Miracle-dz/UniFAR-FTUs huggingface dataset:
Partial Differential Equations

In the following exercises we assume U is open, bounded set, with smooth boundary, and T > 0. Exercise 16 still has some gap to be overcame. The difficult exercise 9 is solved by mimicking a proof in a paper of Brezis-Evans on 2016/07/31. 1. Prove there is at most one smooth solution of this initial/boundary-value problem for the heat equation with Neumann boundary conditions  ut −∆u = f in U × (0, T ) ∂u ∂ν = 0 on ∂U × [0, T ] u = g on U × {t = 0}. (1) Proof. Consider E(t) = ∫ U (v(x, t)) dx, where v solves the heat equation with all corresponding boundary conditions are zero. Then we see E ′(t) ≤ 0 and E(0) = 0 ≤ E(t), ∀t. So E(t) ≡ 0 and hence v ≡ 0. 1 2 Prove the following weak maximum principle for (1) with f = 0: If g ≥ 0 in U , then u ≥ 0 in UT . Proof. The proof strategy is the same as the Dirichlet problem except the contradiction argument for ∂U needs the Hopf boundary lemma. ∗Department of Math., National Taiwan University. Email: d04221001@ntu.edu. tw 1 Note that we may assume g ≥ c > 0 by considering u+ c and letting c→ 0 in the final step. Consider u ,δ(x, t) = u(x, t)+ t+δ[|x|+h(x)], where , δ > 0 and h(x) is the harmonic function which equals to −|x| on ∂U (whose existence can be proved by Perron’s method or using the Riesz representation theorem with elliptic regularity theory as Chapter 6). Since |x| + h(x) is bounded (due to its continuity on U), u ,δ(x, 0) ≥ c − δ‖|x| + h(x)‖∞ ≥ 0 for all small δ > 0 (independent of ). Suppose inf(x,t)∈UT u ,δ(x, t) < 0. By continuity of u ,δ, we know the infimum is attained at (x0, t0). It’s obvious that t0 > 0. If (x0, t0) ∈ U × (0, T ], then 0 ≥ (∂t −∆)u ,δ(x0, t0) = − 2nδ > 0 for all small δ > 0. So (x0, t0) ∈ ∂U × (0, T ]. However, the minimality at (x0, t0) implies ∂u ,δ ∂ν (x0, t0) ≤ 0. However, the Hopf boundary lemma implies ∂ ∂ν (|x| 2 + h(x))(x0) > 0 and hence ∂u ,δ ∂ν (x0, t0) > 0. Therefore inf(x,t)∈UT u ,δ(x, t) ≥ 0 and hence inf(x,t)∈UT u(x, t) ≥ 0 by letting , δ → 0. 2. Proof. Since d dt 1 2 ‖u(·, t)‖2L2 = − ∫ U u(−∆u) ≤ −λ1‖u‖2L2 , basic Gronwall’s inequality shows the desired inequality. 3. Proof. 0 = ∫ UT (ut + Lu)v d(x, t) = ∫ U ∫ T 0 utv dt dx+ ∫ U ∫ T 0 (Lu)v dt dx = ∫ U u(x, T )v(x, T )− u(x, 0)v(x, 0) dx− ∫ UT u(vt − L∗u) d(x, t) 4. Proof. Let {wk} be the family of all eigenfunctions of −∆ with zero Dirichlet boundary conditions. It’s an orthogonal basis of H 0 (U) (See Evans, step 3 in page 357. By Poincaré’s inequality, β‖um‖2H1 ≤ ‖Dum‖2L2 = m ∑ k=1 dm ∫
Risk Level and Variation in Social Support Access Among Justice-Involved Youth

Social support is important for the average incarcerated person, although variation exists. The amount of support received and whether improvements in support are made over confinement can vary across numerous factors including sentence length and quality of family relationships. Preliminary evidence suggests that risk level might also be an important factor to consider, though no study has examined this possibility. Accordingly, the current study examines whether access to social support (family contact, willingness to support, treatment participation, and non-family support) differs based on risk level classification (low, moderate, moderate-high, high). Additionally, we assess how risk level is associated with changes in social support during confinement. Using a sample of incarcerated youth, results show that access to social support, and to a lesser extent changes in social support during confinement, differ across risk level. The findings have important implications for juvenile justice system responses and efforts to promote support.
A Comparison of Kinetic and Hydrodynamic Models of an Expanding Ion-Exosphere THOMAS E. I--IoLZER 1 Department o) Applied Physics and In)or,nation Science Institute )or Pure and Applied Physical Sciences

The results of calculations based on kinetic and hydrodynamic models of a collisionless ion-exosphere are compared, and it is found that the density and velocity of an ionic constituent flowing supersonically throughout the exosphere can be adequately described by ordinary hydrodynamic equations with isotropic pressure. This suggests that the basic flow characteristics of the polar wind can be determined by hydrodynamic equations throughout the polar ionosphere. We derive a set of CGL-type hydrodynamic equations (for anisotropic pressure) that include the effects of energy and momentum transfer processes (e.g., collisions), and we show that such equations are applicable to problems of polar wind and solar wind temperature anisotropies, as well as to related flow problems.
Hygroscopicity and CCN Activity of Water‐Soluble Extracts From the Arctic Aerosols in Winter to Early Summer

The hygroscopicity and cloud condensation nuclei (CCN) activity of water‐soluble extracts from Arctic lower tropospheric aerosols during winter to summer were investigated under sub‐ and super‐saturated conditions, with chemical composition and air mass origin. The κ values from the hygroscopic growth factor at 80%–90% relative humidity (κHTDMA) during winter, spring, and summer were 0.34 ± 0.06, 0.41 ± 0.05, and 0.25 ± 0.02, respectively, whereas those derived from CCN activation diameter at 0.29%–0.59% supersaturation (κCCNC) were 0.42 ± 0.03, 0.43 ± 0.05, and 0.34 ± 0.11, respectively. The hygroscopicity and CCN activity showed clear seasonal variations following changes in composition that are linked to natural, anthropogenic, and biogenic sources. During winter and spring, κ was high when highly hygroscopic components such as sea salts, sulfate, and highly‐oxidized/aged particles were dominant due to long‐range atmospheric transport and photochemical reactions. In contrast, κ was significantly lower in summer when water‐insoluble (22%) and water‐soluble organic matter (OM) (17%) were dominated with high biogenic activity associated with ice‐edge zones. The κCCNC and κHTDMA values agreed well within 8% and the surface tension agreed with that of pure water within 10%. The κ for OM (κOM) estimated from chemical composition during spring and summer was on average 0.04 ± 0.06 (up to ∼0.17). This result suggests that highly hygroscopic components such as sea salt and sulfate mainly controlled particle hygroscopicity and CCN activity in winter, but water‐soluble OM could also contribute in spring and summer.
Catching up with the European Union’s recovery and resilience agenda: green transition reforms in the Italian National Recovery and Resilience Plan

ABSTRACT Drawing on Europeanization and policy change scholarships, and with the aim of understanding the extent to which Next Generation EU has become a vector of policy change in the country, this article analyses policy reforms concerning the ecological transition, which Italy has designed within the framework of its National Recovery and Resilience Plan (NRRP). More specifically, the study provides a descriptive analysis of the two versions of the plan (Conte’s proposal and Draghi’s final version) in order to unveil the nature and direction of policy change embedded in particular in its final version, as compared to previous Italian ecological policies. The empirical research presented in the article shows that the logics of path-dependence and policy layering have prevailed in the design of the green pillar of the NRRP, as it introduced a limited number of policy reforms; lacked specific implementation targets, and was developed in conformity with a highly fragmented and top-down style of policy making: one that was typical of previous Italian environmental and climate policies.
Cardiac Sodium/Hydrogen Exchanger (NHE11) as a Novel Potential Target for SGLT2i in Heart Failure: A Preliminary Study

Despite the reduction of cardiovascular events, including the risk of death, associated with sodium/glucose cotransporter 2 inhibitors (SGLT2i), their basic action remains unclear. Sodium/hydrogen exchanger (NHE) has been proposed as the mechanism of action, but there are controversies related to its function and expression in heart failure (HF). We hypothesized that sodium transported-related molecules could be altered in HF and modulated through SGLT2i. Transcriptome alterations in genes involved in sodium transport in HF were investigated in human heart samples by RNA-sequencing. NHE11 and NHE1 protein levels were determined by ELISA; the effect of empagliflozin on NHE11 and NHE1 mRNA levels in rats’ left ventricular tissues was studied through RT-qPCR. We highlighted the overexpression of SLC9C2 and SCL9A1 sodium transport genes and the increase of the proteins that encode them (NHE11 and NHE1). NHE11 levels were correlated with left ventricular diameters, so we studied the effect of SGLT2i on its expression, observing that NHE11 mRNA levels were reduced in treated rats. We showed alterations in several sodium transports and reinforced the importance of these channels in HF progression. We described upregulation in NHE11 and NHE1, but only NHE11 correlated with human cardiac dysfunction, and its levels were reduced after treatment with empagliflozin. These results propose NHE11 as a potential target of SGLT2i in cardiac tissue.
THEORETICAL STUDIES OF THE HEAT TRANSFER PROCESS AND THE HYDRODYNAMICS OF A CONDENSATE FILM BY NUMERICAL EXPERIMENT

The paper presents an analysis of the mathematical description of the process of film condensation of vapors from steam-gas mixtures, taking into account the dependence of the processes of heat transfer and the hydrodynamics of the condensate film. It is established that the presence of an insignificant fraction of non-condensing gases sharply reduces the surface temperature of the condensate film. The numerical experiment has confirmed that the loss of stability of the film and its deceleration are associated with the temperature gradient on its surface. It is established that the influence of non-isothermicity on the characteristics of transfer processes can significantly change their flow mode even with small fluctuations in functional parameters. The influence of temperature on the condensate density leads to a certain increase in the average and local Nusselt numbers. As a result of the conducted studies, it was found that the dependence of the condensate density on temperature contributes to an increase in the average and local Nusselt number. This is confirmed by its fundamentally small effect on heat exchange under condensation conditions for liquids in the following range  ~ 10 10   4 2  .
The physics of interacting electrons in disordered systems

Introduction Electronic states of an isolated impurity Historical development of localization theories in impurity bands Electron-electron interactions in impurity bands Cluster calculations for interacting electron systems Renormalization approach for disordered electron systems Hopping conduction and related phenomena Two-dimensional disordered systems Two-dimensional systems in magnetic fields Interacting electrons in the quantum Hall regime - The fractional quantum Hall effect Index
Intention–behaviour relationship within community running clubs: examining the moderating influence of leisure constraints and facilitators within the environment

ABSTRACT While intention is considered the proximal determinant of behaviour, scholars point to an intention–behaviour gap, calling for research that accounts for the environmental moderators that constrain or facilitate the intention–behaviour relationship. Adopting an ecological perspective, the study examined the potential moderating influence of five leisure constraints and facilitators on the running intention–behaviour relationship within the context of community running clubs. Employing a repeated measures design with members of four community running clubs across the Midwest United States, results of the multi-level modelling indicate running intentions and time significantly predict actual running behaviours. The findings provide no evidence that the leisure constraints and facilitators (leisure constraints, leader autonomy support, perceived motivational climate, club operations and club programmes) influenced the relationship between running intention and running behaviour. These results add further evidence to support the intention–behaviour relationship and suggest this relationship may fluctuate across contexts.
Elastico-Viscous Flow between Two Rotating Discs of Different Transpiration for High Reynolds Numbers (RESEARCH NOTE)

The flow in an elastico-viscous fluid between two co-axial infinite rotating porous discs is considered for high cross flow Reynolds number. The discs are rotating with different angular velocity and the injection rate of the fluid at one disc is different from the suction rate of other disc. The effect of suction parameters on the velocity components have been investigated numerically and solved by iterative methods/finite difference methods and depicted graphically. This study has immense practical utility especially when the fluid is non-Newtonian. The results are applicable in the chemical industry using fluids of higher Reynolds numbers.
A Low-density Wind-tunnel Study of Shock-wave Structure and Relaxation Phenomena in Gases

Note presenting an investigation of the profiles and thicknesses of normal shock waves of moderate strength in terms of the variation of the equilibrium temperature of an insulated transverse cylinder in free-molecule flow. The shock waves were produced in a steady state in the jet of a low-density wind tunnel, at initial Mach numbers of 1.72 and 1.82 in helium and 1.78, 1.85, 1.90, 1.98, 3.70, and 3.91 in air. Results regarding the free-molecule probe response in a uniform stream, operation of normal shock holders, normal-shock-wave profiles, shock-wave thicknesses, evidence of wire-size effect on shock-wave profile, and the potential of experimental error are provided.
Homotopy classification of nondegenerate quasiperiodic curves on the 2-sphere

We classify the curves on S with fixed monodromy operator and nowhere vanishing geodesic curvature. The number of connected components of the space of such curves turns out to be 2 or 3 depending on the corresponding monodromy. This allows us to classify completely symplectic leaves of the Zamolodchikov algebra, the next case after the Virasoro algebra in the natural hierarchy of the Poisson structures on the spaces of linear differential equations.
Fat Infiltration of Multifidus Muscle Is Correlated with Neck Disability in Patients with Non-Specific Chronic Neck Pain

Abstract Background: Chronic non-specific neck pain (CINP) is common, but the etiology remains unclear. This study aimed to examine the relationship between cervical muscle composition (cervical multifidus and longus capitis/longus colli), morphometry, range of movement, muscle function, and disability severity (Neck Disability Index) in patients with CINP. Methods: From September 2020 to July 2021, subjects underwent cervical MRI and clinical tests (cervical range of motion, cranio-cervical flexion test, neck flexor, and extensor muscle endurance). MRI analysis comprised muscle cross-sectional area, volume, and fat infiltration of multifidus and longus colli between C4 and C7 levels. Results: Twenty-five participants were included. Multiple linear regression analysis indicated that NDI was positively correlated with the volume percentage of fat infiltration of the multifidus (B = 0.496), negatively correlated with fat-free muscle volume of the multifidus normalized by subject height (B = −0.230), and accounted for 32% of the variance. There was no relationship between neck disability and longus capitis/longus colli morphology. We also found no relationship between neck disability scores, neck flexor or extensor muscle endurance, or the outcome motor control test of craniocervical flexion (p > 0.05). Conclusions: Neck disability was moderately correlated with the percentage of fat volume in the multifidus muscle and fat-free volume of the multifidus. There was no relationship between NDI scores and muscle function test outcomes or any fat or volume measures pertaining to the longus colli muscle.
Does Bullying Occur behind Closed Doors? Agreement of Bullying Reports between Parents and Children and Its Differential Associations with Child Outcomes

The present study was aimed at examining the level of agreement between parent and child perceptions of sibling and peer bullying (victimization and perpetration), and investigating whether any differential associations with child emotional and conduct problems could be identified across raters. The actor-partner interdependence model (APIM) was utilized in order to statistically account for the non-independence of the parent-child dyad. The study was based on a sample of 142 parent-child dyads (children: Mage = 12.3 years; parents: Mage = 44.4 years) and employed an online survey design. Bullying experiences and child outcomes were assessed via parent- and self-report. Intraclass correlation analysis revealed a moderate level of agreement between parent- and child-reports of sibling and peer bullying victimization. Low to moderate levels of parent-child agreement emerged for sibling perpetration and low agreement for peer perpetration. Moreover, APIMs found that parent- and child-reports of bullying were differentially associated with child adjustment. The results of this study suggest that child- and parent data each add additional and unique information into the big picture. Thus, our findings argue for the utility of integrating parent and child perspectives simultaneously in research and clinical practice, in order to uncover the complex reality of child functioning in the context of the family system.
Effect of Irrigation and Nitrogen Topdressing at Different Leaf Ages on the Length and Growth of Wheat Leaves, Leaf Sheaths, and Internodes

The lengths of leaves, leaf sheaths, and internodes are the main factors affecting individual plant types. An ideotype is a basis for developing a high-yielding population structure. Water and nitrogen (N) fertilizer can directly affect the growth of a plant’s organs. To evaluate the effects of irrigation and nitrogen application on the length and growth of wheat leaves, leaf sheaths, and internodes, we carried out a 5 year field experiment in the high yield wheat fields of the North China Plain. Five treatments (T2–T6) were applied, and irrigation was carried out in springtime at the appearance of the second leaf (T2), the third leaf (T3), the fourth leaf (T4), the fifth leaf (T5) and the sixth leaf (T6). The results showed that the irrigation and N topdressing periods had different effects on the leaves, leaf sheaths, and internodes. The lengths of the upper three leaves gradually increased with the progression of the irrigation and N topdressing. The increases in the lengths of the leaf sheath were similar and followed the irrigation and N topdressing pattern at four stages of leaves in the spring: n-1, n-2, n-3 and n-4. The most effective growth of the internodes was achieved by irrigation and N topdressing at the n + 2 and n + 3 stages. The vertical spacing among the upper three leaves increased with irrigation and N topdressing at the appearance of the top second (or flag) leaf. Differences in temperature and precipitation over the years either weakened or enhanced the differences in the plants’ organ lengths with the different treatments. However, the orders of treatments did not alter organ length in different years. Earlier irrigation and N topdressing treatments (T2, T3, and T4) showed an inhibitory effect on the leaves and leaf sheaths during the early growth stage. The inhibitory effect was more evident in the later-emerged leaves and leaf sheaths than in those that emerged earlier. However, irrigation and N fertilization increased the final length of the organs by improving the growth rate during the rapid incremental phase (RIP) and the slow incremental phase (SIP). Although the most significant extensions of the lengths of leaves, leaf sheaths, and internodes were achieved by irrigation and N topdressing before the organs entered the RIP, the specific growth stages were different among the three organs. These results can provide a reference for directly regulating the development of wheat organs and constructing an ideotype.
The Transcranial Light Therapy Improves Synaptic Plasticity in the Alzheimer’s Disease Mouse Model

Alzheimer’s disease (AD) is the main cause of dementia worldwide. Emerging non-invasive treatments such as photobiomodulation target the mitochondria to minimize brain damage, improving cognitive functions. In this work, an experimental design was carried out to evaluate the effect of transcranial light therapy (TLTC) on synaptic plasticity (SP) and cognitive functions in an AD animal model. Twenty-three mice were separated into two general groups: an APP/PS1 (ALZ) transgenic group and a wild-type (WT) group. Each group was randomly subdivided into two subgroups: mice with and without TLTC, depending on whether they would undergo treatment with TLTC. Cognitive function, measured through an object recognition task, showed non-significant improvement after TLTC. SP, on the other hand, was evaluated using four electrophysiological parameters from the Schaffer-CA1 collateral hippocampal synapses: excitatory field potentials (fEPSP), paired pulse facilitation (PPF), long-term depression (LTD), and long-term potentiation (LTP). An improvement was observed in subjects treated with TLTC, showing higher levels of LTP than those transgenic mice that were not exposed to the treatment. Therefore, the results obtained in this work showed that TLTC could be an efficient non-invasive treatment for AD-associated SP deficits.
Atom-canonicity, relativized representations and omitting types for clique guarded semantics and guarded logics

We study atom canonicity for several varieties of cylindric like algebras that contain properly the variety of representable algebras. The algebras in such varieties have relativized representations, and we thereby obtain many omitting types theorems, both negative and positive for finite variable fragments and / or modifications of first order logic. Negative results are obtained when we keep usual syntax and relativize models (so that they witness commutativity of quantifiers only locally) and positive ones are obtained when we weaken 'commutativity of quantifiers' in the syntax and relativize semantics differently. Such algebras have weak neat embedding properties, too, in the sense that they embed into neat reducts of algebras in higher dimensions possibly finite. In the second part of the paper various notions of representability originally formulated for atom structure are lifted in an obvious way to the algebra level, like weak and strong representability. Such classes of algebras (that are atomic) are characterized completely via neat embeddings. Finally, several model theoretic questions on such classes consisting of algebras having weak neat embedding properties and relativized representations, like decidability of their equational or universal theory, their finite axiomatizability if first order definable, are posed and answered.