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We consider a parabolic nonlinear equation on the Heisenberg group. Applying the Gidas–Spruck type estimates, we prove that under suitable conditions, the equation does not have positive solutions. As an application of the nonexistence result, we provide optimal universal estimates for positive solutions.

For any parameter vector (r,t,ω) in ℍ := ℝ × ℝn × ℝn, the representation matrix S(r,t,ω) of the Heisenberg group (ℍ,∗) is extended to a new form SB,W,A(r,t,ω) with nonsingular matrices B,W,A which are related to the directional translation 풯Bt : f → f(⋅− Bt), directional modulation ℳWω : f(x) →eix′Wω and dilation 풟A : f →|det(A)|− 1 2f(A−1⋅) if B′W = I n, the identity matrix of order n. In the case B′W≠I n, an intrinsic group structure under the matrix SB,W,A(r,t,ω) is investigated and a unitary operator representation is offered. [ABSTRACT FROM AUTHOR]

In this paper, we consider a nonlinear discontinuous sub-elliptic system with drift on the Heisenberg group, where the coefficients in system are discontinuous and satisfy the vanishing mean oscillation condition and growth conditions with the growth index $ 2 2 < p < ∞ , and the non-homogeneous terms satisfy the controllable growth conditions and the natural growth conditions, respectively. The partial Hölder regularity to weak solutions and the partial Morrey regularity to horizontal gradients of weak solutions are proved by the $ \mathcal {A} $ A -harmonic approximation. One of the difficulties in this paper is how to reasonably define the weak solution and avoid the requirement of integrability of Tu. In fact, we used $ T={X_i}{X_{n+i}}-{X_{n+i}}{X_i} $ T = X i X n + i − X n + i X i . [ABSTRACT FROM AUTHOR]

We prove the convergence of a wide class of continued fractions, including generalized continued fractions over quaternions and octonions. Fractional points in these systems are not bounded away from the unit sphere, so that the iteration map is not uniformly expanding. We bypass this problem by analyzing digit sequences for points that converge to the unit sphere under iteration, expanding on previous methods of Dani–Nogueira. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence and uniqueness of mild solutions to the fractional Navier–Stokes equations related to time derivative of order α ∈ (0 , 1) . And the mild solution is associated with the sublaplacian provided by the left invariant vector fields on the Heisenberg group. We demonstrate that when the nonlinear external force term matches the applicable conditions, the global mild solution can be obtained by using improved Ascoli–Arzela theorem and Schaefer's fixed point theorem. [ABSTRACT FROM AUTHOR]

This research focuses on introducing a novel algorithm for RGB image encryption. The proposed approach is a hybrid fusion of mathematical principles, namely group theory and chaos theory. It leverages the favorable characteristics of the Heisenberg subgroup of G L 2 n + 1 (Z) and a 4D chaotic laser system derived from the Lorenz-Haken equations. The algorithm aims to provide robust encryption by truncating four sequences from the 4D chaotic map. The initial Heisenberg matrix is generated using two input parameters, while the second matrix is constructed using two additional parameters.To enhance encryption efficiency, four square boxes of dimensions 16 × 16 are designed specifically for the 4D laser system. Each box contains cells that represent the rank of values relative to similar values in the same stream. Furthermore, the source image is divided into three monochromatic components. The foreground of the Heisenberg matrix is multiplied by each channel extracted from the source image. The encryption process proceeds with digram substitution, wherein two-pixel couples are replaced using the values stored in the four box squares. The first value is selected from box 1, followed by the second value from box 4. The remaining values needed to complete the rectangle are retrieved from boxes 2 and 3. For the subsequent digram, a circular rotation of the boxes is performed. The first value is obtained from box 2, the second value from box 3, and so forth, ensuring a continuous circular pattern. Finally, a second multiplication is applied, involving the second plane of the Heisenberg matrix and the result obtained in the previous stage. Experimental results demonstrate the algorithm's effectiveness, exhibiting resistance to differential attacks, statistical attacks, and superior performance compared to existing encryption techniques. [ABSTRACT FROM AUTHOR]

In this paper, using variational methods and critical point theory we establish the existence of multiple solutions for a class of elliptic equations on Carnot groups depending on one real positive parameter and involving a subcritical nonlinearity. Some recent results are extended and improved. [ABSTRACT FROM AUTHOR]

A geodetically convex set in the Heisenberg group H n , n ≥ 1 is defined to be a set with the property that a geodesic joining any two points in the set lies completely in it. Here we classify the geodetically convex sets to be either an empty set, a singleton set, an arc of a geodesic or the whole space H n . We also show that a geodetically convex function on H n is a constant function. These results generalize the known results of H 1 to higher dimensional Heisenberg group. [ABSTRACT FROM AUTHOR]

This paper is primarily devoted to a class of interpolation inequalities of Hardy and Rellich types on the Heisenberg group H n . Consequently, several weighted Hardy type, Heisenberg–Pauli–Weyl uncertainty principle and Hardy–Rellich type inqualities are established on H n . Moreover, new weighted Sobolev type embeddings are derived. Finally, an integral inequality for vector fields in a domain of the Heisenberg group is obtained, leading to several specific weighted Hardy type inequalities by making careful choices of vector fields. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Cauchy problem for the degenerate parabolic equations on the Heisenberg groups with power law non-linearities. We obtain Fujita-type critical exponents, which depend on the homogeneous dimension of the Heisenberg groups. The analysis includes the case of porous medium equations. Our proof approach is based on methods of nonlinear capacity estimates specifically adapted to the nature of the Heisenberg groups. We also use the Kaplan eigenfunctions method in combination with the Hopf-type lemma on the Heisenberg groups. [ABSTRACT FROM AUTHOR]

Abstract In this work, we study the following Schrödinger-Poisson system { − Δ H u + μ ϕ u = λ u − γ , in Ω , − Δ H ϕ = u 2 , in Ω , u > 0 , in Ω , u = ϕ = 0 , on ∂ Ω , $$ \textstyle\begin{cases} -\Delta _{H}u+\mu \phi u=\lambda u^{-\gamma}, &\text{in } \Omega , \\ -\Delta _{H}\phi =u^{2}, &\text{in } \Omega , \\ u>0, &\text{in } \Omega , \\ u=\phi =0, &\text{on } \partial \Omega , \end{cases} $$ where Δ H $\Delta _{H}$ is the Kohn-Laplacian on the first Heisenberg group H 1 $\mathbb{H}^{1}$ , and Ω ⊂ H 1 $\Omega \subset \mathbb{H}^{1}$ is a smooth bounded domain, μ = ± 1 $\mu =\pm 1$ , 0 < γ < 1 $0 0 $\lambda >0$ are some real parameters. For the above system, we prove the existence and uniqueness of positive solution for μ = 1 $\mu =1$ and each λ > 0 $\lambda >0$ . Multiple solutions of the system are also considered for μ = − 1 $\mu =-1$ and λ > 0 $\lambda >0$ small enough using the critical point theory for nonsmooth functional.

In this article, we study the recent development of the qualitative uncertainty principle on certain Lie groups. In particular, we consider that if the Weyl transform on certain step-two nilpotent Lie groups is of finite rank, then the function has to be zero almost everywhere as long as the nonvanishing set for the function has finite measure. Further, we consider that if the Weyl transform of each Fourier–Wigner piece of a suitable function on the Heisenberg motion group is of finite rank, then the function has to be zero almost everywhere whenever the nonvanishing set for each Fourier–Wigner piece has finite measure. [ABSTRACT FROM AUTHOR]

In this paper, we prove new Liouville-type properties of fully nonlinear subelliptic equations modelled on the Heisenberg group. A key tool that can be of independent interest is a strong maximum principle for solutions of such differential equations. [ABSTRACT FROM AUTHOR]

We study the nonholonomic motion of a point particle on the Heisenberg group around the fixed “sun” placed at the origin whose potential is given by the fundamental solution of the sub-Laplacian. In contrast with several recent papers that approach this problem as a variational one (hence a control problem) we study the equations of dynamical motion which are non-variational in nonholonomic mechanics. We find three independent first integrals of the system and show that its bounded trajectories are wound up around certain surfaces of the fourth order. We also describe some particular cases of trajectories. [ABSTRACT FROM AUTHOR]

In the Heisenberg group ℍ1 with a sub-Finsler structure, an (X,Y )-Lipschitz surface which is complete, oriented, connected and stable must be a vertical plane. In particular, the result holds for entire intrinsic graphs of Euclidean Lipschitz functions. [ABSTRACT FROM AUTHOR]

In this work, we study the following Schrödinger-Poisson system { − Δ H u + μ ϕ u = λ u − γ , in Ω , − Δ H ϕ = u 2 , in Ω , u > 0 , in Ω , u = ϕ = 0 , on ∂ Ω , where Δ H is the Kohn-Laplacian on the first Heisenberg group H 1 , and Ω ⊂ H 1 is a smooth bounded domain, μ = ± 1 , 0 < γ < 1 , and λ > 0 are some real parameters. For the above system, we prove the existence and uniqueness of positive solution for μ = 1 and each λ > 0 . Multiple solutions of the system are also considered for μ = − 1 and λ > 0 small enough using the critical point theory for nonsmooth functional. [ABSTRACT FROM AUTHOR]

A class of critical Kirchhoff-Poisson systems on Heisenberg group was studied. Due to the existence of critical and non-local terms, the space embedding was not compact. Under the assumptions that the nonlinear terms were appropriate, the space compactness was overcame by the variational method and established the existence of at least one solution to this system. Furthermore, using deformation lemma and topological degree theory, it was proved that the solution was a sign-changing solution. [ABSTRACT FROM AUTHOR]

This paper is concerned with a PDE-based approach to the horizontally quasiconvex (h-quasiconvex, for short) envelope of a given continuous function in the Heisenberg group. We provide a characterization for upper semicontinuous, h-quasiconvex functions in terms of the viscosity subsolution to a first-order nonlocal Hamilton-Jacobi equation. We also construct the corresponding envelope of a continuous function by iterating the nonlocal operator. One important step in our arguments is to prove the uniqueness and existence of viscosity solutions to the Dirichlet boundary problem for the nonlocal Hamilton-Jacobi equation. Applications of our approach to the h-convex hull of a given set in the Heisenberg group are discussed as well. [ABSTRACT FROM AUTHOR]

Let L = - Δ ℍ n + V {L=-{\Delta}_{\mathbb{H}^{n}}+V} be a Schrödinger operator on Heisenberg groups ℍ n {\mathbb{H}^{n}} , where Δ ℍ n {{\Delta}_{\mathbb{H}^{n}}} is the sub-Laplacian, the nonnegative potential V belongs to the reverse Hölder class B 풬 / 2 {B_{\mathcal{Q}/2}} . Here 풬 {\mathcal{Q}} is the homogeneous dimension of ℍ n {\mathbb{H}^{n}} . In this article, we introduce the fractional heat semigroups { e - t ⁢ L α } t > 0 {\{e^{-tL^{\alpha}}\}_{t>0}} , α > 0 {\alpha>0} , associated with L. By the fundamental solution of the heat equation, we estimate the gradient and the time-fractional derivatives of the fractional heat kernel K α , t L ⁢ ( ⋅ , ⋅ ) {K_{\alpha,t}^{L}(\,\cdot\,,\cdot\,)} , respectively. As an application, we characterize the space BMO L γ ⁢ ( ℍ n ) {\mathrm{BMO}_{L}^{\gamma}(\mathbb{H}^{n})} via { e - t ⁢ L α } t > 0 {\{e^{-tL^{\alpha}}\}_{t>0}} . [ABSTRACT FROM AUTHOR]

In this paper, the following conformable fractional sub-elliptic equation is investigated: \[ \begin{cases} -\Delta_H^{\alpha} u = f(\xi,u,\nabla_H^\alpha u), & {\rm in} \ \Omega,\\ u > 0, & {\rm in} \ \Omega,\\ u=0, & {\rm on} \ \partial \Omega. \end{cases} \] { − Δ H α u = f (ξ , u , ∇ H α u) , in Ω , u > 0 , in Ω , u = 0 , on ∂ Ω. Ω represents the bounded domain in space $ \mathbb {H}^n $ H n , $ \alpha \in (0,1) $ α ∈ (0 , 1) , and f>0 is a bounded function. This paper is organized into three sections. In the first section, we present some of the features of the Heisenberg Laplacian. To prove the solution to the above equation, we use the Arzelà–Ascoli theorem and Bolzano–Weierstrass theorem in the second section. We then prove the uniqueness and cylindrical symmetry of the solution in the third section. [ABSTRACT FROM AUTHOR]

Let n ∈ N and H n be the Heisenberg group of dimension 2 n + 1 . Let Ω be a bounded open subset of H n and p ∈ (1 , Q) , where Q is the homogeneous dimension of H n . Within an appropriate framework, we prove an interior pointwise gradient estimate for a weak solution to the problem - div H (A (x , X u)) = g - div H f in Ω , u = 0 on ∂ Ω , where X represents the horizontal gradient on the Heisenberg group. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of at least one bounded weak solution for Kohn–Spencer Laplacian with a weight depending on the solution and convection term of the form \[ -div_{\mathbb{H}^n}(\nu(\xi,u) |D_{\mathbb{H}^n}u|^{p-2}_{\mathbb{H}^n}D_{\mathbb{H}^n}u)=f(\xi,u,D_{\mathbb{H}^n} u) \]−divHn(ν(ξ,u)|DHnu|Hnp−2DHnu)=f(ξ,u,DHnu) in a bounded domain $ \Omega \subset \mathbb {H}^n $ Ω⊂Hn. We show the set of solutions is uniformly bounded by a special Moser's iteration. [ABSTRACT FROM AUTHOR]

(1) Suppose μ is a smooth measure on a smooth hypersurface of positive Gaussian curvature in R 2 n . If n ≥ 2 , then W (μ) , the Weyl transform of μ is a compact operator, and if p > n ≥ 6 , then W (μ) belongs to the p-Schatten class. (2) There exist Schatten class operators with linearly dependent quantum translates. [ABSTRACT FROM AUTHOR]

In this paper, we study the rough fractional Hausdorff operator on variable exponent Morrey–Herz spaces in the setting of the Heisenberg group. We define Morrey–Herz spaces with three variable exponents and then give sufficient and necessary conditions for the boundedness of the rough fractional Hausdorff operator. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence and multiplicity of solutions for the following Kirchhoff-Choquard type equation involving the fractional p-Laplacian on the Heisenberg group: ... where (-Δ)ps is the fractional p-Laplacian on the Heisenberg group ℍN, M is the Kirchhoff function, V (ξ) is the potential function, 0 < s < 1,1 < p < N/s, μ > 0, f (ξ, u) is the nonlinear function, 0 < λ < Q, Q = 2N + 2, and Qλ* = 2Q-λ/Q-2 is the Sobolev critical exponent. Using the Krasnoselskii genus theorem, the existence of infinitely many solutions is obtained if μ is sufficiently large. In addition, using the fractional version of the concentrated compactness principle, we prove that problem has m pairs of solutions if μ is sufficiently small. As far as we know, the results of our study are new even in the Euclidean case. [ABSTRACT FROM AUTHOR]

In this paper, we study a class of dispersive wave equations on the Heisenberg group H n . Based on the group Fourier transform on H n , the properties of the Laguerre functions and the stationary phase lemma, we establish the decay estimates for a class of dispersive semigroup on H n given by e it ϕ (L) , where ϕ : R + → R is smooth, and L is the sub-Laplacian on H n . Finally, using the duality arguments, we apply the obtained results to derive the Strichartz inequalities for the solutions of some specific equations, such as the fractional Schrödinger equation, the fractional wave equation and the fourth-order Schrödinger equation. [ABSTRACT FROM AUTHOR]

We compute the point spectrum of the complex twisted Laplacian. We also refine a Theorem proved by Thangavelu. [ABSTRACT FROM AUTHOR]

Our research is divided into two parts. The first part studies the asymptotic behavior of a class of nonlocal diffusion problems in the fractional Sobolev–Slobodeckiĭ-spaces in the Heisenberg group by employing the theory of monotone operators and pullback attractors. In the second part of the manuscript, under some appropriate conditions, we study the well-posedness of a fractional time problem using fractional flows. To the best of our knowledge, this is the first time that fractional diffusion problems have been studied in the Heisenberg group. [ABSTRACT FROM AUTHOR]

In this paper, we characterize the system of left translates {L(2k,l,m)g: k, l,m ∈ Z}, g ∈ L²(H), to be a frame sequence or a Riesz sequence in terms of the twisted translates of the corresponding function gλ. Here, H denotes the Heisenberg group and gλ the inverse Fourier transform of g with respect to the central variable. This type of characterization for a Riesz sequence allows us to find some concrete examples. We also study the structure of the oblique dual of the system of left translates {L(2k,l,m)g: k, l, m ∈ Z} on H. This result is also illustrated with an example. [ABSTRACT FROM AUTHOR]

In this article, the Liouville theorems for the Kirchhoff-type parabolic equations on the Heisenberg group were established. The main technique for proving the result relies on the method of test functions.

We deal with multiplicity of solutions to the following Schrödinger-Poisson-type system in this article: ΔHu−μ1ϕ1u=∣u∣2u+Fu(ξ,u,v),inΩ,−ΔHv+μ2ϕ2v=∣v∣2v+Fv(ξ,u,v),inΩ,−ΔHϕ1=u2,−ΔHϕ2=v2,inΩ,ϕ1=ϕ2=u=v=0,on∂Ω,\left\{\begin{array}{ll}{\Delta }_{H}u-{\mu }_{1}{\phi }_{1}u={| u| }^{2}u+{F}_{u}\left(\xi ,u,v),\hspace{1.0em}& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega ,\\ -{\Delta }_{H}v+{\mu }_{2}{\phi }_{2}v={| v| }^{2}v+{F}_{v}\left(\xi ,u,v),\hspace{1.0em}& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega ,\\ -{\Delta }_{H}{\phi }_{1}={u}^{2},\hspace{1.0em}-{\Delta }_{H}{\phi }_{2}={v}^{2},\hspace{1.0em}& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega ,\\ {\phi }_{1}={\phi }_{2}=u=v=0,\hspace{1.0em}& \hspace{0.1em}\text{on}\hspace{0.1em}\hspace{0.33em}\partial \Omega ,\end{array}\right. where ΔH{\Delta }_{H} is the Kohn-Laplacian and Ω\Omega is a smooth bounded region on the first Heisenberg group H1{{\mathbb{H}}}^{1}, μ1{\mu }_{1}, and μ2{\mu }_{2} are some real parameters, and F=F(x,u,v),Fu=∂F∂uF=F\left(x,u,v),{F}_{u}=\frac{\partial F}{\partial u}, Fv=∂F∂u{F}_{v}=\frac{\partial F}{\partial u} satisfying natural growth conditions. By the limit index theory and the concentration compactness principles, we prove that the aforementioned system has multiplicity of solutions for μ1,μ2

In this article, we investigate the Kirchhoff-Schrödinger-Poisson type systems on the Heisenberg group of the following form: $ \begin{equation*} \left\{ \begin{array}{lll} {-(a+b\int_{\Omega}|\nabla_{H} u|^{p}d\xi)\Delta_{H, p}u-\mu\phi |u|^{p-2}u} = \lambda |u|^{q-2}u+|u|^{Q^{\ast}-2}u &\mbox{in}\ \Omega, \\ -\Delta_{H}\phi = |u|^{p} &\mbox{in}\ \Omega, \\ u = \phi = 0 &\mbox{on}\ \partial\Omega, \end{array} \right. \end{equation*} $ where $ a, b $ are positive real numbers, $ \Omega\subset \mathbb{H}^N $ is a bounded region with smooth boundary, $ 1 < p < Q $, $ Q = 2N + 2 $ is the homogeneous dimension of the Heisenberg group $ \mathbb{H}^N $, $ Q^{\ast} = \frac{pQ}{Q-p} $, $ q\in(2p, Q^{\ast}) $ and $ \Delta_{H, p}u = \mbox{div}(|\nabla_{H} u|^{p-2}\nabla_{H} u) $ is the $ p $-horizontal Laplacian. Under some appropriate conditions for the parameters $ \mu $ and $ \lambda $, we establish existence and multiplicity results for the system above. To some extent, we generalize the results of An and Liu (Israel J. Math., 2020) and Liu et al. (Adv. Nonlinear Anal., 2022).

In this paper, we study $ n $-dimensional Hardy operator and its dual in mixed radial-angular spaces on Heisenberg group and obtain their sharp bounds by using the rotation method. Furthermore, the sharp bounds of $ n $-dimensional weighted Hardy operator and weighted Cesàro operator are also obtained.

In this paper, we study the structure of truncated connections on the multidimensional Heisenberg group endowed with a left-invariant sub-Riemannian structure. We find that sub-Riemannian geodesic lines are parabolas whose orthogonal projections onto the corresponding contact planes are straight lines. In addition to such parabolas, some straight lines lying in contact planes are also geodesics. [ABSTRACT FROM AUTHOR]

Here, we consider an anisotropic equation \[ -\Delta _{{\mathbb {H}^n},\overrightarrow {p}}u +a(q)|u|^{p^–2}u=\lambda w(q)|u|^{m-2}u-h(q)|u|^{l-2}u, \] in the Heisenberg group {\mathbb {H}^n}, where the operator \Delta _{{\mathbb {H}^n},\overrightarrow {p}} is the horizontal anisotropic p-Laplacian on the Heisenberg group and is defined in the sequel. By the variational methods, we prove the existence of the entire weak solutions. [ABSTRACT FROM AUTHOR]

Let H denote the 3-dimensional Heisenberg Lie group. The main purpose of this paper is to classify all possible linear control systems on the homogeneous spaces of H through its closed subgroups. We expose a detailed study on the control behavior (i.e., controllability property and control sets) of some particular dynamics evolving on non simply connected homogeneous (state) spaces of dimension two and three. [ABSTRACT FROM AUTHOR]

In this paper, we study rough Hausdorff operators on variable exponent Lebesgue spaces in the setting of the Heisenberg group. We prove the boundedness of rough Hausdorff operators by giving some sufficient conditions. [ABSTRACT FROM AUTHOR]

A flag domain in ℝ 3 is a subset of ℝ 3 of the form { (x , y , t) : y < A ⁢ (x) } , where A : ℝ → ℝ is a Lipschitz function. We solve the Dirichlet and Neumann problems for the sub-elliptic Kohn–Laplacian △ ♭ = X 2 + Y 2 in flag domains Ω ⊂ ℝ 3 , with L 2 -boundary values. We also obtain improved regularity for solutions to the Dirichlet problem if the boundary values have first order L 2 -Sobolev regularity. Our solutions are obtained as sub-elliptic single and double layer potentials, which are best viewed as integral operators on the first Heisenberg group. We develop the theory of these operators on flag domains, and their boundaries. [ABSTRACT FROM AUTHOR]

In this paper, we define the notion of Bott connection in the Heisenberg group (ℍ 3 , g) and derive the expression of the Bott connection by using the Levi-Civita connection. Moreover, we derive the expressions of killing vector fields by using the killing equation and obtain some explicit formulas for killing magnetic curves associated to the Bott connection. Furthermore, we give some examples of killing magnetic curves. [ABSTRACT FROM AUTHOR]

The left-invariant sub-Lorentzian problem on the Heisenberg group is considered. An optimal synthesis is constructed, the sub-Lorentzian distance and spheres are described. [ABSTRACT FROM AUTHOR]

We establish the second variation of sub-Riemannian surface measure for minimal non-horizontal submanifolds of a sub-Riemannian stratified Lie group. We obtain some applications for codimension one. Furthermore, we present a new proof of the fact that the hyperbolic paraboloid is stable in the Heisenberg group. [ABSTRACT FROM AUTHOR]

Topological states of matter can be classified only in terms of global topological invariants. These global topological invariants are encoded in terms of global observable topological phase factors in the state vectors of electrons. In condensed matter, the energy spectrum of the Hamiltonian operator has a band structure, meaning that it is piecewise continuous. The energy in each continuous piece depends on the quasi-momentum which varies in the Brillouin zone. Thus, the Brillouin zone of quasi-momentum variables constitutes the base localization space of the energy eigenstates of electrons. This is a continuous topological parameter space bearing the homotopy of a torus. Since the base localization space has the homotopy of a torus, if we vary the quasi-momentum in a direction, when the edge of the zone is reached, we obtain a closed path. Then, if we lift this loop from the base space to the sections of the sheaf-theoretic fibration induced by the localization of the energy eigenfunctions, we obtain a global topological phase factor which encodes the topological structure of the Brillouin zone. Because it is homotopically equivalent to a torus, the global phase factor turns out to be quantized, taking integer values. The experimental significance of this model stems from the recent discovery that there are observable global topological phase factors in fairly ordinary materials. In this communication, we show that it is the unitary representation theory of the discrete Heisenberg group in terms of commutative modular symplectic variables, giving rise to a joint commutative representation space endowed with an integral and Z 2 -invariant symplectic form that articulates the specific form of the topological conditions characterizing both the quantum Hall effect and the spin quantum Hall effect under a unified sheaf-theoretic cohomological framework. [ABSTRACT FROM AUTHOR]

In this paper, we concern with the following noncooperative Choquard-Kirchhoff system: K (| ∇ H u | 2 2) Δ H u = (∫ Ω | u (η) | Q λ ∗ | η - 1 ξ | λ d η) | u | Q λ ∗ - 2 u + F u (ξ , u , v) in Ω , - K (| ∇ H v | 2 2) Δ H v = (∫ Ω | v (η) | Q λ ∗ | η - 1 ξ | λ d η) | v | Q λ ∗ - 2 v + F v (ξ , u , v) in Ω , u = 0 , v = 0 in H N \ Ω , where Ω is a smooth bounded domain in H N , K is the Kirchhoff function, and Q λ ∗ = 2 Q - λ Q - 2 is the Hardy-Littlewood-Sobolev critical exponent on the Heisenberg group. With the help of the Limit index theory due to Li and a version of the concentration-compactness principle for the Choquard equation on the Heisenberg group, we obtain that the multiplicity of solutions for a class of noncooperative elliptic system with homogeneous Dirichlet boundary conditions involving the Hardy-Littlewood-Sobolev critical exponent on the Heisenberg group. [ABSTRACT FROM AUTHOR]

We prove the existence of four-dimensional compact manifolds admitting some non-Einstein Lorentzian metrics, which are critical points for all quadratic curvature functionals. For this purpose, we consider left-invariant semi-direct extensions G S = H ⋊ exp (R S) of the Heisenberg Lie group H, for any S ∈ s p (1 , R) , equipped with a family g a of left-invariant metrics. After showing the existence of lattices in all these four-dimensional solvable Lie groups, we completely determine when g a is a critical point for some quadratic curvature functionals. In particular, some four-dimensional solvmanifolds raising from these solvable Lie groups admit non-Einstein Lorentzian metrics, which are critical for all quadratic curvature functionals. [ABSTRACT FROM AUTHOR]

We introduce and study model-theoretic connected components of rings as an analogue of model-theoretic connected components of definable groups. We develop their basic theory and use them to describe both the definable and classical Bohr compactifications of rings. We then use model-theoretic connected components to explicitly calculate Bohr compactifications of some classical matrix groups, such as the discrete Heisenberg group ${\mathrm {UT}}_3({\mathbb {Z}})$ , the continuous Heisenberg group ${\mathrm {UT}}_3({\mathbb {R}})$ , and, more generally, groups of upper unitriangular and invertible upper triangular matrices over unital rings. [ABSTRACT FROM AUTHOR]