Modified Bessel Functions Research Articles

Performance analysis of subcarrier intensity modulation using rectangular QAM over Malaga turbulence channels with integer and non‐integer β

AbstractIn this paper, we derive the performances of optical wireless communication system utilizing adaptive subcarrier intensity modulation over the Malaga turbulent channel. More specifically, analytical closed‐form solutions and asymptotic results are derived for average bit error rate, achievable spectral efficiency, outage probability, and ergodic capacity by utilizing series expansion identity of modified Bessel function. Our asymptotic and analytical results based on series solutions with finite numbers highly matched to the numerical results. By exploiting the inherent nature of fading channel, the proposed adaptive scheme enhances the spectral efficiency without additional transmit power while satisfying the required bit error rate criterion. Copyright © 2016 John Wiley & Sons, Ltd.

A unified framework for performance evaluation over generalized fading channels

In this paper, we present a unified framework to analyze the performance of the average bit error probability (BEP) and the outage probability over generalized fading channels. Specifically, we assume that the probability density function (PDF) of the instantaneous signal-to-noise ratio $$\zeta $$ź is given by the product of: power function, exponential function, and the modified Bessel function of the first kind, i.e., $$f_{\zeta }(\zeta )=\zeta ^{\lambda -1}exp\left( -a\zeta ^{\beta }\right) I_{v}\left( b\zeta ^{\beta }\right) $$fź(ź)=źź-1exp-aźβIvbźβ. Based on this PDF, we obtain a novel closed-form expression for the average BEP over such channels perturbed by an additive white generalized Gaussian noise (AWGGN). Note that other well-known noise types can be deduced from the AWGGN as special cases such as Gaussian noise, Laplacian noise, and impulsive noise. Furthermore, we obtain a novel closed-form expression for the outage probability. As an example of such channels, and without loss of generality, we analyze the performance of the average BEP and the outage probability over the $$\eta $$ź---$$\mu $$μ fading channels. Analytical results accompanied with Monte-Carlo simulations are provided to validate our analysis.

Stationary analysis of a fluid queue driven by an M / M / 1 queue with working vacation

This paper studies a fluid queueing model driven by an M / M / 1 queue subject to working vacation. The underlying system of differential difference equations that governs the process is solved using continued fraction and generating function methodologies. Explicit expressions for the joint steady-state probabilities of the state of the background queueing model and the content of the buffer are obtained in terms of modified Bessel function of the first kind. Numerical illustrations are added to depict the variations of the state probabilities and the buffer content distributions against varying values of the parameters subject to the stability conditions.

Mehler-Heine asymptotics for multiple orthogonal polynomials

Mehler-Heine asymptotics describe the behavior of orthogonal polynomials near the edges of the interval where the orthogonality measure is supported. For Jacobi polynomials and Laguerre polynomials this asymptotic behavior near the hard edge involves Bessel functions $J_\alpha$. We show that the asymptotic behavior near the endpoint of the interval of (one of) the measures for multiple orthogonal polynomials involves a generalization of the Bessel function. The multiple orthogonal polynomials considered are Jacobi-Angelesco polynomials, Jacobi-Pi\~neiro polynomials, multiple Laguerre polynomials, multiple orthogonal polynomials associated with modified Bessel functions (of the first and second kind), and multiple orthogonal polynomials associated with Meijer $G$-functions.

Transient analysis of a two-heterogeneous servers queue with system disaster, server repair and customers’ impatience

A two-heterogeneous servers queue with system disaster, server failure and repair is considered. In addition, the customers become impatient when the system is down. The customers arrive according to a Poisson process and service time follows exponential distribution. Each customer requires exactly one server for its service and the customers select the servers on fastest server first basis. Explicit expressions are derived for the time-dependent system size probabilities in terms of the modified Bessel function, by employing the generating function along with continued fraction and the identity of the confluent hypergeometric function. Further, the steady-state probabilities of the number of customers in the system are deduced and finally some important performance measures are obtained.

A linear model of population dynamics

The Malthus process of population growth is reformulated in terms of the probability [Formula: see text] to find exactly [Formula: see text] individuals at time [Formula: see text] assuming that both the birth and the death rates are linear functions of the population size. The master equation for [Formula: see text] is solved exactly. It is shown that [Formula: see text] strongly deviates from the Poisson distribution and is expressed in terms either of Laguerre’s polynomials or a modified Bessel function. The latter expression allows for considerable simplifications of the asymptotic analysis of [Formula: see text].

A three-dimensional semi-analytical solution for predicting drug release through the orifice of a spherical device

Three-dimensional solute transport was investigated for a spherical device with a release hole. The governing equation was derived using the Fick’s second law. A mixed Neumann-Dirichlet condition was imposed at the boundary to represent diffusion through a small region on the surface of the device. The cumulative percentage of drug released was calculated in the Laplace domain and represented by the first term of an infinite series of Legendre and modified Bessel functions of the first kind. Application of the Zakian algorithm yielded the time-domain closed-form expression. The first-order solution closely matched a numerical solution generated by Mathematica®. The proposed method allowed computation of the characteristic time. A larger surface pore resulted in a smaller effective time constant. The agreement between the numerical solution and the semi-analytical method improved noticeably as the size of the orifice increased. It took four time constants for the device to release approximately ninety-eight of its drug content.

A new type of sharp bounds for ratios of modified Bessel functions

The bounds for the ratios of first and second kind modified Bessel functions of consecutive orders are important quantities appearing in a large number of scientific applications. We obtain new bounds which are accurate in a large region of parameters and which are sharper than previous bounds. The new bounds are obtained by a qualitative analysis of the Riccati equation satisfied by these ratios. A procedure is considered in which the bounds obtained from the analysis of the Riccati equation are used to define a new function satisfying a new Riccati equation which yields sharper bounds. Similar ideas can be applied to other functions.

On the Kernel of the Cooray–Rubinstein Formula in the Time Domain

The kernel of the Cooray–Rubinstein formula in the time domain involves a modified Bessel function of order zero. In this letter, approximations of the kernel are proposed, which do not use any special function, and, hence, are able to simplify the numerical evaluation in view of lightning-induced voltages calculations.

Fast electromagnetic modeling in cylindrically layered media excited by eccentred magnetic dipole

AbstractWe developed a fast algorithm to calculate a response of cylindrically layered media excited by the vertical magnetic dipole eccentred with respect to the axis of symmetry. The algorithm calculates response in the range of frequencies typical for induction and dielectric logging. The media conductivity and dielectric constant are described by piecewise‐constant functions. The corresponding boundary value problem is solved by method of separation of variables. Fourier transform is applied to Maxwell equations and boundary conditions to express field components through Fourier transforms of vertical components of an electrical and magnetic field. In addition, an expansion of vertical components into an infinite series with respect to angular harmonics is used to reduce the original problem to a series of 1‐D problems that only depend on the radial coordinate. The solution to each 1‐D radial problem for the angular harmonics is presented as a linear combination of modified Bessel functions. Finally, inverse Fourier transformation is applied to the angular harmonics of vertical components to derive electrical and magnetic field of the original boundary value problem. We provide detailed discussion on the elements that are critical for the numerical implementation of the algorithm: a proper normalization, convergence, and integration. Specifically, we show how to perform integration in the complex plane by avoiding intersection of the integration pass with the cuts located on the Riemann surface. Numerical results show the usefulness of the algorithm for solving inverse problems and for studying the effect of eccentricity in induction and dielectric logging.

Couple on a rotating permeable sphere in a couple stress fluid

Abstract In this paper, the flow generated by the slow steady rotation of a permeable sphere with thin surface about its axis of symmetry in an incompressible couple stress fluid which is otherwise at rest is studied. Expressions for the velocity and couple stress tensor components are obtained in terms of modified Bessel functions and Gagenbauer’s polynomials. The couple experienced by the permeable sphere due to the internal and external flow is evaluated. The more is couple stress offered by the fluid (i.e. the lesser values of couple stress parameter), the more the fluid particles are thrown away from the sphere. It is noticed that the internal flow is similar to viscous flow and the couple acting on the sphere due to internal flow is zero. The problem differs from that of micro-polar fluids.

Transient heat conduction in an infinite medium subjected to multiple cylindrical heat sources: An application to shallow geothermal systems

In this paper, we introduce analytical solutions for transient heat conduction in an infinite solid mass subjected to a varying single or multiple cylindrical heat sources. The solutions are formulated for two types of boundary conditions: a time-dependent Neumann boundary condition, and a time-dependent Dirichlet boundary condition. We solve the initial and boundary value problem for a single heat source using the modified Bessel function, for the spatial domain, and the fast Fourier transform, for the temporal domain. For multiple heat sources, we apply directly the superposition principle for the Neumann boundary condition, but for the Dirichlet boundary condition, we conduct an analytical coupling, which allows for the exact thermal interaction between all involved heat sources. The heat sources can exhibit different time-dependent signals, and can have any distribution in space. The solutions are verified against the analytical solution given by Carslaw and Jaeger for a constant Neumann boundary condition, and the finite element solution for both types of boundary conditions. Compared to these two solutions, the proposed solutions are exact at all radial distances, highly elegant, robust and easy to implement.

A class of linear viscoelastic models based on Bessel functions

In this paper we investigate a general class of linear viscoelastic models whose creep and relaxation memory functions are expressed in Laplace domain by suitable ratios of modified Bessel functions of contiguous order. In time domain these functions are shown to be expressed by Dirichlet series (that is infinite Prony series). It follows that the corresponding creep compliance and relaxation modulus turn out to be characterized by infinite discrete spectra of retardation and relaxation time respectively. As a matter of fact, we get a class of viscoelastic models depending on a real parameter $\nu > -1$. Such models exhibit rheological properties akin to those of a fractional Maxwell model (of order $1/2$) for short times and of a standard Maxwell model for long times.

SOME INEQUALITIES AND ABSOLUTE MONOTONICITY FOR MODIFIED BESSEL FUNCTIONS OF THE FIRST KIND

Abstract. By employing a refined version of the P´olya type integralinequality and other techniques, the authors establish some inequalitiesand absolute monotonicity for modified Bessel functions of the first kindwith nonnegative integer order. 1. Main resultsIt is well known that modified Bessel functions of the first kind I ±ν (z) aresolutions of the differential equationz 2 d 2 wdz 2 +zdwdz−z 2 +ν 2 w = 0.They are holomorphic functions of z throughout the z-plane cut along thenegative real axis, and are entire functions of ν for fixed z 6= 0. When ν = ±n,I ν (z) are entire functions of z. In [1, p. 375, 9.6.7], it is listed thatI ν (z) =X ∞k=0 1k!Γ(ν +k +1)z2 2k+ν , z ∈ C, ν ∈ R\{−1,−2,...},whereΓ(z) = lim n→∞ n!n z Q nk=0 (z +k), z ∈ C\{0,−1,−2,...}is the classical gamma function, see [1, p. 255, 6.1.2].On [12, p. 63], the following three double inequalities are derived:1− z/21+ z/2 0, ν ≥ −12; Received July 18, 2015.2010 Mathematics Subject Classification. Primary 33C10; Secondary 26A48, 26D15,44A10.Key words and phrases. inequality, absolute monotonicity, complete monotonic function,modified Bessel function of the first kind, P´olya type integral inequality.

Расчет присоединенной длины концевого отверстия канала без фланца при выполнении проектных акустических разработок

The analysis of known calculated dependences for determining the relative cut-in length of the end hole of the channel
 without clamping shoulder (end correction) needed to perform applicative acoustic calculation, in particular by the design
 of jet noise mufflers. The accuracy of the results obtained by calculating with the known dependencies is estimated.
 The method of calculating of the end correction with machine method is offered, the basis of which was used by the
 development of analytical dependence for calculating the values of the cut-in length of the holes without clamping shoulder
 with high accuracy adopted in this article as a reference. These values are calculated in Matlab system with relative and
 absolute accuracy 10–7 and 10–12, in particular, it was found that when ka=0l/a=0,6127, but not 0.6133 as it was previously
 thought in the world acoustics. A computer approach of calculation of the modified Bessel functions for large values of the
 argument was developed. A correction of design equations which are widely used in acoustic engineering development,
 which increases the accuracy of calculating the cut-in length of the end channels holes without clamping shoulder. The
 results are intended for use in both basic acoustic investigations and in their numerous practical applications.

Modified Dini functions: monotonicity patterns and functional inequalities

We deduce some new functional inequalities, like Tur\'an type inequalities, Redheffer type inequalities, and a Mittag-Leffler expansion for a special combination of modified Bessel functions of the first kind, called modified Dini functions. Moreover, we show the complete monotonicity of a quotient of modified Dini functions by introducing a new continuous infinitely divisible probability distribution. The key tool in our proofs is a recently developed infinite product representation for a special combination of Bessel functions of the first, which was very useful in determining the radius of convexity of some normalized Bessel functions of the first kind.

Some new results on the eigenvalues of complex non-central Wishart matrices with a rank-1 mean

Let W be an n × n complex non-central Wishart matrix with m ( ≥ n ) degrees of freedom and a rank-1 mean. In this paper, we consider three problems related to the eigenvalues of W . To be specific, we derive a new expression for the cumulative distribution function (c.d.f.) of the minimum eigenvalue ( λ min ) of W . The c.d.f. is expressed as the determinant of a square matrix, the size of which depends only on the difference m − n . This further facilitates the analysis of the microscopic limit of the minimum eigenvalue. The microscopic limit takes the form of the determinant of a square matrix with its entries expressed in terms of the modified Bessel functions of the first kind. We also develop a moment generating function based approach to derive the probability density function of the random variable tr ( W ) / λ min , where tr ( ⋅ ) denotes the trace of a square matrix. Moreover, we establish that, as m , n → ∞ with m − n fixed, tr ( W ) / λ min scales like n 3 . Finally, we find the average of the reciprocal of the characteristic polynomial det [ z I n + W ] , | arg z | < π , where I n and det [ ⋅ ] denote the identity matrix of size n and the determinant, respectively.

Analytical solution for time-dependent potentials in a fiber stimulated by an external electrode.

This study provides an analytical solution for time-dependent potentials in a 3D cylindrical fiber stimulated by an extracellular point electrode. The membrane is passive and represented by surface resistance and surface capacitance. Separation of variables solution expresses intracellular and extracellular potentials as sums involving modified Bessel functions; the coefficients ([Formula: see text] and [Formula: see text]) depend on time. In contrast to previous analytical solutions, where [Formula: see text] and [Formula: see text] had to be determined numerically, here [Formula: see text] and [Formula: see text] are given by explicit formulas that resemble the formulas for potentials in a fiber stimulated by a transverse electric field. The comparison of the 3D analytical solution with the 1D cable model shows that the cable model approximates transmembrane potential with the error below 5% when the distance between the electrode and the fiber is 0.2-4mm and when the stimulus is longer than 3.3ms. For stimuli between 0.43 and 3.3ms, the range of fiber-electrode distances with error below 5% shrinks, and it disappears completely for stimuli shorter than 0.43ms. Thus, our study shows that the applicability of the 1D cable model may be more limited than previously considered.

Bounds for the product of modified Bessel functions

In this note our aim is to present some monotonicity properties of the product of modified Bessel functions of first and second kind. Certain bounds for the product of modified Bessel functions of first and second kind are also obtained. These bounds improve and extend known bounds for the product of modified Bessel functions of first and second kind of order zero. A new Tur\'an type inequality is also given for the product of modified Bessel functions, and some open problems are stated, which may be of interest for further research.

Universal scaling of the logarithmic negativity in massive quantum field theory

We consider the logarithmic negativity, a measure of bipartite entanglement, in a general unitary 1 + 1-dimensional massive quantum field theory, not necessarily integrable. We compute the negativity between a finite region of length r and an adjacent semi-infinite region, and that between two semi-infinite regions separated by a distance r. We show that the former saturates to a finite value, and that the latter tends to zero, as . We show that in both cases, the leading corrections are exponential decays in r (described by modified Bessel functions) that are solely controlled by the mass spectrum of the model, independently of its scattering matrix. This implies that, like the entanglement entropy (EE), the logarithmic negativity displays a very high level of universality, allowing one to extract information about the mass spectrum. Further, a study of sub-leading terms shows that, unlike the EE, a large-r analysis of the negativity allows for the detection of bound states.