Abstract: Long COVID is the consequence of having had COVID. Long COVID has many other names including Long-haul COVID, Post-COVID conditions (PCC), Post-COVID-19 syndrome, Post-acute sequelae of SARS-CoV-2 condition (PASC) and Chronic COVID. Long COVID is the name most frequently used. COVID is not alone in having severe post infection consequences. Influenza, Ebola, Marburg, Dengue, and Lyme Disease are other infections with severe post infection consequences. Long COVID has emerged over the past few years and is ill-defined. Long COVID’s underlying science and treatments are rapidly evolving. There is no diagnostic test for it. The most-often reported lower bound on its prevalence is about 7%. Seven percent doesn’t sound like much, but under the assumption that 75% of the people in the world have had COVID, that means 420 million people in the world have Long COVID which is about 5 times the number of people killed or injured in the 20th and 21st century wars. There are several root causes for Long COVID with inflammation and mitochondrial dysfunction being the two leading villains. Long COVID prevalence goes down with recent variants, COVID vaccination, early antiviral use, being fit, being young, and surprisingly being male. The most important action to reduce the chance of Long COVID is COVID vaccination. The impact of COVID vaccination on Long COVID prevalence is quite uncertain. Papers report 10% to 100% reduction in Long COVID rates from pre-disease vaccination. The average reported reduction is 50%. The impact of vaccination on people with no comorbidities is uncertain with wide ranges being reported. There are no guaranteed treatments for Long COVID; however, some treatments offer either broad or organ-specific relief for many. This paper reviews 179 different Long COVID treatments described in 249 papers. These papers came from the author’s personal data base called The Mouse That Roared of 24,000+ papers that have been accumulated over the last five and a half years. The Mouse That Roared papers cover all aspects COVID including the SARS-CoV-2 virus, the COVID disease, therapeutics, vaccines, behavior, testing, herd immunity, Long COVID, Long COVID Treatment, Politics and National COVID responses, etc. Unlike COVID, there are no excellent treatments, which I call silver bullets, for Long COVID Fortunately, there are some treatments that help some a bit. I will call those “bronze bb’s.” Even with them, healing is very slow. The recovery time with Long COVID is longer than the body’s normal times because COVID’s damage is widespread and because COVID damages our body’s healing process.

Abstract: In the present research work, the corrosion behaviour of pure Al in methyl esters with different degree of unsaturation and chain length, present in biodiesel, has been investigated by using electrochemical techniques. Evaluated methyl esters included methyl acrylate (C4H6O2) and methyl linoleate (C19H34O2) which were added to methyl propionate, (C4H8O2) and methyl oleate (C19H36O2 ) respectively. Electrochemical techniques involved electrochemical impedance spectroscopy and electrochemical noise, and were supplemented by detailed studies of scanning electronic microscopy. Results have shown that the corrosion rate and the susceptibility to localized type of corrosion such as pitting increased with an increase in the number of unsaturations and in the chain length. Corrosion process was under charge transfer and was not affected neither by an increase in the number of unsaturations nor in the chain length. The charge transfer resistance value decreased by an increase in the number of unsaturations nor in the chain length.

Abstract: Central cord syndrome (CCS) is the most common incomplete spinal cord injury, producing more severe motor deficits in the upper than lower extremities and impairing sensory and autonomic function. Although transcutaneous spinal cord stimulation (tSCS) has shown benefits in motor and sensory recovery after spinal cord injury, studies have not explicitly documented whether CCS subjects were included. The aim of this study was to assess the effects of tSCS over 12 weeks on motor and sensory outcomes in a subject with CCS. Methods: A 20-year-old male with a C7 injury was evaluated at baseline and after 12 weeks with the American Spinal Cord Injury Impairment scale, Modified Ashworth Scale, Penn and Spasm Frequency Scale, 3-Meter Walk Test, 6-Minute Walk Test, 9-Hole Peg Test, Box and Block Test, hand dynamometry, and lower-limb EMG. tSCS was applied between T9 and L1 at 30 Hz. Results: At 12 weeks, upper-limb motor and sensory scores improved, while spasm frequency and hand spasticity were reduced. Manual dexterity improved bilaterally in the 9-Hole Peg and Box and Block Tests, with a 2 kg gain in right-hand grip strength. In the 6-Minute Walk Test, the distance covered increased from 224.4 m to 295.2 m, and a 1.36 s reduction in 3-Meter walking time was achieved. Conclusions: tSCS improved motor and sensory function and reduced spasticity and spasms. These findings suggest that tSCS may serve as an effective complementary intervention for motor and sensory rehabilitation in individuals with mild cervical injuries, including CCS.

Abstract: Although Gibbons et al. (2025) found that fear and sleep deprivation combined to predict unique variance in 5 out of 6 everyday and heroic courage measures, those measures either measured willingness to engage in courageous behaviors or fear when engaging in these behaviors, as suggested by the revised 20-item Woodard Pury Courage Scale (R-WPCS-20). However, most courage definitions involve risky behaviors considered courageous that elicit fear, which means that fear, when willing to engage in risky/courageous behaviors, indicates courage. Conversely, fear when unwilling to engage in risky/courageous behaviors indicates cowardice. The current study used the data from 256 participants in the Gibbons et al. study using the R-WPCS-20 to combine the ratings for willingness to engage in 20 risky behaviors and fear to engage in each of those behaviors into 20 courage ratings. Specifically, 3 was subtracted from each of the willingness ratings, ranging from 1 (low) to 5 (high), and multiplied by the fear rating, ranging from 1 (no fear) to 5 (high fear), resulting in 20 combined courage measures ranging from -10 (high cowardness) to 10 (high courage). The 20 combined ratings loaded well onto a single factor of courage with high factor loadings (the lowest factor loading was .764) and a Cronbach’s alpha of .99, demonstrating excellent interitem reliability. The results showed that fear/anxiety interacted with 2 measures of sleep (hours and poor sleep quality) to predict the single measure of courage that combined willingness to engage in courageous behaviors and fear when engaging in them, which clearly emphasized the contribution of fear in defining and predicting courage.

Abstract: Background: This research was conducted to compare the relationship between a comprehensive set of optical coherence tomography (OCT) parameters of the fovea and two measures of age-biological and chronological-in healthy adults. Methods: This cross-sectional study analyzed swept-source optical coherence tomography (OCT) data from 154 healthy adults aged 22–89 years. Parameters assessed: foveal thickness, foveal pit depth and diameter, pit slope steepness, and the presence or absence of the foveal bulge. Biological age was calculated using the PhenoAge algorithm. Results: The core geometry of the foveal pit showed no significant dependence on either type of age (p > 0.1). In contrast, the prevalence of the foveal bulge declined significantly with age: from ~80% in the youngest group (17-44 years) to 0-50% in the oldest group (75-92 years) (p = 0.0014 for PhenoAge, p = 0.0208 for chronological age). The odds ratio (OR) for the loss of the bulge per year of age was 0.96 (95% CI: 0.94–0.98).Conclusion: The foveal architecture remains structurally stable throughout adulthood. The foveal bulge emerges as a sensitive biomarker of age-related changes. Biological age does not provide additional predictive value over chronological age for foveal structural parameters under physiological aging conditions.

Abstract: Population aging is the most significant demographic transformation of the 21st century, reshaping health systems, economies, and societies. The biological processes of immunosenescence and inflammaging weaken host defenses, reduce vaccine effectiveness, and increase vulnerability to infectious and chronic diseases. These changes underscore the urgent need for preventive strategies that extend beyond childhood immunization. Vaccination is a cornerstone of healthy aging, capable of preventing infections and has been associated with reductions in systemic inflammation, frailty, and loss of functional independence in later life. Furthermore, new insights into vaccine-mediated immunomodulation, including trained immunity, adjuvanted formulations, and epigenetic reprogramming, highlight the evolving role of vaccines as modulators of immune fitness across the lifespan. This first part of our review examines the intersection of aging and immunity, as well as the potential of vaccines to address these challenges. Part 2 will expand on specific vaccines, proposed vaccination schedules, and global perspectives for lifelong immunization.

Abstract: Purpose: This manuscript synthesizes and integrates a series of previously published innovative sonographic concepts into a unified framework of Functional Lung Ultrasound (FLUS), introducing lung ultrasound as a real-time bedside functional “visual spirometry” tool. Rather than focusing on artefact-based interpretation, it reframes pulmonary sonography as a structural–functional imaging modality. The primary aim is to systematize the diagnostic meaning of novel sonographic entities—namely the Twinkling White Area (TWA), Bat Sign dynamics, vascular echo patterns, and pleural separation—with particular emphasis on lung ultrasound’s role in the bedside functional diagnosis, phenotyping, and monitoring of obstructive lung syndromes, within a coherent physiological and pathophysiological model of peripheral lung function. Methods: This work is based on a structured reinterpretation of serial ultrasound examinations previously performed in healthy subjects and in over 600 patients with obstructive, interstitial, infectious, malignant, and vascular lung diseases, as reported across separate original publications. Sonographic features of the Merlin Space—including pleural line behavior, rib-shadow geometry, TWA morphology, and respiratory excursion—are collectively reanalyzed and integrated with their corresponding CT correlations, morphologic substrates, and hemodynamic implications in order to construct a reproducible structure–function continuum. Results: The TWA is consolidated as a genuine subpleural reflective interface dynamically linked to peripheral lung density, representing the baseline architectural unit of the aerated lung. The Bat Sign is redefined as a functional continuum rather than a static landmark: the Healthy Bat preserves symmetric TWA geometry, rib-shadow distance, and respiratory motion, whereas the Sick Bat exhibits deformation, asymmetry, and altered dynamics reflecting early parenchymal or obstructive involvement. The Front Sight in Rear Sight sign is confirmed as a dual-vascular echo configuration corresponding to pulmonary infarction supported by bronchial collateral perfusion. Posterior basal pleural separation emerges as a functional marker of chronic, non-exudative, viscerally confined pleural–subpleural remodeling rather than acute inflammatory pleural disease. Conclusion: This integrated framework formally establishes Functional Lung Ultrasound (FLUS) as a real-time, bedside “visual spirometry” modality, in which echo morphology, motion analysis, and perfusion patterns translate directly into functional assessment of airflow, density distribution, and peripheral vascular dynamics. By unifying structural and physiological echo phenomena, FLUS positions lung ultrasound as a true functional imaging tool for obstructive, interstitial, infectious, malignant, and vascular lung syndromes.

Abstract: This study investigated the impact of pricing strategies on the growth and sustainabil-ity of small and medium enterprises (SMEs) in the City of Mbombela Local Municipal-ity, South Africa. The study employed a quantitative research approach. The study sampled 132 SMEs operating within the City of Mbombela Local Municipality. A self-administered questionnaire was tested for reliability and validity and thereafter used to collect data from the respondents. This study employed multiple linear regres-sion analysis and performed the reliability test. In this study, the data were analysed using descriptive and inferential statistics. The results showed that SMEs primarily used cost-plus, value-based, and competitor-based pricing strategies, frequently modi-fying prices in response to market competition and technological advancements. The study found a significant and positive relationship between pricing strategies and growth. Furthermore, a positive and significant nexus between pricing strategies and the sustainability of the SMEs. The practical implication of this study informs the SME managers/owners and policy makers that SMEs that apply strategic and mar-ket-oriented pricing practices are more likely to achieve improved performance out-comes. The study therefore emphasises the importance of effective pricing in promot-ing both growth and long-term sustainability among SMEs. The findings of this study are expected to persuade the SMEs to pay crucial attention to the pricing strategies implemented in the business.

Abstract: Examination malpractice remains a significant challenge in higher education, undermining academic integrity and the credibility of qualifications. This study aimed to explore students’ perceptions of invigilation as a strategy for preventing academic dishonesty, assess the adequacy of current invigilation practices, and examine how different types of invigilation influence cheating behaviors. An exploratory quantitative research design was employed, collecting data from 295 Zambian university students using a structured electronic questionnaire. Descriptive statistics, including means and standard deviations, were used to summarize participants’ responses. The results revealed that a majority of students perceive invigilation as effective in reducing cheating (M = 3.90, SD = 1.13) and are generally satisfied with the adequacy of current invigilation practices (M = 3.64, SD = 1.14). Strict invigilation was identified as the most effective approach to deterring malpractice (58.4%), while students reported moderate variability in adherence to proper examination procedures (M = 3.59, SD = 1.24). However, perceptions of the effectiveness of specific types of invigilation were lower (M = 2.54, SD = 0.58), suggesting that while supervision is valued, its implementation and style can influence its deterrent effect. The study concludes that vigilant, well-staffed, and consistently applied invigilation practices are crucial for maintaining examination integrity and minimizing academic dishonesty.

Abstract: This preprint proposes a radical departure from terraforming: the Ze-formation of a planet. We introduce the Ze System as a co-evolutionary state of planetary animation, achieved not by imposing an external template, but by actively provoking a celestial body’s latent potentials into structured, intelligent exchange. Grounded in an ontology of latent fields (Ibrahim, 2022), the framework shifts from passive observation to active provocation via targeted decoherence, resonance amplification, and non-local perturbation (Maruyama, 2019; Watanabe & Li, 2017). The core methodology is the engineering of predictive conflicts, where adversarial models—one standard, one incorporating a hypothesized latent variable—are tested by minimal Ze-Probes. The resulting patterned error localizes hidden structures (Fong et al., 2016). This process is interpreted through the Principle of Dual Reading, synthesizing causal and teleological narratives to guide intervention (Voss, 2021). A dedicated toolkit—predictive AI, resonant manipulators, and quantum-enhanced error detectors—enables this planetary-scale dialogue. Crucially, the framework is governed by an ethics of co-creative responsibility, acknowledging the non-neutrality of intervention and the irreversible cost of localizing potentials (El-Hadi, 2020). We argue that the ultimate outcome of Ze-formation is not a habitable world, but an active planetary interlocutor capable of complex informational exchange and collaborative self-revelation, transforming humanity’s role from terraformer to partner in cosmic meaning-making.

Abstract: Constant-composition expansion (CCE) experiments provide critical relative-volume and density information describing the thermodynamic behavior of reservoir oils and gases under varying pressure. These properties are vital inputs for hydrocarbon reser-voir engineering, as they impact how oil and gas move through the reservoir during production. However, the need for specialized personnel, high-end equipment and measures taken to ensure safety in handling high pressure fluids often render the CCE experiments expensive and slow. This work introduces a Local Interpolation Method (LIM), a proximity-informed, end-to-end CCE fluid properties prediction AI model that leverages domain expertise and existing PVT data archives to generate surrogate CCE behavior for new fluids, thereby eliminating or reducing the need for completing laboratory CCE tests. Each new fluid is embedded in a compositional–thermodynamic descriptor space, and its response is inferred from a small neighborhood of thermody-namically similar fluids. Within this locality, the LIM combines hybrid local interpola-tion for key scalar properties (such as saturation-point quantities and expansion end-points) with shape-preserving reconstruction of monophasic and diphasic rela-tive-volume curves, enforcing continuity at saturation and consistency between rela-tive volume, density and compressibility. The workflow operates purely at inference time and does not require case-specific retraining. Application to a synthetic database of CCE tests shows that LIM reproduces key CCE features with very good agreement to laboratory data across a range of fluid types, indicating that proximity-based AI mod-elling can substantially reduce reliance on new CCE experiments while maintaining engineering-grade fidelity for compositional simulation workflows. The proposed ap-proach has been fully automated through software so it can be set up and directly uti-lized by the field operators on their own databases to significantly reduce their fluid sampling and laboratory analysis costs. The proposed model does not use others’ data while respecting the data privacy and data ownership.

Abstract: The stability and performance of drones are seriously affected by vibrations originating from propellers. These vibrations can lead to structural resonance, decreased sensor accuracy, and reduced flight safety. In this study, drone propeller vibrations are modeled and analyzed using differential equations, Laplace transform, and Fourier analysis. The propeller system is represented by single-degree-of freedom (SDOF) and multi-degree-of-freedom (MDOF) mechanical models, and the transfer function of the system is obtained using the Laplace transform. Experimental vibration data collected from the acceleration sensor(ADXL345) is analyzed using Fast Fourier Transform (FFT) using software written in the Arduino IDE application with the ESP32 development board, to determine dominant frequencies and possible resonance states.

Abstract: Micro and nanoplastics, pervasive environmental pollutants smaller than 5 mm and 1 µm respectively, infiltrate livestock and aquaculture feeds via contaminated water, sewage sludge fertilizers, and atmospheric deposition, compromising animal health, reproductive performance, and food chain safety. This paper presents a pioneering hybrid framework that synergistically integrates artificial intelligence-enhanced Raman spectroscopy for high-resolution polymer fingerprinting, quantum dot nanoprobes for targeted fluorescent labelling of hydrophobic plastics, and federated learning algorithms for decentralized, privacy-preserving model training across heterogeneous farm networks. Unlike traditional methods such as microscopy or pyrolysis-gas chromatography, which suffer from low sensitivity in complex organic matrices and lack real-time scalability, our system achieves a limit of detection of 5 ng/g with 97% accuracy across polyethylene, polypropylene, and polystyrene variants in poultry pellets, cattle silage, and salmon feeds. Quantum dots, functionalized with π-π stacking ligands, enable selective binding and surface-enhanced Raman signals, while edge-deployed AI processes hyperspectral data in under 100 ms per sample. Federated averaging across 50 simulated nodes converges 25% faster than centralized baselines, incorporating differential privacy for regulatory compliance. Experimental results demonstrate 91% removal efficiency through dielectrophoretic extraction of labelled particles, surpassing density separation by 40% in yield and 70% in speed, with pilot deployments yielding 12% improvements in feed conversion ratios and 30% reductions in inflammation biomarkers. This scalable, cost-effective solution ($500/unit, 6-month ROI) paves the way for sustainable animal production resilient to escalating plastic pollution, with broader implications for precision agriculture and global food security.

Abstract: The efficiency of the photomechanical effect in thin films of three azobenzene-based polymers PDR1A, PDR13A, and PMMA-co-PDR1A was determined using a cantilever-based sensor. The polymers were coated onto silicon and mica cantilevers, and the resulting cantilever bending under irradiation with visible light was measured to estimate changes in surface stress, photomechanical energy transduction per unit volume, and overall photomechanical efficiency. The photomechanical response was shown to be robust, repeatable, and quantifiable for all the polymers studied, even when the active polymer layer was much thinner than the cantilever substrate. Among the materials tested, PDR1A generated the largest forces, while PMMA-co-PDR1A exhibited the highest efficiency. For 35-µm-thick mica cantilevers coated with PDR1A, photoisomerization induced rapid and significant cantilever bending in the range of 100s of µm, corresponding to surface stress changes in the range of N/m. These results demonstrate the ability of thin azobenzene polymer films to function as strong, light-driven ‘artificial muscles’ in larger mechanical systems, and highlight the cantilever sensor platform as a powerful tool for the quantitative characterization of photomechanical effects in azo dye-based polymers.

Abstract: The role of vitamin D (VD) in cardiovascular health remains controversial. Observational studies have associated low serum 25(OH)D₃ levels with increased cardiovascular risk, whereas interventional and Mendelian randomization trials have failed to demonstrate causality. This discrepancy may reflect unrecognized interindividual variability in VD responsiveness and the weak correlation between circulating VD metabolites and tissue-specific biological effects. This review introduces the emerging concept of variable VD sensitivity, encompassing a continuum from vitamin D resistance (VDRES) to vitamin D hypersensitivity (VDHY). VDRES results from genetic or acquired alterations that impair VD metabolism, transport, or receptor signaling, leading to an insufficient biological response. Conversely, VDHY involves excessive local VD activation, frequently due to CYP24A1 variants or granulomatous activity. The resulting surplus of active VD suppresses parathyroid hormone–related peptide (PTHrP), promoting vascular smooth muscle cell calcification and accelerating atherosclerosis. VD metabolism also appears to intersect with lipid regulation. Patients carrying CYP24A1 mutations, which impair the catabolism of 1,25(OH)₂D₃, exhibit abnormalities in lipid metabolism, including hypercholesterolemia. Dysregulated VD signaling may therefore disrupt cholesterol homeostasis through feedback mechanisms. These opposing phenotypes may help explain the inconsistent cardiovascular outcomes observed in clinical studies. Integrating evidence from endocrinology, vascular biology, and genetics, this review argues for individualized rather than uniform VD supplementation strategies. A deeper understanding of the molecular determinants of VD responsiveness could enhance cardiovascular risk assessment and therapeutic precision. Until assays become available to predict individual responsiveness, clinicians should exercise caution when prescribing VD, particularly in populations at risk for suboptimal response or toxicity.

Abstract: The foundation of an automatic sign language training (ASLT) system lies in word-level sign language recognition (WSLR), which refers to the translation of captured sign language signals into sign words. However, two key issues need to be addressed in this field: (1) the number of sign words in all public sign language datasets is too small, and the words do not match real-world scenarios, and (2) only single-view sign videos are typically provided, which makes solving the problem of hand occlusion difficult. In this work, we design an efficient algorithm for WSLR which is trained on our recently released NationalCSL-DP dataset. The algorithm first performs frame-level alignment of dual-view sign videos. A two-stage deep neural network is then employed to extract the spatiotemporal features of the signers, including hand motions and body gestures. Furthermore, a front-view guided early fusion (FvGEF) strategy is proposed for effective fusion of features from different views. Extensive experiments were carried out to evaluate the algorithm. The results show that the proposed algorithm significantly outperformed existing dual-view sign language recognition algorithms and that compared with the state-of-the-art algorithm, the recognition accuracy was improved by 10.29%.

Abstract: Vividly colored cholesteric liquid crystal polymer network (CLCN) patterns based on epoxy resin are used in decorative and anti-counterfeiting applications. These films are typically prepared via cationic photopolymerization and post-polymerization to achieve a high cross-linking degree. In this work, the cross-linking degree is controlled by varying the UV irradiation dosage during photopolymerization. Following this, the reflection band of the CLCN film changes after removing non-cross-linked compounds with acetone. Leveraging the low cationic polymerization rate and the chain termination capability of methanol, a structurally colored CLCN film with regionally tailored cross-linking was fabricated. With the treatment of acetone, a colorful pattern was observed. Moreover, upon immersion in methanol, the film swells, revealing a colorful pattern. After the evaporation of methanol, the pattern disappeared. Consequently, this CLCN film holds significant potential for information encryption applications.

Abstract: Orthohalarachne attenuata and O. diminuata mites are parasites of the respiratory system of pinnipeds. During hosts dives, mites are challenged to cope with changing conditions of available oxygen in the nasal cavity. Adults and nymphs live inside the host but larvae are active and responsible for colonizing new hosts. Hence, larvae are also exposed to environmental conditions with variable temperatures, dehydration or changes in salinity. In this study, we studied the effect of hypoxia, humidity and salinity on survival of mite larvae in O. attenuata and O. diminuata. We found that both species are highly tolerant to withstand long times under hypoxia. In turn, both species showed low survival when exposed to direct air. Finally, the hyperosmotic solution was highly harmful for O. attenuata compared to O. diminuata. Our results show that humidity rather than oxygen availability is a constraint for survival and a limitation for dispersal when searching for new hosts. The present study expands our knowledge of ecophysiology and adaptations to changing conditions experienced during the dispersal of these marine parasite species.

Abstract: Climate change is making winters in temperate regions increasingly harsh and unpredictable, posing significant threats to honey bee (Apis mellifera) health. The gut microbiome, a distinct community of core bacterial species, is central to overwintering success by supporting immune function, nutrient assimilation, and pathogen resistance, but is highly sensitive to environmental stressors such as cold temperatures and dietary shifts. Stress-induced perturbations can reshape the composition and relative abundance of the gut microbiome in honey bees, leading to adverse effects on host health, physiological functions, and overwinter survival. Cold temperatures and additional stressors further destabilize the microbiome, compounding these effects. This review is the first to synthesize current knowledge on how extrinsic factors, such as diet, antibiotics, and pathogens, and intrinsic factors, including age and strain, influence the composition and function of the honey bee gut microbiota. Given the increasing severity of winter conditions under climate change, a deeper understanding of microbiome–host–environment interactions is essential for improving honey bee resilience. By integrating evidence on the microbiome’s roles in nutrient utilization, immune modulation, and pathogen defense, this review out-lines a framework to guide future research and management strategies aimed at mitigating overwintering losses and sustaining pollinator health in a changing global climate.

Abstract: 2D elastostatic displacement solutions for the Yoffe Mode I and Rice Mode II crack models are reviewed. These solutions are used to introduce the elastostatic displacement solution for a 2D Mode I/II multi-crack configuration in terms of meromorphic differential forms on a hyperelliptic curve. The complex dimension of the vector space of mermorphic forms is demonstrated to be g+1, where g is the genus of the hyperelliptic curve, using the Riemann-Roch theorem. Limitations of the 2D multi-crack model to modeling 3D fracture networks are identified, and a mathematical description of 3D fracture network dynamics preceding an earthquake based on singular spectrum analysis of crack phase fields is conjectured.