Upper-half Space Research Articles

Thermoelastic fracture of two-dimensional hexagonal quasicrystal media weakened by a penny-shaped crack subjected to uniformly antisymmetric heat fluxes

This study focuses on the thermoelastic fracture behavior of quasicrystals, a class of solids that exhibit unique properties between traditional crystals and amorphous materials. The complex atomic structure of quasicrystals poses challenges in understanding their fracture mechanisms under thermoelastic loading conditions. Central to our investigation is an analytical examination of a penny-shaped crack in an infinite three-dimensional body composed of two-dimensional hexagonal quasicrystals, where the upper and lower crack surfaces are applied to a pair of uniformly antisymmetric heat fluxes. This crack problem requires simultaneous consideration of thermal-phason-phonon multiple field coupling. According to the symmetry of field variables with respect to the crack plane, the thermal-phason-phonon coupled crack problem is transformed into a mixed boundary value problem in the upper half space. The extended displacement discontinuities, encompassing both phonon and phason displacement discontinuities, as well as the temperature discontinuity, are chosen as the basic unknown variables to construct the boundary integral-differential equations governing the mixed boundary value problem. Based on these boundary governing equations and Fabrikant's potential theory method, the problem with the crack surface subjected to uniform antisymmetric heat fluxes is solved. The solutions of thermal-phason-phonon fields on the crack plane, and in the full space are given in closed-form. Numerical results are employed to validate the obtained analytical solutions and visually illustrate the spatial distribution of thermal-phonon-phason coupling fields in the vicinity of the crack. The study provides fundamental insights into the behavior of cracks in quasicrystals under thermal loading, with potential implications for the design of new materials and structures.

Analytical solutions to Mode I penny-shaped crack problems in two-dimensional hexagonal quasicrystals with piezoelectric effect

The paper studies a penny-shaped crack in an infinite three-dimensional body of two-dimensional hexagonal quasicrystal media with piezoelectric effect. The crack surfaces are applied combined electric and normal phonon loadings. Such a Model I crack problem is transformed into a mixed boundary value problem in the upper half-space, which is analytically solved using Fabrikant's potential theory method. The boundary integral-differential equations governing Model I crack problems are presented for two-dimensional hexagonal piezoelectric quasicrystals. The normal phonon displacement discontinuity and electric potential discontinuity across crack surfaces are taken as the unknown variables of boundary governing equations. Analytical solutions of all field variables are derived not only for the crack plane but also for the full space. Solutions in integral form are provided for the penny-shaped crack under arbitrarily distributed electric and normal phonon loadings. Closed-form solutions in terms of elementary functions are given for concentrated point loadings and uniformly distributed loadings, respectively. Key fracture mechanics parameters, such as crack surface extended displacements (i.e., normal phonon displacement, electric potential), crack tip extended stresses (i.e., normal phonon stress, electric displacement) distribution, and corresponding extended stress intensity factors, are clearly derived. Numerical results are utilized to verify the present analytical solutions and graphically illustrate the distribution of phonon-phason-electric coupling fields around the crack. The present solution can serve as a benchmark for both experimental and numerical investigations.

Skyglow from ground-reflected radiation: model improvements

ABSTRACT Light from ground-based sources directed into the upper hemisphere can be partially controlled, for example, through suitable lamp shades or by reducing the number of luminaires and their lumen output. However, ground-reflected radiation is pervasive in artificially lit urban environments and cannot be entirely avoided. This component of upward-directed light is typically modelled using Lambertian diffuse reflection. Here, we demonstrate that the current analytical models for ground-reflected radiation can be improved by incorporating additional components, such as reflections from vertically oriented surfaces and vegetation. Our findings indicate that near edges of cities, the contribution of the reflected radiation to the overall skyglow is slightly higher than recent models predict. However, at medium-to-long distances the skyglow increases by 50–200 per cent compared to what these models suggest. This is because non-horizontal surfaces reflect more light towards small angles above the horizontal, contrasting with the outcomes predicted by the basic Lambertian framework. Consequently, light escaping from ground-based sources can propagate more effectively over longer distances, even when there is a complete cutoff of light emitted from luminaires in the upper half-space. These findings have significant implications for skyglow modelling. Furthermore, it also limits the available options for implementing measures to reduce skyglow levels at astronomical observatories located well beyond city boundaries.

$$L^1-$$decay of higher-order norms of solutions to the Navier–Stokes equations in the upper-half space

$$L^1-$$decay of higher-order norms of solutions to the Navier–Stokes equations in the upper-half space

On the Dirichlet problem for the elliptic equations with boundary values in variable Morrey-Lorentz spaces

Under a metric measure space setting, we show that a solution to the elliptic equation with a non-negative potential, defined on the upper half-space, is in the essentially-bounded-Morrey-Lorentz space with variable exponent if and only if it can be represented as the Poisson integral of a variable Morrey-Lorentz function, where the doubling property, the pointwise upper bound on the heat kernel, the mean value property and the Liouville property are assumed. This extends the known result of Stein-Weiss in 1971 from the Euclidean space to the metric measure space, and from the Lebesgue function to the Morrey-Lorentz function with variable exponent.

Programmable metasurfaces with high angular stability for arbitrarily-polarized obliquely-incident waves

Programmable metasurfaces show enormous potential in real-time electromagnetic (EM) manipulation and their stable responses to variable EM waves including incident angle and polarization changes are crucial for related applications. However, the demonstrated programmable metasurfaces are commonly considered in normal-incident waves with fixed polarization; how to achieve stable performance in more realistic scenarios of wide-angle and full-polarized wave incidences is still a challenge. Here, we propose and realize a wide-angle and full-polarization programmable metasurface, which can generate stabilized amplitude and phase responses at both transverse electric (TE) and transverse magnetic (TM) oblique incidences. These are achieved by introducing metallic walls around the metasurface element to reduce element coupling, which greatly improves the angular stability of its reflection responses. The mechanisms of angular insensitivity are analyzed comprehensively, and as a proof of concept, obliquely-incident beam steering, large-angle beam scanning and oblique vortex beam emitting are demonstrated on this programmable metasurface. Both the simulation and experimental results demonstrate that the programmable metasurface can operate well at both TE and TM wide-angle oblique incidences in the upper half space. Our work offers an actual route for improving the angular stability and polarization insensitivity of metasurfaces, which could push them one step closer towards more complicated applications.

A Selberg Trace Formula for GL3(Fp)∖GL3(Fq)/K

In this paper, we prove a discrete analog of the Selberg Trace Formula for the group GL3(Fq). By considering a cubic extension of the finite field Fq, we define an analog of the upper half-space and an action of GL3(Fq) on it. To compute the orbital sums, we explicitly identify the double coset spaces and fundamental domains in our upper half space. To understand the spectral side of the trace formula, we decompose the induced representation ρ=IndΓG1 for G=GL3(Fq) and Γ=GL3(Fp).

Transference of shear horizontal waves in a functionally graded piezoelectric structure

A brief analysis of the transference of shear horizontal wave has been carried out in a functionally graded half-spaces bonded imperfectly. The top half-space is functionally graded piezoelectric material whereas the bottom half-space is elastic substrate and top half-space is graded quadratically. To obtain the solution in upper half-space, the partial differential equation is transformed into an ordinary differential equation using the variable-separable method, and WKB asymptotic technique is employed to derive analytic solutions for the functionally graded half-space, enabling the determination of dispersion relations in the form of a determinant. Reflection and transmission coefficients, representing ratios of wave amplitudes, are derived for an imperfect boundary. Numerical analysis focuses on a specific model and the graphical interpretation of results is facilitated using Mathematica-7.

Sharp affine weighted L 2 Sobolev inequalities on the upper half space

Abstract We establish some sharp affine weighted L 2 Sobolev inequalities on the upper half space, which involves a divergent operator with degeneracy on the boundary. Moreover, for some certain exponents cases, we also characterize the extremal functions and best constants. Our approach only relies on the L 2 structure of gradient norm, affine invariance and a class of weighted L 2 Sobolev inequality on the upper half space. This is a simple approach which does not depend on the geometric structure of Euclidean space such as Brunn–Minkowski theory on convex geometry.

The stationary Gierer–Meinhardt system in the upper half-space: existence, nonexistence and asymptotics

We study the existence, nonexistence and asymptotics of positive solutions to the Gierer–Meinhardt system -Δu+λu=upvq+ρ(x),u>0inR+N:=RN-1×(0,∞),-Δv+λv=umvs,v>0inR+N,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} {\\left\\{ \\begin{array}{ll} \\displaystyle -\\Delta u+\\lambda u=\\frac{u^p}{v^q}+\\rho (x)\\,, u>0 &{} \ ext{ in } {\\mathbb {R}}^N_+:={\\mathbb {R}}^{N-1}\ imes (0, \\infty ), \\\\ \\displaystyle -\\Delta v+\\lambda v=\\frac{u^m}{v^s} \\,, v>0 &{} \ ext{ in } {\\mathbb {R}}^N_+, \\end{array}\\right. } \\end{aligned}$$\\end{document}where N≥3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N\\ge 3$$\\end{document}, λ>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda >0$$\\end{document} and ρ∈C(R+N¯)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\rho \\in C({\\overline{{\\mathbb {R}^N_+}}})$$\\end{document}, ρ>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\rho >0$$\\end{document}. The solutions (u, v) are assumed to satisfy the homogeneous Neumann boundary condition ∂u∂xN=∂v∂xN=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\displaystyle \\frac{\\partial u}{\\partial x_N}=\\frac{\\partial v}{\\partial x_N}=0$$\\end{document} on ∂R+N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\partial {\\mathbb {R}}^N_+$$\\end{document} and u(x)→0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$u(x)\\rightarrow 0$$\\end{document}, v(x)→0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$v(x)\\rightarrow 0$$\\end{document} as x∈R+N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$x\\in {\\mathbb {R}}^N_+$$\\end{document}, |x|→∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$|x|\\rightarrow \\infty $$\\end{document}. Under various conditions on the exponents m,p,q,s>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$m,p,q,s>0$$\\end{document} and the data ρ(x)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\rho (x)$$\\end{document} we obtain qualitative properties of the solutions (u, v). In particular, we derive the existence of a solution (u, v) with u(x) having the minimal asymptotic behaviour u(x)≃Φλ(x)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$u(x)\\simeq \\Phi _\\lambda (x)$$\\end{document} as |x|→∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$|x|\\rightarrow \\infty $$\\end{document}, where Φλ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Phi _\\lambda $$\\end{document} denotes the fundamental solution of -Δ+λI\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$-\\Delta +\\lambda I$$\\end{document} in R+N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}^N_+}$$\\end{document}. The approach combines integral representations of solutions and various integral estimates with fixed point arguments.

Fundamental properties of Cauchy–Szegő projection on quaternionic Siegel upper half space and applications

Abstract We investigate the Cauchy–Szegő projection for quaternionic Siegel upper half space to obtain the pointwise (higher order) regularity estimates for Cauchy–Szegő kernel and prove that the Cauchy–Szegő kernel is nonzero everywhere, which further yields a non-degenerated pointwise lower bound. As applications, we prove the uniform boundedness of Cauchy–Szegő projection on every atom on the quaternionic Heisenberg group, which is used to give an atomic decomposition of regular Hardy space H p {H^{p}} on quaternionic Siegel upper half space for 2 3 < p ≤ 1 {\frac{2}{3}<p\leq 1} . Moreover, we establish the characterisation of singular values of the commutator of Cauchy–Szegő projection based on the kernel estimates. The quaternionic structure (lack of commutativity) is encoded in the symmetry groups of regular functions and the associated partial differential equations.

Bounds for the Bergman Kernel and the Sup-Norm of Holomorphic Siegel Cusp Forms

Abstract We prove “polynomial in $k$” bounds on the size of the Bergman kernel for the space of holomorphic Siegel cusp forms of degree $n$ and weight $k$. When $n=1,2$ our bounds agree with the conjectural bounds, while the lower bounds match for all $n \ge 1$. For an $L^{2}$-normalized Siegel cusp form $F$ of degree $2$, our bound for its sup-norm is $O_{\epsilon } (k^{9/4+\epsilon })$. Further, we show that in any compact set $\Omega $ (which does not depend on $k$) contained in the Siegel fundamental domain of $\textrm {Sp}(2,\mathbb {Z})$ on the Siegel upper half space, the sup-norm of $F$ is $O_{\Omega }(k^{3/2 - \eta })$ for some $\eta>0$, going beyond the “generic” bound in this setting.

Stein–Weiss type inequality on the upper half space and its applications

In this paper, we establish some Stein–Weiss type inequalities with general kernels on the upper half space and study the extremal functions of the optimal constant. Furthermore, we also investigate the regularity, asymptotic estimates, symmetry and non-existence results of positive solutions of corresponding Euler–Lagrange system. As an application, we derive some Liouville type results for the Hartree type equations on the half space.

Cylindrical singular minimal surfaces

We construct a foliation of the upper half space in \mathbb{R}^3 consisting of minimizing cylindrical \alpha -singular minimal surfaces when \alpha<0 . Furthermore, we discuss stability results for the \alpha -catenaries when \alpha>0 and relate these to the geodesics of a conformal metric.

Modulo-addition operation enables terahertz programmable metasurface for high-resolution two-dimensional beam steering.

Electrically controlled terahertz (THz) beamforming antennas are essential for various applications such as wireless communications, security checks, and radar to improve coverage and information capacity. The emerging programmable metasurface provides a flexible, cost-effective platform for THz beam steering. However, scaling such arrays to achieve high-gain beam steering faces several technical challenges. Here, we propose a pixelated liquid crystal THz metasurface with a crossbar structure, thereby increasing the array scale to more than 3000. The coding pattern on the programmable device is generated by the modulo-addition of the coding sequences on the top and bottom layers. We experimentally demonstrate the programmable liquid crystal metasurface capable of active beam deflection in the upper half-space. This scale-up of programmable devices opens exciting opportunities in pencil beamforming, high-speed information processing, and optical computing.

The Boundedness of Forelli–Rudin Type Operators on the Siegel Upper Half Space

The Boundedness of Forelli–Rudin Type Operators on the Siegel Upper Half Space

On the Dirichlet problem for the Schrödinger equation in the upper half-space

On the Dirichlet problem for the Schrödinger equation in the upper half-space

A compact star-shaped leaky-wave antenna for half-space steering

ABSTRACT In this paper, a compact leaky-wave planer antenna is proposed. The proposed antenna fabricated by the printed circuit board technology can scan a large part of its upper half-space. The proposed antenna consists of a metallic six-pointed star on a regular hexagon. The back of the hexagon is wholly covered with a conductor. The structure’s detailed design and analysis have been done using the dispersion diagram. In the proposed antenna, changing the frequency can steer the main beam in elevation at six-step angles of azimuth. The simulation and measurement results confirm the theory well. The leaky wave structure is an attractive and economical method to scan half-space.

Modified Quaternionic Analysis on the Ball

A natural generalization of the classical complex analysis to higher dimensions is the theory of monogenic functions (see, e.g. Brackx et al. in Clifford analysis, Pitman, Boston, 1982). Unfortunately the powers of the underlying variable are not in this system. But they are in the modified system introduced by the first author in Leutwiler (Complex Var Theory Appl 20:19–51, 1992). Since this modified system lives in the upper half space {mathbb {R}}_{+}^{3} it is natural to transplant it—with the so-called Cayley mapping—to the unit ball B(0, 1) in {mathbb {R}}^{3}. It is this transplanted system which we are going to investigate in this article. It represents the counterpart to the theory of (H)-solutions studied in Leutwiler (Complex Var Theory Appl 20:19–51, 1992).

Study of SH-wave in a pre-stressed anisotropic magnetoelastic layer sandwich by heterogeneous semi-infinite media

A brief study of SH-wave has been encountered in a pre-stressed anisotropic magnetoelastic layer loaded by heterogeneous half-spaces. The semi-infinite media are heterogeneous with different heterogeneity parameters and the upper half-space is taken exponentially varying with depth while the lower medium varying linearly. The solutions are more apparent with analytical methods clubbed with Mathematica-7 to get the numerical results for a particular model with available data. The study involves obtaining the closed-form expression for reflection and transmission coefficients with the continuous boundary conditions at interfaces. The terms for reflection and transmission coefficients manifest the notable dependencies of material coefficients with inhomogeneity parameters, layer thickness, and the known components of wave propagation, i.e., phase velocity, wave number, and incidence angle. The two special cases have been considered, i.e., one when the layer vanishes means only two heterogeneous half-spaces and the other two isotropic half-spaces that lead to validation of results with well-known reference. The numerical results are shown by graphs which make the result transpicuous to interpret physically.