Non-integer Values Research Articles

Angular momenta, helicity, and other properties of dielectric-fiber and metallic-wire modes

Spin and orbital angular momenta (AM) of light are well studied for free-space electromagnetic fields, even nonparaxial. One of the important applications of these concepts is the information transfer using AM modes, often via optical fibers and other guiding systems. However, the self-consistent description of the spin and orbital AM of light in optical media (including dispersive and metallic cases) was provided only recently [K.Y. Bliokh et al., Phys. Rev. Lett. 119, 073901 (2017)]. Here we present the first accurate calculations, both analytical and numerical, of the spin and orbital AM, as well as the helicity and other properties, for the full-vector eigenmodes of cylindrical dielectric and metallic (nanowire) waveguides. We find remarkable fundamental relations, such as the quantization of the canonical total AM of cylindrical guided modes in the general nonparaxial case. This quantization, as well as the noninteger values of the spin and orbital AM, are determined by the generalized geometric and dynamical phases in the mode fields. Moreover, we show that the spin AM of metallic-wire modes is determined, in the geometrical-optics approximation, by the transverse spin of surface plasmon-polaritons propagating along helical trajectories on the wire surface. Our work provides a solid platform for future studies and applications of the AM and helicity properties of guided optical and plasmonic waves.

Skein relations for spin networks, modifed

An alternative framework underlying connection between tensor [Formula: see text]-calculus and spin networks is suggested. New sign convention for the inner product in the dual spinor space leads to a simpler and direct set of initial rules for the diagrammatic recoupling methods. Yet, it preserves the standard chromatic graph evaluations. In contrast with the standard formulation, the background space is that of symmetric tensor spaces, which seems to be in accordance with the representation theory of [Formula: see text]. An example of Apollonian disk packing is shown to be a source of spin networks. The graph labeling is extended to non-integer values, resulting in the complex values of chromatic evaluations.

The effects of different noise types and mobility on error rate of digital modulation schemes over millimeter-wave Weibull fading channels

Millimeter (mm-wave) communication is a prominent candidate to support the evolution towards fifth generation (5G) wireless systems. As such, in this paper, we study the impact of different noise type and mobility on the performance of coherent binary digital modulation schemes in mm-wave Weibull fading channels. To this end, exact and new closed-form expressions are derived for the bit error rate of coherent digital modulation schemes in millimeter wave Weibull fading channels in the presence of additive non-Gaussian noise. In addition, new and exact closed-form expressions are obtained for the symbol error rate of square M-ary quadrature amplitude modulation scheme (M-AQM). The derived expressions take into consideration the mobility of the wireless receives. Besides, they are valid for integer and non-integer values of the fading and noise shaping parameters. Analytical results are supported by Monte-Carlo simulations to validate the accuracy of the obtained results.

Thermodynamic Justification for the Parabolic Model for Reactivity Indicators with Respect to Electron Number and a Rigorous Definition for the Electrophilicity: The Essential Role Played by the Electronic Entropy.

The temperature-dependence of the Helmholtz free energy with respect to the number of electrons is analyzed within the framework of the Grand Canonical Ensemble. At the zero-temperature limit, the Helmholtz free energy behaves as a Heaviside function of the number of electrons; however, as the temperature increases, the profile smoothens and exhibits a minimum value at noninteger positive values of the fractional electronic charge. We show that the exact average electronic energy as a function of the number of electrons does not display this feature at any temperature, since this behavior is solely due to the electronic entropy. Our mathematical analysis thus indicates that the widely used parabolic interpolation model should not be viewed as an approximation for the average electronic energy, but for the dependence of the Helmholtz free energy upon the number of electrons, and this analysis is corroborated by numerical results. Finally, an electrophilicity index is defined for the Helmholtz free energy showing that, for a given chemical species, there exists a temperature value for which this quantity is equivalent to the electrophilicity index defined within the parabolic interpolation of the electronic energy as a function of the number of electrons. Our formulation suggests that the convexity property of the energy versus the number of electrons together with the entropic contribution does not allow for an analogous nucleophilicity index to be defined.

Performance analysis of outage probability and error rate of square M-QAM in mobile wireless communication systems over generalized α-μ fading channels with non-Gaussian noise

This paper derives new and exact closed-form expressions for the average symbol error rate (SER) of square M-ary quadrature amplitude modulation (M-QAM) in wireless communication systems over the α-μ fading channels subject to an additive non-Gaussian noise. The obtained expressions take into account static and mobile wireless receivers. In addition, a closed-form expression for the outage probability in mobile networks is obtained. Please note that all derived expressions in this paper a valid for integer and non-integer values of the fading parameters. Analytical results are presented to study the impact of noise shaping parameter, severity of fading, and mobility on the average SER. Monte-Carlo simulations results are also provided to validate the accuracy of the analytical results.

Nucleation model for mesopore-confined water freezing kinetics

Cyclic freezing of water is the main reason of microstructure damages in porous materials exploited in cold climate. It is usually assumed that water freezes much faster than temperature changes, therefore the kinetic effects are neglected. Nevertheless, in the reality such assumption might be often violated. In this paper the kinetics of water freezing in mesopores of two silica gels with dominant pore diameter equal 11 and 36 nm has been investigated. To obtain the freezing rate functions the differential scanning calorimetry test with multiple cooling rate program has been performed. The activation energy is calculated using both, differential and local integral methods. It is found that the activation energy is proportional to the reciprocal temperature. The most suitable kinetic model has been selected by means of two methods: the classical procedure recommended by ICTAC KC and the linear model-fitting method. The former method indicates that two models are appropriate: the empirical Šesták–Berggren model and physical Johnson-Mehl-Avrami model. The SB model demonstrates high accordance with the experimental data, whereas the JMA model allows to approximate the experimental data with high accuracy only if the activation energy is calculated using differential approach. The application range of linear model-fitting method has been extended with the fractional numbers for JMA model parameter. The best correlation between experimental data and theoretical model is obtained for the non-integer values of JMA parameter. It indicates that nucleation and nuclei growth for both silica gels are of fractional dimensions.

Fractional Airy beams.

Airy beams possess a number of properties that ensure their multifunction and high relevance in many applications. This fact stimulates scientists to search for new modifications and generalizations of classical Airy beams. Several generalizations of the Airy functions are known, on the basis of both the modification of the differential equation and the variations in the integral representation. In this paper we propose and investigate a new type of Airy beams-fractional Airy beams (FrAiB). They are based on the generalization of the integral representation and are close to the Olver functions, but we are considering a wider range of the power-law dependence of the argument, including non-integer (fractional) values of the power. A theoretical and numerical analysis of the FrAiBs, as well as their symmetrized variants, was performed. The properties of FrAiBs, such as being non-diffracting and autofocusing, were numerically investigated by means of the fractional Fourier transform, describing the beam transformations by paraxial optical systems. We believe that new beams can be useful for laser manipulation techniques and lensless laser patterning.

New analytical expressions for the performance metrics of wireless communication system over Weibull/Lognormal composite fading

A realistic propagation scenario is more frequently characterized by a composite multipath/shadowed fading. In this work, a class of Weibull/Lognormal (W-LN) composite fading channel has been analysed and investigated. New analytical approximations to various important performance metrics of wireless communication systems such as the average symbol error probability (SEP), the outage probability, the amount of fading (AF), and the channel capacity are derived. Further, an asymptotic analysis of the composite fading channel is carried out and closed form expressions of the average SEP with and without maximal ratio combining (MRC) diversity along with the coding and diversity gain are presented. The solutions are presented in terms of Fox H-function and are valid for both the integer and non-integer values of the multipath and shadowing parameters. The results are validated via Monte-Carlo simulations and exact numerical results.

Design of a state observer to approximate signals by using the concept of fractional variable-order derivative

This article proposes a state observer to find a model for a given signal, i.e. to approximate a treated signal. The design of the state observer is based on a dynamical system of equations which is generated from the increasing-order differentiation of a n-th order Fourier series. This dynamical system is set in state space representation by considering that the Fourier series is the first state and the rest of the states are the successive derivatives of the series. The purpose of the state observer is the recursive estimation of the states in order to recover the coefficients from them. This set of coefficients produces the best fit between the dynamical system and the signal. The dynamical system used for the observer conception shall be, together with the estimated coefficients, the model that will describe the signal behavior. The special feature of the proposed observer is the order of the differential equations of the model on which it is based, dα(t)ν(t)/dtα(t), which can take integer and non-integer values, i.e. α(t)∈(0,1]. Even more important, α(t) can be a smooth function such that α(t)∈(0,1] in the interval t∈[0,T]. The procedure to design the state observer of variable-order as well as some examples of its use in engineering applications are presented.

Performance Evaluation of Wireless Communication Systems over Composite $${\varvec{\alpha}}{-}{\varvec{\mu}}/$$ α - μ / Gamma Fading Channels

The composite $$\alpha {-}\mu /$$ gamma distribution is considered in this paper. Specifically, we derived closed-form expressions for the power probability density function (PDF) and the cumulative density function (CDF). We then use the PDF and the CDF to derive novel closed-form expressions for the outage probability, the average symbol error rate, and the average channel capacity over the composite $$\alpha {-}\mu /$$ gamma fading channels. All derived expressions are valid for integer and non-integer values of the fading parameters. Some representative numerical examples are provided to study the impact of the fading and shadowing parameters on the system performance. Furthermore, the numerical results are compared with Monte-Carlo simulations. Both results demonstrate excellent agreement which validates our analysis.

Fractally Fourier decimated homogeneous turbulent shear flow in noninteger dimensions.

Time evolution of the fully resolved incompressible homogeneous turbulent shear flow in noninteger Fourier dimensions is numerically investigated. The Fourier dimension of the flow field is extended from the integer value 3 to the noninteger values by projecting the Navier-Stokes equation on the fractal set of the active Fourier modes with dimensions 2.7≤d≤3.0. The results of this study revealed that the dynamics of both large and small scale structures are nontrivially influenced by changing the Fourier dimension d. While both turbulent production and dissipation are significantly hampered as d decreases, the evolution of their ratio is almost independent of the Fourier dimension. The mechanism of the energy distribution among different spatial directions is also impeded by decreasing d. Due to this deficient energy distribution, turbulent field shows a higher level of the large-scale anisotropy in lower Fourier dimensions. In addition, the persistence of the vortex stretching mechanism and the forward spectral energy transfer, which are three-dimensional turbulence characteristics, are examined at changing d, from the standard case d=3.0 to the strongly decimated flow field for d=2.7. As the Fourier dimension decreases, these forward energy transfer mechanisms are strongly suppressed, which in turn reduces both the small-scale intermittency and the deviation from Gaussianity. Besides the energy exchange intensity, the variations of d considerably modify the relative weights of local to nonlocal triadic interactions. It is found that the contribution of the nonlocal triads to the total turbulent kinetic energy exchange increases as the Fourier dimension increases.

Estimating Renyi Entropy of Discrete Distributions

It was shown recently that estimating the Shannon entropy $H(p)$ of a discrete $k$ -symbol distribution $p$ requires $\Theta (k/\log k)$ samples, a number that grows near-linearly in the support size. In many applications, $H(p)$ can be replaced by the more general Renyi entropy of order $ \alpha $ and $ H_{\alpha }(p)$ . We determine the number of samples needed to estimate $ H_{\alpha }(p)$ for all $ \alpha $ , showing that $ \alpha requires a super-linear, roughly $k^{1/ \alpha }$ samples, noninteger $ \alpha >1$ requires a near-linear $k$ samples, but, perhaps surprisingly, integer $\alpha >1$ requires only $\Theta (k^{1-1/\alpha })$ samples. Furthermore, developing on a recently established connection between polynomial approximation and estimation of additive functions of the form $\sum _{x}f (p_{x})$ , we reduce the sample complexity for noninteger values of $\alpha $ by a factor of $\log k$ compared with the empirical estimator. The estimators achieving these bounds are simple and run in time linear in the number of samples. Our lower bounds provide explicit constructions of distributions with different Renyi entropies that are hard to distinguish.

Induced QCD I: theory

We explore an alternative discretization of continuum SU(N_c) Yang-Mills theory on a Euclidean spacetime lattice, originally introduced by Budzcies and Zirnbauer. In this discretization the self-interactions of the gauge field are induced by a path integral over N_b auxiliary boson fields, which are coupled linearly to the gauge field. The main progress compared to earlier approaches is that N_b can be as small as N_c. In the present paper we (i) extend the proof that the continuum limit of the new discretization reproduces Yang-Mills theory in two dimensions from gauge group U(N_c) to SU(N_c), (ii) derive refined bounds on N_b for non-integer values, and (iii) perform a perturbative calculation to match the bare parameter of the induced gauge theory to the standard lattice coupling. In follow-up papers we will present numerical evidence in support of the conjecture that the induced gauge theory reproduces Yang-Mills theory also in three and four dimensions, and explore the possibility to integrate out the gauge fields to arrive at a dual formulation of lattice QCD.

A HYBRID INTEGER DATA ENVELOPMENT ANALYSIS BASED ON AN ALTERNATIVE APPROACH OF SUPER SLACK BASED MEASURE FOR MEASURING SUPER EFFICIENCY AND INEFFICIENCY OF DECISION MAKING UNITS

Conventional data envelopment analysis (DEA) models assume that all values of input and output variables are non-integers. However, in many situations of real life DEA application, some of the input and/or output variables are to be integers.The rounding process of the non-integer values to their nearest integers can cause inaccurate efficiency and performance measurement. Thus, it is important to develop a non-radial hybrid integer DEA model. Moreover, the projection of decision making units (DMUs) in the existing super efficiency slack based measure hybrid integer model may not be strongly Pareto-efficient.Thus, our hybrid integer DEA model is developed based on an alternative super efficiency slack based measure DEA model for ranking efficient and inefficient DMUs.The developed model can project efficient DMUs resulted from super efficiency strongly Pareto-efficient by identifying input excesses and output shortfalls of non-integer and integer slack variables. It then discriminates inefficient DMUs whose efficiency scores resulted from super efficiency slack based measure model equal to one.An application to measure the efficiency of 41 academic departments at National Cheng Kung University in Taiwan (NCKU) is used to test the applicability of developed models. The results showed that 51.20% of the departments (DMUs) were super efficiency. Moreover, the result also showed that most of the inefficient departments were caused by their input excesses of expenses and teaching space, and output shortfalls associating with publications and grants.

Quantum Spectral Curve and the Numerical Solution of the Spectral Problem in AdS 5 /CF T 4

We developed an efficient numerical algorithm for computing the spectrum of anomalous dimensions of the planar ${\cal N}=4$ Super-Yang--Mills at finite coupling. The method is based on the Quantum Spectral Curve formalism. In contrast to Thermodynamic Bethe Ansatz, worked out only for some very special operators, this method is applicable for generic states/operators and is much faster and precise due to its Q-quadratic convergence rate. To demonstrate the method we evaluate the dimensions $\Delta$ of twist operators in $sl(2)$ sector directly for any value of the spin $S$ including non-integer values. In particular, we compute the BFKL pomeron intercept in a wide range of the 't Hooft coupling constant with up to $20$ significant figures precision, confirming two previously known from the perturbation theory orders and giving prediction for several new coefficients. Furthermore, we explore numerically a rich branch cut structure for complexified spin $S$.

Spectrum-sensing in cognitive radio networks over composite multipath/shadowed fading channels

This paper analyzes the performance of spectrum-sensing-based energy detection (ED) in cognitive radio networks (CRNs) over generalized fading channels. The fading channel is modeled by the extended generalized-K (EGK) distribution. Exact and accurate analytical expressions for the average detection probability with different detection schemes, such as, single channel, diversity reception, and cooperative spectrum sensing, are derived and evaluated. Obtained expressions can be reduced to other well-known fading channels such as, Weibull, Nakagami-m, and Rayleigh. It is shown that the analytical framework can be used for both integer and non-integer values of the fading/shadowing parameters. The impact of key fading/shadowing parameters on the performance of energy detectors is discussed with receiver operating characteristics (ROC) curves. The accuracy of the derived analytical expressions is corroborated via Monte-Carlo simulation results.

From Fractal Geometry to Fractal Analysis

Euclidian geometry pertained only to the artificial realities of the first, second and third dimensions. Fractal geometry is a new branch of mathematics that proves useful in representing natural phenomena whose dimensions (fractal dimensions) are non-integer values. Fractal geometry was conceived in the 1970s, and mainly developed by Benoit Mandelbrot. In fractal geometry fractals are normally the results of an iterative or recursive construction using corresponding algorithm. Fractal analysis is a nontraditional mathematical and experimental method derived from Mandelbrot’s Fractal Geometry of Nature, Euclidean geometry and calculus. The main aims of the present study are: 1) to address the dimensional imbalances in some texts on fractal geometry, proving that logarithm of a physical quantity (e.g. length of a segment) is senseless; 2) to define the modified capacity dimension, calculate its value for Koch fractal set and show that such definition satisfies basic demands of physics, before all the dimensional balance; and 3) to calculate theoretically the fractal dimension of a circle of unit radius. A quantitative determination of the similarity using the set of Koch fractals is carried out. An important result is the relationship between the modified capacity dimension and fractal dimension obtained using the log-log method. The text includes some important modifications and advances in fractal theory. It is important to notice that these modifications and quantifications do not affect already known facts in fractal geometry and fractal analysis.

Phase Retrieval from Incomplete Magnitude Information via Total Variation Regularization

The phase retrieval problem has drawn considerable attention, as many optical detection devices can only measure magnitudes of the Fourier transform of the underlying object (signal or image). This paper addresses the phase retrieval problem from incomplete data, where only partial magnitudes of Fourier transform are obtained. In particular, we consider structured illuminated patterns in holography and find that noninteger values used in designing such patterns often yield better reconstruction than the conventional integer-valued ones. Furthermore, we demonstrate theoretically and numerically that three diffracted sets of (complete) magnitude data are sufficient to recover the object. To compensate for incomplete information, we incorporate a total variation regularization a priori to guarantee that the reconstructed image satisfies some desirable properties. The proposed model can be solved efficiently by an alternative directional multiplier method with provable convergence. Numerical experiments valid...

Bir Kontrole Karşı Anakütle Oranlarının Sıralı Hipotezi İçin Bir Test Prosedürü

Objective: This paper aims to present a novel procedure for testing a set of population proportions against an ordered alternative with a control. Material and Methods: The distribution of the test statistic for the proposed test was determined theoretically and through Monte-Carlo experiments. The efficiency of the proposed test method was compared with the classical Chi-square test of homogeneity of population proportions using their empirical Type I error rates and powers at various sample sizes. Results: The new test statistic that was developed for testing a set of population proportions against an ordered alternative with a control was found to have a Chi-square distribution with non-integer values degrees of freedom v that depend on the number of population groups k being compared. Table of values of v for comparing up to 26 population groups was constructed while an expression was developed to determine v for cases where k > 26. Further results showed that the new test method is capable of detecting the superiority of a treatment, for instance a new drug type, over some of the existing ones in situations where only the qualitative data on users' preferences of all the available treatments (drug types) are available. The new test method was found to be relatively more powerful and consistent at estimating the nominal Type I error rates (α), especially at smaller sample sizes than the classical Chi-square test of homogeneity of population proportions. Conclusion: Conclusion: Conclusion: Conclusion: The new test method proposed here could find applications in pharmacology where a newly developed drug might be expected to be more preferred by users than some of the existing ones. This kind of test problem can equally exist in medicine, engineering and humanities in situations where only the qualitative data on users' preferences of a set of treatments or systems are available.

BER analysis over alpha-mu fading channel using proposed novel MGF

In this paper, closed-form expressions of Bit Error Rate BER for different modulation techniques such as BPSK, BFSK, DQPSK, BDPSK and MPSK over alpha-mu α-μ fading distribution are presented, followed by proposing a novel Moment Generating Function MGF. Specifically, we have proposed 'A Novel MGF for Alpha-mu Fading Distribution'. The proposed MGF is a modification of the MGF in Magableh and Matalgah's 2009 article as proposed MGF is valid for all values of alpha, whereas, the MGF presented in Magableh and Matalgah's article is valid only for non-integer values of alpha. Further, various closed-form expressions of BER using the proposed MGF are presented in this paper. Also, BER of various modulation schemes based on proposed MGF has been compared with similar results presented in Shin and Lee's 2004 article as special case of alpha-mu distribution. This shows perfect agreement of our results with that of Shin and Lee.