Dimensional Hypersphere Research Articles

Why Mature Galaxies Seem to Have Filled the Universe Shortly After the Big Bang — A New Cosmological Model, that Predicted the JWST Observations

Recent observations from the first dataset, provided by NASA's James Webb Space Telescope (JWST) of six massive galaxies, at a time in the early universe, seem to defy conventional cosmological models, as they appear to be as mature and developed as our own local group. Such unexpected discoveries JUSTIFY A RADICALLY NOVEL MODEL OF COSMOLOGY. TO QUOTE JOEL LEJA, ASSISTANT PROFESSOR OF ASTRONOMY AND ASTROPHYSICS AT PENN STATE ''IT TURNS OUT WE FOUND SOMETHING SO UNEXPECTED IT ACTUALLY CREATES PROBLEMS FOR SCIENCE. IT CALLS THE WHOLE PICTURE OF EARLY GALAXY FORMATION INTO QUESTION''. This article provides an alternative mathematical model of cosmological redshifting (\(z\)), which actually predicted such mature galaxies in a 2022 preprint, prior to these recent observations. As well, this model also predicts discrepancies between theoretical and observed galaxy rotation curves with apparent increased energy density. The Azimuthal Projection Model of the Universe is conceptualized as an \(\mathbb{R}^5\) spacetime, with a four spatial dimensional hypersphere azimuthally projected onto a three-dimensional spatial sphere. This simple parsimonious model requires only a few assumptions, excluding dark energy to satisfy the Cosmological Constant \(\Lambda\), and is shown to match the Universal expansion rate, as established from supernova cosmology survey points. This novel model conceives the universe as a higher dimensional dynamic with spacetime as a projection, rather than as an arrow from the absolute beginning of the Big Bang. Redshifting is alternatively proposed as azimuthal angular projections of wavelengths \(\lambda\). Accelerated Universal Expansion is alternatively proposed as azimuthal projections of meridians, asymptotical to a horizon, and Lambert's cosine law of luminous intensity. A radical implication of this model is that azimuthal angular projections are positional dependent, and thus it's conceivable that apparent distances between galaxies vary with the location of the observer (see figure 4). Supportive mathematical evidence is described from the Hubble Tension; Discrepancies between visible spectra redshifting of cepheid variables (the most recent calculation is \(Ho=74.03 \pm 1.42km/sec/Mpc\)), and from temperature fluctuations in the Cosmic Microwave Background (CMB) (which are calculated to be \(Ho=68.7 \pm 1.3km/sec/Mpc\)), which resolves the discrepancy by recalibrating redshift data from supernova Cosmology survey points.

Optimal general Hermite-Hadamard-type inequalities in a ball and their applications in multidimensional numerical integration

In this paper, we are interested in the problem of approximation of a definite integral over a ball of a given function f in d-dimensional space when, rather than function evaluations, a number of integrals over certain (d−1)-dimensional hyperspheres are only available. In this context several families of ‘extended’ multidimensional integration formulas based on a weighted sum of integrals over some hyperspheres can be defined. The special cases include multivariate analogues of the well-known midpoint rule and the trapezoidal rule. Basic properties of these families are derived, in particular, we show that they all satisfy a multivariate version of Hermite–Hadamard inequality. As an immediate consequence of this inequality, we derive explicit expressions of the best constants, which appear in their optimal error estimates. Theoretical and numerical results show that the proposed method reaches at least the second order of approximation. We present several numerical examples to illustrate various features of these new cubature formulas.

The curvature-induced gauge potential and the geometric momentum for a particle on a hypersphere

A particle that is constrained to freely move on a hyperspherical surface in an N≥2 dimensional flat space experiences a curvature-induced gauge potential, whose form was given long ago (Ohnuki and Kitakado, 1993). We demonstrate that the momentum for the particle on the hypersphere is the geometric one including the gauge potential and its components obey the commutation relations pi,pj=−iħJij/r2, in which ħ is the Planck’s constant, and pi (i,j=1,2,3,…N) denotes the i-th component of the geometric momentum, and Jij specifies the ij-th component of the generalized angular momentum containing both the orbital part and the coupling of the generators of continuous rotational symmetry group SO(N−1) and curvature, and r denotes the radius of the N−1 dimensional hypersphere.

Geodesic projection of the von Mises–Fisher distribution for projection pursuit of directional data

We investigate geodesic projections of von Mises–Fisher (vMF) distributed directional data. The vMF distribution for random directions on the (p−1)-dimensional unit hypersphere Sp−1⊂Rp plays the role of multivariate normal distribution in directional statistics. For one-dimensional circle S1, the vMF distribution is called von Mises (vM) distribution. Projections onto geodesics are one of main ingredients of modeling and exploring directional data. We show that the projection of vMF distributed random directions onto any geodesic is approximately vM-distributed, albeit not exactly the same. In particular, the distribution of the geodesic-projected score is an infinite scale mixture of vM distributions. Approximations by vM distributions are given along various asymptotic scenarios including large and small concentrations (κ→∞,κ→0), high-dimensions (p→∞), and two important cases of double-asymptotics (p,κ→∞, κ∕p→c or κ∕p→λ), to support our claim: geodesic projections of the vMF are approximately vM. As one of potential applications of the result, we contemplate a projection pursuit exploration of high-dimensional directional data. We show that in a high dimensional model almost all geodesic-projections of directional data are nearly vM, thus measures of non-vM-ness are a viable candidate for projection index.

Generation and Frame Characteristics of Predefined Evenly-Distributed Class Centroids for Pattern Classification

Predefined evenly-distributed class centroids (PEDCC) can be widely used in models and algorithms of pattern classification, such as CNN classifiers, classification autoencoders, clustering, and semi-supervised learning, etc. Its basic idea is to predefine the class centers, which are evenly-distributed on the unit hypersphere in feature space, to maximize the inter-class distance. The previous method of generating PEDCC uses an iterative algorithm based on a charge model. The generated class centers will have some errors with the theoretically evenly-distributed points, and the generation time is long. This paper takes advantage of regular polyhedron in high-dimensional space and the evenly distributed points on the n dimensional hypersphere to generate PEDCC mathematically. Then, we discussed the basic and extended characteristics of the frames formed by PEDCC, and some meaningful conclusions are obtained. Finally, the effectiveness of the new algorithm and related conclusions are proved by experiments. The mathematical analysis and experimental results of this paper can provide a theoretical tool for using PEDCC to solve the key problems in the field of pattern recognition, such as interpretable supervised/unsupervised learning, incremental learning, uncertainty analysis and so on.

Spectral neural network potentials for binary alloys

In this work, we present a numerical implementation to compute the atom-centered descriptors introduced by Bartok et al. [Phys. Rev. B 87, 184115 (2013)] based on the harmonic analysis of the atomic neighbor density function. Specifically, we focus on two types of descriptors, the smooth SO(3) power spectrum with the explicit inclusion of a radial basis and the SO(4) bispectrum obtained through mapping the radial component onto a polar angle of a four dimensional hypersphere. With these descriptors, various interatomic potentials for binary Ni–Mo alloys are obtained based on linear and neural network regression models. Numerical experiments suggest that both descriptors produce similar results in terms of accuracy. For linear regression, the smooth SO(3) power spectrum is superior to the SO(4) bispectrum when a large band limit is used. In neural network regression, better accuracy can be achieved with even less number of expansion components for both descriptors. As such, we demonstrate that spectral neural network potentials are feasible choices for large scale atomistic simulations.

About the geographic grid of the three-dimensional hypersphere

According to the Poincaré conjecture (1904) proved by Grigory Perelman (2002-2003) that any simply connected compact three- dimensional manifold without edges is homeomorphic to a three- dimensional hypersphere [1], to solve the problems of visualizing four- dimensional objects in three-dimensional space [2], it is proposed to choose a suitable manifold, in in this case, a ball, establishing a homeomorphism between objects located in different spaces by technological means of cartography. As a result of this work, it seems possible to build a dynamic video of the population distribution process on a map of the globe, which provides informational four-dimensional data flow, following the ideas embodied in 4D Anatomy [3]. The proposed technology opens up new ways of visualizing four-dimensional space This work was performed within the framework of the state assignment of the ICM MG SB RAS (project 0315-2019-0003).

Generating Two-Qubit Werner States Using Superpositions of Dimer and Random States in Hilbert Space

Werner states are unitarily invariant states in the d dimensional Hilbert space. These states are entangled (EPR correlated) but still admits a hidden variable model. In this paper, we superpose a 2N qubit Bell state which are pairwise entangled with a 2N qubit completely random real pure state. For large N limits, the two-qubit reduced density matrix (both qubits are in the same Bell state of the superposed state) very closely resembles a Werner state. The random state is sampled from the surface of a 22N dimensional hypersphere of unit radius or equivalently sampled from a normal distribution of zero mean and unit variance. The quantitative analysis of entanglement measures such as concurrence and block entropy also reinforce our claim.

Typicality in quasispecies evolution in high dimensions.

We study quasispecies and closely related evolutionary dynamics like the replicator-mutator equation in high dimensions. In particular, we show that under certain conditions, the average fitness of almost all quasispecies of a given dimension becomes independent of mutational probabilities in high dimensional sequence spaces. This result is a consequence of concentration of measure on a high dimensional hypersphere and its extension to Lipschitz functions known as the Levy's Lemma. Our results naturally extend to other functional capabilities that can be described as Lipschitz functions and whose input parameters are the frequencies of individual constituents of the quasispecies. In order to show this, we give a generalization of Levy's Lemma and discuss possible biological consequences of our work.

Optimum sensor placement for localization of a hazardous source under log normal shadowing

We consider the problem of optimum sensor placement for localizing a hazardous source located inside an $ N $-dimensional hypersphere centered at the origin with a known radius $ r_1 $. All one knows about the probability density function (pdf) of the source location is that it is spherically symmetric, i.e. it is a function only of the distance from the center. The sensors must be placed at a safe distance of at least $ r_2>r_1 $ from the center, to avoid damage. Localization must be effected from the strength of a signal emanating from the source, as received by a set of sensors that do not lie on an $ (N-1) - $ dimensional hyperplane. Under the assumption that this signal strength experiences log normal shadowing, we characterize non-coplanar sensor positions that optimize three distinguished parameters associated with the underlying Fisher Information Matrix (FIM): maximizing its smallest eigenvalue, maximizing its determinant, and minimizing the trace of its inverse. We show that all three have the same set of optimizing solutions and involve placing the sensors on the surface of the hypersphere of radius $ r_2. $ As spherical symmetry of the pdf precludes uniqueness we provide certain canonical optimizing solutions where the $ i $-th sensor position $ x_i = Q^{i-1}x_1 $, with $ Q $ an orthogonal matrix. We provide necessary and sufficient conditions on $ Q $ and $ x_1 $ for $ x_i $ to be non-coplanar and optimizing. In addition, we provide a geometrical interpretation of these solutions. We observe the $ N $-dimensional solutions for $ N>3 $ have implications for optimal design of sensing matrices in certain compressed sensing problems.

Moments of the inverse participation ratio for the Laplacian on finite regular graphs

We investigate the first and second moments of the inverse participation ratio (IPR) for all eigenvectors of the Laplacian on finite random regular graphs with n vertices and degree z. By exactly diagonalizing a large set of z-regular graphs, we find that as n becomes large, the mean of the inverse participation ratio on each graph, when averaged over a large ensemble of graphs, approaches the numerical value 3. This universal number is understood as the large-n limit of the average of the quartic polynomial corresponding to the IPR over an appropriate -dimensional hypersphere of . For a large, but not exhaustive ensemble of graphs, the mean variance of the inverse participation ratio for all graph Laplacian eigenvectors deviates from its continuous hypersphere average due to large graph-to-graph fluctuations that arise from the existence of highly localized modes.

A geometric framework for statistical analysis of trajectories with distinct temporal spans.

Analyzing data representing multifarious trajectories is central to the many fields in Science and Engineering; for example, trajectories representing a tennis serve, a gymnast's parallel bar routine, progression/remission of disease and so on. We present a novel geometric algorithm for performing statistical analysis of trajectories with distinct number of samples representing longitudinal (or temporal) data. A key feature of our proposal is that unlike existing schemes, our model is deployable in regimes where each participant provides a different number of acquisitions (trajectories have different number of sample points or temporal span). To achieve this, we develop a novel method involving the parallel transport of the tangent vectors along each given trajectory to the starting point of the respective trajectories and then use the span of the matrix whose columns consist of these vectors, to construct a linear subspace in R m . We then map these linear subspaces (possibly of distinct dimensions) of R m on to a single high dimensional hypersphere. This enables computing group statistics over trajectories by instead performing statistics on the hypersphere (equipped with a simpler geometry). Given a point on the hypersphere representing a trajectory, we also provide a "reverse mapping" algorithm to uniquely (under certain assumptions) reconstruct the subspace that corresponds to this point. Finally, by using existing algorithms for recursive Fréchet mean and exact principal geodesic analysis on the hypersphere, we present several experiments on synthetic and real (vision and medical) data sets showing how group testing on such diversely sampled longitudinal data is possible by analyzing the reconstructed data in the subspace spanned by the first few principal components.

Bold-Independent Computational Entropy Assesses Functional Donut-Like Structures in Brain fMRI Images.

We introduce a novel method for the measurement of information level in fMRI (functional Magnetic Resonance Imaging) neural data sets, based on image subdivision in small polygons equipped with different entropic content. We show how this method, called maximal nucleus clustering (MNC), is a novel, fast and inexpensive image-analysis technique, independent from the standard blood-oxygen-level dependent signals. MNC facilitates the objective detection of hidden temporal patterns of entropy/information in zones of fMRI images generally not taken into account by the subjective standpoint of the observer. This approach befits the geometric character of fMRIs. The main purpose of this study is to provide a computable framework for fMRI that not only facilitates analyses, but also provides an easily decipherable visualization of structures. This framework commands attention because it is easily implemented using conventional software systems. In order to evaluate the potential applications of MNC, we looked for the presence of a fourth dimension's distinctive hallmarks in a temporal sequence of 2D images taken during spontaneous brain activity. Indeed, recent findings suggest that several brain activities, such as mind-wandering and memory retrieval, might take place in the functional space of a four dimensional hypersphere, which is a double donut-like structure undetectable in the usual three dimensions. We found that the Rényi entropy is higher in MNC areas than in the surrounding ones, and that these temporal patterns closely resemble the trajectories predicted by the possible presence of a hypersphere in the brain.

Five dimensional binary hard hypersphere mixtures: A Monte Carlo study

Additive binary mixtures of five dimensional hyperspheres were investigated by Monte Carlo simulations. Both equal packing fraction and equal mole fraction systems with diameter ratios of 0.4 and 0.5 were examined. A range of total densities were studied, spanning low to moderate density fluids. The pair correlation functions and the equations of state were determined and compared with molecular dynamics data and a variety of theoretical predictions. A significant result of the equal packing fraction simulations was the discovery of how quickly the larger hyperspheres reorganized into a dense fluid after a random initial placement. In the equal mole fraction case, the pair correlation functions for the larger hypersphere agree with the pair correlation function of a pure fluid at an appropriately scaled density. The theoretical results for the equation of state compare well to the Monte Carlo calculations for all but the highest densities studied.

Unscented von Mises–Fisher Filtering

We introduce the unscented von Mises–Fisher filter (UvMFF), a nonlinear filtering algorithm for dynamic state estimation on the $n$ -dimensional unit hypersphere. Estimation problems on the unit hypersphere occur in computer vision, e.g., when using omnidirectional cameras, as well as in signal processing. As approaches in literature are limited to very simple system and measurement models, we propose a deterministic sampling scheme on the unit hypersphere, which allows us to handle nonlinear system and measurement models. The proposed approach can be seen as a hyperspherical variant of the unscented Kalman filter (UKF). The advantages of the novel method are shown by means of simulations.

Affine hypersurfaces with self congruent center map

In this paper, we study locally strictly convex affine hypersurfaces for which the center map is centroaffine congruent with the original hypersurface. By the equiaffine support function ρ, we show that the hypersurface is locally isometric to a warped product R×|ρ|N, where the gradient direction of ρ is along R. As a main result, we complete the classification when gradρ is the eigenvector of affine shape operator, which shows how to explicitly construct such hypersurfaces starting from one (or two) low dimensional affine hypersphere(s).

Fourier and Gegenbauer Expansions for a Fundamental Solution of Laplace's Equation in Hyperspherical Geometry

For a fundamental solution of Laplace's equation on the $R$-radius $d$-dimensional hypersphere, we compute the azimuthal Fourier coefficients in closed form in two and three dimensions. We also compute the Gegenbauer polynomial expansion for a fundamental solution of Laplace's equation in hyperspherical geometry in geodesic polar coordinates. From this expansion in three-dimensions, we derive an addition theorem for the azimuthal Fourier coefficients of a fundamental solution of Laplace's equation on the 3-sphere. Applications of our expansions are given, namely closed-form solutions to Poisson's equation with uniform density source distributions. The Newtonian potential is obtained for the 2-disc on the 2-sphere and 3-ball and circular curve segment on the 3-sphere. Applications are also given to the superintegrable Kepler-Coulomb and isotropic oscillator potentials.

Sub- and super-fidelity as bounds for quantum fidelity

We derive several bounds on fidelity between quantum states. In particular we show that fidelity is bounded from above by a simple to compute quantity we call super--fidelity. It is analogous to another quantity called sub--fidelity. For any two states of a two--dimensional quantum system (N=2) all three quantities coincide. We demonstrate that sub-- and super--fidelity are concave functions. We also show that super--fidelity is super--multiplicative while sub--fidelity is sub--multiplicative and design feasible schemes to measure these quantities in an experiment.Super--fidelity can be used to define a distance between quantum states. With respect to this metric the set of quantum states forms a part of a N^2-1 dimensional hypersphere.

Sub- and super-fidelity as bounds for quantum fidelity

We derive several bounds on fidelity between quantum states. In particular we show thatfidelity is bounded from above by a simple to compute quantity we call super-fidelity.It is analogous to another quantity called sub-fidelity. For any two states of a two-dimensional quantum system (N = 2) all three quantities coincide. We demonstratethat sub- and super-fidelity are concave functions. We also show that super-fidelity issuper-multiplicative while sub-fidelity is sub-multiplicative and design feasible schemesto measure these quantities in an experiment. Super-fidelity can be used to define adistance between quantum states. With respect to this metric the set of quantum statesforms a part of a N2 - 1 dimensional hypersphere.

The entropy of a mixture of probability distributions

If a message can have n different values and all values are equally probable, then the entropy of the message is log(n). In the present paper, we investigate the expectation value of the entropy, for arbitrary probability distribution. For that purpose, we apply mixed probability distributions. The mixing distribution is represented by a point on an infinite dimensional hypersphere in Hilbert space. During an `arbitrary' calculation, this mixing distribution has the tendency to become uniform over a flat probability space of ever decreasing dimensionality. Once such smeared-out mixing distribution is established, subsequent computing steps introduce an entropy loss expected to equal $\\frac{1}{m+1} + \\frac{1}{m+2} + ... + \\frac{1}{n}$, where n is the number of possible inputs and m the number of possible outcomes of the computation.