Dimensional Sphere Research Articles

Spherical radial basis function approximation of some physical quantities measured

The article treats the spherical interpolation and approximation, i.e., a way of measured data processing in case when the scalar data sampling is performed at nodes on the unit 2D sphere surface in the 3D Euclidean space (in general, on the (d−1)-dimensional sphere surface in the d-dimensional space). We use the spherical radial basis function interpolation with an inverse multiquadric basis function and a second degree polynomial in Cartesian coordinates as a trend.The formulae of this type can be useful in the interpretation of various physical measurements and have wide applications in geosciences, e.g., in the treatment of anisotropy of magnetic susceptibility data measured. We present the advantages of the formula chosen on numerical examples. However, in practical computation with high number of sampling nodes, the matrix of the system for determining interpolation coefficients may be ill-conditioned. Then the standard double precision LU factorization does not give a reliable solution due to the effect of round-off error accumulation and some more sophisticated means of solving the system are to be used.

Simplicial volume and essentiality of manifolds fibered over spheres

AbstractWe study the question when a manifold that fibers over a sphere can be rationally essential, or have positive simplicial volume. More concretely, we show that mapping tori of manifolds (whose fundamental groups can be quite arbitrary) of dimension with non‐zero simplicial volume are very common. This contrasts the case of fiber bundles over a sphere of dimension : we prove that their total spaces are rationally inessential if , and always have simplicial volume 0. Using a result by Dranishnikov, we also deduce a surprising property of macroscopic dimension, and we give two applications to positive scalar curvature and characteristic classes, respectively.

A note on Fourier eigenfunctions in four dimensions

In this note, we exhibit a weakly holomorphic modular form for use in constructing a Fourier eigenfunction in four dimensions. Such auxiliary functions may be of use to the D4 checkerboard lattice and the four dimensional sphere packing problem.

Measurement of the pattern shifts for HR-EBSD with larger lattice rotations

High-resolution electron backscattering diffraction (HR-EBSD) was used to measure rotations and elastic strains by matching diffraction patterns based on cross-correlation. However, the subset-based phase correlation algorithm was unable to determine pattern shifts accurately when large rotations occurred. In this paper, a new matching algorithm was proposed to measure pattern shifts and recover the elastic strain and lattice rotation with finite deformation theory. The algorithm was implemented in two steps: (a) Integral pixel matching: The pixel-related information of the Kikuchi patterns was mapped to the original three-dimensional sphere to obtain the image projected in parallel by using the feature points as the pattern center through the transformation of its spatial coordinates. The correlation between the images projected in parallel before and after deformation was then obtained. The locations of the integral pixels were determined by the peaks of the surface of correlation obtained by traversing all pixels in the search area. (b) subpixel refinement: the locations of subpixels were obtained by FAGN with an appropriate shape function involving rotation and translation. The algorithm was applied to dynamic simulated test sets, and its results were compared with those of the first-pass cross-correlation and the second-pass cross-correlation method with remapping. The proposed method was more robust in the case of rotation and solved the problem that displacement vectors could not be accurately measured when a larger lattice rotation occurred. The mean errors of the measured displacement, rotation, and strain components were 0.02 pixel, 0.5×10−4rad, and 1×10−4, respectively. Compared with the second-pass cross-correlation method, the angle of rotation was more precisely extracted.

The Lettered Barriada: Workers, Archival Power, and the Politics of Knowledge in Puerto Rico

The Lettered Barriada: Workers, Archival Power, and the Politics of Knowledge in Puerto Rico

Algebraic (super-)integrability from commutants of subalgebras in universal enveloping algebras

Starting from a purely algebraic procedure based on the commutant of a subalgebra in the universal enveloping algebra of a given Lie algebra, the notion of algebraic Hamiltonians and the constants of the motion generating a polynomial symmetry algebra is proposed. The case of the special linear Lie algebra is discussed in detail, where an explicit basis for the commutant with respect to the Cartan subalgebra is obtained, and the order of the polynomial algebra is computed. It is further shown that, with an appropriate realization of , this provides an explicit connection with the generic superintegrable model on the -dimensional sphere and the related Racah algebra R(n). In particular, we show explicitly how the models on the two-sphere and three-sphere and the associated symmetry algebras can be obtained from the quadratic and cubic polynomial algebras generated by the commutants defined in the enveloping algebra of and , respectively. The construction is performed in the classical (or Poisson-Lie) context, where the Berezin bracket replaces the commutator.

Algebraic dimension and complex subvarieties of hypercomplex nilmanifolds

A nilmanifold is a (left) quotient of a nilpotent Lie group by a cocompact lattice. A hypercomplex structure on a manifold is a triple of complex structure operators satisfying the quaternionic relations. A hypercomplex nilmanifold is a compact quotient of a nilpotent Lie group equipped with a left-invariant hypercomplex structure. Such a manifold admits a whole 2-dimensional sphere S2 of complex structures induced by quaternions. We prove that for any hypercomplex nilmanifold M and a generic complex structure L∈S2, the complex manifold (M,L) has algebraic dimension 0. A stronger result is proven when the hypercomplex nilmanifold is abelian. Consider the Lie algebra of left-invariant vector fields of Hodge type (1,0) on the corresponding nilpotent Lie group with respect to some complex structure I∈S2. A hypercomplex nilmanifold is called abelian when this Lie algebra is abelian. We prove that all complex subvarieties of (M,L) for generic L∈S2 on a hypercomplex abelian nilmanifold are also hypercomplex nilmanifolds.

Three-dimensional flower spheres MoSe2/NiSe heterostructure with fast kinetic and stable structure for durable sodium-ion storage

Due to the large size of Na+ and the slow redox kinetics, the energy storage performance of sodium ion batteries (SIBs) is far from satisfactory. Heterojunction with layered structure has great potential in advanced material of SIBs. Herein, the 3D flower spheres MoSe2/NiSe heterostructure with fast dynamic properties was proposed, which shows enhancement in the adsorption and storage capacity of Na+ with a low migration energy barrier. Unique structure design can further effectively alleviate the volume expansion during sodiation/desodiation process. Furthermore, the DFT calculation results reveal the charge transfer mechanism between NiSe and MoSe2, as well as improved intrinsic electric structure of MoSe2/NiSe. In consequence, the designed MoSe2/NiSe possesses a good cyclability over 3000 cycles with a capacity of 172 mAh/g at 5.0 A/g, and the full cell with Na3V2(PO4)3/C cathode shows high-rate performance of 276 mAh/g at 2.0 A/g, which solidly proves the feasibility of the designed MoSe2/NiSe anode in the practical application. This work shows that the heterojunction can achieve long-lasting sodium electrochemistry and expand to other energy storage fields.

Generalised Wendland functions for the sphere

In this paper, we compute the spherical Fourier expansion coefficients for the restriction of the generalised Wendland functions from d-dimensional Euclidean space to the (d − 1)-dimensional unit sphere. We use results from the theory of special functions to show that they can be expressed in a closed form as a multiple of a certain 3F2 hypergeometric function. We present tight asymptotic bounds on the decay rate of the spherical Fourier coefficients and, in the case where d is odd, we are able to provide the precise asymptotic rate of decay. Numerical evidence suggests that this precise asymptotic rate also holds when d is even and we pose this as an open problem. Finally, we observe a close connection between the asymptotic decay rate of the spherical Fourier coefficients and that of the corresponding Euclidean Fourier transform.

Wedge holography in flat space and celestial holography

In this paper, we study codimension two holography in flat spacetimes, based on the idea of the wedge holography. We propose that a region in a $d+1$ dimensional flat spacetime surrounded by two end of the world-branes, which are given by $d$ dimensional hyperbolic spaces, is dual to a conformal field theory (CFT) on a $d-1$ dimensional sphere. Similarly, we also propose that a $d+1$ dimensional region in the flat spacetime bounded by two $d$ dimensional de Sitter spaces is holographically dual to a CFT on a $d-1$ dimensional sphere. Our calculations of the partition function, holographic entanglement entropy and two point functions, support these duality relations and imply that such CFTs are non-unitary. Finally, we glue these two dualities along null surfaces to realize a codimension two holography for a full Minkowski spacetime and discuss a possible connection to the celestial holography.

Deconvolution of spherical data corrupted with unknown noise

We consider the deconvolution problem for densities supported on a (d−1)-dimensional sphere with unknown center and unknown radius, in the situation where the distribution of the noise is unknown and without any other observations. We propose estimators of the radius, of the center, and of the density of the signal on the sphere that are proved consistent without further information. The estimator of the radius is proved to have almost parametric convergence rate for any dimension d. When d=2, the estimator of the density is proved to achieve the same rate of convergence over Sobolev regularity classes of densities as when the noise distribution is known.

Гиперболический объём 3-d многообразий, A-многочлены, численные проверки гипотез

We continue our study of the connections between the hyperbolic volume of the complement of a knot in the three dimensional sphere with topological invariants of this knot. This time we pay attention to A(M,L) parametrization for the affine variety with casp, produced by a knot (so-called A-polynomials). Then, using the known expressions of A-polynomials for number of knots we present results of the numerical tests for the conjectures on asymptotics of solutions of q-difference equations connected with the hyperbolic volume of these knots.

Variational analysis in one and two dimensions of a frustrated spin system: chirality and magnetic anisotropy transitions

<abstract><p>We study the energy of a ferromagnetic/antiferromagnetic frustrated spin system where the spin takes values on two disjoint circles of the 2-dimensional unit sphere. This analysis will be carried out first on a one-dimensional lattice and then on a two-dimensional lattice. The energy consists of the sum of a term that depends on nearest and next-to-nearest interactions and a penalizing term related to the spins' magnetic anisotropy transitions. We analyze the asymptotic behaviour of the energy, that is when the system is close to the helimagnet/ferromagnet transition point as the number of particles diverges. In the one-dimensional setting we compute the $ \Gamma $-limit of scalings of the energy at first and second order. As a result, it is shown how much energy the system spends for any magnetic anistropy transition and chirality transition. In the two-dimensional setting, by computing the $ \Gamma $-limit of a scaling of the energy, we study the geometric rigidity of chirality transitions.</p></abstract>

Generation of Homogeneous Populations of Cortical Interneurons from Human Pluripotent Stem Cells.

Cortical interneurons (cINs), especially those that are derived from the medial ganglionic eminence (MGE) during early development, are associated with various neuropsychiatric disorders. Human pluripotent stem cell (hPSC)-derived cINs can provide unlimited cell sources for studying disease mechanisms and developing novel therapeutics. Here, we describe an optimized method to generate homogeneous cIN populations based on three-dimensional (3D) cIN sphere generation. This optimized differentiation system could sustain generated cINs relatively long term without compromising their survival or phenotypes.

Facile fabrication of Fe3O4@Mg(OH)2 magnetic composites and their application in Cu(ii) ion removal.

In this study, we fabricated magnetic Fe3O4@Mg(OH)2 composites through the seed deposition technique to achieve Cu(ii) ion removal from aqueous solutions. As indicated by the characterization results, three-dimensional flower-like spheres composed of external Mg(OH)2 were formed, with nano-Fe3O4 particles uniformly embedded in the "flower petals" of the spheres. The efficacy of Fe3O4@Mg(OH)2-3 in Cu(ii) ion removal was examined through batch experiments. The impact of solution pH on removal efficiency was examined, and the pseudo-second-order model and the Langmuir model provided good fits to the adsorption kinetics and isotherm data, respectively. Remarkably, Fe3O4@Mg(OH)2-3 exhibited a significant removal capacity of 1051.65 mg g-1 for Cu(ii) ions. Additionally, the composite displayed a notable saturation magnetization value of 17.3 emu g-1, facilitating isolation from sample solutions through external magnetic fields after Cu(ii) ion absorption. At the solid-liquid interface, a mechanism involving ion exchange between Mg(ii) and Cu(ii) cations was realized as the mode of Cu(ii) ion removal. The composites' effective adsorption properties and rapid magnetic separation highlighted their suitability for use in treating copper-contaminated water.

Geometrical Images in Lexis and Phraseology of Modern Russian Language

The article reviews geometrical objects as a factor forming the image system of modern Russian language, and explores the lexical units motivated by the concept of geometrical objects and phraseological units with geometrical components. This approach reveals multiplicity of geometrical objects participating in the process: there are three-dimensional (sphere, spatial angle, etc.), two-dimensional (circle), one-dimensional (line, etc.) and geometrical objects without dimension (point). When considering the language material the author uses the concept of the inner form of the word. Numerous examples are given demonstrating the process of geometric metaphorization. The analysis has shown that circle and spatial angle demonstrate high activity in forming lexical and phraseological units under consideration. Many of those geometrical images have an ancient origin. The article also touches on the issue of national characteristics of the perception of angles and circles. Some examples given by the author indicate the further development of language phenomena motivated by the concept of those geometrical objects in our time. Modern trends in the development of geometric imagery in the Russian language have been identified. The role of such geometric objects as a point, line, edge, etc. in the modern process of enriching the figurative structure of the Russian language is noted. The issues of lexical compatibility of the studied words are considered, the tendency of its change and expansion is traced. Modern trends of development of geometrical imagery in Russian language as a systematic process reflecting essential features of national view of the world are revealed.

Fabrication of dendritic PdCu alloy supported on 3D N-doped hollow graphene for efficient ethanol electrooxidation.

Fabricating highly efficient Pd-based nanocatalysts with a well-defined structure is desired for the commercialization of direct ethanol fuel cell (DEFC). Herein, a series of hierarchical three-dimensional N-doped hollow graphene spheres (NHGS) supported dendritic PdCu alloy catalysts PdxCu(d)-NHGS (x: Cu/Pd theoretical molar ratio of 4, 2, and 1) are assembled by one-pot ascorbic acid reduction-immobilization method. Aiming to maximize the Pd utilization and realize the efficient ethanol electrooxidation, this novel electrocatalyst offers potent activity sites and promotes electron and ion kinetics simultaneously. Characterization indicates that the as-obtained Pd4Cu(d) alloy nanoparticles with average sizes of approximately 55 nm are evenly dispersed on the NHGS supporting materials obtained by using the SiO2 nanospheres template strategy. Three catalysts all exhibit enhanced electrocatalytic activity, of which the Pd4Cu(d)-NHGS shows the highest mass current activity (2683 mA mgPd-1), which is 2.59 times of the commercial Pd/C toward ethanol electrooxidation in alkaline medium. Based on the results, we believed that the Pd4Cu(d)-NHGS could exhibit extensive application prospect in alkaline DEFC.

Two Shaky Pillars of Quantum Computing

Superposition and entanglement are two theoretical pillars quantum computing rests upon. In the g-qubit theory quantum wave functions are identified by points on the surface of three-dimensional sphere S3. That gives different, more physically feasible explanation of what superposition and entanglement are. The core of quantum computing scheme should be in manipulation and transferring of wave functions on S3 as operators acting on observables and formulated in terms of geometrical algebra. In this way quantum computer will be a kind of analog computer keeping and processing information by sets of objects possessing infinite number of degrees of freedom, contrary to the two value bits or two-dimensional Hilbert space elements, qubits.

Analytic weakly mixing diffeomorphisms on odd dimensional spheres

We present an approximation by conjugation scheme to obtain real-analytic diffeomorphisms of odd dimensional spheres that are weakly mixing with respect to the volume.

An Index Theorem for Quarter-Plane Toeplitz Operators via Extended Symbols and Gapped Invariants Related to Corner States

In this paper, we discuss index theory for Toeplitz operators on a discrete quarter-plane of two-variable rational matrix function symbols. By using Gohberg–Kreĭn theory for matrix factorizations, we extend the symbols defined originally on a two-dimensional torus to some three-dimensional sphere and derive a formula to express their Fredholm indices through extended symbols. Variants for families of (self-adjoint) Fredholm quarter-plane Toeplitz operators and those preserving real structures are also included. For some bulk-edge gapped single-particle Hamiltonians of finite hopping range on a discrete lattice with a codimension-two right angle corner, topological invariants related to corner states are provided through extensions of bulk Hamiltonians.