Boundary Sphere Research Articles

Singular extension of critical Sobolev mappings under an exponential weak-type estimate

Given m∈N∖{0} and a compact Riemannian manifold N, we construct for every map u in the critical Sobolev space Wm/(m+1),m+1(Sm,N), a map U:B1m+1→N whose trace is u and which satisfies an exponential weak-type Sobolev estimate. The result and its proof carry on to the extension to a half-space of maps on its boundary hyperplane and to the extension to the hyperbolic space of maps on its boundary sphere at infinity.

Polytopality of Simple Games

The Bier sphere Bier ( G ) = Bier ( K ) : = K ∗ Δ K ° and the canonical fan Fan ( Γ ) = Fan ( K ) are combinatorial/geometric companions of a simple game G = ( P , Γ ) (equivalently the associated simplicial complex K), where P is the set of players, Γ ⊆ 2 P is the set of wining coalitions, and K : = 2 P ∖ Γ is the simplicial complex of losing coalitions. We propose and study a general “Steinitz problem” for simple games as the problem of characterizing which games G are polytopal (canonically polytopal) in the sense that the corresponding Bier sphere Bier ( G ) (fan Fan ( Γ ) ) can be realized as the boundary sphere (normal fan) of a convex polytope. We characterize (roughly) weighted majority games as the games Γ which are canonically (pseudo) polytopal (Theorems 1.1 and 1.2) and show, by an experimental/theoretical argument (Theorem 1.4), that simple games such that Bier ( G ) is nonpolytopal do not exist in dimension 3. This should be compared to the fact that asymptotically almost all simple games are nonpolytopal and a challenging open problem is to find a nonpolytopal simple game with the smallest number of players.

Grey Galaxies’ as an endpoint of the Kerr-AdS superradiant instability

Kerr-AdSd+1 black holes for d ≥ 3 suffer from classical superradiant instabilities over a range of masses above extremality. We conjecture that these instabilities settle down into Grey Galaxies (GGs) — a new class of coarse-grained solutions to Einstein’s equations which we construct in d = 3. Grey Galaxies are made up of a black hole with critical angular velocity ω = 1 in the ‘centre’ of AdS, surrounded by a large flat disk of thermal bulk gas that revolves around the centre of AdS at the speed of light. The gas carries a finite fraction of the total energy, as its parametrically low energy density and large radius are inversely related. GGs exist at masses that extend all the way down to the unitarity bound. Their thermodynamics is that of a weakly interacting mix of Kerr-AdS black holes and the bulk gas. Their boundary stress tensor is the sum of a smooth ‘black hole’ contribution and a peaked gas contribution that is delta function localized around the equator of the boundary sphere in the large N limit. We also construct another class of solutions with the same charges; ‘Revolving Black Holes (RBHs)’. RBHs are macroscopically charged SO(d, 2) descendants of AdS-Kerr solutions, and consist of ω = 1 black holes revolving around the centre of AdS at a fixed radial location but in a quantum wave function in the angular directions. RBH solutions are marginally entropically subdominant to GG solutions and do not constitute the endpoint of the superradiant instability. Nonetheless, we argue that supersymmetric versions of these solutions have interesting implications for the spectrum of supersymmetric states in, e.g. 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 Yang-Mills theory.

Disoriented homology and double branched covers

AbstractThis paper provides a convenient and practical method to compute the homology and intersection pairing of a branched double cover of the 4-ball.To projections of links in the 3-ball, and to projections of surfaces in the 4-ball into the boundary sphere, we associate a sequence of homology groups, called the disoriented homology. We show that the disoriented homology is isomorphic to the homology of the double branched cover of the link or surface. We define a pairing on the first disoriented homology group of a surface and show that this is equal to the intersection pairing of the branched cover. These results generalize work of Gordon and Litherland, for embedded surfaces in the 3-sphere, to arbitrary surfaces in the 4-ball. We also give a generalization of the signature formula of Gordon–Litherland to the general setting.Our results are underpinned by a theorem describing a handle decomposition of the branched double cover of a codimension-2 submanifold in the n-ball, which generalizes previous results of Akbulut–Kirby and others.

Acoustic radiation force of a sphere with a hemispherically split boundary condition in a plane wave

In this paper, an analytical expression of the acoustic radiation force (ARF) for a spherical particle with a hemispherically split impedance boundary in a plane wave is deduced. Numerical calculations are carried out by considering the effect of the magnitude and phase of the acoustic impedance on the ARF. Computation results show that the increase in the magnitude of the acoustic impedance results in an overall decrease in the ARF, whereas the phase of the acoustic impedance results in a decrease in the ARF in the low frequency region. As the frequency increases, the positive phase angle leads to a decrease in the ARF, and the negative phase angle causes the ARF to increase rapidly. For a hemispherically split impedance sphere, the values of the ARF range from those of the rigid sphere and uniform impedance sphere. The finite-element models for the calculation of the ARF of a hemispherically split impedance boundary sphere are established and the correctness of the analytical theory is proved by numerical comparison. This work is expected to contribute theoretical support to the acoustic manipulation of particles with a nonuniform hemispherically split structure.

Coupling a Global Heliospheric Magnetohydrodynamic Model to a Magnetofrictional Model of the Low Corona

Abstract Recent efforts coupling our Sun-to-Earth magnetohydrodynamics (MHD) model and lower-corona magnetofrictional (MF) model are described. Our Global Heliospheric MHD (GHM) model uses time-dependent three-component magnetic field data from the lower-corona MF model as time-dependent boundary values. The MF model uses data-assimilation techniques to introduce the vector magnetic field data from the Solar Dynamics Observatory/Helioseismic and Magnetic Imager, hence as a whole this simulation coupling structure is driven with actual observations. The GHM model employs a newly developed interface boundary treatment that is based on the concept of characteristics, and it properly treats the interface boundary sphere set at a height of the sub-Alfvénic lower corona (1.15 R ⊙ in this work). The coupled model framework numerically produces twisted nonpotential magnetic features and consequent eruption events in the solar corona in response to the time-dependent boundary values. The combination of our two originally independently developed models presented here is a model framework toward achieving further capabilities of modeling the nonlinear time-dependent nature of magnetic field and plasma, from small-scale solar active regions to large-scale solar wind structures. This work is a part of the Coronal Global Evolutionary Model project for enhancing our understanding of Sun–Earth physics to help improve space weather capabilities.

Volume decay and concentration of high-dimensional Euclidean balls – a PDE and variational perspective

Abstract It is a well-known fact – which can be shown by elementary calculus – that the volume of the unit ball in ℝ n {\mathbb{R}^{n}} decays to zero and simultaneously gets concentrated on the thin shell near the boundary sphere as n ↗ ∞ {n\nearrow\infty} . Many rigorous proofs and heuristic arguments are provided for this fact from different viewpoints, including Euclidean geometry, convex geometry, Banach space theory, combinatorics, probability, discrete geometry, etc. In this note, we give yet another two proofs via the regularity theory of elliptic partial differential equations and calculus of variations.

Polytopal Bier Spheres and Kantorovich–Rubinstein Polytopes of Weighted Cycles

The problem of deciding if a given triangulation of a sphere can be realized as the boundary sphere of a simplicial, convex polytope is known as the ‘Simplicial Steinitz problem’. It is known by an indirect and non-constructive argument that a vast majority of Bier spheres are non-polytopal. Contrary to that, we demonstrate that the Bier spheres associated to threshold simplicial complexes are all polytopal. Moreover, we show that all Bier spheres are starshaped. We also establish a connection between Bier spheres and Kantorovich–Rubinstein polytopes by showing that the boundary sphere of the KR-polytope associated to a polygonal linkage (weighted cycle) is isomorphic to the Bier sphere of the associated simplicial complex of “short sets”.

Classification of Harmonic Functions in the Exterior of a Unit Sphere and Its Application to the Description of the External Magnetic Field of the Earth

The magnetic fields of a subterranean source observed on the Earth’s surface or from the orbit of a satellite are studied. We analyze the space of all harmonic functions g in the external spherical domain, which are defined as series in a standard basis. The functions decrease at infinity; their gradients are normal to the field of directions specified at a boundary sphere, and the field of directions is determined by the gradient of a basis function f. The results can be used in the solution of the general problem in which the functions f and g are assumed to be independent of two spherical coordinates on the boundary sphere. Specific calculations were made only under the assumption of the independence of both g and f of the longitude coordinate. The known results are refined for the case in which f is the dipole or quadrupole harmonic, and a new result is obtained for the case in which f is the octupole harmonic.

How to prove the Baum–Connes conjecture for the groups [formula omitted] ?

We present a programme of proof of the Baum–Connes conjecture with coefficients for G=Sp(n,1) or F4(-20), i.e. simple Lie groups of real rank one having Kazhdan’s property (T). We use the geometry of the boundary sphere to produce a G-Fredholm module, together with a homotopy to the trivial representation through uniformly bounded representations. The strip of uniformly bounded representations of M. Cowling plays here the role of the complementary series. We explain how, modulo some conjectural estimates, this construction would prove the conjecture.

Українська гуманістика сьогодні: культурологічний аспект

The article focuses onto the theoretical developments of Ukrainian specialists in cultural studies, which were provided during the last decade. It is noted that the active development of cultural knowledge requires both the fixation of already worked out problems and the definition of new problems in the logics of further research process. It is shown that during the 2010-2020 period the theoretical interest of scientists was directed to the argumentation of specific principles of cultural analysis, which helps to distinguish “Cultural Studies” among other “structural elements” of humanism, in particular, socio-political knowledge, philosophy, aesthetics, history and theory of religion, art criticism, etc. The “boundary sphere” is outlined, where the theoretical interests of cultural studies intercross with other Humanities. The importance of generalization and systematization of those theoretical spheres that are in consideration of modern Ukrainian scientists is pointed out and those “white spots” are more clearly delineated that year by year are in the focus of researchers’ attention. Besides, basing onto the publications that appear during 2019, especially those that reflect the conceptual basis of future PhD or doctoral dissertations, it can be argued that scientists mostly “push off” from those groundworks that were to some extent stated previously in 2018-2019

Warsaw discs and semicomputability

We examine conditions under which a semicomputable set in a computable metric space is computable. Topology plays an important role in the description of such conditions. Motivated by the known result that a semicomputable cell is computable if its boundary sphere is computable, we investigate semicomputable Warsaw discs and their boundary Warsaw circles. We prove that a semicomputable Warsaw disc is computable if its boundary Warsaw circle is semicomputable.

Model of Solar Wind in the Heliosphere at Low and High Latitudes

The spatial distributions of the magnetic field, plasma density, and current at distances of (20–400)RS from the Sun (where RS is the solar radius) are investigated within a stationary axisymmetric MHD model of the solar wind (SW) at all latitudes in the inertial frame of reference with the origin at the center of the Sun. The model takes into account differential (with respect to the heliolatitude) rotation of the Sun and full corotation of plasma inside a boundary sphere of radius 20RS, which breaks down beyond this sphere. Self-consistent distributions of the plasma density, current, and magnetic field in the SW are obtained by numerically solving a set of time-independent MHD equations in spherical coordinates. It is demonstrated that the calculated results do not contradict observational data and describe a gradual transition from the fast SW at high heliolatitudes to the slow SW at low heliolatitudes, as well as the steepening of the profiles of the main SW characteristics with increasing distance from the Sun. The obtained dependences extend understanding of the SW structure at low and high latitudes and agree with the well-known Parker model in the limit of a small Ampere force.

Loop gravity string

In this work we study canonical gravity in finite regions for which we introduce a generalisation of the Gibbons-Hawking boundary term including the Immirzi parameter. We study the canonical formulation on a spacelike hypersuface with a boundary sphere and show how the presence of this term leads to an unprecedented type of degrees of freedom coming from the restoration of the gauge and diffeomorphism symmetry at the boundary. In the presence of a loop quantum gravity state, these boundary degrees of freedom localize along a set of punctures on the boundary sphere. We demonstrate that these degrees of freedom are effectively described by auxiliary strings with a 3-dimensional internal target space attached to each puncture. We show that the string currents represent the local frame field, that the string angular momenta represent the area flux and that the string stress tensor represents the two dimensional metric on the boundary of the region of interest. Finally, we show that the commutators of these broken diffeomorphisms charges of quantum geometry satisfy at each puncture a Virasoro algebra with central charge $c=3$. This leads to a description of the boundary degrees of freedom in terms of a CFT structure with central charge proportional to the number of loop punctures. The boundary $SU(2)$ gauge symmetry is recovered via the action of the $U(1)^3$ Kac-Moody generators (associated with the string current) in a way that is the exact analog of an infinite dimensional generalization of the Schwinger spin-representation. We finally show that this symmetry is broken by the presence of background curvature.

A nonlinear semi-concurrent multiscale method for fractures

A nonlinear semi-concurrent multiscale method for the modelling of crack propagation evolving from micro-structure for non-linear material behaviour is developed. The present method is based on an asymptotic expansion homogenization combined with the semi-concurrent finite element modelling approach. A modified periodic boundary conditions and sphere grains generation procedure are devised for non-linear material model with post-failure stage. The statistical convergence of the microscale RVE model regarding the RVE characteristic size and composition material distributions is studied. It was found that convergence of the macroscopic material properties is determined by the reproduction of the periodic geometry of the RVEs. The present method is validated by the experimental results for brittle material. The location and sequence of crack initiation captured by the present method is compared with the experimental data. The experimental phenomenon of the wing-cracks initiation location and crack branching pattern are successfully captured by the present method. The results show that the present method is an effective for the modelling of dynamic damage evolution for brittle materials.

Maximal open radius for Strebel point

Let $$T(S)$$ be the Teichmuller space of a hyperbolic Riemann surface $$S$$ . As is well known, when $$T(S)$$ is infinite-dimensional, the set $$\mathcal {SP}$$ of Strebel points is open and dense. Given a Strebel point $$[f]$$ in $$T(S)$$ , we prove that the the maximal radius of the open ball contained in $$\mathcal {SP}$$ and centered at $$[f]$$ is $$\frac{1}{4}\log \frac{K_0([f])}{H([f])}$$ . It is surprising that the boundary sphere of the maximal open ball is contained in $$\mathcal {SP}$$ . As a consequence, any open Strebel ball has no non-Strebel points as its boundary points. The infinitesimal version is also obtained.

Chemical potential in the first law for holographic entanglement entropy

Entanglement entropy in conformal field theories is known to satisfy a first law. For spherical entangling surfaces, this has been shown to follow via the AdS/CFT correspondence and the holographic prescription for entanglement entropy from the bulk first law for Killing horizons. The bulk first law can be extended to include variations in the cosmological constant $\Lambda$, which we established in earlier work. Here we show that this implies an extension of the boundary first law to include varying the number of degrees of freedom of the boundary CFT. The thermodynamic potential conjugate to $\Lambda$ in the bulk is called the thermodynamic volume and has a simple geometric formula. In the boundary first law it plays the role of a chemical potential. For the bulk minimal surface $\Sigma$ corresponding to a boundary sphere, the thermodynamic volume is found to be proportional to the area of $\Sigma$, in agreement with the variation of the known result for entanglement entropy of spheres. The dependence of the CFT chemical potential on the entanglement entropy and number of degrees of freedom is similar to how the thermodynamic chemical potential of an ideal gas depends on entropy and particle number.

Deformation retracts of neighborhood complexes of stable Kneser graphs

In 2003, A. Bjorner and M. de Longueville proved that the neighborhood complex of the stable Kneser graph SGn,k is homotopy equivalent to a k-sphere. Further, for n = 2 they showed that the neighborhood complex deformation retracts to a subcomplex isomorphic to the associahedron. They went on to ask whether or not, for all n and k, the neighborhood complex of SGn,k contains as a deformation retract the boundary complex of a simplicial polytope. Our purpose is to give a positive answer to this question in the case k = 2. We also find in this case that, after partially subdividing the neighborhood complex, the resulting complex deformation retracts onto a subcomplex arising as a polyhedral boundary sphere that is invariant under the action induced by the automorphism group of SGn,2.

Estimating a multivariate normal mean with a bounded signal to noise ratio under scaled squared error loss

For normal models with \(X \sim N_p(\theta, \sigma^{2} I_{p}), \;\; S^{2} \sim \sigma^{2}\chi^{2}_{k}, \;\mbox{independent}\), we consider the problem of estimating θ under scale invariant squared error loss ||d − θ||2/σ2, when it is known that the signal-to-noise ratio \({\left\|\theta\right\|}/{\sigma}\) is bounded above by m. Risk analysis is achieved by making use of a conditional risk decomposition and we obtain in particular sufficient conditions for an estimator to dominate either the unbiased estimator δUB(X) = X, or the maximum likelihood estimator \(\delta_{ML}(X,S^2)\), or both of these benchmark procedures. The given developments bring into play the pivotal role of the boundary Bayes estimator δBU,0 associated with a prior on (θ,σ2) such that θ|σ2 is uniformly distributed on the (boundary) sphere of radius mσ and a non-informative 1/σ2 prior measure is placed marginally on σ2. With a series of technical results related to δBU,0; which relate to particular ratios of confluent hypergeometric functions; we show that, whenever \(m \leq \sqrt{p}\) and p ≥ 2, δBU,0 dominates both δUB and δML. The finding can be viewed as both a multivariate extension of p = 1 result due to Kubokawa (Sankhyā 67:499–525, 2005) and an unknown variance extension of a similar dominance finding due to Marchand and Perron (Ann Stat 29:1078–1093, 2001). Various other dominance results are obtained, illustrations are provided and commented upon. In particular, for \(m \leq \sqrt{p/2}\), a wide class of Bayes estimators, which include priors where θ|σ2 is uniformly distributed on the ball of radius mσ centered at the origin, are shown to dominate δUB.

Boundary Restricted Non-Overlapping Sphere Tree for Unified Multidimensional Solid Object Index

针对现有空间索引剖分结构复杂、节点重叠率高及对多维实体对象检索及运算支撑较弱等问题,构建了一种边界约束的非相交球实体对象多维统一空间索引;利用球的几何代数外积表达,提出了基于求交算子的直线-平面和直线-球面的相交判定与交点提取方法,建立了多维实体对象体元化剖分方法及包含边界约束的非相交离散球实体填充算法,实现了实体对象空间均匀、非重叠的分割,并在填充球的个数、重叠率以及对象逼近近似度等约束条件上获得了较好的平衡.定义了最小外包球生成与更新的迭代算法与包含球体积修正的批量 Neural Gas 层次聚类算法,在尽可能保证球树各分支平衡性的前提下,实现了索引层次体系的稳健构建.利用几何代数下球对象间几何关系计算的内蕴性与参数更新的动态性,实现了索引结构的动态生成与更新,进而设计了实体对象表面及其内部任意位置及区域的检索策略及基于实体索引的空间关系计算方法.基于不同实体对象的模拟实验显示,基于几何代数的实体对象索引可以有效实现多维实体对象表面及其内部任意位置及区域的快速检索,并能在有限时间内以较高的精度实现多维实体对象最近邻距离和动态实体对象相交状态的检索.相对于常用球树索引,所提出的索引方法在填充率、节点重叠率、填充误差、体元个数、层次球个数、体积百分比和时间占用等方面均具有明显优势,且不同分辨率剖分条件下的索引结构及空间关系计算精度具有更高的稳健性,可运用于具有较强时间约束下复杂多维动态场景中对象检索与空间关系计算.;To solve leakages as complex subdivision structures, high nodes overlap probability and poorsupporting on multi-dimensional entity object retrievals and the computation of existing spatial indexes. This paperpresents a boundary restricted non-overlapping sphere tree for a unified multidimensional solid object index. Byusing the outer product expression of a sphere in Geometric Algebra, an approach is explored for intersectionjudgments and point extractions between lines and planes and lines and spherical surfaces based on meet operators.The element subdivision of multi-dimensional object voxelization and the boundary restricted non-overlappingsphere-filling algorithm is developed to balance the conditions (e.g. granularity, non-overlapping subdivision ofobject voxelization), the duplication and approximate degrees of approaching objects. Furthermore, generating andupdating minimum boundary sphere and batch Neural Gas hierarchical clustering algorithm is also presented, andcontains a volume adjusted, index level system steady with the balance of each branch of a sphere tree. Next, withthe advantage of a clear geometric meaning, simple geometric relations calculation among spheres and dynamicalupdateable parameters, index structure can be dynamically generated and updated. Finally, the unifiedmultidimensional solid object index, a query mechanism of any location and range on and in the solid objects isproposed. The updated minimum boundary sphere construction, algorithm, and the volume adjusted adaptive batchNeural Gas hierarchical cluster algorithm are defined to quickly, robustly, relatively, and uniformly classify thefilling sphere. The simulation of different physical objects and performance comparison with a commonly usedsphere tree indexes suggest that the proposed index can effectively query any position or regions on and in the solidobjects, and support the nearest linkage distance and dynamical overlapping query under limited time restrictionswith high precision.