Half Space Research Articles

Reflection and transmission of plane wave at the interface between two distinct nonlocal triclinic micropolar generalized thermoelastic half spaces under DPL and LS theory

Reflection and transmission of plane wave at the interface between two distinct nonlocal triclinic micropolar generalized thermoelastic half spaces under DPL and LS theory

A Liouville-type theorem and one-dimensional symmetry of solutions for elliptic equations with general gradient nonlinearity

In this paper, we are concerned with the following elliptic equation involving a general gradient nonlinearity:−Δv=μ|v|p−1v+G(∇v)inΩ, where n≥1, either Ω=Rn is the whole space or Ω=R+n={x=(x1,…,xn)∈Rn:xn>0} is the half space, μ≥0, p∈R and G:Rn→R satisfies some suitable conditions. One particular case of G is G(γ)=∑i=1kai|γ|pi with ai>0 and pi>1 for any 1≤i≤k. When Ω=Rn, we establish a Bernstein estimate to prove some Liouville-type results of the equation under certain conditions on μ and p. When Ω=R+n, by combining the Bernstein estimates with a moving planes argument, we obtain the one-dimensional symmetry of the solution v with v=0 on ∂R+n. We note that the exponent p is allowed to be negative in our results. Finally, we obtain a gradient estimate in some special cases with μ>0 and p>1.

On initial-boundary value problem of the stochastic Navier–Stokes equations in the half space

We study the initial-boundary value problem of the stochastic Navier–Stokes equations in half-space. We prove the existence of weak solutions in standard Besov space-valued random processes when the initial data belong to the critical Besov space.

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.

Mesh-free kinematic shakedown analysis of cohesive soils

A novel mesh-free solution is proposed for kinematic shakedown analysis of cohesive soils under repeating loads. For this purpose, the continuous velocity field in the mathematical expression of Koiter’s theorem is discretized by the Radial Point Interpolation Method (RPIM), as a mesh-free approach. The strain rate smoothing technique is implemented in conjunction with the RPIM to satisfy the admissibility conditions at the entire problem domain. Using the nodal integration and the discretized velocity field, the kinematic shakedown problem is expressed as a nonlinear optimization problem. The optimization problem is solved by separation of plastic and non-plastic/rigid zones using a repetitive algorithm. Eventually, the efficiency of the proposed approach is elucidated by solving examples of a strip footing resting on cohesive soil and a cohesive half space pavement under repeating loads.

Thermoelastic diffusion in a half space due to fractional order theory

Based on the theory of fractional thermoelastic diffusion, a two dimensional problem for thermoelastic half space is considered. The upper surface of the half space is taken to be traction free and subjected to ramp type heating. Integral transform technique and a direct approach method are used to obtain the solution in the transformed domain. Numerical inversion techniques are used to obtain the inverse double transforms. The effect of Tempered-Caputo fractional derivative is studied. Numerical results are discussed and represented graphically.

Finding repeatable progressive pass clusters and application in international football

Progressive passing in football (soccer) is a key aspect in creating positive possession outcomes. Whilst this is well established, there is not a consistent way to describe the different types of progressive passes. We expand on the previous literature, providing a complete methodological approach to progressive pass clustering from selection of the number of clusters (k) to risk-reward profiling of these progressive pass types. In this paper the Separation and Concordance (SeCo) framework is utilised to provide a process to analyse k-means clustering solutions in a more repeatable way. The results demonstrate that we can find stable progressive pass clusters in International Football and their efficacy with progressive passes “Mid Central to Mid Half Space” in build-up and “Mid Half Space to Final Central” into the final 3rd having the best balance between risk (turnover) and reward (shot created) in the subsequent possession. This allowed for opposition profiling of player and team patterns in different phases of play, with a case study presented for the teams in the Last 16 of the 2022 World Cup.

The surface counter-terms of the ϕ44 theory on the half space R+×R3

In a previous work, we established perturbative renormalizability to all orders of the massive ϕ44-theory on a half-space also called the semi-infinite massive ϕ44-theory. Five counter-terms which are functions depending on the position in the space, were needed to make the theory finite. The aim of the present paper is to establish that for a particular choice of the renormalization conditions the effective action consists of a part which is independent of the boundary conditions (Dirichlet, Neumann and Robin) plus a boundary term in the case of the Robin and Neumann boundary conditions. The key idea of our method is the decomposition of the correlators into a bulk part, which is defined as the scalar field model on the full space R4 with a quartic interaction restricted to the half-space, plus a remainder which we call “the surface part.” We analyse this surface part and establish perturbatively that the ϕ44 theory in R+×R3 is made finite by adding the bulk counter-terms and two additional counter-terms to the bare interaction in the case of Robin and Neumann boundary conditions. These surface counter-terms are position independent and are proportional to ∫Sϕ2 and ∫Sϕ∂nϕ. For Dirichlet boundary conditions, we prove that no surface counter-terms are needed and the bulk counter-terms are sufficient to renormalize the connected amputated (Dirichlet) Schwinger functions. A key technical novelty as compared to our previous work is a proof that the power counting of the surface part of the correlators is better by one scaling dimension than their bulk counterparts.

Surface wave propagation along a narrow transition layer in a slab Voigt geometry

The dispersion properties of electromagnetic surface waves, which propagate along a slab narrow transition layer separating two magnetoactive plasma uniform half spaces, are studied. Voigt geometry is considered, in which the waves propagate perpendicularly to an external uniform static magnetic field, which in turn is parallel to the interfaces between the plasmas and the transition layer. Electromagnetic power absorption within local resonances inside the transition layer is out of scope of the study. The influence of the plasma particle density profile smoothness on the surface wave dispersion properties is emphasized.

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.

Rayleigh waves in isotropic elastic materials with micro-voids

In this paper, we show that a general method introduced by Fu and Mielke allows to give a complete answer on the existence and uniqueness of a subsonic solution describing the propagation of surface waves in an isotropic half space modelled with the linear theory of isotropic elastic materials with micro-voids. Our result is valid for the entire class of materials admitting real wave propagation which include auxetic materials (negative Poisson’s ratio) and composite materials with negative-stiffness inclusions (negative Young’s modulus). Moreover, the used method allows to formulate a simple and complete numerical strategy for the computation of the solution.

Analytical solution to quickly assess ground displacement for a pressurized or depleted deep reservoir intersected by a fault in a half space

Quick estimates of fluid-induced subsurface deformation are helpful to assess the land uplift/subsidence and to reveal precursors of induced seismicity. Here, we adopt the inclusion theory and Green's function to develop a closed-form solution in a half space for the poroelastic response of a reservoir compartmentalized by an intersecting fault that can be offset and either permeable or impermeable. Simulated results reveal that (1) fault permeability mainly impacts the spatial distribution of displacement while its effect on displacement magnitude is small; (2) ground displacement slightly increases with fault dip while slightly decreases with increasing fault offset; in contrast, reservoir geometry shows a stronger effect than fault geometry: the ground displacement is proportional to the vertical and lateral depth ratios, defined as the ratios of reservoir thickness (h) and width (w) to reservoir depth (D), respectively; (3) the maximum vertical displacement is the double of the horizontal one regardless of fault permeability, fault and reservoir geometries, and mechanical parameters. Comparing the solution in a half space with that in a full space shows that neglecting the free surface underestimates the poroelastic displacement in the overburden. The validity of full-space solutions can be assessed with the product of the lateral and vertical depth ratios, i.e., wh/D2. The full-space solutions become valid when wh/D2 decreases to an intrinsic threshold. This threshold may range from 0.01 to 0.02 for displacement, and its specific value depends on the field background and demands of projects, but can be estimated based on our solution. It is larger for stress than for displacement, and wh/D2 ≤ 0.1 is recommended as a general condition for neglecting the free-surface effects on induced stress. The analytical solution represents a useful tool for estimating ground deformation and for gaining insights of reservoir and fault geometries by analyzing surface deformation patterns.

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.

GHz surface waves in Al/LiTaO3/Si composite: Effect of the Drude electrode on dispersion, attenuation and mode shapes

In fifth generation (5G) mobile communication, radio frequency is in the super high frequency range (3–30 GHz). Thus, different from Ohm law, electrode loss at GHz frequency becomes frequency-dependent. As such, many properties, especially attenuation of piezoelectric acoustic devices, should be reevaluated. In this paper, the attenuation of GHz surface wave propagating in composite structure made of metal electrode, piezoelectric film and elastic half space is investigated based on the loss model of Drude electrode. Specifically, the Al/LiTaO3/Si composite is considered where Rayleigh mode and SH mode are fully coupled. Based on the Stroh-type formalism, the dispersion equations of the composite with the Drude electrode are derived, and compared with the traditional perfect conductor electrode which is under the short circuit boundary condition. The phase velocity, attenuation and wave mode shapes are obtained successfully by the multidimensional moduli ratio convergence method. Numerical results are presented to reveal some interesting intrinsic features, which include two analytical upper bounds of phase velocity, nonexistent validation of intersection points in dispersion curves, and relations among veering, attenuation jumps and wave-mode conversions. These features and relations provide an in-depth insight for GHz wave motion properties in piezoelectric composites.

Rayleigh wave through half space semiconductor solid with temperature dependent properties

The article focuses on Rayleigh wave propagation in a homogeneous isotropic semi-conductor thermoelastic medium rotating with fixed angular frequency with temperature-dependent properties. The elastic constants depend upon the temperature function. The effects of temperature dependency parameter, time derivative, and fractional order are illustrated. By the theory of thermo-elasticity, waves result in the generation of thermal signals that propagate through the medium. A heat conduction model of three-phase lag (3PL) along with fractional order time derivative is used to analyze the thermal signals. The secular equations of Rayleigh waves are derived mathematically at the stress-free, carrier density and thermally insulated boundaries. Some specific properties like velocity, attenuation coefficient, specific heat loss and penetration depth for Rayleigh waves have been evaluated and presented graphically. The secular equations are computed numerically and depicted graphically using Matlab.

Global existence and large-time behavior of solutions to the 1D compressible Navier-Stokes equations with density-dependent viscosity in the half space

Global existence and large-time behavior of solutions to the 1D compressible Navier-Stokes equations with density-dependent viscosity in the half space

Exact Expressions for the Near Field of a Thin Uniform Circular Loop Current with Application to Loops Lying on a Half Space

Exact Expressions for the Near Field of a Thin Uniform Circular Loop Current with Application to Loops Lying on a Half Space

Monotonicity and Liouville type theorems of solutions to fully nonlinear fractional problems in the half space

Monotonicity and Liouville type theorems of solutions to fully nonlinear fractional problems in the half space

HubHSP graph: Capturing local geometrical and statistical data properties via spanning graphs

The computation of a continuous generative model to describe a finite sample of an infinite metric space can prove challenging and lead to erroneous hypothesis, particularly in high-dimensional spaces. In this paper, we follow a different route and define the Hubness Half Space Partitioning graph (HubHSP graph). By constructing this spanning graph over the dataset, we can capture both the geometrical and statistical properties of the data without resorting to any continuity assumption. Leveraging the classical graph-theoretic apparatus, the HubHSP graph facilitates critical operations, including the creation of a representative sample of the original dataset, without relying on density estimation. This representative subsample is essential for a range of operations, including indexing, visualization, and machine learning tasks such as clustering or inductive learning. With the HubHSP graph, we can bypass the limitations of traditional methods and obtain a holistic understanding of our dataset’s properties, enabling us to unlock its full potential.

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We discuss entanglement entropies in 4d interface CFTs based on 4d N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 SYM coupled to 3d N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 degrees of freedom localized on an interface. Focusing on the entanglement between the two half spaces to either side of the interface, we show that applying the Ryu-Takayanagi prescription in general leads to multiple natural entangle- ment entropies. We interpret the different entropies as corresponding to different ways of assigning the 3d degrees of freedom localized on the interface to the two half spaces. We contrast these findings with recent discussions of universal relations for entanglement entropies in 2d interface CFTs and formulate generalized relations for 4d interface CFTs which incorporate our results.