Noninteger Numbers Research Articles

Theory of paraxial optical skyrmions

Vector light beams, characterized by a spatially varying polarization, can exhibit localized structures reminiscent of the skyrmions familiar from the study of magnetic media. We present a theory of such skyrmions within paraxial optics, exploiting mathematical analogies with the study of superfluids, especially the A phase of superfluid He 3 . The key feature is the skyrmion field, which, together with the underlying skyrmion vector potential, determines the properties of the skyrmions and, more generally, the polarization structure of every paraxial vector beam. In addition to structures with integer skyrmion numbers, we find polarization patterns with non-integer skyrmion numbers; these seem to have no analogue in other fields of physics.

On products of idempotents and nilpotents

This article studies the ring structure arising from products of idempotents and nilpotents. Thus the argument is concerned essentially with the one-sided IQNN property of rings. We first prove that if the $2$ by $2$ full matrix ring over a principal ideal domain $F$ of characteristic zero is right IQNN then $F$ contains infinitely many non-integer rational numbers; and that the concepts of right IQNN and right quasi-Abelian are independent of each other. We next introduce a ring property, called {\it right IAN}, as a generalization of both right IQNN and right quasi-Abelian; and provide several kinds of methods to construct right IAN rings. In the procedure, we also show that the right IQNN and right IAN do not go up to polynomial rings.

Improved estimates for the number of non-negative integer matrices with given row and column sums

The number of non-negative integer matrices with given row and column sums features in a variety of problems in mathematics and statistics but no closed-form expression for it is known, so we rely on approximations. In this paper, we describe a new such approximation, motivated by consideration of the statistics of matrices with non-integer numbers of columns. This estimate can be evaluated in time linear in the size of the matrix and returns results of accuracy as good as or better than existing linear-time approximations across a wide range of settings. We show that the estimate is asymptotically exact in the regime of sparse tables, while empirically performing at least as well as other linear-time estimates in the regime of dense tables. We also use the new estimate as the starting point for an improved numerical method for either counting or sampling matrices with given margins using sequential importance sampling. Code implementing our methods is available.

Quantum geometric tensor and the topological characterization of the extended Su–Schrieffer–Heeger model

We investigate the quantum metric and topological Euler number in a cyclically modulated Su–Schrieffer–Heeger (SSH) model with long-range hopping terms. By computing the quantum geometry tensor, we derive exact expressions for the quantum metric and Berry curvature of the energy band electrons, and we obtain the phase diagram of the model marked by the first Chern number. Furthermore, we also obtain the topological Euler number of the energy band based on the Gauss–Bonnet theorem on the topological characterization of the closed Bloch states manifold in the first Brillouin zone. However, some regions where the Berry curvature is identically zero in the first Brillouin zone result in the degeneracy of the quantum metric, which leads to ill-defined non-integer topological Euler numbers. Nevertheless, the non-integer “Euler number” provides valuable insights and an upper bound for the absolute values of the Chern numbers.

Complete and orthonormal sets of exponential-type orbitals with non-integer quantum numbers

Atomic and molecular orbitals show exponential decrease at long range. Complete orthonormal basis sets for atoms should satisfy this criterion. A number of such bases have been used in physics (e.g. Coulomb Sturmians). The challenge of this work is first adapting Slater type Orbitals for this role, as they are not radially orthogonal. Even more important is their generalization to non-integer quantum numbers that have applications for configuration interaction. This generalization requires the whole apparatus of non-integer calculus that is presented using the Riemann–Liouville approach.

One-parameter Darboux-deformed Fibonacci numbers

One-parameter Darboux deformations are established for the simple ordinary differential equation (ODE) satisfied by the continuous generalizations of the Fibonacci sequence recently discussed by Faraoni and Atieh [Symmetry 13, 200 (2021)], who promoted a formal analogy with the Friedmann equation in the FLRW homogeneous cosmology. The method allows the introduction of deformations of the continuous Fibonacci sequences, hence of Darboux-deformed Fibonacci (noninteger) numbers. Considering the same ODE as a parametric oscillator equation, the Ermakov–Lewis invariants for these sequences are also discussed.

Applicability of noninteger bessel type orbital basis sets: numerical and analytical approaches

Recently, usefulness of the noninteger principal quantum numbers for Bessel type orbitals was discussed by Weniger in (Adv Quant Chem 83:209–237, 2021). In this study, we analyzed the applicability and numerical accuracy of basis sets of noninteger Bessel type orbitals to electronic structure calculations. Both numerical and analytical approaches are applied to two-electron atomic systems. The results of the numerical test demonstrated the potential of the noninteger values of principal quantum number for the improving of Bessel type functions approach in use of LCAO methods. Nevertheless, the analytical approach is still not suitable and in development and needs to be investigated further.The performance of the presented basis functions is also compared to the numerical Hartree–Fock results.

ON APPROXIMATION OF CONSTANTS BY POLYNOMIALS WITH INTEGER COEFFICIENTS

In this paper, the exact rate of decrease of best approximations of non-integer numbers by polynomials with integer coefficients is established.

A new efficiency relaxation model for rigorous stage number optimization of distillation columns

Determining the optimal state number in distillation column design is a mixed-integer nonlinear programming (MINLP) problem, which is difficult to solve. One strategy to solve this problem is relaxing integer stage numbers into continuous variables and solving an easier nonlinear programming (NLP) problem. However, the existing relaxation model often converges to non-integer stage numbers or poor local optima in practical applications, which motivates this paper to develop a new relaxation model. The proposed relaxation model employs an interpolation function to relax the discrete decision on the existence of a stage by varying interphase mass and heat transfer rate between zero and equilibrium state. By adjusting a shape parameter in the interpolation function, the interpolation function shape and its penalty extent of non-integer stage can be manipulated. The results of three distillation column optimization cases show that the proposed model can effectively avoid converges to non-integer stage numbers or poor optima.

Heat Transfer in a Fractional Nanofluid Flow through a Permeable Medium

This study examines viscoelastic fractional nanofluid flow through Darcy medium. Memory characteristics due to elasticity are explored with noninteger time derivatives. The unsteady motion of MHD flow is modeled by nonlinear differential equations. Buoyancy forces are incorporated via convection parameters in the flow domain. Fractional relaxation time is considered to control the propagation speed of temperature. A finite difference, along with finite element, a numerical algorithm has been developed for the computation of governing flow equations. Friction coefficient, Sherwood numbers, and Nusselt numbers are computed for the noninteger derivative model. Simulations revealed that noninteger numbers have congruous behavior for concentration, temperature, and velocity fields. It is also noted that heat flux, δ 1 , and mass flux, δ 2 , numbers have contradictory effects on the friction coefficient. Various flows, particularly in polymer industries and electrospinning for the production of nanofibers, can be tackled in a comparable pattern.

Design of a Cryptographic System for Communication Security using Chaotic Signals

Disturbance or corresponding errors of the transmission of information affect the ability of error detection. The chaotic encryption system prevents errors and secures the transmission system safely. The security assures by updating chaotic signals with the parameters of the chaotic circuits which are frequently changed. The data decipher and the encryption by the chaotic signaling system renews and changes the initial condition of a chaotic electric circuit. When most of the decimal portions are less than the threshold, the transmission is accepted, and all the noninteger numbers are rounded to their nearest integers. The criterion allows the error-detection function in the security system that is proposed in this paper. The chaotic encryption system for information is applied to public channels by the authorized individual. Three pictorial examples transmitted in the proposed system successfully demonstrate the security and performance. The new system provides high efficiency in the satellite communication network.

Mathematical model of COVID-19 spread in Turkey and South Africa: theory, methods, and applications

A comprehensive study about the spread of COVID-19 cases in Turkey and South Africa has been presented in this paper. An exhaustive statistical analysis was performed using data collected from Turkey and South Africa within the period of 11 March 2020 to 3 May 2020 and 05 March and 3 of May, respectively. It was observed that in the case of Turkey, a negative Spearman correlation for the number of infected class and a positive Spearman correlation for both the number of deaths and recoveries were obtained. This implied that the daily infections could decrease, while the daily deaths and number of recovered people could increase under current conditions. In the case of South Africa, a negative Spearman correlation for both daily deaths and daily infected people were obtained, indicating that these numbers may decrease if the current conditions are maintained. The utilization of a statistical technique predicted the daily number of infected, recovered, and dead people for each country; and three results were obtained for Turkey, namely an upper boundary, a prediction from current situation and lower boundary. The histograms of the daily number of newly infected, recovered and death showed a sign of lognormal and normal distribution, which is presented using the Bell curving method parameters estimation. A new mathematical model COVID-19 comprised of nine classes was suggested; of which a formula of the reproductive number, well-poseness of the solutions and the stability analysis were presented in detail. The suggested model was further extended to the scope of nonlocal operators for each case; whereby a numerical method was used to provide numerical solutions, and simulations were performed for different non-integer numbers. Additionally, sections devoted to control optimal and others dedicated to compare cases between Turkey and South Africa with the aim to comprehend why there are less numbers of deaths and infected people in South Africa than Turkey were presented in detail.

Analytical evaluation of relativistic molecular integrals: III. Computation and results for molecular auxiliary functions

This work describes the fully analytical method for the calculation of the molecular integrals over Slater-type orbitals with non-integer principal quantum numbers. These integrals are expressed through relativistic molecular auxiliary functions derived in our previous paper (Bagci and Hoggan in Phys Rev E 91(2):023303, 2015). The procedure for computation of the molecular auxiliary functions is detailed. It applies both in relativistic and non-relativistic electronic structure theory. It is capable of yielding highly accurate molecular integrals for all ranges of orbital parameters and quantum numbers.

Advantages of Slater-type spinor orbitals in the Dirac–Hartree–Fock method. Results for hydrogen-like atoms with super-critical nuclear charge

This work presents the formalism for evaluating molecular SCF equations, adapted to four-component Dirac spinors, which in turn reduce to Slater-type orbitals with non-integer principal quantum numbers in the non-relativistic limit. If Slater-type spinor orbitals are used in the algebraic approximation to solve the Dirac equation, the “catastrophe” previously noted for atomic numbers $$Z>137$$, in Dirac equation resolution with a potential corresponding to a point-charge no longer applies. It is observed that, ground-state energy of hydrogen-like atoms reaches the negative-energy continuum $$\left( -mc^2 \right)$$ with super-critical nuclear charge $$Z_{c}$$, about $$Z_{c}=160$$. The difficulty associated with finding relations for molecular integrals over Slater-type spinors which are non-analytic in the sense of complex analysis at $$r = 0$$, is eliminated. Unique numerical accuracy is provided by solving the molecular integrals through Laplace expansion of the Coulomb interaction and prolate spheroidal coordinates. New convergent series representation formulae are derived. The technique draws on previous work by the author. The general formalism is presented in this paper.

Analytical evaluation of relativistic molecular integrals: III. Computation and results for molecular auxiliary functions

This work describes the fully analytical method for calculation of the molecular integrals over Slater-type orbitals with non-integer principal quantum numbers. These integrals are expressed through relativistic molecular auxiliary functions derived in our previous paper [Phys. Rev. E 91, 023303 (2015)]. The procedure for computation of the molecular auxiliary functions is detailed. It applies both in relativistic and non-relativistic electronic structure theory. It is capable of yielding highly accurate molecular integrals for all ranges of orbital parameters and quantum numbers.

Nontrivial magnetic field related phenomena in the singlelayer graphene on ferroelectric substrate (Review Article)

The review is focused on our predictions of nontrivial physical phenomena taking place in the nanostructure single-layer graphene on ferroelectric substrate, which are related with magnetic field. In particular we predicted that 180-degree domain walls in a strained ferroelectric film can induce p-n junctions in a graphene channel and lead to the unusual temperature and gate voltage dependences of the perpendicular modes v┴ of the integer quantum Hall effect. The non-integer numbers and their irregular sequence principally differ from the conventional sequence v┴ = 3/2, 5/3, … The unusual v┴-numbers originate from significantly different numbers of the edge modes, v1 and v2, corresponding to different concentration of carriers in the left (n1) and right (n2) ferroelectric domains of p-n junction boundary. The difference between n1 and n2 disappears with the vanishing of the film spontaneous polarization in a paraelectric phase, which can be varied in a wide temperature range by an appropriate choice of misfit strain originated from the film-substrate lattice mismatch. Next we studied the electric conductivity of the system ferromagnetic dielectric-graphene channel-ferroelectric substrate. The magnetic dielectric locally transforms the band spectrum of graphene by inducing an energy gap in it and making it spin-asymmetric with respect to the free electrons. It was demonstrated, that if the Fermi level in the graphene channel belongs to energy intervals, where the graphene band spectrum, modified by EuO, becomes sharply spin-asymmetric, such a device can be an ideal non-volatile spin filter. The practical application of the system under consideration would be restricted by a Curie temperature of a ferromagnet. Controlling of the Fermi level (e.g., by temperature that changes ferroelectric polarization) can convert a spin filter to a spin valve.

Some Results on Harmonic Type Sums

In this study, we consider the summatory function of convolutions of the Möbius function with harmonic numbers, and we show that these summatory functions are linked to the distribution of prime numbers. In particular, we give infinitely many asymptotics which are consequences of the Riemann hypothesis. We also give quantitative estimate for the moment function which counts non-integer hyperharmonic numbers. Then, we obtain the asymptotic behaviour of hyperharmonics.

Number of particles in fission fragments

In current simulations of fission, the number of protons and neutrons in a given fission fragment is almost always obtained by integrating the total density of particles in the sector of space that contains the fragment. Because of the antisymmetry of the many-body wave function of the whole nucleus, this procedure systematically gives noninteger numbers of particles in the fragments. We introduce a novel sampling method to estimate rigorously the probability of finding $Z$ protons and $N$ neutrons in a fission fragment without resorting to projectors, which can sometimes give unwieldy results. When applied on standard Hartree-Fock-Bogoliubov many-body states, we show that our approach reproduces indeed the results of full particle-number projection. We then estimate the charge and mass number dispersion of several scission configurations in $^{240}$Pu with and without pairing correlations included. We show that odd-even effects in the charge probability naturally occur within our approach, which could explain the well-known odd-even staggering of charge distributions. Our method is applicable either in static calculations of scission configurations such as, e.g., in the macroscopic-microscopic approach or energy density functional theory, but also in explicitly time-dependent density functional theory simulations of fission.

Do cuttlefish have fraction number sense?

Number sense is a key cognitive function in animals. The biological functions of number discrimination have a wide range, including the selection of prey and social interaction. In a previous study, we have shown that cuttlefish are able to distinguish numerical differences among various integers, including 1 vs. 2, 2 vs. 3, 3 vs. 4, and 4 vs. 5. However, it is not known whether cuttlefish are able to discriminate various fractions, that is, various non-integer numbers. In addition, no study on invertebrates has examined fraction number sense. Using the active preying behavior of cuttlefish (Sepia pharaonis), we investigated the spontaneous preference for larger quantity by presenting two-alternative choice between 1 vs. 1.5, 1.5 vs. 2, 2 vs. 2.5, and 2.5 vs. 3. In this context, the quantity1.5 is composed of one large shrimp and one small shrimp, in which the size of the small shrimp is one-half of that of the large shrimp. The result shows that the cuttlefish chose larger quantity in the first three pairs, but they could not distinguish the pair 2.5 vs. 3. Despite that the absolute differences in these pairs are the same (0.5), the relative differences in these pairs decrease (0.5, 0.33, 0.25, and 0.2, respectively). This implies that the perceived difference in quantity is proportional to the initial quantity (Weber's law). Although the present study does not truly differentiate the number difference from the quantity difference, this result does raise the possibility that cuttlefish may be equipped with the primitive concept of fractions, and if so, the perceived just noticeable difference is similar for both integer and fraction number discrimination.

Integer quantum Hall effect in graphene channel with p-n junction at domain wall in a strained ferroelectric film

We revealed that 180° domain walls in a strained ferroelectric film can induce p-n junctions in a graphene channel and lead to the nontrivial temperature and gate voltage dependences of the perpendicular modes of the integer quantum Hall effect (IQHE). In particular, the number of perpendicular modes v⊥, corresponding to the p-n junction across the graphene channel, varies with the gate voltage increase from small integers to higher non-integer numbers, e.g., v⊥ = 1.9, 2, …, 5.1, 6.875, …, 9.1, …, 23,…, 37.4, in the vicinity of the transition temperature from the ferroelectric to paraelectric phase. The non-integer numbers and their irregular sequence principally differ from the sequence of non-integer numbers ν = 3/2, 5/3, … reported earlier. The unusual v⊥-numbers originate from significantly different numbers of the edge modes, ν1 and ν2, corresponding to different concentrations of carriers in the left (n1) and right (n2) domains of the p-n junction boundary. The concentrations n1 and n2 are determined by the gate voltage and spontaneous polarization contributions, and so their difference originates from different directions of the spontaneous polarization in different domains of the strained ferroelectric film. The difference between n1 and n2 disappears with the vanishing of the film spontaneous polarization in a paraelectric phase. The temperature transition from the ferroelectric to paraelectric phase taking place in a strained ferroelectric film can be varied in a wide temperature range by an appropriate choice of misfit strain so that the first plateaus of the predicted IQHE effect can be observed even at room temperatures.