Homogeneous Dimension Research Articles

Results of existence and uniqueness for the Cauchy problem of semilinear heat equations on stratified Lie groups

The aim of this paper is to give existence and uniqueness results for solutions of the Cauchy problem for semilinear heat equations on stratified Lie groups G with the homogeneous dimension N. We consider the nonlinear function behaves like |u|α or |u|α−1u(α>1) and the initial data u0 belongs to the Sobolev spaces Lsp(G) for 1<p<∞ and 0<s<N/p. Since stratified Lie groups G include the Euclidean space Rn as an example, our results are an extension of the existence and uniqueness results obtained by F. Ribaud on Rn to G. It should be noted that our proof is very different from it given by Ribaud on Rn. We adopt the generalized fractional chain rule on G to obtain the estimate for the nonlinear term, which is very different from the paracomposition technique adopted by Ribaud on Rn. By using the generalized fractional chain rule on G, we can avoid the discussion of Fourier analysis on G and make the proof more simple.

Schatten properties of Calderón–Zygmund singular integral commutator on stratified Lie groups

We provide full characterization of the Schatten properties of [Mb,T], the commutator of Calderón–Zygmund singular integral T with symbol b(Mbf(x):=b(x)f(x)) on stratified Lie groups G. We show that, when p is larger than the homogeneous dimension Q of G, the Schatten Lp norm of the commutator is equivalent to the Besov semi-norm BpQ/p of the function b; but when p≤Q, the commutator belongs to Lp if and only if b is a constant. For the endpoint case at the critical index p=Q, we further show that the Schatten LQ,∞ norm of the commutator is equivalent to the Sobolev norm W˙1,Q of b. Our method at the endpoint case differs from existing methods of Fourier transforms or trace formula for Euclidean spaces or Heisenberg groups, respectively, and hence can be applied to various settings beyond.

From hunter-gatherers to food producers: New dental insights into the Nile Valley population history (Late Paleolithic-Neolithic).

This study presents biological affinities between the last hunter-fisher-gatherers and first food-producing societies from the Nile Valley. We investigate odontometric and dental tissue proportion changes between these populations from the Middle Nile Valley and acknowledge the biological processes behind them. Dental remains of 329 individuals from Nubia and Central Sudan that date from the Late Pleistocene to the mid-Holocene are studied. Using 3D imaging techniques, we investigated outer and inner metric aspects of upper central incisors, and first and second upper molars. Late Paleolithic and Mesolithic foragers display homogeneous crown dimensions, dental tissue proportions, and enamel thickness distribution. This contrasts with Neolithic trends for significant differences from earlier samples on inner and outer aspects. Finally, within the Neolithic sample differences are found between Nubian and Central Sudanese sites. Substantial dental variation appears to have occurred around 6000 bce in the Nile Valley, coinciding with the emergence of food-producing societies in the region. Archeological and biological records suggest little differences in dietary habits and dental health during this transition. Furthermore, the substantial variations identified here would have happened in an extremely short time, a few centuries at most. This does not support in situ diet-related adaptation. Rather, we suggest these data are consistent with some level of population discontinuity between the Mesolithic and Neolithic samples considered here. Complex settlement processes could also explain the differences between Nubia and Central Sudan, and with previous results based on nonmetric traits.

Gradient estimates for fundamental solutions of a Schrödinger operator on stratified Lie groups

Let L = − Δ G + Υ \mathcal L=-\Delta _{\mathbb {G}}+\Upsilon be a Schrödinger operator with a nonnegative potential Υ \Upsilon belonging to the reverse Hölder class B Q / 2 B_{Q/2} , where Q Q is the homogeneous dimension of the stratified Lie group G \mathbb {G} . Inspired by Shen’s pioneer work and Li’s work, we study fundamental solutions of the Schrödinger operator L \mathcal L on the stratified Lie group G \mathbb {G} in this paper. By proving an exponential decreasing variant of mean value inequality, we obtain the exponential decreasing upper estimates, the local Hölder estimates and the gradient estimates of the fundamental solutions of the Schrödinger operator L \mathcal L on the stratified Lie group. As two applications, we obtain the De Giorgi-Nash-Moser theory on the improved Hölder estimate for the weak solutions of the Schrödinger equation and a Liouville-type lemma for L \mathcal {L} -harmonic functions on G \mathbb {G} .

Interior a priori estimates for supersolutions of fully nonlinear subelliptic equations under geometric conditions

AbstractIn this note, we prove interior a priori first‐ and second‐order estimates for solutions of fully nonlinear degenerate elliptic inequalities structured over the vector fields of Carnot groups, under the main assumption that is semiconvex along the fields. These estimates for supersolutions are new even for linear subelliptic inequalities in nondivergence form, whereas in the nonlinear setting they do not require neither convexity nor concavity on the second derivatives. We complement the analysis exhibiting an explicit example showing that horizontal regularity of Calderón–Zygmund type for fully nonlinear subelliptic equations posed on the Heisenberg group cannot be in general expected in the range , being the homogeneous dimension of the group.

Pedagogical Complexities in Multicultural History Classrooms in Botswana Senior Secondary Schools

Botswana is a pluralistic society, yet its education system is tailored from a homogenous dimension. This paper interrogates the competence of history teachers in teaching multiculturally diverse classrooms. The study examined the challenges faced by History teachers in multiculturally diverse classrooms and the pedagogical approaches that could be used to combat that state of affairs. The study was pursued using the theoretical framework of Critical Pedagogy which ideologically advances that education within multicultural societies must be democratic and take into account social justice and diversity issues. The study implored the qualitative research methods and was pursued within 12 (N=12) senior secondary schools in 7 (N=7) educational regions in Botswana. A total of 36 (N=36) teachers and 120 (N=120) students participated in the study. The results indicate that the history teachers have no training on how to teach multiculturally diverse classrooms. Such a scenario end up frustrating both teachers and students since a lot of them end up feeling pedagogically isolated in the teaching-learning process. Therefore, there is need to rethink teacher training amongst the many factors which could help ameliorate this state of affairs.

MXene dots as photocatalysts for CO2 hydrogenation

Four MXene dots (MDs) (Ti3C2, Ti2C, V2C and Nb2C) have been prepared by laser ablation of the corresponding MAX phases in aqueous medium and subsequent separation by double centrifugation. These MDs were characterized by atomic force and transmission electron microscopy that show, respectively, thickness between 3 and 7.5 nm corresponding to few-layer nanoparticles and homogeneous lateral dimensions between 2.5 and 5 nm with high crystallinity. IR spectroscopy indicates that Ti3C2 and Ti2C contain OH as surface terminal groups, while XPS indicates the presence of O on the surface for the four samples. MDs present the most intense absorption band in the UV region with onset below 400 nm. These samples exhibit photocatalytic activity for CO2 hydrogenation to CO and methane, following the order Ti3C2>Nb2C>Ti2C>V2C. Isotopic label experiments confirm CO2 as the origin of the photo products. Most of the photoresponse derives from the UV photons, the photocatalytic activity increasing with the temperature in the range from 200 to 300 °C. These results show the possibility of tuning the composition, surface terminal groups, and other structural parameters of MXenes to produce semiconducting materials exhibiting important photocatalytic activity. Considering their versatility, flexible composition, and tunability, MXenes appear as promising 2D materials with intrinsic photoresponse to develop highly efficient photocatalysts.

Lipschitz and Triebel–Lizorkin spaces, commutators in Dunkl setting

We first study the Lipschitz spaces Λdβ associated with the Dunkl metric, β∈(0,1), and prove that it is a proper subspace of the classical Lipschitz spaces Λβ on RN, as the Dunkl metric and the Euclidean metric are non-equivalent. Next, we further show that the Lipschitz spaces Λβ connects to the Triebel–Lizorkin spaces Ḟp,Dα,q associated with the Dunkl Laplacian △D in RN and to the commutators of the Dunkl Riesz transform and the fractional Dunkl Laplacian △D−α/2, 0<α<N (the homogeneous dimension for Dunkl measure), which is represented via the functional calculus of the Dunkl heat semigroup e−t△D. The key steps in this paper are a finer decomposition of the underlying space via Dunkl metric and Euclidean metric to bypass the use of Fourier analysis, and a discrete weak-type Calderón reproducing formula in these new Triebel–Lizorkin spaces Ḟp,Dα,q.

Study on Physical Dimensions of Some Expressions of Petrophysics

This study arose during the survey of a large number of works aiming to prepare a permeability model that presented a better accuracy than those consecrated in the literature for different types of reservoir rocks in the petrophysical area. The methodology used was to appreciate the mathematical expressions replacing physical quantities by their dimensions in the LMT system. After replacing length by L, mass by M and time by T, we compare the first members of the equations with the second to verify the homogeneity of the magnitudes involved in each mathematical formula. When these expressions are tested with experimental data, were identified formulas that did not show equilibrium in dimensions of their magnitudes, where it was observed that the permeability values found did not exhibit units of squared length. It has been used the phase spatial encoding in the Nuclear Magnetic Resonance Tool with experimental data of porosity, irreducible fluid volume, free fluid volume and transverse relaxation time. This distortion led us to conclude that the equations did not present their homogeneous physical dimensions, motivating a deep study in the identification of expressions used in the permeability calculation, in sandstone and carbonate rocks, with no equilibrium in their dimensional equations in any unit system. This review aims to identify the constants used in the equations equilibrium suggesting modifications that can be obtained through pre-established relationships to increase the reliability of the equations previously used.

Leader‐following consensus of a class of heterogeneous uncertain multi‐agent systems with a distributed model reference adaptive control law

SummaryThis paper considers the leader‐following consensus problem of a multi‐agent system whose agents have heterogeneous uncertain dynamics in homogeneous state dimensions. Each agent utilizes a distributed model reference adaptive control (MRAC) law to deal with uncertainties. The nominal part of the MRAC is a distributed static state feedback control law. The reference model is the leader‐following consensus problem of reference agents with homogeneous linear time‐invariant dynamics. This problem becomes an N‐player graphical differential game under a given cost function. The reference agents utilize distributed static state feedback controllers that constitute a Nash equilibrium solution to the graphical differential game. This paper provides the conditions for the distributed MRAC to guarantee that each agent asymptotically tracks the corresponding reference agent; consequently, the multi‐agent system solves the leader‐following consensus problem as the reference model does. These conditions yield a straightforward design method for the MRAC. A numerical example demonstrates an application of the proposed approach to a multi‐robot system with nonlinear dynamics.

A class of semilinear elliptic equations on groups of polynomial growth

In this paper, we study the semilinear elliptic equation−Δu+a(x)|u|p−2u−b(x)|u|q−2u=0 on a Cayley graph of a discrete group of polynomial growth with the homogeneous dimension N≥1, where 2≤p<q<+∞. We first prove the existence of positive solutions to the above equation with constant coefficients a¯,b¯. Then we establish a decomposition of Palais-Smale sequences for the functional with variable coefficients a(x),b(x), which tend to the constants a¯,b¯ at infinity.

Gradient estimates for Schrödinger operators with characterizations of 𝐵𝑀𝑂_{ℒ} on Heisenberg groups

Let L = − Δ H n + V \mathcal {L}=-\Delta _{\mathbb {H}^n}+V be a Schrödinger operator with the nonnegative potential V V belonging to the reverse Hölder class B Q B_{Q} , where Q Q is the homogeneous dimension of the Heisenberg group H n \mathbb {H}^n . In this paper, we obtain pointwise bounds for the spatial derivatives of the heat and Poisson kernels related to L \mathcal {L} . As an application, we characterize the space B M O L ( H n ) BMO_{\mathcal {L}}(\mathbb {H}^n) , associated to the Schrödinger operator L \mathcal {L} , in terms of two Carleson type measures involving the spatial derivatives of the heat kernel of the semigroup { e − s L } s &gt; 0 \{e^{-s\mathcal {L}}\}_{s&gt;0} and the Poisson kernel of the semigroup { e − s L } s &gt; 0 \{e^{-s\sqrt {\mathcal {L}}}\}_{s&gt;0} , respectively. At last, we pose a conjecture about the converse characterization of B M O L ( H n ) BMO_{\mathcal {L}}(\mathbb {H}^n) .

Synthesis and characterization of natural biomaterial composite nanofibers for ocular drug delivery systems

Due to its numerous reported benefits, including biomimetic, immunomodulation, and compatibility with tissue microenvironment, natural bio composite nanomaterials are in demand as an effective replacement in the field of tissue engineering. As a result, in this study, we investigated polysaccharides containing natural materials from Coriandrum sativum(CS) and Trigonella foenum-graecum(TFG) to create a composite nanofiber matrix with polyvinyl alcohol (PVA). Drug-free and drug-containing versions of the natural composite nanofiber have been developed. Zeta potential, Fourier transform infrared spectroscopy (FTIR) and scanning electron microscopy were used to characterize the conductivity, molecular makeup and size of the fibers (SEM). Brunauer-Emmett-Teller (BET) analysis was used to determine the topography of the surface. To determine how natural biomaterials will affect PVA, the stiffness of the composite nanofibers was tested. Drug release from the PVA composite nanofiber matrix using HPLC analysis was conducted on fibers loaded with drugs. Primary Human tenon fibroblast cells were used to test the fibers biocompatibility. SEM analysis revealed that bead-free composite nanofibers had homogeneous dimensions ranging from 150 nm to 200 nm. The presence of heterogeneous molecules from the natural material was revealed by FTIR data in the composite nanofiber. According to the BET analysis, the addition of natural biomaterial significantly altered the surface topography. With the addition of the drug 5-Fluorouracil, the stiffness of PVA nanofibers decreased from 103 ± 6 to 70 ± 15 Kpa. At the same time, the drug improved the mechanical properties of the composite nanofiber mats, increasing stiffness from 150 ± 22 Kpa to 180 ± 17 Kpa. Human Tenon fibroblast Cell toxicity assay showed that there was a difference in the cell viability of the composite nanofibers with the drug. The release kinetics of HPLC data showed that the composite matrix fibre mat had sustained drug release for 4 hrs. Cell viability assay findings revealed that composite nanofibers did not have any significant effect on the cell viability. Further research on the toxicity of the composite material in vivo is required to determine the mechanism. This may be helpful in modulating fibrosis process post trabeculectomy surgery.

IntRinsic: An R Package for Model-Based Estimation of the Intrinsic Dimension of a Dataset

This article illustrates intRinsic, an R package that implements novel state-of-the-art likelihood-based estimators of the intrinsic dimension of a dataset, an essential quantity for most dimensionality reduction techniques. In order to make these novel estimators easily accessible, the package contains a small number of high-level functions that rely on a broader set of efficient, low-level routines. Generally speaking, intRinsic encompasses models that fall into two categories: homogeneous and heterogeneous intrinsic dimension estimators. The first category contains the two nearest neighbors estimator, a method derived from the distributional properties of the ratios of the distances between each data point and its first two closest neighbors. The functions dedicated to this method carry out inference under both the frequentist and Bayesian frameworks. In the second category, we find the heterogeneous intrinsic dimension algorithm, a Bayesian mixture model for which an efficient Gibbs sampler is implemented. After presenting the theoretical background, we demonstrate the performance of the models on simulated datasets. This way, we can facilitate the exposition by immediately assessing the validity of the results. Then, we employ the package to study the intrinsic dimension of the Alon dataset, obtained from a famous microarray experiment. Finally, we show how the estimation of homogeneous and heterogeneous intrinsic dimensions allows us to gain valuable insights into the topological structure of a dataset.

Construction of ionic liquid-filled silica shell microcapsules based on emulsion template and evaluation of their adsorption properties toward 3,4,5-trichlorophenol after various surface functionalization

The development of environmentally friendly and efficient methods for the removal of Phenolic endocrine disruptors (PEDs) from aqueous is essential to protect the environment and human health. In this work, sol–gel method and emulsion template approach were combined and applied to construct 1-Butyl-3-methylimidazolium hexafluorophosphate ([BMIM]PF6) ionic liquid-filled silica-shell microcapsules (SiO2@IL). And amino, halogen and long-chain alkanes surface-functionalized silica-shell encapsulated ionic liquid microcapsules (SiO2-NH2@IL, SiO2-Cl@IL, and SiO2-C16@IL) were successfully formulated by silica layer modification techniques. The microcapsules have homogeneous dimensions, about 1 µm, and high loading, about 92 wt% of ILs. Four microcapsules showed high adsorption properties and excellent regeneration performance for removing 3,4,5-trichlorophenol (3,4,5-TCP), which benefited from the excellent performance of [BMIM]PF6 on the extraction of 3,4,5-TCP and the enhanced hydrophobic and electrostatic effects of the surface modified microcapsules. The maximum adsorption capacity of SiO2@IL, SiO2-NH2@IL, SiO2-Cl@IL and SiO2-C16@IL calculated from the Langmuir model was 1047, 1207, 1108 and 1145 mg g−1 at 25 °C, respectively. The main factors affecting the extraction performance were the hydrophobicity and steric hindrance of [BMIM]PF6. After three regeneration cycles, the adsorption capacity and desorption efficiency of microcapsules were still above 90 %. The functionalization of the shell layer enhances the hydrophobic and electrostatic effects, which is conducive to the adsorption process, and provides some theoretical and data support for the subsequent construction of ionic liquid encapsulation materials for chlorophenol removal.

Exploring the hybridization from policy learning on the forest-certification schemes in China

The policy of forest certification in China is a collaborative tactic for environmental entities to influence local politics based on the declaration of a national “Five-Year Plan” by President Xi Jin-Ping in the 2010s. This is due to the homogeneous impact led by the Forest Stewardship Council (FSC) to hybridize Chinese forestry management with organizational autonomy, policy participation, and power-sharing. While the Chinese public agencies learned professional knowledge and institutions from the IOs, they also required the FSC to embrace the Chinese characteristics of local politics, standards, and the legitimacy of international principles. Two findings from the research are revealing: First, the FSC and IO’s central role of civil empowerment enhanced the China’s power-sharing with local stakeholders for cultivating the China certification scheme; second, the FSC’s principle modification embraced the Chinese characteristics to reform the policy hybridization mechanism under the political culture of Administrative Absorption of Society. The innovative purpose of this research is to employ the homogeneous and heterogeneous dimensions to explore the influential factors of policy hybridization from a network perspective.

Hospital Anxiety and Depression Scale Cannot Reliably Differentiate Anxiety and Depression Following Traumatic Brain Injury

<h3>Research Objectives</h3> To evaluate whether scores on the Hospital Anxiety and Depression Scale (HADS) following traumatic brain injury (TBI) reliably reflect specific factors of anxiety and depression or instead a single factor of general distress. <h3>Design</h3> Survey. Participants completed the HADS at one year post-TBI. A subsample (21%) also completed the Structured Clinical Interview for DSM-IV Axis I Disorders (SCID) at the same time. <h3>Setting</h3> Community. <h3>Participants</h3> 874 adults (mean age at injury = 40 years, 74% male, 10% non-English speaking background) recruited from consecutive admissions to a rehabilitation hospital following a moderate-to-severe TBI. Individuals with other neurological conditions or insufficient cognitive/English ability were excluded. <h3>Interventions</h3> Not applicable. <h3>Main Outcome Measures</h3> HADS. <h3>Results</h3> Bifactor analysis revealed a dominant general distress factor accounting for 84% of the systematic variance in HADS total scores. When holding this general factor constant, the specific anxiety and depression factors accounted for little residual variance in their corresponding subscale scores (12% and 19%, respectively). Item loadings on the general factor changed by only 8% on average when removing the specific factors, suggesting minimal bias in treating the HADS as a unidimensional instrument. Moreover, in a subsample who completed the SCID (n=184), the HADS subscales did not discriminate formal diagnoses of anxiety and depressive disorders (i.e., differences in subscale scores between diagnostic groups non-significant at p > .05). <h3>Conclusions</h3> In this large TBI cohort, HADS total scores predominately reflected variance due to a single underlying latent variable. The subscale scores had low reliability with most of their variance reflecting a general distress factor. We also did not find a relationship between the HADS subscale scores and their corresponding formal diagnostic categories on the SCID. We therefore recommend TBI clinicians and researchers exercise caution in interpreting the individual HADS subscales and consider instead using the total score as a more reliable measure of general distress. Whilst screening tools have advantages, transdiagnostic measures of psychopathology that can yield dissociable, homogenous symptom dimensions should also be validated for TBI. <h3>Author(s) Disclosures</h3> None.

Least energy solutions to quasilinear subelliptic equations with constant and degenerate potentials on the Heisenberg group

Let H n = C n × R $\mathbb {H}^{n}=\mathbb {C}^{n}\times \mathbb {R}$ be the n $n$ -dimensional Heisenberg group, Q = 2 n + 2 $Q=2n+2$ be the homogeneous dimension of H n $\mathbb {H}^{n}$ . In this paper, we investigate the existence of a least energy solution to the Q $Q$ -subLaplacian Schrödinger equation with either a constant V = γ $V=\gamma$ or a degenerate potential V $V$ vanishing on a bounded open subset of H n $\mathbb {H}^n$ : − div H ∇ H u Q − 2 ∇ H u + V ( ξ ) u Q − 2 u = f u $$\begin{equation} -\mathrm{div}_{\mathbb {H}}{\left({\left|\nabla _{\mathbb {H}}u\right|}^{Q-2} \nabla _{\mathbb {H}}u\right)} +V(\xi ) {\left|u\right|}^{Q-2}u=f{\left(u\right)} \end{equation}$$ (0.1)with the non-linear term f $f$ of maximal exponential growth exp ( α t Q Q − 1 ) $\exp (\alpha t^{\frac{Q}{Q-1}})$ as t → + ∞ $t\rightarrow +\infty$ . Since the Pólya–Szegö-type inequality fails on H n $\mathbb {H}^n$ , the coercivity of the potential has been a standard assumption in the literature for subelliptic equations to exclude the vanishing phenomena of Palais–Smale sequence on the entire space H n $\mathbb {H}^n$ . Our aim in this paper is to remove this strong assumption. To this end, we first establish a sharp critical Trudinger–Moser inequality involving a degenerate potential on H n $\mathbb {H}^n$ . Second, we prove the existence of a least energy solution to the above equation with the constant potential V ( ξ ) = γ > 0 $V(\xi )=\gamma >0$ . Third, we establish the existence of a least energy solution to the Q $Q$ -subelliptic equation (0.1) involving the degenerate potential which vanishes on some open bounded set of H n $\mathbb {H}^{n}$ . We develop arguments that avoid using any symmetrization on H n $\mathbb {H}^n$ where the Pólya–Szegö inequality fails. Fourth, we also establish the existence of a least energy solution to (0.1) when the potential is a non-degenerate Rabinowitz type potential but still fails to be coercive. Our results in this paper improve significantly on the earlier ones on quasilinear Schrödinger equations on the Heisenberg group in the literature. We note that all the main results and their proofs in this paper hold on stratified groups with the same proofs.

Regularity for Quasi-Linear p-Laplacian Type Non-Homogeneous Equations in the Heisenberg Group

When 2−1/Q&lt;p≤2, we establish the Cloc0,1 and Cloc1,α-regularities of weak solutions to quasi-linear p-Laplacian type non-homogeneous equations in the Heisenberg group Hn, where Q=2n+2 is the homogeneous dimension of Hn.

Liouville‐type theorem for finite Morse index solutions to the Choquard equation involving Δλ‐Laplacian

In this paper, we are concerned with the following Choquard‐type equation: where , and is a sub‐elliptic operator of the form Under some general conditions of , we prove that the equation has no nontrivial solution which is stable outside a compact set in the subcritical case , where is the homogeneous dimension of associated to . In addition, for the critical case , we shall show that any solution of the equation above, which is stable outside a compact set, has finite energy. More precisely, we show that