Group Hn Research Articles

Second-Order Regularity for Degenerate Parabolic Quasi-Linear Equations in the Heisenberg Group

In the Heisenberg group Hn, we obtain the local second-order HWloc2,2-regularity for the weak solution u to a class of degenerate parabolic quasi-linear equations ∂tu=∑i=12nXiAi(Xu) modeled on the parabolic p-Laplacian equation. Specifically, when 2≤p≤4, we demonstrate the integrability of (∂tu)2, namely, ∂tu∈Lloc2; when 2≤p<3, we demonstrate the HWloc2,2-regularity of u, namely, XXu∈Lloc2. For the HWloc2,2-regularity, when p≥2, the range of p is optimal compared to the Euclidean case.

Realizing regular representations of finite groups

Given a regular representation of a finite group G and a positive integer number n, we construct a (finite) topological space X such that its group of homotopy classes of self-homotopy equivalences E(X) and its group of homeomorphisms Aut(X) are isomorphic to G, and the action of G on the n-th homology group Hn(X) is the regular representation. We also discuss other representations.

On the identities and cocharacters of the algebra of 3 × 3 matrices with orthosymplectic superinvolution

Let M1,2(F) be the algebra of 3×3 matrices with orthosymplectic superinvolution ⁎ over a field F of characteristic zero. We study the ⁎-identities of this algebra through the representation theory of the group Hn=(Z2×Z2)∼Sn. We decompose the space of multilinear ⁎-identities of degree n into the sum of irreducibles under the Hn-action in order to study the irreducible characters appearing in this decomposition with non-zero multiplicity. Moreover, by using the representation theory of the general linear group, we determine all the ⁎-polynomial identities of M1,2(F) up to degree 3.

A Stone–von Neumann equivalence of categories for smooth representations of the Heisenberg group

The classical Stone–von Neumann theorem relates the irreducible unitary representations of the Heisenberg group Hn to non-trivial unitary characters of its center Z, and plays a crucial role in the construction of the oscillator representation for the metaplectic group. In this paper we extend these ideas to non-unitary and non-irreducible representations, thereby obtaining an equivalence of categories between certain representations of Z and those of Hn. Our main result is a smooth equivalence, which involves the fundamental ideas of du Cloux on differentiable representations and smooth imprimitivity systems for Nash groups. We show how to extend the oscillator representation to the smooth setting and give an application to degenerate Whittaker models for representations of reductive groups. We also include an algebraic equivalence, which can be regarded as a generalization of Kashiwara’s lemma from the theory of D-modules.

The prescribed mean curvature equation for t-graphs in the sub-Finsler Heisenberg group [formula omitted]

We study the sub-Finsler prescribed mean curvature equation, associated to a strictly convex body K0⊆R2n, for t-graphs on a bounded domain Ω in the Heisenberg group Hn. When the prescribed datum H is constant and strictly smaller than the Finsler mean curvature of ∂Ω, we prove the existence of a Lipschitz solution to the Dirichlet problem for the sub-Finsler CMC equation by means of a Finsler approximation scheme.

Krylov complexity for Jacobi coherent states

We develop computational tools necessary to extend the application of Krylov complexity beyond the simple Hamiltonian systems considered thus far in the literature. As a first step toward this broader goal, we show how the Lanczos algorithm that iteratively generates the Krylov basis can be augmented to treat coherent states associated with the Jacobi group, the semi-direct product of the 3-dimensional real Heisenberg-Weyl group H1, and the symplectic group, Sp(2, ℝ) ≃ SU(1, 1). Such coherent states are physically realized as squeezed states in, for example, quantum optics [1]. With the Krylov basis for both the SU(1, 1) and Heisenberg-Weyl groups being well understood, their semi-direct product is also partially analytically tractable. We exploit this to benchmark a scheme to numerically compute the Lanczos coefficients which, in principle, generalizes to the more general Jacobi group Hn ⋊ Sp(2n, ℝ).

Effect of Preadministration of Nalmefene on Sufentanil-Induced Cough During Induction of General Anesthesia in Patients Undergoing Breast Surgery: A Double-Blind Randomized Controlled Trial.

This study was designed to investigate the effects of preadministration of nalmefene before general anesthesia induction on sufentanil-induced cough (SIC) in patients undergoing breast surgery. A total of 105 patients scheduled for elective breast surgery under general anesthesia were selected and randomly assigned into three groups: normal saline (Group C), low-dose nalmefene 0.1 μg·kg-1 (Group LN), and high-dose nalmefene 0.25 μg·kg-1 (Group HN). Sufentanil 0.5 μg·kg-1 was injected intravenously within 2 s after 5 min of intervention. The count and severity of cough within 2 min after sufentanil injection, as well as the time to first cough, were recorded. In addition, we also collected intraoperative hemodynamic data, postoperative pain scores, the incidence of receiving rescue analgesics, and side effects up to 24 h after surgery. Compared to Group C, the incidence of SIC was significantly lower in Group LN and HN (64.7% vs 30.3% and 14.7%, respectively; P < 0.001), but no significant difference was observed between the two groups (P=0.126). Compared to Group C, the risk factors decreased by 53.4% (95% confidence interval [CI] =0.181-0.735, P=0.008) in Group LN and by 75.9% (95% CI=0.432-0.898, P=0.001) in Group HN. Of the patients with SIC, less frequent SIC within 2 min after induction and a lower proportion of severe coughs were observed than Group C (P < 0.05), and no difference was detected between Group LN and HN. Additionally, the onset time to the first SIC did not differ significantly between the groups. Intraoperative hemodynamic data, postoperative pain scores, and side effects in the first 24 h did not differ among the groups. Preadministration of nalmefene prior to induction of general anesthesia effectively suppressed SIC in patients undergoing breast surgery, without affecting intraoperative hemodynamic fluctuation and postoperative pain intensity.

Horizontal p-Kirchhoff equation on the Heisenberg group

We study the existence and multiplicity of solutions both for the isotropic and the anisotropic horizontal p-Kirchhoff equations on the Heisenberg group Hn, via Krasnoselskii's genus. Finally, an interesting question for the anisotropic embedding in the subelliptic literature is presented and it is conjectured that the exponent p‾⁎:=QnQ∑i=1n1pi−n can be the effective critical exponent in the setting of anisotropic horizontal Sobolev space.

Generalized Schrödinger operators on the Heisenberg group and Hardy spaces

Let L=−ΔHn+μ be a generalized Schrödinger operator on the Heisenberg group Hn, where ΔHn is the sub-Laplacian, and μ is a nonnegative Radon measure satisfying certain conditions. In this paper, we first establish some estimates of the fundamental solution and the heat kernel of L. Applying these estimates, we then study the Hardy spaces HL1(Hn) introduced in terms of the maximal function associated with the heat semigroup e−tL; in particular, we obtain an atomic decomposition of HL1(Hn), and prove the Riesz transform characterization of HL1(Hn). The dual space of HL1(Hn) is also studied.

Effects of Surface Textures Created Using Additive Manufacturing on Shear Bond Strength Between Resin and Zirconia.

This investigation aimed to assess the impact of additive manufacturing-generated surface textures on zirconia bond strength. Zirconia samples (n = 144) fabricated using digital light-processing (DLP) technology were categorized into 6 groups according to the type of surface conditioning (group NN: no designs, no air abrasion; group NY: no designs, with air abrasion; group GN: groove designs, no air abrasion; group GY: groove designs with air abrasion; group HN: hexagon grid, no air abrasion; group HY: hexagon grid, with air abrasion). Composite resin cylinders were cemented to the treated zirconia surfaces with dual-curing, self-adhesive resin cement (Clearfil SA Luting). The shear bond strength (SBS) was tested after water storage for 3 days or 3 days with an additional 10,000 thermocycles. The zirconia samples fabricated using DLP technology have high accuracy. The SBS of the NY, GY, and HY groups did not significantly differ after 3 days, and neither did the SBS of the NN, GN, and HN groups. The NN, NY, and HY groups exhibited reduced SBS compared to their initial values following artificial aging, while the SBS of the remaining three groups were not diminished. The GY group obtained the highest SBS value after aging. Printing grooves with air abrasion can improve the bond strength.

A Detailed Study of Kirchhoff-type Critical Elliptic Equations and p-Sub-Laplacian Operators within the Heisenberg Group Hn Framework

A Detailed Study of Kirchhoff-type Critical Elliptic Equations and p-Sub-Laplacian Operators within the Heisenberg Group Hn Framework

Weighted estimates for bilinear fractional integral operator on the Heisenberg group

In this article, we consider an analogue of Kenig and Stein's bilinear fractional integral operator on the Heisenberg group Hn. We completely characterize exponents α,β and γ such that the operator is bounded from Lp(Hn,|x|αp)×Lq(Hn,|x|βq) to Lr(Hn,|x|−γr). Also, analogous sharp results are obtained on the Euclidean space.

Parabolic Anderson model on Heisenberg groups: The Itô setting

In this note we focus our attention on a stochastic heat equation defined on the Heisenberg group Hn of order n. This equation is written as ∂tu=12Δu+uW˙α, where Δ is the hypoelliptic Laplacian on Hn and {W˙α;α>0} is a family of Gaussian space-time noises which are white in time and have a covariance structure generated by (−Δ)−α in space. Our aim is threefold: (i) Give a proper description of the noise Wα; (ii) Prove that one can solve the stochastic heat equation in the Itô sense as soon as α>n2; (iii) Give some basic moment estimates for the solution u(t,x).

Analogues of theorems of Chernoff and Ingham on the Heisenberg group

We prove an analogue of Chernoff’s theorem for the Laplacian Δℍ on the Heisenberg group ℍn. As an application, we prove Ingham type theorems for the group Fourier transform on ℍn and also for the spectral projections associated to the sublaplacian.

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.

Hecke relations, cosets and the classification of 2d RCFTs

We systemically study the Hecke relations and the c = 8k coset relations among 2d rational conformal field theories (RCFTs) with up to seven characters. We propose that the characters of any 2d RCFT — unitary or non-unitary — satisfying a holomorphic modular linear differential equation (MLDE) can be realized as either a Hecke image or the coset of a Hecke image with respect to a c = 8k theory. Benefited from the recent results on holomorphic modular bootstrap, we check this proposal for all admissible theories with up to five characters. We also find many new interesting Hecke relations. For example, the characters of WZW models (E6)2, (E7)2, ( {E}_{7frac{1}{2}} )2 can be realized as the Hecke images T13, T19, T19 of Virasoro minimal models Msub(7, 6), M(5, 4), Meff(13, 2) respectively. Besides, we find the characters associated to the second largest Fisher group Fi23 and the Harada-Norton group HN can be realized as the Hecke images T23, T19 of the product theories Meff(5, 2) ⊗ Meff(7, 2) and Meff(7, 2)⊗2 respectively. Mathematically, our study provides a great many interesting examples of vector-valued modular functions up to rank seven.

Restoring Impaired Fertility Through Diet: Observations of Switching From High-Fat Diet During Puberty to Normal Diet in Adulthood Among Obese Male Mice.

BackgroundObesity is associated with a decrease in testicular function, yet the effects and mechanisms relative to different stages of sexual development remain unclear. The aim of this study is to determine whether high-fat diet-induced obesity impairs male fertility during puberty and in adulthood, and to ascertain its underlying mechanisms. This study aims to further reveal whether restoring to a normal diet can improve impaired fertility.MethodsMale mice were divided into 6 groups: the group N and H exposed to a normal diet or high-fat diet during puberty. The group NN or NH were further maintained a normal diet or exposed to high-fat diet in adulthood, the group HH or HN were further maintained high-fat diet or switched to normal diet in adulthood. Metabolic parameters, fertility parameters, testicular function parameters, TUNEL staining and testicular function-related proteins were evaluated, respectively.ResultsThe fertility of the mice in the high-fat diet group was impaired, which validated by declines in pregnancy rates and litter weight loss. Further analysis demonstrated the increased level of oxidative stress, the increased number of spermatogenic cell apoptosis and decreased number of sperm and decreased acrosome integrity. The expression of steroidogenic acute regulatory (StAR) and spermatogenesis related proteins (WT-1) decreased. Fertility among the HN group recovered, accompanied by the recovery of metabolism, fertility and testicular function parameters, StAR and WT-1 expression.ConclusionsThe findings suggest that high-fat diet-induced obesity impairs male fertility during puberty and in adulthood. The loss of acrosome integrity, the increase of oxidative stress, the increase of cells apoptosis and the down-regulation of StAR and WT-1 may be the underlying mechanisms. Switching from high-fat diets during puberty to normal diets in adulthood can improve male fertility.

Effect of prenatal different auditory environment on learning ability and fearfulness in chicks.

ObjectiveEarly environmental enrichment in life can improve cognition in animals. The effect of prenatal auditory stimulation on learning ability and fear level in chick embryos remained unexplored. Therefore, this study investigated the effect of prenatal auditory stimulation on the learning ability and fear level of chicks.MethodsA total of 450 fertilized eggs were randomly divided into 5 groups, including control group (C), low-sound intensity music group (LM), low-sound intensity noise group (LN), high-sound intensity noise group (HN) and high-sound intensity music group (HM). From the 10th day of embryonic development until hatching, group LM and group LN received 65 to 75 dB of music and noise stimulation. Group HN and group HM received 85 to 95 dB of noise and music stimulation, and group C received no additional sound. At the end of incubation, the one-trial passive avoidance learning (PAL) task and tonic immobility (TI) tests were carried out, and the serum corticosterone (CORT) and serotonin (5-HT) concentrations were determined.ResultsThe results showed that compared with the group C, 65 to 75 dB of music and noise stimulation did not affect the PAL avoidance rate (p>0.05), duration of TI (p>0.05) and the concentration of CORT (p>0.05) and 5-HT (p>0.05) in chicks. However, 85 to 95 dB of music and noise stimulation could reduce duration of TI (p<0.05) and the concentration of CORT (p<0.05), but no significant effect was observed on the concentration of 5-HT (p>0.05) and PAL avoidance rate (p>0.05).ConclusionTherefore, the prenatal auditory stimulation of 85 to 95 dB can effectively reduce the fear level of chicks while it does not affect the learning ability.

Sampling in spaces of entire functions of exponential type in [formula omitted

In this paper we consider the question of sampling for spaces of entire functions of exponential type in several variables. The novelty resides in the growth condition we impose on the entire functions, that is, that their restriction to a hypersurface is square integrable with respect to a natural measure. The hypersurface we consider is the boundary bU of the Siegel upper half-space U and it is fundamental that bU can be identified with the Heisenberg group Hn. We consider entire functions in Cn+1 of exponential type with respect to the hypersurface bU whose restriction to bU are square integrable with respect to the Haar measure on Hn. For these functions we prove a version of the Whittaker–Kotelnikov–Shannon Theorem. Instrumental in our work are spaces of entire functions in Cn+1 of exponential type with respect to the hypersurface bU whose restrictions to bU belong to some homogeneous Sobolev space on Hn. For these spaces, using the group Fourier transform on Hn, we prove a Paley–Wiener type theorem and a Plancherel–Pólya type inequality.

Nodal solutions for Q-Laplacian problem with exponential nonlinearities on the Heisenberg group

This paper deals with the existence of nodal solutions (sign-changing) to the following elliptic equation involving Q-Laplacian:{−ΔQu=λf(ξ,u)inΩ,u=0on∂Ω, where ΔQ(⋅):=divHn(|∇Hn(⋅)|HnQ−2∇Hn(⋅)) is the Q-Laplacian operator, Ω is an open, smooth bounded domain in the Heisenberg group Hn, Q=2n+2 is the homogeneous dimension of Hn, λ is a positive parameter, and nonlinear term f(ξ,u) behaves like exp⁡(β|s|QQ−1) as s→∞. Under suitable conditions on f, together with the constraint variational method and the quantitative deformation lemma, we obtain the existence, energy estimates and the convergence property of the least energy sign-changing solutions for the above problem in subcritical and critical exponential growth cases.