars for surveillance.

Blow up in a periodic semilinear heat equation

Blow up in a one-dimensional semilinear heat equation is studied using a combination of numerical and analytical tools. The focus is on problems periodic in the space variable and starting out from a nearly flat, positive initial condition. Novel results include asymptotic approximations of the solution on different timescales that are, in combination, valid over the entire space and time interval right up to and including the blow-up time. Both the asymptotic analysis and the numerical methods benefit from a well-known reciprocal substitution that transforms the problem into one that does not blow up but remains bounded. This allows for highly accurate computations of blow-up times and the solution profile at the critical time, which are then used to confirm the asymptotics. The approach also makes it possible to continue a solution numerically beyond the singularity. The specific post-blow-up dynamics are believed to be presented here for the first time.