Radius Of Hypersphere Research Articles

Toward a geometrical foundation for physics

We present the conjecture that the fundamental notions of physics, including such basic concepts as spin, charge(s), mass and spacetime, emerge from a mathematical theory having to do with fluctuations in the metrics of 2-manifolds. This involves looking at a set of 2-manifolds whose metrics depend on a global parameter \(t\), where each manifold is also associated with a set of six parameters whose values change continuously along \(t\) depending on the metrics and associated parameters of all the other manifolds. More intuitively, and specifically, what we do is define objects that generate oscillations on affine planes in \(R^3\), parameterized by the global parameter \(t\), where the plane associated with each object changes continuously along \(t\) depending on the oscillations that are generated by similar objects on other affine planes. We study some typical structures that arise when the oscillations’ amplitudes tend to zero, and show that the state of each such structure at any value of \(t\) can be associated with a real parameter \(0 \leqslant \beta < 1\), and that a special type of interaction with other similar structures exists which results in readjusting the value of \(\beta \). We go on to think of \(R^3\) at any value of \(t\) as corresponding to a hypersphere in \(R^4\) via the stereographic projection, where \(t\) is identified as the logarithm of hypersphere radius in the foliation of \(R^4\) in concentric hyperspheres. This leads to interpreting some curves arising from the structures we study as corresponding to certain \(S^2\) curves. We use both sides of that last correspondence to derive the free Dirac equation as the classical equation of motion of an abstract point particle, where our typical structures and their interactions are understood in connection with electrons interacting with an em radiation field, and the emergent description of any point in spacetime typically involves an infinite number of discrete values of \(t\). The Dirac equation is thus derived from the Frenet equations of our special family of curves, and the physical meaning is determined by the identification of physical and geometrical entities required by the derivation. We propose that the origin of the infinitesimal oscillations in our theory has to do with the axiomatization of the Euclidean plane and the construction of the real numbers. We briefly discuss the relevance of these ideas to the strong interaction and to gravity.

Soft-Input Soft-Output List Sphere Detection with a Probabilistic Radius Tightening

In this paper, we present a low-complexity list sphere detection algorithm for achieving near-optimal a posteriori probability (APP) detection in an iterative detection and decoding (IDD). Motivated by the fact that the list sphere decoding searching a fixed number of candidates is computationally inefficient in many scenarios, we design a criterion to search lattice points with non-vanishing likelihood and then derive a hypersphere radius satisfying this condition. Further, in order to exploit the original sphere constraint as it is instead of using necessary conditioned version, we combine a probabilistic tree pruning strategy and the proposed list sphere search. Two features, tightened hypersphere radius and probabilistic tree pruning, collaborate and improve the search efficiency in a complementary fashion. Through simulations on 4x4 MIMO system, we show that the proposed method provides substantial reduction in complexity while achieving negligible performance loss over the conventional list sphere detection.

Design and Implementation of Binary Neural Network Classification Learning Algorithm

In this paper a Binary Neural Network Learning (BNN-CLA)[1] is analyzed and implemented for solving multi class problem normally the classifier are construct by combining the outputs of several binary ones. The BNNC offers high degree of parallelism in hidden layer formation for all multiple classes to reduce the time for learning. The learning method is an iterative process to optimize the classifier parameters. In this approach, overlapping problem is tackled to enhance the performance of classifier by changing hyper-sphere radius. Exhaustive testing is carried out. Accuracies and number of neuron are evaluated and compare with BNNC[4]. The method have been tested on Fisher's well known Iris data data set and experimental result shown the classification ability improved by using FCLA algorithm. While comparing with BNNC[4] in most cases accuracies improved in BNNL because of elimination of samples which are lying in overlapping region of classes. Thus tacking overlapping issue improved performance of this classifier. keyword: Semi-Supervised classification, BNN Geometrical Expansion, Hyper sphere, overlapped classes.

Chaotic behavior of pulsating heat pipes

Chaotic behavior of closed loop pulsating heat pipes (PHPs) was studied. The PHPs were fabricated by capillary tubes with outer and inner diameters of 2.0 and 1.20 mm. FC-72 and deionized water were used as the working fluids. Experiments cover the following data ranges: number of turns of 4, 6, and 9, inclination angles from 5° (near horizontal) to 90° (vertical), charge ratios from 50% to 80%, heating powers from 7.5 to 60.0 W. The nonlinear analysis is based on the recorded time series of temperatures on the evaporation, adiabatic, and condensation sections. The present study confirms that PHPs are deterministic chaotic systems. Autocorrelation functions (ACF) are decreased versus time, indicating prediction ability of the system is finite. Three typical attractor patterns are identified. Hurst exponents are very high, i.e., from 0.85 to 0.95, indicating very strong persistent properties of PHPs. Curves of correlation integral versus radius of hypersphere indicate two linear sections for water PHPs, corresponding to both high frequency, low amplitude, and low frequency, large amplitude oscillations. At small inclination angles near horizontal, correlation dimensions are not uniform at different turns of PHPs. The non-uniformity of correlation dimensions is significantly improved with increases in inclination angles. Effect of inclination angles on the chaotic parameters is complex for FC-72 PHPs, but it is certain that correlation dimensions and Kolmogorov entropies are increased with increases in inclination angles. The optimal charge ratios are about 60–70%, at which correlation dimensions and Kolmogorov entropies are high. The higher the heating power, the larger the correlation dimensions and Kolmogorov entropies are. For most runs, large correlation dimensions and Kolmogorov entropies correspond to small thermal resistances, i.e., better thermal performance, except for FC-72 PHPs at small inclination angles of θ < 15°.

A new maximal-margin spherical-structured multi-class support vector machine

Support vector machines (SVMs), initially proposed for two-class classification problems, have been very successful in pattern recognition problems. For multi-class classification problems, the standard hyperplane-based SVMs are made by constructing and combining several maximal-margin hyperplanes, and each class of data is confined into a certain area constructed by those hyperplanes. Instead of using hyperplanes, hyperspheres that tightly enclosed the data of each class can be used. Since the class-specific hyperspheres are constructed for each class separately, the spherical-structured SVMs can be used to deal with the multi-class classification problem easily. In addition, the center and radius of the class-specific hypersphere characterize the distribution of examples from that class, and may be useful for dealing with imbalance problems. In this paper, we incorporate the concept of maximal margin into the spherical-structured SVMs. Besides, the proposed approach has the advantage of using a new parameter on controlling the number of support vectors. Experimental results show that the proposed method performs well on both artificial and benchmark datasets.

Chaotic characteristics in an evaporator with a vapor–liquid–solid boiling flow

In this paper, quantitative and qualitative investigations on the unstable fluctuations of wall temperature signals in a multiple-tube evaporator with a vapor–liquid–solid boiling flow were carried out by employing the nonlinear analysis tools including reconstructed phase space analysis, correlation dimension analysis and Kolmogorov entropy analysis besides the traditional analysis methods such as power spectral analysis and autocorrelation analysis. The values of Kolmogorov entropy are positive and finite, and the values of correlation dimension are from 1.5 to 2.0. The analysis results indicate that the physical parameter fluctuations of the system with a vapor–liquid–solid three-phase boiling flow are chaotic. Based on the estimates of correlation dimension, it can be found that at least two independent variables are needed in order to describe the flow boiling behavior of such system. The curve shapes of correlation integral versus radius of hypersphere vary with the variations of multi-phase flow regimes or states. So do the curve shapes of slope or Kolmogorov entropy versus radius of hypersphere. And the multi-phase flow regimes or states at given operation conditions in such system may be identified or characterized by the shape variations of these curves. In the estimations of chaotic invariants including correlation dimension and Kolmogorov entropy at the same given operation condition, the multi-value phenomena are found. The phenomena reflect the multi-scale behavior occurred in such multi-phase flow boiling systems.

Territory covered by N random walkers on fractal media: the Sierpinski gasket and the percolation aggregate.

We address the problem of evaluating the number S(N)(t) of distinct sites visited up to time t by N noninteracting random walkers all starting from the same origin in fractal media. For a wide class of fractals (of which the percolation cluster at criticality and the Sierpinski gasket are typical examples) we propose, for large N and after the short-time compact regime, an asymptotic series for S(N)(t) analogous to that found for Euclidean media: S(N)(t) approximately S(N)(t)(1-Delta). Here S(N)(t) is the number of sites (volume) inside a hypersphere of radius L[ln(N)/c]1/v where L is the root-mean-square chemical displacement of a single random walker, and v and c determine how fast 1-Gamma(t)(l) (the probability that a given site at chemical distance l from the origin is visited by a single random walker by time t) decays for large values of l/L: 1-Gamma(t)(l) approximately exp[-c(l/L)(v)]. For the fractals considered in this paper, v=d(l)w/((d(l)w)-1), d(l)w being the chemical-diffusion exponent. The corrective term Delta is expressed as a series in ln(-n)(N)ln(m) ln(N) (with n> or =1 and 0< or =m< or =n), which is given explicitly up to n=2. This corrective term contributes substantially to the final value of S(N)(t) even for relatively large values of N.

Vibrationally excited states and fragmentation geometries of NeN and ArN clusters, N=3–6, using hyperspherical coordinates

We calculate the ground state and a class of zero orbital angular momentum (L=0) vibrationally excited state energies for NeN and ArN clusters using an adiabatic hyperspherical representation to solve the nuclear Schrödinger equation. The Schrödinger equation in the hyperangular coordinates is solved for a sequence of fixed hyperradii by diffusion Monte Carlo techniques, which determines the lowest effective potential curve. We monitor structural properties such as the pair and angle distribution as a function of the hyperspherical radius. These structural studies allow us to identify configurational changes as the N atom cluster fragments into an (N−1)-atom cluster plus an atom. We also determine separately the ground state of the full 3N-dimensional nuclear Schrödinger equation for the ground state, and compare the resulting structural properties with those calculated in the adiabatic hyperspherical approximation.

Analytical functions for the calculation of hyperspherical potential curves of atomic systems

We present angular basis functions for the Schr\"odinger equation of two-electron systems in hyperspherical coordinates. By using the hyperspherical adiabatic approach, the wave functions of two-electron systems are expanded in analytical functions, which generalizes the Jacobi polynomials. We show that these functions, obtained by selecting the diagonal terms of the angular equation, allow efficient diagonalization of the Hamiltonian for all values of the hyperspherical radius. The method is applied to the determination of the ${}^{1}{S}^{e}$ energy levels of the ${\mathrm{Li}}^{+}$ and we show that the precision can be improved in a systematic and controllable way.

Monte Carlo hyperspherical description of helium cluster excited states

The J=0 many-body Schrödinger equation for HeN4 clusters with N=3–10 is solved numerically by combining Monte Carlo methods with the adiabatic hyperspherical approximation. We find ground state and excited state energies for these systems with an adiabatic separation scheme that reduces the problem to motion in a one-dimensional effective potential curve as a function of the hyperspherical radius R. We predict the number of J=0 bound states for these clusters, and also the He+HeN−1 elastic scattering lengths up to N=10. For N=5–10, these are the first such calculations reported.

Territory covered by N random walkers.

The problem of evaluating the number of distinct sites S(N)(t) covered up to time t by N random walkers is revisited. For the nontrivial time regime and for N>>1 we show how to get the asymptotic behavior of S(N)(t) and we calculate the main and first two corrective terms. The mth corrective term decays mildly as 1/ln(m) N. For d-dimensional (d=1,2,3) simple cubic lattices, the main term is the volume of the hypersphere of radius [(ln N(2))2Dt/d](1/2), D being the diffusion constant, and the corrective terms account for the roughening of the surface of the set of visited sites.

Photo double-ionization of helium : A new approach combining R matrix and semiclassical techniques in an hyperspherical framework

We introduce a new method for computing photo double ionization (PDI) cross sections for two electron atoms. It is formulated in terms of the hyperspherical radius R and relies upon a combination of R matrix techniques in the inner region R≤R0 with a semi classical approximation for the R motion in the outer region. We present a first application of this method to the PDI of He within a model of reduced dimensionality where r1=r2. It demonstrates the validity of our numerical scheme and provides a first quantitative estimate of the energy domain of validity of the Wannier mechanism.

Highly excited states for the helium atom in the hyperspherical adiabatic approach

The hyperspherical adiabatic approach is used to obtain the highly excited series and of the helium atom. The introduction of appropriate asymptotic conditions at large values of the hyperspherical radius results in a stable algorithm that allows the calculation of the full atomic spectrum with precision of a few parts per million. Comparison with the variational calculations available in the literature shows that the accuracy of the results improves with increasing principal quantum number. We present the energies up to n = 31 which is the typical value used in multiphoton excitation experiments.

Theoretical investigation of the Ar+H2+(0⩽v⩽4, j=0)→ArH++H nonadiabatic reaction dynamics

The title reaction is investigated using a semiclassical coupled wave packet method where the hyperspherical radius ρ is treated classically and the other coordinates quantally. Dynamical calculations are performed in a coplanarlike approximation using eight coupled electronic states. State-to-state reaction cross sections are obtained in the energy range 0.3 eV⩽Ecoll⩽5 eV for five different initial rovibrational states. The internal energy of the ArH+ product is found to be very high, especially at low collision energy. A comprehensive analysis of the reaction mechanisms is presented.

Parametric Worst-Case Analysis by Prabi Method: Application to Flexible Space Structures

In this paper, a new stability robustness analysis for discrete time systems with real parametric uncertainties is proposed. This analysis provides an upper bound of the stability hypersphere radius but also the parametric worst-case i.e. the direction in parametric space which drives straightforwardly the system to instability. This method is based on duality between the quality of the fictitious closed-loop identification of uncertain parameters and the closed-loop stability robustness w.r.t to these parameters.

Theoretical investigation of the Ar+(J) + H2 → ArH+ + H reaction: semiclassical coupled wavepacket treatment

The title reaction is investigated using a semiclassical coupled wavepacket method where the hyperspherical radius ϱ is treated classically and the other coordinates quantally. The wavefunction is expanded over electronic states, whose potential energy surfaces and couplings are determined within a DIMZO framework. Dynamical calculations are performed in a coplanar-like approximation using eight coupled electronic states. State to state reaction cross sections are obtained in the energy range 0.3 eV ≤ E ≤ 5 eV. A comprehensive analysis of the reaction mechanisms is presented. It shows that the collision can be viewed as a two step process: the sharing of the system between the electronic adiabatic states, followed by the reaction along the uncoupled ground adiabatic potential energy surface. Two reaction mechanisms are exhibited, one of them occurring at low collision energy only.

Semiclassical coupled wave packet study of the nonadiabatic collisions Ar+(J)+H2: Zero angular momentum case

The title reaction is investigated for total angular momentum ℐ=0 using a semiclassical coupled wave packet method where the Smith–Whitten-type hyperspherical angles θ and φ are treated quantally, and the hyperspherical radius ρ is treated classically. The wave function is expanded over an electronic basis set which includes 28 states. The diabatic potential energy surfaces are determined by DIMZO calculations. Probabilities for reaction, charge transfer, collision induced dissociation, dissociative charge transfer, and fine structure transitions are obtained in the energy range 0.3 eV≤E≤30 eV. A comprehensive analysis of the reaction mechanisms is presented.

Crossover from geometrical to stochastic fractal statistics for translationally invariant random distributions of independent particles in n-dimensional Euclidean space

The paper deals with the statistics of translationally invariant random distributions of independent particles in Euclidean space. The geometrical fractal model used in astrophysics and hydrodynamics assumes that the number of particles enclosed in a hypersphere of radius r obeys a Poisson law v N ( N!) −1exp(− v), where the average number of particles is a power function of radius r: v ∼ r d f and d f is the dimension of a fractal structure embedded in the Euclidean space considered. A statistical fractal distribution is introduced by assuming that the probability density of the distance between two nearest particles has a long tail of the inverse power law type φ 0( r) ∼ r −(1 + H r ) as r → ∞, where H r > 0 is a statistical fractal exponent. The distribution of the number of particles enclosed in a hypersphere of radius r is also Poissonian but the average number of points increases logarithmically rather than algebraically with the radius r: v∼ H r ln( r r 0 ) as r → ∞. The spatial distribution of points corresponding to this statistical fractal model is much rarer than in the geometrical fractal case. An alternative approach is derived by assuming that the probability density of the distance between two particles has a very broad logarithmic tail φ 0(r) dr ∼ d[ ln m( r r 0 )] [ ln m( r r 0 )] 1 + H , H > 0, r → ∞ where ln m( r r 0 ) = ln … ln( r r 0 ) is the mth iterated logarithm of r r 0 . For such a logarithmic statistical fractal the number of particles increases much more slowly with the radius r than in the ‘pure’ statistical fractal case: v ∼ H ln m + 1( r r 0 ) as r → ∞. By using a heuristic approach two general probabilistic models are derived which include both the geometrical and statistical fractal models as particular cases; these models predict power law dependences of the average number of particles for small systems and logarithmic dependences for large systems, respectively. The significance of the generalized models is elucidated by using the notion of a specific fractal hypervolume \\ ̃ gw which corresponds to a given particle. For the generalized model derived from the pure statistical fractal \\ ̃ gw is made up of two additive contributions: a constant one which determines the behavior of the model for small systems and a linearly increasing contribution with the total fractal hypervolume which is predominant for large systems. A similar structure of \\ ̃ gw exists for the generalized model leading to the logarithmic statistical fractal regime.

Energy sharing and angular distribution in the double photoionization of helium.

The double photoionization of an atom is considered in the near-threshold domain. The angular correlation pattern is analyzed for an arbitrary sharing of the excess energy by the electrons. The motion over the hyperspherical radius R is treated semiclassically which casts the problem in a nonstationary form. For motion over the hyperspherical angle \ensuremath{\alpha} the set of classical trajectories is considered. They converge to the same (R\ensuremath{\rightarrow}\ensuremath{\infty}) limit which corresponds to a chosen energy-sharing ratio. For each \ensuremath{\alpha}(R) trajectory the quantum wave packet (over the correlation angle ${\mathrm{\ensuremath{\theta}}}_{12}$) is propagated from the inner zone to the region of free-electron motion. At the border of the reaction zone a realistic boundary condition is imposed. The cross sections exhibit a weak dependence on the energy-sharing ratio.

Largest hypersphere of stability for polynomials with perturbed coefficients

Introduction T HIS study is a continuation of an earlier study in which a method for finding the radius of the largest hypersphere of stability for a nominally stable system in the parameter space was introduced. Closed-form solutions to find such a radius for a second-order system were reported. In this Note we intend to apply the same method used in Ref. 1, i.e., the Lagrange multiplier approach, to Hurwitz polynomials under coefficient perturbations to obtain closed-form solutions for thirdand fourth-order polynomials. Consider the characteristic equation of a nominally stable linear system