Sphere Of Unit Radius Research Articles

The differential game “Cossacks–robbers” on time scales

In finite-dimensional Euclidean space, we address the problem of simple pursuit of a group of evaders by a group of pursuers in a given time scale with equal opportunities for all participants. The set of controls of each participant is a sphere of unit radius with its center at the origin. The goal of the group of pursuers is to catch all evaders. The goal sets are the origin. The goal of the evaders is the opposite one, namely, for at least one evader to avoid capture. Conditions for solvability of the local and global problems of evasion and the upper and lower estimates of the minimal number of evaders avoiding a given number of pursuers from any initial positions are obtained.

Random Motions at Finite Velocity on Non-Euclidean Spaces

In this paper, random motions at finite velocity on the Poincaré half-plane and on the unit-radius sphere are studied. The moving particle at each Poisson event chooses a uniformly distributed direction independent of the previous evolution. This implies that the current distance d(P0,Pt) from the starting point P0 is obtained by applying the hyperbolic Carnot formula in the Poincaré half-plane and the spherical Carnot formula in the analysis of the motion on the sphere. We obtain explicit results of the conditional and unconditional mean distance in both cases. Some results for higher-order moments are also presented for a small number of changes of direction.

On the Sensitivity of the Gamma of the Quantum Cryptographic System AKM2017 to Changes in the Session Key

The quantum cryptographic system AKM2017 is considered. The results of the analysis of the dependence of the degree of difference between the encryption and decryption gamut on the degree of difference between the corresponding session keys are presented. The equality of these degrees of distinction is revealed and substantiated. For an arbitrarily fixed encryption session key, the distribution of session decryption keys by classes is revealed and described, depending on the value of the degree of difference between the encryption and decryption gamuts. One class is made up of all session decryption keys, leading to the same value of the degree of difference between the encryption and decryption gamuts. A geometric interpretation of the specified distribution by classes is given in the form of placement along circles (class is a circle) on the surface of a sphere of unit radius centered at the origin of the Euclidean rectangular coordinate system in a three-dimensional linear space over the field of real numbers. The stated results can be used to solve the problems of optimizing the values of the parameters of accuracy and reliability of the functioning of variants of practical implementations of the quantum cryptographic system AKM2017, for example, when setting up a session decryption key, which makes it possible to guarantee a predetermined small value of the mathematical expectation of the number of incorrectly decrypted binary plain text.

Analytical solutions for three-dimensional Coulomb transition matrix at negative energy and integer values of interaction parameter

With the use of a stereographic projection of momentum space into a four-dimensional sphere of unit radius, the possibility of analytical solution of the three-dimensional two-body Lippmann–Schwinger equation with the Coulomb interaction at negative energy has been studied. Simple analytical expressions for the three-dimensional Coulomb transition matrix in the case of the repulsive Coulomb interaction and positive integer values of the Coulomb parameter have been obtained. The worked out method has also been applied to the generalized three-dimensional Coulomb transition matrix in the case of the attractive Coulomb interaction and negative integer values of the Coulomb parameter.

Deformation Broadening and the Fine Structure of Spectral Lines in Optical Spectra of Dielectric Crystals Containing Rare-Earth Ions

The procedure of calculation of the spectral line shape in optical spectra of rare-earth ions in crystals with the inclusion of random deformations of an elastically anisotropic crystal lattice caused by point defects is developed. The distribution function of components of the random strain tensor in the case of a low defect concentration is obtained as the generalized six-dimensional Lorentz distribution. The distribution function parameters are represented by the integral functional of the strain tensor components on a sphere of unit radius containing an isotropic point defect in its center. The numerical calculations of the strain tensors induced by point defects and the parameters of the distribution functions of random strains in LiLuF4 and LaAlO3 crystals have been performed. The calculated envelope with the doublet structure corresponding to the Γ2(3H4) → Γ34(3H5) singlet–doublet transition in the absorption spectrum of Pr3+ ions in the LiLuF4 crystal agrees well with the data of the measurements.

Definition of service angle of android’s robot hand by method of small movements of gripper’s axis synthesis by speed vector

The paper presents a generalized method for determining the service solid angle based on the assigned gripper axis orientation with a stationary grip center. Motion synthesis in this work is carried out in the vector of velocities. As an example, a solid angle of the android robot arm is determined, this angle being formed by the longitudinal axis of a gripper. The nature of the method is based on the study of sets of configuration positions, defining the end point positions of the unit radius sphere sweep, which specifies the service solid angle. From this the spherical curve specifying the shape of the desired solid angle was determined. The results of the research can be used in the development of control systems of autonomous android robots.

Деформационное уширение и тонкая структура спектральных линий в оптических спектрах диэлектрических кристаллов, содержащих редкоземельные ионы

AbstractThe procedure of calculation of the spectral line shape in optical spectra of rare-earth ions in crystals with the inclusion of random deformations of an elastically anisotropic crystal lattice caused by point defects is developed. The distribution function of components of the random strain tensor in the case of a low defect concentration is obtained as the generalized six-dimensional Lorentz distribution. The distribution function parameters are represented by the integral functional of the strain tensor components on a sphere of unit radius containing an isotropic point defect in its center. The numerical calculations of the strain tensors induced by point defects and the parameters of the distribution functions of random strains in LiLuF_4 and LaAlO_3 crystals have been performed. The calculated envelope with the doublet structure corresponding to the Γ_2(^3 H _4) → Γ_34(^3 H _5) singlet–doublet transition in the absorption spectrum of Pr^3+ ions in the LiLuF_4 crystal agrees well with the data of the measurements.

Fuzzy circle and new fuzzy sphere through confining potentials and energy cutoffs

Guided by ordinary quantum mechanics we introduce new fuzzy spheres of dimensions d=1,2: we consider an ordinary quantum particle in D=d+1 dimensions subject to a rotation invariant potential well V(r) with a very sharp minimum on a sphere of unit radius. Imposing a sufficiently low energy cutoff to `freeze' the radial excitations makes only a finite-dimensional Hilbert subspace accessible and on it the coordinates noncommutative \`a la Snyder; in fact, on it they generate the whole algebra of observables. The construction is equivariant not only under rotations - as Madore's fuzzy sphere -, but under the full orthogonal group O(D). Making the cutoff and the depth of the well dependent on (and diverging with) a natural number L, and keeping the leading terms in 1/L, we obtain a sequence S^d_L of fuzzy spheres converging (in a suitable sense) to the sphere S^d as L diverges (whereby we recover ordinary quantum mechanics on S^d). These models may be useful in condensed matter problems where particles are confined on a sphere by an (at least approximately) rotation-invariant potential, beside being suggestive of analogous mechanisms in quantum field theory or quantum gravity.

Modal Analysis of Fluid Flows: An Overview

Modal Analysis of Fluid Flows: An Overview

Hydrodynamics of superfluid quantum space: particle of spin-1/2 in a magnetic field

The modified Navier-Stokes equation describing the velocity field in the superfluid quantum space is loaded by the external Lorentz force introducing electromagnetic fields. In order to open the path for getting the \Schrodinger-Pauli equation describing the behavior of a particle with spin-1/2 in the magnetic field we need to extend the continuity equation to take into account conservation of spin flows on the 3D sphere. This extension includes conservation of the density distribution function in 6D space, that is a multiplication of the 3D Euclidean space by the 3D sphere of unit radius. The special unitary group SU(2) underlies the rotations of the spin on this sphere. This group is isomorphic to the group of quaternions containing the real 4x4 matrices of norm 1. Transition to the quaternion group opens up the way to the possibility of describing the spin-1/2 behavior in a magnetic field as a motion of a spin flag on the 2D sphere. Maxwell's electromagnetic field theory manifests itself in the quaternion group basis by the natural manner.

Uncoupled thermoelasticity solutions applied on beam dumps

In particle accelerators the process of beam absorption is vital. At CERN particle beams are accelerated at energies of the order of TeV. In the event of a system failure or following collisions, the beam needs to be safely absorbed by dedicated protecting blocks. The thermal shock caused by the rapid energy deposition within the absorbing block causes thermal stresses that may rise above critical levels. The present paper provides a convenient expression of such stresses under hypotheses described hereafter. The temperature field caused by the beam energy deposition is assumed to be Gaussian. Such a field models a non-diffusive heat deposition. These effects are described as thermoelastic as long as the stresses remain below the proportional limit and can be analytically modeled by the coupled equations of thermoelasticity. The analytical solution to the uncoupled thermoelastic problem in an infinite domain is presented herein and matched with a finite unit radius sphere. The assumption of zero diffusion as well as the validity of the match with a finite geometry is quantified such that the obtained solutions can be rigorously applied to real problems. Furthermore, truncated series solutions, which are not novel, are used for comparison purposes. All quantities are nondimensional and the problem reduces to a dependence of five dimensionless parameters. The equations of elasticity are presented in the potential formulation where the shear potential is assumed to be nil due to the source being a gradient and the absence of boundaries. Nevertheless equivalent three-dimensional stresses are computed using the compressive potential and optimized using standard analytical optimization methods. An alternative algorithm for finding the critical points of the three-dimensional stress function is presented. Finally, a case study concerning the proton synchrotron booster dump is presented where the aforementioned analytical solutions are used and the preceding assumptions verified.

Reflecting Random Flights

We consider random flights in $\mathbb{R}^d$ reflecting on the surface of a sphere $\mathbb{S}^{d-1}_R,$ with center at the origin and with radius $R,$ where reflection is performed by means of circular inversion. Random flights studied in this paper are motions where the orientation of the deviations are uniformly distributed on the unit-radius sphere $\mathbb{S}^{d-1}_1$. We obtain the explicit probability distributions of the position of the moving particle when the number of changes of direction is fixed and equal to $n\geq 1$. We show that these distributions involve functions which are solutions of the Euler-Darboux-Poisson equation. The unconditional probability distributions of the reflecting random flights are obtained by suitably randomizing $n$ by means of a fractional-type Poisson process. Random flights reflecting on hyperplanes according to the optical reflection form are considered and the related distributional properties derived.

Geometry of n-dimensional Euclidean space Gaussian enfoldments

In this study the geometric features and relationships of the points contained into a Gaussian enfoldment of n-dimensional Euclidean space are analyzed. Euclidean distances and angles are described by means of a simple formulation, which demonstrates the topological change underwent by n-dimensional Euclidean spaces upon Gaussian enfoldment, transforming the Euclidean points into enfoldment points lying in a closed sphere of unit radius. This property relates Gaussian enfoldments with the holographic electronic density theorem.

Spherical nonlinear correlations for global invariant three-dimensional object recognition

We define a nonlinear filtering based on correlations on unit spheres to obtain both rotation- and scale-invariant three-dimensional (3D) object detection. Tridimensionality is expressed in terms of range images. The phase Fourier transform (PhFT) of a range image provides information about the orientations of the 3D object surfaces. When the object is sequentially rotated, the amplitudes of the different PhFTs form a unit radius sphere. On the other hand, a scale change is equivalent to a multiplication of the amplitude of the PhFT by a constant factor. The effect of both rotation and scale changes for 3D objects means a change in the intensity of the unit radius sphere. We define a 3D filtering based on nonlinear operations between spherical correlations to achieve both scale- and rotation-invariant 3D object recognition.

Studying the morphometric characteristics of nuclear pore complexes of a sensory neuron using methods of spherical stochastic geometry

The properties of a mathematical model of a nuclear pore complex are studied using a random cap process on a two-dimensional Euclidean sphere of unit radius. The results obtained make it possible to calculate stereometric characteristics of a nuclear pore complexes from electron-diffraction patterns.

Detection of three-dimensional objects under arbitrary rotations based on range images

In this paper a unique map or signature of three dimensional objects is defined. The map is obtained locally, for every possible rotation of the object, by the Fourier transform of the phase-encoded range-image at each specific rotation. From these local maps, a global map of orientations is built that contains the information about the surface normals of the object. The map is defined on a unit radius sphere and permits, by correlation techniques, the detection and orientation evaluation of three dimensional objects with three axis translation invariance from a single range image.

Maximal Functions with Mitigating Factors in the Plane

Given a smooth, compactly supported hypersurface S in Rn that does not pass through the origin, and denoting by tS the surface dilated by a factor t>0, we can consider the averaging operator defined for functions f∈S, the Schwartz class of functions, by A t f ( x ) = ∫ t s f ( x - y ) d σ ( y ) where dσ is Lebesgue measure on S. We can now define a maximal average operator, M f ( x ) = sup t > 0 | A t f ( x ) | In the case where S is Sn−1, the sphere of unit radius in Rn, we are looking at Stein's spherical maximal function, as treated in his paper [10]. Stein proved that M is a bounded operator on the Lp spaces if and only if p>n/(n−1) when n>3. Subsequently, Bourgain [2] showed that if S is any compactly supported smooth curve with non-vanishing Gaussian curvature, then M will be bounded on Lp, if and only if p>2, thus dealing with the case of the circular maximal function in the plane. (For related results, see also [6] and [7]). In the case where S is a curve whose curvature vanishes to order at most m−2 at a single point, Iosevich [4] showed that M is bounded on Lp for p>m, and unbounded if p = m. If we study curves given by Γ(s) = (s, γ(s)+1), s∈[0, 1], for some suitably smooth γ, where γ(0) = γ′(0) = … = γ(m−1)(0) ≠ γ(m)(0)>0, then we can reinterpret his results as follows. Define M k f ( x ) = sup t > 0 | ∫ 2 - k 2 1 - k f ( x - t Γ ( s ) ) d s | for Schwartz functions f. Iosevich proved that ‖ M k ‖ L p − L p ⩽ c p 2 − k ( 1 − m / p ) for p>2. If we note that κ(s), the curvature of the curve Γ(s) is approximately 2−k(m−2) whenever s∈[2−k, 21−k], then we have that the operator M σ f ( x ) = sup t > 0 | ∫ 0 1 f ( x - t Γ ( s ) ) ( κ ( s ) ) σ d s | is bounded on Lp for some p>2, if σ is sufficiently large, since ‖ M σ ‖ L p - L p ⩽ C ∑ k ⩾ 0 2 - k ( m - 2 ) σ ‖ M k ‖ L p - L p ⩽ C ∑ k ⩾ 0 2 - k ( ( m - 2 ) σ + 1 - m / p ) which is finite so long as σ>(m/p−1) (m−2)−1. If we want to choose σ independent of m>2, the type of the curve, such that Mσ is bounded on Lp for some fixed p>2, then clearly we can take σ = 1/p. In this paper we show that Mσ will be bounded on Lp for p>max{σ−1, s} for a class of infinitely flat, convex curves in the plane. Counterexamples will show that this is the best possible result, in that there exist flat curves for which Mσ is unbounded for 2<p⩽σ−1.

Preliminary communication Low frequency dielectric relaxation in the smectic C* phase of a ferroelectric liquid crystal

Dielectric measurements were made on the ferroelectric liquid crystal Felix 018/100 manufactured by Hoechst, Germany, over the temperature range 30 to 65degreeC (smectic C* phase), frequency range 0.1Hz to 100kHz, with bias voltages of 0, 1, 3 and 10 V, and in a dielectric cell with a spacing of 4 times the helical pitch. Plots of the dielectric loss versus log (frequency) show the usual monotonic increase in the loss with decreasing frequency, as well as the usual loss peak at approximately 1kHz. Plots of the log (dielectric loss) against log (frequency) at low frequencies, have slopes varying from -0.75 to -0.89 when the temperature increases from 30 to 65degreeC. Following the suggestion of Scaife, transforming the complex permittivity data to the complex polarizability of a sphere of unit radius in a vacuum, and plotting the loss polarizability against log (frequency), shows two distinct and separate loss peaks. The sums of the two loss peaks appear to be independent of temperature and bias voltage, even though both depend on these variables.

High orders of Weyl series: resurgence for odd balls

The semiclassical Weyl series for the d-dimension, unit-radius sphere quantum billiard is studied. A conjecture of Berry and Howls (1994 447 527-55) on the late terms of such series for two-dimensional billiards is seen to survive for general (integer) dimension. The conjecture postulates a leading-order, factorial-on-power approximation for the late terms of the Weyl series in terms of the length of a periodic orbit of the classical system. The expansions manifest a difference between odd and even dimensions. The dominating orbit is the diametral, length-4 path in the even-dimension spheres, echoing the known result for the circle billiard. However, when d is odd, it is the next-longest orbit. This surprise can be traced to an `accidental' symmetry in a postulated hyperasymptotic remainder term. Higher-order asymptotic correction terms are found confirming the resurgent link of the Weyl series to the low orders of the oscillatory periodic orbit corrections. From the structure of the latter, it is possible to make further conjectures on the late terms of the periodic orbit corrections themselves. A factorial-on-power behaviour is also found, but now involving the differences between p-bounce orbits and associated whispering-gallery modes.

Precise determination of the critical threshold and exponents in a three-dimensional continuum percolation model

We present a large-scale computer simulation of the prototypical three-dimensional continuum percolation model consisting of a distribution of overlapping (spatially uncorrelated) spheres. By using simulations of up to particles and studying the finite-size scaling of various effective percolation thresholds, we obtain a value of . This value is significantly smaller than the values obtained for simulations that have been carried out using smaller systems. Employing this value of and systems of size L = 160 (relative to a sphere of unit radius), we also obtain estimates of the critical exponents , and for the continuum system and show that the values are different than those obtained using previous values of .