#### Sample records for bessel functions

1. Bessel functions

2014-01-01

After presenting the theory in engineers' language without the unfriendly abstraction of pure mathematics, several illustrative examples are discussed in great detail to see how the various functions of the Bessel family enter into the solution of technically important problems. Axisymmetric vibrations of a circular membrane, oscillations of a uniform chain, heat transfer in circular fins, buckling of columns of varying cross-section, vibrations of a circular plate and current density in a conductor of circular cross-section are considered. The problems are formulated purely from physical considerations (using, for example, Newton's law of motion, Fourier's law of heat conduction electromagnetic field equations, etc.) Infinite series expansions, recurrence relations, manipulation of expressions involving Bessel functions, orthogonality and expansion in Fourier-Bessel series are also covered in some detail. Some important topics such as asymptotic expansions, generating function and Sturm-Lioville theory are r...

2. Noncommutative Bessel symmetric functions

Novelli, Jean-Christophe; Thibon, Jean-Yves

2006-01-01

The consideration of tensor products of 0-Hecke algebra modules leads to natural analogs of the Bessel J-functions in the algebra of noncommutative symmetric functions. This provides a simple explanation of various combinatorial properties of Bessel functions.

3. Integrals of Bessel functions

Babusci, D.; Dattoli, G.; Germano, B.; Martinelli, M. R.; Ricci, P. E.

2011-01-01

We use the operator method to evaluate a class of integrals involving Bessel or Bessel-type functions. The technique we propose is based on the formal reduction of these family of functions to Gaussians.

4. Conformable Fractional Bessel Equation and Bessel Functions

Gökdoğan, Ahmet; Ünal, Emrah; Çelik, Ercan

2015-01-01

In this work, we study the fractional power series solutions around regular singular point x=0 of conformable fractional Bessel differential equation and fractional Bessel functions. Then, we compare fractional solutions with ordinary solutions. In addition, we present certain property of fractional Bessel functions.

5. Unified Bessel, Modified Bessel, Spherical Bessel and Bessel-Clifford Functions

Yaşar, Banu Yılmaz; Özarslan, Mehmet Ali

2016-01-01

In the present paper, unification of Bessel, modified Bessel, spherical Bessel and Bessel-Clifford functions via the generalized Pochhammer symbol [ Srivastava HM, Cetinkaya A, K{\\i}ymaz O. A certain generalized Pochhammer symbol and its applications to hypergeometric functions. Applied Mathematics and Computation, 2014, 226 : 484-491] is defined. Several potentially useful properties of the unified family such as generating function, integral representation, Laplace transform and Mellin tran...

6. Some integrals involving Bessel functions

Glasser, M. Lawrence; Montaldi, Emilio

1993-01-01

A number of new definite integrals involving Bessel functions are presented. These have been derived by finding new integral representations for the product of two Bessel functions of different order and argument in terms of the generalized hypergeometric function with subsequent reduction to special cases. Connection is made with Weber's second exponential integral and Laplace transforms of products of three Bessel functions.

7. Cohomology and Bessel functions Theory

Mekhfi, Mustapha

2002-01-01

By studying cohomological quantum mechanics on the punctured plane,we were led to identify (reduced) Bessel functions with homotopic loops living on the plane.This identification led us to correspondence rules between exponentials and Bessel functions.The use of these rules makes us retrieve known but also new formulas in Bessel functions theory.

8. q-Bessel-Macdonald functions

Rogov, V. -B. K.

2000-01-01

The modified q-Bessel functions and the q-Bessel-Macdonald functions of the first and second kind are introduced. Their definition is based on representations as power series. Recurrence relations, the q-Wronskians, asymptotic decompositions and q-integral representations are received. In addition, the q-Bessel-Macdonald function of kind 3 is determined by its q-integral representation.

9. Symmetric q-Bessel functions

Giuseppe Dattoli

1996-05-01

Full Text Available q analog of bessel functions, symmetric under the interchange of q and q^ −1 are introduced. The definition is based on the generating function realized as product of symmetric q-exponential functions with appropriate arguments. Symmetric q-Bessel function are shown to satisfy various identities as well as second-order q-differential equations, which in the limit q → 1 reproduce those obeyed by the usual cylindrical Bessel functions. A brief discussion on the possible algebraic setting for symmetric q-Bessel functions is also provided.

10. Generalized Bessel functions of the first kind

Baricz, Árpád

2010-01-01

In this volume we study the generalized Bessel functions of the first kind by using a number of classical and new findings in complex and classical analysis. Our aim is to present interesting geometric properties and functional inequalities for these generalized Bessel functions. Moreover, we extend many known inequalities involving circular and hyperbolic functions to Bessel and modified Bessel functions.

11. A weak kernel formula for Bessel functions

Chai, Jingsong

2015-01-01

In this paper, we prove a weak kernel formula of Bessel functions attached to irreducible generic representations of p-adic $GL(n)$. As an application, we show that the Bessel function defined by Bessel distribution coincides with the Bessel function defined via uniqueness of Whittaker models on the open Bruhat cell.

12. Van der Corput inequalities for Bessel functions

Baricz, Árpád; Laforgia, Andrea; Pogány, Tibor K.

2014-01-01

In this note we offer some log-concavity properties of certain functions related to Bessel functions of the first kind and modified Bessel functions of the first and second kind, by solving partially a recent conjecture on the log-convexity/log-concavity properties for modified Bessel functions of the first kind and their derivatives. Moreover, we give an application of the mentioned results by extending two inequalities of van der Corput to Bessel and modified Bessel functions of the first k...

13. Numerical analysis for the moments of Bessel functions and Bessel-trigonometric functions

Wang, Yinkun; Ying LI; Luo, Jianshu

2016-01-01

The moments of Bessel functions and Bessel-trigonometric functions play a basic role in many practical problems and numerical analysis. This paper presents a complete analysis for these moments based on the recursive relations of Bessel functions. To evaluate the moments of Bessel functions numerically, a fast and efficient scheme is also proposed to approximate the integral of Bessel function of the first kind and of zero order. The moments of Bessel-trigonometric functions are proved to be ...

14. Extension of Oppenheim's Problem to Bessel Functions

Zhu Ling

2007-01-01

Full Text Available Our aim is to extend some trigonometric inequalities to Bessel functions. Moreover, we deduce the hyperbolic analogue of these trigonometric inequalities, and we extend these inequalities to modified Bessel functions.

15. Extension of Oppenheim's Problem to Bessel Functions

Ling Zhu

2008-01-01

Full Text Available Our aim is to extend some trigonometric inequalities to Bessel functions. Moreover, we deduce the hyperbolic analogue of these trigonometric inequalities, and we extend these inequalities to modified Bessel functions.

16. On two-dimensional Bessel functions

Korsch, H. J.; Klumpp, A.; Witthaut, D.

2006-01-01

The general properties of two-dimensional generalized Bessel functions are discussed. Various asymptotic approximations are derived and applied to analyze the basic structure of the two-dimensional Bessel functions as well as their nodal lines.

17. Extension of Oppenheim's Problem to Bessel Functions

Ling Zhu; Árpád Baricz

2007-01-01

Our aim is to extend some trigonometric inequalities to Bessel functions. Moreover, we deduce the hyperbolic analogue of these trigonometric inequalities, and we extend these inequalities to modified Bessel functions.

18. Numerical Calculation of Bessel Functions

Schwartz, Charles

2012-01-01

A new computational procedure is offered to provide simple, accurate and flexible methods for using modern computers to give numerical evaluations of the various Bessel functions. The Trapezoidal Rule, applied to suitable integral representations, may become the method of choice for evaluation of the many Special Functions of mathematical physics.

19. Theory of Bessel Functions of High Rank - I: Fundamental Bessel Functions

Qi, Zhi

2014-01-01

In this article we introduce a new category of special functions called fundamental Bessel functions arising from the Voronoi summation formula for $\\mathrm{GL}_n (\\mathbb{R})$. The fundamental Bessel functions of rank one and two are the oscillatory exponential functions $e^{\\pm i x}$ and the classical Bessel functions respectively. The main implements and subjects of our study of fundamental Bessel functions are their formal integral representations and Bessel equations.

20. Tur\\'an determinants of Bessel functions

Baricz, Árpád

2011-01-01

In this paper our aim is to survey the Tur\\'an type inequalities and related problems for the Bessel functions of the first kind. Moreover, we extend the known higher order Tur\\'an type inequalities for Bessel functions of the first kind to real parameters and we deduce new closed integral representation formulae for the second kind Neumann type series of Bessel functions of the first kind occurring in the study of Tur\\'an determinants of Bessel functions of the first kind. At the end of the paper we prove a Tur\\'an type inequality for the Bessel functions of the second kind.

1. Tur\\'an determinants of Bessel functions

Baricz, Árpád; Pogány, Tibor K.

2011-01-01

In this paper first we survey the Tur\\'an type inequalities and related problems for the Bessel functions of the first kind. Then we extend the known higher order Tur\\'an type inequalities for Bessel functions of the first kind to real parameters and we deduce new closed integral representation formulae for the second kind Neumann type series of Bessel functions of the first kind occurring in the study of Tur\\'an determinants of Bessel functions of the first kind. At the end of the paper we p...

2. Functional inequalities for modified Bessel functions

Baricz, Árpád; Vuorinen, Matti

2010-01-01

In this paper our aim is to show some mean value inequalities for the modified Bessel functions of the first and second kinds. Our proofs are based on some bounds for the logarithmic derivatives of these functions, which are in fact equivalent to the corresponding Tur\\'an type inequalities for these functions. As an application of the results concerning the modified Bessel function of the second kind we prove that the cumulative distribution function of the gamma-gamma distribution is log-concave. At the end of this paper several open problems are posed, which may be of interest for further research.

3. Redheffer type inequalities for modified Bessel functions

Khaled Mehrez

2016-01-01

In this short note, we give new proofs of Redheffer’s inequality for modified Bessel functions of first kind published by Ling Zhu (2011). In addition, using the Grosswald formula we prove new Redheffer type inequality for the modified Bessel functions of the second kind.

4. Bessel functions and equations of mathematical physics

Epelde García, Markel

2016-01-01

The aim of this dissertation is to introduce Bessel functions to the reader, as well as studying some of their properties. Moreover, the final goal of this document is to present the most well- known applications of Bessel functions in physics.

5. K-Bessel functions in two variables

Hacen Dib

2003-01-01

The Bessel-Muirhead hypergeometric system (or 0F1-system) in two variables (and three variables) is solved using symmetric series, with an explicit formula for coefficients, in order to express the K-Bessel function as a linear combination of the J-solutions. Limits of this method and suggestions for generalizations to a higher rank are discussed.

6. Interlacing of real zeros of Bessel functions

Palmai, Tamas

2010-01-01

We unify the known three distinct inequality sequences [Abramowitz 9.5.2] of real zeros of Bessel functions into a single, generalized one. This result is triggered by a uniqueness proof concerning a particular inverse scattering problem.

7. Jackson Integral Representations of Modified $q$-Bessel Functions and $q$-Bessel-Macdonald Functions

Olshanetsky, M. A.; Rogov, V-B. K.

1996-01-01

The $q$ analog of Modified Bessel functions and Bessel-Macdonald functions, were defined in our previous work (q-alg/950913) as general solutions of a second order difference equations. Here we present a collection of their representations by the Jackson q-integral.

8. Bessel functions for root systems via the trigonometric setting

Ørsted, Bent; Said, S.B.

2005-01-01

In this paper, we study generalized Bessel functions related to root systems and give explicit formulas in several cases.......In this paper, we study generalized Bessel functions related to root systems and give explicit formulas in several cases....

9. Generalized Bessel functions in tunnelling ionization

Reiss, H R

2003-01-01

We develop two new approximations for the generalized Bessel function that frequently arises in the analytical treatment of strong-field processes, especially in non-perturbative multiphoton ionization theories. Both these new forms are applicable to the tunnelling environment in atomic ionization, and are analytically much simpler than the currently used low-frequency asymptotic approximation for the generalized Bessel function. The second of the new forms is an approximation to the first, and it is the second new form that exhibits the well-known tunnelling exponential.

10. Tur\\'an type inequalities for general Bessel functions

Baricz, Árpád; Ponnusamy, Saminathan; Singh, Sanjeev

2015-01-01

In this paper some Tur\\'an type inequalities for the general Bessel function, monotonicity and bounds for its logarithmic derivative are derived. Moreover we find the series representation and the relative extrema of the Tur\\'anian of general Bessel functions. The key tools in the proofs are the recurrence relations together with some asymptotic relations for Bessel functions.

11. Bessel functions and local converse conjecture of Jacquet

Chai, Jingsong

2014-01-01

In this paper, we prove a kernel formula of Bessel functions attached to irreducible smooth supercuspidal representations of p-adic $GL(n)$. We also show that the Bessel function defined by Bessel distribution coincides with the Bessel function defined via uniqueness of Whittaker models on the open Bruhat cell. As an application we give a proof of the local converse conjecture of Jacquet.

12. Approximation of Analytic Functions by Bessel's Functions of Fractional Order

Soon-Mo Jung

2011-01-01

Full Text Available We will solve the inhomogeneous Bessel's differential equation x2y″(x+xy′(x+(x2-ν2y(x=∑m=0∞amxm, where ν is a positive nonintegral number and apply this result for approximating analytic functions of a special type by the Bessel functions of fractional order.

13. Approximation of Analytic Functions by Bessel's Functions of Fractional Order

Soon-Mo Jung

2011-01-01

We will solve the inhomogeneous Bessel's differential equation x2y″(x)+xy′(x)+(x2-ν2)y(x)=∑m=0∞amxm, where ν is a positive nonintegral number and apply this result for approximating analytic functions of a special type by the Bessel functions of fractional order.

14. Monotonicity and bounds on Bessel functions

Larry Landau

2000-07-01

Full Text Available survey my recent results on monotonicity with respect to order of general Bessel functions, which follow from a new identity and lead to best possible uniform bounds. Application may be made to the "spreading of the wave packet" for a free quantum particle on a lattice and to estimates for perturbative expansions.

15. A linear combination of modified Bessel functions

Shitzer, A.; Chato, J. C.

1971-01-01

A linear combination of modified Bessel functions is defined, discussed briefly, and tabulated. This combination was found to recur in the analysis of various heat transfer problems and in the analysis of the thermal behavior of living tissue when modeled by cylindrical shells.

16. Some Inequalities for Modified Bessel Functions

Laforgia Andrea

2010-01-01

Full Text Available We denote by and the Bessel functions of the first and third kinds, respectively. Motivated by the relevance of the function , , in many contexts of applied mathematics and, in particular, in some elasticity problems Simpson and Spector (1984, we establish new inequalities for . The results are based on the recurrence relations for and and the Turán-type inequalities for such functions. Similar investigations are developed to establish new inequalities for .

17. Extension of Frame's type inequalities to Bessel and modified Bessel functions

Mehrez, Khaled

2016-01-01

Our aim is to extend some trigonometric inequalities to Bessel functions. Moreover, we extend the hyperbolic analogue of these trigonometric inequalities. As an application of these results we present a generalization of Cusa-type inequality to modified Bessel function. Our main motivation to write this paper is a recent publication of Chen and S\\'andor, which we wish to complement.

18. Exponential generating functions for the associated Bessel functions

Similar to the associated Legendre functions, the differential equation for the associated Bessel functions Bl,m(x) is introduced so that its form remains invariant under the transformation l → -l - 1. A Rodrigues formula for the associated Bessel functions as squared integrable solutions in both regions l l,m(x) may be taken into account as the union of the increasing (decreasing) infinite sequences with respect to l. It is shown that two new different types of exponential generating functions are attributed to the associated Bessel functions corresponding to these rearranged sequences

19. Three-particle integrals with Bessel functions

Frolov, A. M.; Wardlaw, D. M.

2014-02-01

Analytical formulas for some useful three-particle integrals are derived. Many of these integrals include Bessel and/or trigonometric functions of one and two interparticle (relative) coordinates r 32, r 31, and r 21. The formulas obtained in such an analysis allow us to consider three-particle integrals of more complicated functions of relative/perimetric coordinates. In many actual problems such three-particle integrals can be found in matrix elements of the Hamiltonian and other operators.

20. Generalized Bessel transform of $(\\beta, \\gamma)$-generalized Bessel Lipschitz functions

2015-01-01

In this paper, we prove an analog of Younis’s theorem 5.2 in~[4] for the generalized Fourier-Bessel transform on the Half line for functions satisfying the $(\\beta, \\gamma)$-generalized Bessel Lipschitz condition in the space $\\mathrm{L}^{2}_{\\alpha,n}$.

1. Zeros of Bessel functions: monotonicity, concavity, inequalities

Andrea Laforgia

2007-12-01

Full Text Available We present a survey of the most important inequalities and monotonicity, concavity (convexity results of the zeros of Bessel functions. The results refer to the deﬁnition Jνκ of the zeros of Cν (x = Jν (x cosα −Yν (x sinα, formulated in [6], where κ is a continuous variable. Sometimes, also the Sturm comparison theorem is an important tool of our results.

2. Cross-product of Bessel functions: monotonicity patterns and functional inequalities

Baricz, Árpád; Ponnusamy, Saminathan; Singh, Sanjeev

2015-01-01

In this paper we study the Dini functions and the cross-product of Bessel functions. Moreover, we are interested on the monotonicity patterns for the cross-product of Bessel and modified Bessel functions. In addition, we deduce Redheffer-type inequalities, and the interlacing property of the zeros of Dini functions and the cross-product of Bessel and modified Bessel functions. Bounds for logarithmic derivatives of these functions are also derived. The key tools in our proofs are some recently...

3. Some Unified Integrals Associated with Generalized Bessel-Maitland Function

Abouzaid, M. S.; Abusufian, A. H.; K. S. Nisar

2016-01-01

Generalized integral formulas involving the generalized Bessel-Maitland function are considered and it expressed in terms of generalized Wright hypergeometric functions. By assuming appropriate values of the parameters in the main results, we obtained some interesting results of ordinary Bessel function.

4. Exponential generating functions for the associated Bessel functions

Fakhri, H [Department of Theoretical Physics and Astrophysics, Physics Faculty, University of Tabriz, PO Box 51666-16471, Tabriz (Iran, Islamic Republic of); Mojaveri, B; Nobary, M A Gomshi [Department of Physics, Faculty of Science, Razi University, Kermanshah (Iran, Islamic Republic of)], E-mail: hfakhri@tabrizu.ac.ir, E-mail: bmojaveri@raziu.ac.ir, E-mail: mnobary@razi.ac.ir

2008-09-26

Similar to the associated Legendre functions, the differential equation for the associated Bessel functions B{sub l,m}(x) is introduced so that its form remains invariant under the transformation l {yields} -l - 1. A Rodrigues formula for the associated Bessel functions as squared integrable solutions in both regions l < 0 and l {>=} 0 is presented. The functions with the same m but with different positive and negative values of l are not independent of each other, while the functions with the same l + m (l - m) but with different values of l and m are independent of each other. So, all the functions B{sub l,m}(x) may be taken into account as the union of the increasing (decreasing) infinite sequences with respect to l. It is shown that two new different types of exponential generating functions are attributed to the associated Bessel functions corresponding to these rearranged sequences.

5. Exponential generating functions for the associated Bessel functions

Fakhri, H.; Mojaveri, B.; Gomshi Nobary, M. A.

2008-09-01

Similar to the associated Legendre functions, the differential equation for the associated Bessel functions Bl,m(x) is introduced so that its form remains invariant under the transformation l → -l - 1. A Rodrigues formula for the associated Bessel functions as squared integrable solutions in both regions l = 0 is presented. The functions with the same m but with different positive and negative values of l are not independent of each other, while the functions with the same l + m (l - m) but with different values of l and m are independent of each other. So, all the functions Bl,m(x) may be taken into account as the union of the increasing (decreasing) infinite sequences with respect to l. It is shown that two new different types of exponential generating functions are attributed to the associated Bessel functions corresponding to these rearranged sequences.

6. New addition formula for the little $q$-Bessel functions

Bouzeffour, Fethi

2013-01-01

Starting from the addition formula for little $q$-Jacobi polynomials, we derive a new addition formula for the little $q$-Bessel functions. The result is obtained by the use of a limit transition. We also establish a product formula for little $q$-Bessel functions with a positive and symmetric kernel.

7. K-Bessel functions associated to 3-rank Jordan algebra

Hacen Dib

2005-01-01

Using the Bessel-Muirhead system, we can express the K-Bessel function defined on a Jordan algebra as a linear combination of the J-solutions. We determine explicitly the coefficients when the rank of this Jordan algebra is three after a reduction to the rank two. The main tools are some algebraic identities developed for this occasion.

8. Composition Formulas of Bessel-Struve Kernel Function

K. S. Nisar; Mondal, S.R; P Agarwal

2016-01-01

The object of this paper is to study and develop the generalized fractional calculus operators involving Appell’s function F3(·) due to Marichev-Saigo-Maeda. Here, we establish the generalized fractional calculus formulas involving Bessel-Struve kernel function Sαλz,  λ,z∈C to obtain the results in terms of generalized Wright functions. The representations of Bessel-Struve kernel function in terms of exponential function and its relation with Bessel and Struve function are also discussed. The...

9. Inequalities involving modified Bessel functions of the first kind II

2007-08-01

The intrinsic properties, including logarithmic convexity (concavity), of the modified Bessel functions of the first kind and some other related functions are obtained. Several inequalities involving functions under discussion are established.

10. Some unified integrals associated with Bessel-Struve kernel function

K.S. Nisar; P. Agarwal; Jain, S

2016-01-01

In this paper, we discuss the generalized integral formula involving Bessel-Struve kernel function $S_{\\alpha }\\left( \\lambda z\\right)$, which expressed in terms of generalized Wright functions. Many interesting special cases also obtained in this study.

11. Evaluation of a Family of Bessel Function Integrals

Birrell, Jeremiah

2015-01-01

We investigate a family of integrals involving modified Bessel functions that arise in the context of neutrino scattering. Recursive formulas are derived for evaluating these integrals and their asymptotic expansions are computed. We prove in certain cases that the asymptotic expansion yields the exact result after a finite number of terms. In each of these cases we derive a formula that bounds the order at which the expansion terminates. The method of calculation developed in this paper is applicable to similar families of integrals that involve Bessel or modified Bessel functions.

12. Bounds for Tur\\'anians of modified Bessel functions

Baricz, Árpád

2012-01-01

Motivated by some applications in applied mathematics, biology, chemistry, physics and engineering sciences, new tight Tur\\'an type inequalities for modified Bessel functions of the first and second kind are deduced. These inequalities provide sharp lower and upper bounds for the Tur\\'anian of modified Bessel functions of the first and second kind, and in most cases the relative errors of the bounds tend to zero as the argument tends to infinity. The chief tools in our proofs are some ideas of Gronwall [19], an integral representation of Ismail [28,29] for the quotient of modified Bessel functions of the second kind, results of Hartman and Watson [24,26,59] and some recent results of Segura [52]. As applications of the main results some sharp Tur\\'an type inequalities are presented for the product of modified Bessel functions of the first and second kind and it is shown that this product is strictly geometrically concave.

13. The radius of convexity of normalized Bessel functions

Baricz, Árpád; Szász, Róbert

2015-01-01

The radius of convexity of two normalized Bessel functions of the first kind are determined in the case when the order is between $-2$ and $-1.$ Our methods include the minimum principle for harmonic functions, the Hadamard factorization of some Dini functions, properties of the zeros of Dini functions via Lommel polynomials and some inequalities for complex and real numbers.

14. Bessel functions in mass action modeling of memories and remembrances

Freeman, Walter J. [Department of Molecular and Cell Biology, University of California, Berkeley, CA 94720-3206 (United States); Capolupo, Antonio [Dipartimento di Fisica, E.R. Caianiello Universitá di Salerno, and INFN Gruppo collegato di Salerno, Fisciano 84084 (Italy); Kozma, Robert [Department of Mathematics, Memphis University, Memphis, TN 38152 (United States); Olivares del Campo, Andrés [The Blackett Laboratory, Imperial College London, Prince Consort Road, London SW7 2BZ (United Kingdom); Vitiello, Giuseppe, E-mail: vitiello@sa.infn.it [Dipartimento di Fisica, E.R. Caianiello Universitá di Salerno, and INFN Gruppo collegato di Salerno, Fisciano 84084 (Italy)

2015-10-02

Data from experimental observations of a class of neurological processes (Freeman K-sets) present functional distribution reproducing Bessel function behavior. We model such processes with couples of damped/amplified oscillators which provide time dependent representation of Bessel equation. The root loci of poles and zeros conform to solutions of K-sets. Some light is shed on the problem of filling the gap between the cellular level dynamics and the brain functional activity. Breakdown of time-reversal symmetry is related with the cortex thermodynamic features. This provides a possible mechanism to deduce lifetime of recorded memory. - Highlights: • We consider data from observations of impulse responses of cortex to electric shocks. • These data are fitted by Bessel functions which may be represented by couples of damped/amplified oscillators. • We study the data by using couples of damped/amplified oscillators. • We discuss lifetime and other properties of the considered brain processes.

15. An integral estimate of Bessel function and its application

DING Yong; LI Ran

2008-01-01

In this paper the authors give a new integral estimate of the Bessel function, which parameterized Marcinkiewicz integral μΩρ with variable kernels is of type (2, 2), where the kernel function Ω does not have any smoothness on the unit sphere in Rn.

16. Inequalities and Asymptotic Formulae Related to Generalizations of the Bessel Functions

Paneva-Konovska, Jordanka

2010-10-01

We consider some families of 3-index generalizations of the Bessel functions of first kind and study the behaviour of such families in domains of the complex plane. We also prove asymptotic formulae for "large" values of indices of these functions. Similar theorems have also been obtained by the author for the Bessel and Bessel-Maitland functions.

17. The asymptotic behavior of q-exponentials and q-Bessel functions

Rogov, V. -B. K.

2006-01-01

The connections between q-Bessel functions of three types and q-exponential of three types are established. The q-exponentials and the q-Bessel functions are represented as the Laurent series. The asymptotic behaviour of the q-exponentials and the q-Bessel functions is investigated.

18. CHEBYSHEV APPROXIMATION OF THE SECOND KIND OF MODIFIED BESSEL FUNCTION OF ORDER ZERO

张璟; 周哲玮

2004-01-01

The second kind of modified Bessel function of order zero is the solutions of many problems in engineering. Modified Bessel equation is transformed by exponential transformation and expanded by J. P. Boyd' s rational Chebyshev basis.

19. Uniform bounds for expressions involving modified Bessel functions

Gaunt, Robert E.

2013-01-01

In this paper, we obtain uniform bounds for a number of expressions that involve derivatives and integrals of modified Bessel functions. These uniform bounds are motivated by the need to bound such expressions in the study of variance-gamma and product normal approximations via Stein's method.

20. Further properties of the zeros of Bessel functions

Carla Giordano

1987-11-01

Full Text Available New monotonicity and convexity properties for the zeros cνk  (k=1,2,... of the Bessel functions are proved. New inequalities for cνk are also given. These inequalities are useful for small values of ν.

1. Three-particle integrals with the Bessel functions

Frolov, Alexei M

2012-01-01

Analytical formulas for some useful three-particles integrals are derived. Many of these integrals include Bessel and trigonometric functions of one interparticle (relative) coordinate $r_{ij}$. Also, we consider the matrix elements of the Uehling potential which describes the lowest order correction to the vacuum polarization between two interacting electric charges.

2. Monotonic sequences related to zeros of Bessel functions

Lorch, Lee; Muldoon, Martin

2008-12-01

In the course of their work on Salem numbers and uniform distribution modulo 1, A. Akiyama and Y. Tanigawa proved some inequalities concerning the values of the Bessel function J 0 at multiples of π, i.e., at the zeros of J 1/2. This raises the question of inequalities and monotonicity properties for the sequences of values of one cylinder function at the zeros of another such function. Here we derive such results by differential equations methods.

3. Optimized Kaiser-Bessel Window Functions for Computed Tomography.

Nilchian, Masih; Ward, John Paul; Vonesch, Cedric; Unser, Michael

2015-11-01

Kaiser-Bessel window functions are frequently used to discretize tomographic problems because they have two desirable properties: 1) their short support leads to a low computational cost and 2) their rotational symmetry makes their imaging transform independent of the direction. In this paper, we aim at optimizing the parameters of these basis functions. We present a formalism based on the theory of approximation and point out the importance of the partition-of-unity condition. While we prove that, for compact-support functions, this condition is incompatible with isotropy, we show that minimizing the deviation from the partition of unity condition is highly beneficial. The numerical results confirm that the proposed tuning of the Kaiser-Bessel window functions yields the best performance. PMID:26151939

4. An integral estimate of Bessel function and its application

2008-01-01

In this paper the authors give a new integral estimate of the Bessel function,which is an extension of Calder(?)n-Zygmund’s result.As an application of this result,we prove that the parameterized Marcinkiewicz integralμ_Ω~p with variable kernels is of type (2,2),where the kernel functionΩdoes not have any smoothness on the unit sphere in R~n.

5. Expansion Formulae for the Kampe De Feriet Function Involving Bessel Function

1971-01-01

Full Text Available In this paper some integrals involving a Kampe de Feriet; function have been evaluated. These have been used to establish some expansion formulae for the Kampe de Feriet function involving Bessel function.

6. On Tur\\'an type inequalities for modified Bessel functions

Baricz, Árpád

2010-01-01

In this note our aim is to point out that certain inequalities for modified Bessel functions of the first and second kind, deduced recently by Laforgia and Natalini, are in fact equivalent to the corresponding Tur\\'an type inequalities for these functions. Moreover, we present some new Tur\\'an type inequalities for the aforementioned functions and we show that their product is decreasing as a function of the order, which has application in the study of stability of radially symmetric solutions in a generalized FitzHugh-Nagumo equation in two spatial dimensions. At the end of this note a conjecture is posed, which may be of interest for further research.

7. One some differential subordination involving the Bessel-Struve kernel function

Saiful R. Mondal; Dhuian, Mohamed Al

2016-01-01

In this article we study the inclusion properties of the Bessel-Struve kernel functions in the Janowski class. In particular, we find the conditions for which the Bessel-Struve kernel functions maps the unit disk to right half plane. An open problem in this aspect are also given. The third order differential subordination involving the Bessel-Struve kernel is also considered. The results are derived by defining suitable classes of admissible functions. One of the recurrence relation of the Be...

8. The Bessel Period Functional on SO(5): The Nontempered Case

Qiu, Yannan

2013-01-01

For automorphic representations in the nontempered cuspidal spectrum of $\\mathrm{SO}_5$, we establish a precise Bessel period formula, in which the square of the global Bessel period is decomposed as an Euler product of regularized local Bessel period integrals of matrix coefficients.

9. APPLICATION OF THE BESSEL-HYBRID FUNCTIONS FOR THE LINEAR FREDHOLM-VOLTERRA INTEGRO-DIFFERENTIAL EQUATIONS

2015-01-01

In this paper a collocation method based on the Bessel-hybrid functions is used for approximation of the solution of linear Fredholm-Volterra integro-differential equations (FVIDEs) under mixed conditions. First, we explain the properties of Bessel-hybrid functions, which are combination of block-pulse functions and Bessel functions of first kind. The method is based upon Bessel-hybrid approximations, so that to obtain the operational matrixes and approximation of functions we use the tran...

10. The Bessel-Struve intertwining operator on ℂ and mean-periodic functions

A. Gasmi

2004-01-01

Full Text Available We give a description of all transmutation operators from the Bessel-Struve operator to the second-derivative operator. Next we define and characterize the mean-periodic functions on the space ℋ of entire functions and we characterize the continuous linear mappings from ℋ into itself which commute with Bessel-Struve operator.

11. Ultradiscrete limit of Bessel function type solutions of the Painlev\\'{e} III equation

Isojima, Shin

2014-01-01

An ultradiscrete analog of the Bessel function is constructed by taking the ultradiscrete limit for a $q$-difference analog of the Bessel function. Then, a direct relationship between a class of special solutions for the ultradiscrete Painlev\\'{e} III equation and those of the discrete Painlev\\'{e} III equation which have a determinantal structure is established.

12. On the subclasses associated with the Bessel-Struve kernel functions

Saiful R. Mondal; Mohammed, Al Dhuain

2016-01-01

The article investigate the necessary and sufficient conditions for the normalized Bessel-struve kernel functions belonging to the classes $\\mathcal{T}_\\lambda(\\alpha)$ , $\\mathcal{L}_\\lambda(\\alpha)$. Some linear operators involving the Bessel-Struve operator are also considered.

13. K-Bessel functions associated to a 3-rank Jordan algebra

Hacen Dib

2005-01-01

Full Text Available Using the Bessel-Muirhead system, we can express the K-Bessel function defined on a Jordan algebra as a linear combination of the J-solutions. We determine explicitly the coefficients when the rank of this Jordan algebra is three after a reduction to the rank two. The main tools are some algebraic identities developed for this occasion.

14. Amos-type bounds for modified Bessel function ratios☆

Hornik, Kurt; Grün, Bettina

2013-01-01

We systematically investigate lower and upper bounds for the modified Bessel function ratio Rν=Iν+1/Iν by functions of the form Gα,β(t)=t/(α+t2+β2) in case Rν is positive for all t>0, or equivalently, where ν≥−1 or ν is a negative integer. For ν≥−1, we give an explicit description of the set of lower bounds and show that it has a greatest element. We also characterize the set of upper bounds and its minimal elements. If ν≥−1/2, the minimal elements are tangent to Rν in exactly one point 0≤t≤∞, and have Rν as their lower envelope. We also provide a new family of explicitly computable upper bounds. Finally, if ν is a negative integer, we explicitly describe the sets of lower and upper bounds, and give their greatest and least elements, respectively. PMID:24926105

15. Amos-type bounds for modified Bessel function ratios.

Hornik, Kurt; Grün, Bettina

2013-12-01

We systematically investigate lower and upper bounds for the modified Bessel function ratio [Formula: see text] by functions of the form [Formula: see text] in case [Formula: see text] is positive for all [Formula: see text], or equivalently, where [Formula: see text] or [Formula: see text] is a negative integer. For [Formula: see text], we give an explicit description of the set of lower bounds and show that it has a greatest element. We also characterize the set of upper bounds and its minimal elements. If [Formula: see text], the minimal elements are tangent to [Formula: see text] in exactly one point [Formula: see text], and have [Formula: see text] as their lower envelope. We also provide a new family of explicitly computable upper bounds. Finally, if [Formula: see text] is a negative integer, we explicitly describe the sets of lower and upper bounds, and give their greatest and least elements, respectively. PMID:24926105

16. On a new class of integrals involving Bessel functions of the first kind

P. Agarwal

2014-06-01

Full Text Available In recent years, several integral formulas involving a variety of special functions have been developed by many authors. Also many integral formulas containing the Bessel function $J_\ 17. A Limit Relation for Dunkl-Bessel Functions of Type A and B Margit Rösler; Michael Voit 2008-01-01 We prove a limit relation for the Dunkl-Bessel function of type$B_N$with multiplicity parameters$k_1$on the roots$\\pm e_i$and$k_2$on$\\pm e_i\\pm e_j$where$k_1$tends to infinity and the arguments are suitably scaled. It gives a good approximation in terms of the Dunkl-type Bessel function of type$A_{N-1}$with multiplicity$k_2$. For certain values of$k_2$an improved estimate is obtained from a corresponding limit relation for Bessel functions on matrix cones. 18. Zeros of combinations of Bessel functions and the mean charge of graphene nanodots Beneventano, C. G.; Fialkovsky, I. V.; Santangelo, E. M. 2016-04-01 We establish some properties of the zeros of sums and differences of contiguous Bessel functions of the first kind. As a by-product, we also prove that the zeros of the derivatives of Bessel functions of the first kind of different orders are interlaced the same way as the zeros of the Bessel functions themselves. As a physical motivation, we consider gated graphene nanodots subject to Berry-Mondragon boundary conditions. We determine the allowed energy levels and calculate the mean charge at zero temperature. We discuss its dependence on the gate ( chemical) potential in detail and also comment on the effect of temperature. 19. Superiority of Bessel function over Zernicke polynomial as base function for radial expansion in tomographic reconstruction A K Chattopadhyay; C V S Rao 2003-07-01 Here we describe the superiority of Bessel function as base function for radial expansion over Zernicke polynomial in the tomographic reconstruction technique. The causes for the superiority have been described in detail. The superiority has been shown both with simulated data for Kadomtsev’s model for saw-tooth oscillation and real experimental x-ray data from W7-AS Stellarator. 20. q-Sumudu Transforms of q-Analogues of Bessel Functions Faruk Uçar 2014-01-01 The main purpose of this paper is to evaluate q-Sumudu transforms of a product of q-Bessel functions. Interesting special cases of theorems are also discussed. Further, the results proved in this paper may find certain applications of q-Sumudu transforms to the solutions of the q-integrodifferential equations involving q-Bessel functions. The results may help to extend the q-theory of orthogonal functions. 1. Bessel harmonic analysis and approximation of functions on the half-line Platonov, Sergei S. 2007-10-01 We study problems of approximation of functions on \\lbrack 0, +\\infty) in the metric of L_p with power weight using generalized Bessel shifts. We prove analogues of direct Jackson theorems for the modulus of smoothness of arbitrary order defined in terms of generalized Bessel shifts. We establish the equivalence of the modulus of smoothness and the K-functional. We define function spaces of Nikol'skii-Besov type and describe them in terms of best approximations. As a tool for approximation, we use a certain class of entire functions of exponential type. In this class, we prove analogues of Bernstein's inequality and others for the Bessel differential operator and its fractional powers. The main tool we use to solve these problems is Bessel harmonic analysis. 2. Bessel harmonic analysis and approximation of functions on the half-line We study problems of approximation of functions on [0,+∞) in the metric of Lp with power weight using generalized Bessel shifts. We prove analogues of direct Jackson theorems for the modulus of smoothness of arbitrary order defined in terms of generalized Bessel shifts. We establish the equivalence of the modulus of smoothness and the K-functional. We define function spaces of Nikol'skii-Besov type and describe them in terms of best approximations. As a tool for approximation, we use a certain class of entire functions of exponential type. In this class, we prove analogues of Bernstein's inequality and others for the Bessel differential operator and its fractional powers. The main tool we use to solve these problems is Bessel harmonic analysis 3. On Generalized Fractional Kinetic Equations Involving Generalized Bessel Function of the First Kind Dinesh Kumar 2015-01-01 Full Text Available We develop a new and further generalized form of the fractional kinetic equation involving generalized Bessel function of the first kind. The manifold generality of the generalized Bessel function of the first kind is discussed in terms of the solution of the fractional kinetic equation in the paper. The results obtained here are quite general in nature and capable of yielding a very large number of known and (presumably new results. 4. Diffraction pattern of modulated structures described by Bessel functions Wolny, Janusz; Buganski, Ireneusz; Strzalka, Radoslaw 2016-05-01 We performed detailed analysis of 1D modulated structure (MS) with harmonic modulation within the statistical approach. By applying two-mode Fourier transform, we were able to derive analytically the structure factor for MS with single harmonic modulation component. We confirmed in a very smooth way that ordinary Bessel functions of the first kind define envelopes tuning the intensities of the diffraction peaks. This applies not only to main reflections of the diffraction pattern but also to all satellites. In the second part, we discussed in details the similarities between harmonically modulated structures with multiple modulations and 1D model quasicrystal. The Fourier expansion of the nodes' positions in the Fibonacci chain gives direct numerical definition of the atomic arrangement in MS. In that sense, we can define 1D quasicrystal as a MS with infinite number of harmonic modulations. We prove that characteristic measures (like v(u) relation typical for statistical approach and diffraction pattern) calculated for MS asymptotically approach their counterparts for 1D quasicrystal as large enough number of modulation terms is taken into account. 5. APPLICATION OF THE BESSEL-HYBRID FUNCTIONS FOR THE LINEAR FREDHOLM-VOLTERRA INTEGRO-DIFFERENTIAL EQUATIONS YADOLLAH ORDOKHANI 2015-12-01 Full Text Available In this paper a collocation method based on the Bessel-hybrid functions is used for approximation of the solution of linear Fredholm-Volterra integro-differential equations (FVIDEs under mixed conditions. First, we explain the properties of Bessel-hybrid functions, which are combination of block-pulse functions and Bessel functions of first kind. The method is based upon Bessel-hybrid approximations, so that to obtain the operational matrixes and approximation of functions we use the transfer matrix from Bessel-hybrid functions to Taylor polynomials. The matrix equations correspond to a system of linear algebraic equations with the unknown Bessel-hybrid coefficients. Present results and comparisons demonstrate our estimate have good degree of accuracy. 6. Characterization of Banach valued BMO functions and UMD Banach spaces by using Bessel convolutions Betancor, Jorge J; Rodríguez-Mesa, Lourdes 2011-01-01 In this paper we consider the space$BMO_o(\\mathbb{R},X)$of bounded mean oscillations and odd functions on$\\mathbb{R}$taking values in a UMD Banach space$X$. The functions in$BMO_o(\\mathbb{R},X)$are characterized by Carleson type conditions involving Bessel convolutions and$\\gamma$-radonifying norms. Also we prove that the UMD Banach spaces are the unique Banach spaces for which certain$\\gamma$-radonifying Carleson inequalities for Bessel-Poisson integrals of$BMO_o(\\mathbb{R},X)$functions hold. 7. An exactly solvable model of an oscillator with nonlinear coupling and zeros of Bessel functions Dodonov, V. V.; Klimov, A. B. 1993-01-01 We consider an oscillator model with nonpolynomial interaction. The model admits exact solutions for two situations: for energy eigenvalues in terms of zeros of Bessel functions, that were considered as functions of the continuous index; and for the corresponding eigenstates in terms of Lommel polynomials. 8. Physical Applications of a Simple Approximation of Bessel Functions of Integer Order Barsan, V.; Cojocaru, S. 2007-01-01 Applications of a simple approximation of Bessel functions of integer order, in terms of trigonometric functions, are discussed for several examples from electromagnetism and optics. The method may be applied in the intermediate regime, bridging the "small values regime" and the "asymptotic" one, and covering, in this way, an area of great… 9. Study Hankel Transforms and Properties of Bessel Function via Entangled State Representation Transformation in Quantum Mechanics FAN Hong-Yi; WANG Yong 2006-01-01 In Phys. Lett. A 313 (2003) 343 we have found that the self-reciprocal Hankel transformation (HT) is embodied in quantum mechanics by a transform between two entangled state representations of continuum variables. In this work we study Hankel transforms and properties of Bessel function via entangled state representations' transformation in quantum mechanics. 10. Remarks on the paper: "Bounds for functions involving ratios of modified Bessel functions" Segura, Javier 2011-01-01 A recent paper by C.G. Kokologiannaki published in J. Math. Anal. Appl. \\cite{Kolo:2012:BFI} gives some properties for ratios of modified Bessel functions and, in particular, some bounds. These bounds are said to improve the range of some inequalities in \\cite{Segura:2011:BRM} or to be sharper. Unfortunately, Kokologiannaki made some mistakes in the comparison and no improvement or extension is made over the results in \\cite{Segura:2011:BRM}. We explain the errors in these comments and show that the bounds given in \\cite{Kolo:2012:BFI} are already contained in \\cite{Segura:2011:BRM}. 11. Physical applications of a simple approximation of Bessel functions of integer order Applications of a simple approximation of Bessel functions of integer order, in terms of trigonometric functions, are discussed for several examples from electromagnetism and optics. The method may be applied in the intermediate regime, bridging the 'small values regime' and the 'asymptotic' one, and covering, in this way, an area of great physical interest. The examples that illustrate our approach are accessible to the undergraduate student 12. Modeling Delays of Microwave Transistors and Transmission Lines by the 2nd Order Bessel Function K. Ulovec 2007-04-01 Full Text Available At present, most of simulation programs can characterize gate delays of microwave transistors. However, the delay is mostly approximated by means of first-order differential equations. In the paper, a more accurate way is suggested which is based on an appropriate second-order differential equation. Concerning the transmission line delay, majority of the simulation programs use both Branin (for lossless lines and LCRG (for lossy lines models. However, the first causes extreme simulation times, and the second causes well-known spurious oscillations in the simulation results. In the paper, an unusual way for modeling the transmission line delay is defined, which is also based on the second-order Bessel function. The proposed model does not create the spurious oscillations and the simulation times are comparable with those obtained with the classical models. Properties of the implementation of the second-order Bessel function are demonstrated by analyses of both digital and analog microwave circuits. 13. A new type of sharp bounds for ratios of modified Bessel functions Ruiz-Antolin, D.; J. Segura 2016-01-01 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 shaper 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 ... 14. Solved problems in analysis as applied to gamma, beta, Legendre and Bessel functions Farrell, Orin J 2013-01-01 Nearly 200 problems, each with a detailed, worked-out solution, deal with the properties and applications of the gamma and beta functions, Legendre polynomials, and Bessel functions. The first two chapters examine gamma and beta functions, including applications to certain geometrical and physical problems such as heat-flow in a straight wire. The following two chapters treat Legendre polynomials, addressing applications to specific series expansions, steady-state heat-flow temperature distribution, gravitational potential of a circular lamina, and application of Gauss's mechanical quadrature 15. Complete Monotonicity of a Difference Between the Exponential and Trigamma Functions and Properties Related to a Modified Bessel Function Qi, Feng; Berg, Christian 2013-01-01 In the paper, the authors find necessary and sufficient conditions for a difference between the exponential function αeβ/t, α, β > 0, and the trigamma function ψ (t) to be completely monotonic on (0,∞). While proving the complete onotonicity, the authors discover some properties related to the fi...... first order modified Bessel function of the first kind I1, including inequalities, monotonicity, unimodality, and convexity.... 16. On infinite series concerning zeros of Bessel functions of the first kind Giusti, Andrea; Mainardi, Francesco 2016-06-01 A relevant result independently obtained by Rayleigh and Sneddon on an identity on series involving the zeros of Bessel functions of the first kind is derived by an alternative method based on Laplace transforms. Our method leads to a Bernstein function of time, expressed by Dirichlet series, that allows us to recover the Rayleigh-Sneddon sum. We also consider another method arriving at the same result based on a relevant formula by Calogero. Moreover, we also provide an electrical example in which this sum results to be extremely useful in order to recover the analytical expression for the response of the system to a certain external input. 17. Bessel Beams McDonald, Kirk T 2000-01-01 Scalar Bessel beams are derived both via the wave equation and via diffraction theory. While such beams have a group velocity that exceeds the speed of light, this is a manifestation of the "scissors paradox" of special relativty. The signal velocity of a modulated Bessel beam is less than the speed of light. Forms of Bessel beams that satisfy Maxwell's equations are also given. 18. Product Bessel Distributions of the First and Second Kinds Saralees Nadarajah 2007-01-01 A new Bessel function distribution is introduced by taking the product of a Bessel function pdf of the first kind and a Bessel function pdf of the second kind. Various particular cases and expressions for moments are derived. 19. Characterizing emitted electromagnetic radiation: the vector-spherical-harmonic/Bessel-function description General expressions for the dependence of the polarization of the radiation emitted in an atomic electronic transition upon the angle of emission are derived on the basis of a vector-spherical-harmonic/Bessel-function description of the photon. The concept of a photon of a specific multipolarity, i.e. quadrupole, octupole, etc, and identifiable as such, is found to be unviable. The angular distribution and dependence upon distance from the source of the transverse and parallel components of the electric and magnetic fields arising from E1, E2, M1 and M2 atomic electronic emissions are elucidated. (paper) 20. Characterizing emitted electromagnetic radiation: the vector-spherical-harmonic/Bessel-function description Grinter, Roger 2014-04-01 General expressions for the dependence of the polarization of the radiation emitted in an atomic electronic transition upon the angle of emission are derived on the basis of a vector-spherical-harmonic/Bessel-function description of the photon. The concept of a photon of a specific multipolarity, i.e. quadrupole, octupole, etc, and identifiable as such, is found to be unviable. The angular distribution and dependence upon distance from the source of the transverse and parallel components of the electric and magnetic fields arising from E1, E2, M1 and M2 atomic electronic emissions are elucidated. 1. Transcendentality of zeros of higher dereivatives of functions involving Bessel functions Martin E. Muldoon 1995-09-01 Full Text Available C.L. Siegel established in 1929 [Ges. Abh., v.1, pp. 209-266] the deep results that (i all zeros of Jv(x and JÃ¢Â€Â²v(x are transcendental when v is rational, xÃ¢Â‰Â 0, and (ii JÃ¢Â€Â²v(x/Jv(x is transcendental when v is rational and x algebraic. As usual, Jv(x is the Bessel function of first kind and order v. Here it is shown that simple arguments permit one to infer from Siegel's results analogous but not identical properties of the zeros of higher derivatives of xÃ¢ÂˆÂ’uJv(x when ÃŽÂ¼ is algebraic and v rational. In particular, JÃ¢Â€Â´1(Ã‚Â±3=0 while all other zeros of JÃ¢Â€Â´1(x and all zeros of JÃ¢Â€Â´v(x, v2Ã¢Â‰Â 1, xÃ¢Â‰Â 0, are transcendental. Further, J0(4(Ã‚Â±3=0 while all other zeros of J0(4(x, xÃ¢Â‰Â 0, and of Jv(4(x, vÃ¢Â‰Â 0, xÃ¢Â‰Â 0, are transcendental. All zeros of Jv(n(x, xÃ¢Â‰Â 0, are transcendental, n=5,Ã¢Â€Â¦,18, when v is rational. For most values of n, the proofs used the symbolic computation package Maple V (Release 1. 2. Keldysh theory re-examined: Application of the generalized Bessel functions Bauer, J H 2015-01-01 A derivation of the ionization rate for the hydrogen-like ion in the strong linearly polarized laser field is presented. This derivation utilizes the famous Keldysh probability amplitude in the length gauge (in the dipole approximation) and without Coulomb effects in the final state of the ionized electron. No further approximations are being made, because the amplitude has been expanded in the double Fourier series in a time domain (with the help of the generalized Bessel functions). Thus, our theory has no other limitations characteristic of the original Keldysh theory. We compare our "exact" theory with the original Keldysh one, studying photoionization energy spectra and total ionization rates. We show breakdown of the original Keldysh theory for higher frequencies. In the barrier-suppresion regime the "exact" Keldysh theory gives results closer to well-known numerical or other analytical results. 3. n阶变型Bessel函数两个定理的证明%Proof of Two Theorem Composed of n Order Modified Bessel Functions 蓝新华 2011-01-01 Base on the properties of the modified Bessel function, we proof the orthogonality of functions systems of two n order modified Bessel functions, give the expression of their lengths.%利用n阶变型Bessel函数的相关性质，证明了由n阶变型Bessel函数组成的两个函数系具有正交性，并给出了长度表达式。 4. Mikusi\\'nski's Operational Calculus with Algebraic Foundations and Applications to Bessel Functions Bengochea, Gabriel; G, Gabriel López 2013-01-01 We construct an operational calculus supported on the algebraic operational calculus introduced by Bengochea and Verde. With this operational calculus we study the solution of certain Bessel type equations. 5. On hyperbolic Bessel processes and beyond Jakubowski, Jacek; Wiśniewolski, Maciej 2013-01-01 We investigate distributions of hyperbolic Bessel processes. We find links between the hyperbolic cosine of hyperbolic Bessel processes and functionals of geometric Brownian motion. We present an explicit formula for the Laplace transform of the hyperbolic cosine of a hyperbolic Bessel process and some other interesting probabilistic representations of this Laplace transform. We express the one-dimensional distribution of a hyperbolic Bessel process in terms of other, known and independent pr... 6. Creation of matter wave Bessel beams Ryu, C.; Henderson, K. C.; Boshier, M. G. 2013-01-01 Bessel beams are plane waves with amplitude profiles described by Bessel functions. They are important because of their property of limited diffraction and their capacity to carry orbital angular momentum. Here we report the creation of a Bessel beam of de Broglie matter waves. The Bessel beam is produced by the free evolution of a thin toroidal atomic Bose-Einstein condensate (BEC) which has been set into rotational motion. By attempting to stir it at different rotation rates, we show that t... 7. The generalized Abel-Plana formula. Applications to Bessel functions and Casimir effect One of the most efficient methods to obtain the vacuum expectation values for the physical observables in the Casimir effect is based on using the Abel-Plana summation formula. This allows us to derive the regularized quantities in a manifestly cutoff independent way and present them in the form of strongly convergent integrals. However, the application of Abel-Plana formula, in its usual form, is restricted by simple geometries when the eigenmodes have a simple dependence on quantum numbers. The author generalized the Abel-Plana formula which essentially enlarges its application range. Based on this generalization, formulae have been obtained for various types of series over the zeros of some combinations of Bessel functions and for integrals involving these functions. It has been shown that these results generalize the special cases existing in literature. Further, the derived summation formulae have been used to summarize series arising in the mode summation approach to the Casimir effect for spherically and cylindrically symmetric boundaries. This allows us to extract the divergent parts from the vacuum expectation values for the local physical observables in a manifestly cutoff independent way. The present paper reviews these results. Some new considerations are also added. (author) 8. Revisiting the orthogonality of Bessel functions of the first kind on an infinite interval The rigorous proof of the orthogonality integral ∫0∞ρ Jν(kρ)Jν(k′ρ) dρ=((δ(k−k′))/k), for ν>=−1, is laborious and requires the use of mathematical techniques that, probably, are unfamiliar to most physics students, even at the graduate level. In physics, we are used to the argument that it may be proved by the use of Hankel transforms. However, the logic of the matter is the opposite, i.e., the existence of the inverse Hankel transform is a consequence of the orthogonality integral. The goal of this work is to prove this integral without circular reasoning. In this paper, using elementary properties of Bessel functions, we give a simple analytical derivation of this integral for the case where ν is an integer, zero, or half-integer not less than −1/2. Then, using the asymptotic behaviour of Jν(x), we extend the result to any ν>=−1. This work is of a pedagogical nature. Therefore, to add educational value to the discussion, we do not skip the details of the calculations. (paper) 9. Intensity transformation of vector Bessel beams using a multilayer system Novitsky, Andrey V. 2008-01-01 We theoretically investigate the generation of vector Bessel beams of the order m using a phase shifted superposition of TE and TM electromagnetic Bessel beams. Such Bessel beams are characterized by the intensity profile described by the superposition of squared Bessel functions of the orders m-1 and m+1. 10. The generalized Abel-Plana formula with applications to Bessel functions and casimir effect One of the most efficient methods for the evaluation of the vacuum expectation values for physical observables in the Casimir effect is based on using the Abel-Plana summation formula. This enables to derive the renormalized quantities in a manifestly cutoff independent way and to present them in the form of strongly convergent integrals. However, applications of the Abel- Plana formula, in its usual form, are restricted by simple geometries when the eigenmodes have a simple dependence on quantum numbers. The author generalized the Abel-Plana formula which essentially enlarges its application range. Based on this generalization, formulae have been obtained for various types of series over the zeros of combinations of Bessel functions and for integrals involving these functions. It has been shown that these results generalize the special cases existing in literature. Further, the derived summation formulae have been used to summarize series arising in the direct mode summation approach to the Casimir effect for spherically and cylindrically symmetric boundaries, for boundaries moving with uniform proper acceleration, and in various braneworld scenarios. This allows to extract from the vacuum expectation values of local physical observables the parts corresponding to the geometry without boundaries and to present the boundary-induced parts in terms of integrals strongly convergent for the points away from the boundaries. As a result, the renormalization procedure for these observables is reduced to the corresponding procedure for bulks without boundaries. The present paper reviews these results. We also aim to collect the results on vacuum expectation values for local physical observables such as the field square and the energy-momentum tensor in manifolds with boundaries for various bulk and boundary geometries. (author) 11. On hyperbolic Bessel processes and beyond Wisniewolski, Maciej 2011-01-01 We investigate distributions of hyperbolic Bessel processes. We find links between the hyperbolic cosinus of the hyperbolic Bessel processes and the functionals of geometric Brownian motion. We present an explicit formula of Laplace transform of hyperbolic cosinus of hyperbolic Bessel processes and some interesting different probabilistic representations of this Laplace transform. We express the one-dimensional distribution of hyperbolic Bessel process in terms of other, known and independent processes. We present some applications including a new proof of Bougerol's identity and it's generalization. We characterize the distribution of the process being hyperbolic sinus of hyperbolic Bessel processes. 12. The Bessel Numbers and Bessel Matrices Sheng Liang YANG; Zhan Ke QIAO 2011-01-01 In this paper,using exponential Riordan arrays,we investigate the Bessel numbers and Bessel matrices.By exploring links between the Bessel matrices,the Stirling matrices and the degenerate Stirling matrices,we show that the Bessel numbers are special case of the degenerate Stirling numbers,and derive explicit formulas for the Bessel numbers in terms of the Stirling numbers and binomial coefficients. 13. Bessel Weighted Asymmetries Avakian, Harut [Thomas Jefferson National Accelerator Facility (TJNAF), Newport News, VA (United States); Gamberg, Leonard [Pennsylvania State Univ., University Park, PA (United States); Rossi, Patrizia [Thomas Jefferson National Accelerator Facility (TJNAF), Newport News, VA (United States); Prokudin, Alexei [Pennsylvania State Univ., University Park, PA (United States) 2016-05-01 We review the concept of Bessel weighted asymmetries for semi-inclusive deep inelastic scattering and focus on the cross section in Fourier space, conjugate to the outgoing hadron’s transverse momentum, where convolutions of transverse momentum dependent parton distribution functions and fragmentation functions become simple products. Individual asymmetric terms in the cross section can be projected out by means of a generalized set of weights involving Bessel functions. The procedure is applied to studies of the double longitudinal spin asymmetry in semi-inclusive deep inelastic scattering using a new dedicated Monte Carlo generator which includes quark intrinsic transverse momentum within the generalized parton model. We observe a few percent systematic offset of the Bessel-weighted asymmetry obtained from Monte Carlo extraction compared to input model calculations, which is due to the limitations imposed by the energy and momentum conservation at the given energy and hard scale Q2. We find that the Bessel weighting technique provides a powerful and reliable tool to study the Fourier transform of TMDs with controlled systematics due to experimental acceptances and resolutions with different TMD model inputs. 14. Simple analytical approximations to the integrals of the Bessel functions Jν: application to the transmittance of a circular aperture Two accurate, yet simple, analytic approximations to the integral of the Bessel function J0 are presented. These first and second-order approximations are obtained by improving on the recently developed method known as two-point quasi-rational approximations. The accuracy of the first-order approximant is better than 0.05. The second-order approximant is practically indistinguishable from the true integral, even for very large values of the argument (overall accuracy is better than 0.002 05). Our approximants are, in addition, analytic and therefore replace with significant advantages both the well known power series and the asymptotic formulae of the integral. Approximants to the transmittance function of a plane wave through a circular aperture are derived, a problem which arises in diffraction theory and particle scattering. The second-order approximant to the transmittance is analytic too, and can be evaluated for small and large values of the argument, just with a hand-calculator. Its accuracy is better than 0.0011. As an extension, two first-order approximations to the integrals of the Bessel functions Jν, of fractional order ν, are derived. (author) 15. Free Bessel laws Banica, Teodor; Belinschi, Serban; Capitaine, Mireille; Collins, Benoit 2007-01-01 We introduce and study a remarkable family of real probability measures$\\pi_{st}$, that we call free Bessel laws. These are related to the free Poisson law$\\pi$via the formulae$\\pi_{s1}=\\pi^{\\boxtimes s}$and$\\pi_{1t}=\\pi^{\\boxplus t}$. Our study includes: definition and basic properties, analytic aspects (supports, atoms, densities), combinatorial aspects (functional transforms, moments, partitions), and a discussion of the relation with random matrices and quantum groups. 16. Computational Diffusion Magnetic Resonance Imaging Based on Time-Dependent Bloch NMR Flow Equation and Bessel Functions. Awojoyogbe, Bamidele O; Dada, Michael O; Onwu, Samuel O; Ige, Taofeeq A; Akinwande, Ninuola I 2016-04-01 Magnetic resonance imaging (MRI) uses a powerful magnetic field along with radio waves and a computer to produce highly detailed "slice-by-slice" pictures of virtually all internal structures of matter. The results enable physicians to examine parts of the body in minute detail and identify diseases in ways that are not possible with other techniques. For example, MRI is one of the few imaging tools that can see through bones, making it an excellent tool for examining the brain and other soft tissues. Pulsed-field gradient experiments provide a straightforward means of obtaining information on the translational motion of nuclear spins. However, the interpretation of the data is complicated by the effects of restricting geometries as in the case of most cancerous tissues and the mathematical concept required to account for this becomes very difficult. Most diffusion magnetic resonance techniques are based on the Stejskal-Tanner formulation usually derived from the Bloch-Torrey partial differential equation by including additional terms to accommodate the diffusion effect. Despite the early success of this technique, it has been shown that it has important limitations, the most of which occurs when there is orientation heterogeneity of the fibers in the voxel of interest (VOI). Overcoming this difficulty requires the specification of diffusion coefficients as function of spatial coordinate(s) and such a phenomenon is an indication of non-uniform compartmental conditions which can be analyzed accurately by solving the time-dependent Bloch NMR flow equation analytically. In this study, a mathematical formulation of magnetic resonance flow sequence in restricted geometry is developed based on a general second order partial differential equation derived directly from the fundamental Bloch NMR flow equations. The NMR signal is obtained completely in terms of NMR experimental parameters. The process is described based on Bessel functions and properties that can make it 17. Properties of Bessel Function Solution to Kepler's Equation with Application to Opposition and Conjunction of Earth-Mars Ebaid, Abdelhalim; Al-Blowy, Ahmed B. 2016-05-01 In this article, a simple approach is suggested to calculate the approximate dates of opposition and conjunction of Earth and Mars since their opposition on August 28, 2003 (at perihelion of Mars). The goal of this article has been achieved via using accurate analytical solution to Kepler's equation in terms of Bessel function. The periodicity property of this solution and its particular values at specified times are discussed through some lemmas. The mathematical conditions of opposition and conjunction of the two planets are formulated. Moreover, the intervals of opposition and conjunction have been determined using the graphs of some defined functions. The calculations reveal that there are nine possible oppositions and conjunctions for Earth and Mars during 20 years started on August 28, 2003. The dates of such oppositions and conjunctions were approximately determined and listed in Tables. It is found that our calculations differ few days from the published real dates of Earth-Mars oppositions due to the neglected effects of the gravitational attraction of other planets in the Solar system on the motion of two planets. The period of 20 years can be extended for any number of years by following the suggested analysis. Furthermore, the current approach may be extended to study the opposition and conjunction of the Earth and any outer planet. 18. Terahertz plasmonic Bessel beamformer Monnai, Yasuaki; Shinoda, Hiroyuki [Graduate School of Frontier Sciences, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656 (Japan); Jahn, David; Koch, Martin [Department of Physics, Philipps University of Marburg, Renthof 5, 35032 Marburg (Germany); Withayachumnankul, Withawat [School of Electrical and Electronics Engineering, The University of Adelaide, Adelaide, SA 5005 (Australia) 2015-01-12 We experimentally demonstrate terahertz Bessel beamforming based on the concept of plasmonics. The proposed planar structure is made of concentric metallic grooves with a subwavelength spacing that couple to a point source to create tightly confined surface waves or spoof surface plasmon polaritons. Concentric scatterers periodically incorporated at a wavelength scale allow for launching the surface waves into free space to define a Bessel beam. The Bessel beam defined at 0.29 THz has been characterized through terahertz time-domain spectroscopy. This approach is capable of generating Bessel beams with planar structures as opposed to bulky axicon lenses and can be readily integrated with solid-state terahertz sources. 19. One-dimensional Schr\\"odinger equation with non-analytic potential$V(x)= -g^2\\exp (-|x|)$and its exact Bessel-function solvability Sasaki, Ryu 2016-01-01 Exact solvability (ES) of one-dimensional quantum potentials$V(x)$is a vague concept. We propose that beyond its most conventional range the ES status should be attributed also to many less common interaction models for which the wave functions remain piecewise proportional to special functions. The claim is supported by constructive analysis of a toy model$V(x)= -g^2\\exp (-|x|)$. The detailed description of the related bound-state and scattering solutions of Schr\\"{o}dinger equation is provided in terms of Bessel functions which are properly matched in the origin. 20. Scale space smoothing, image feature extraction and bessel filters Mahmoodi S.; Gunn S. 2011-01-01 The Green function of Mumford-Shah functional in the absence of discontinuities is known to be a modified Bessel function of the second kind and zero degree. Such a Bessel function is regularized here and used as a filter for feature extraction. It is demonstrated in this paper that a Bessel filter does not follow the scale space smoothing property of bounded linear filters such as Gaussian filters. The features extracted by the Bessel filter are therefore scale invariant. Edges, blobs, and j... 1. Nonlinear Bessel beams Johannisson, P; Lisak, M; Marklund, M; Johannisson, Pontus; Anderson, Dan; Lisak, Mietek; Marklund, Mattias 2003-01-01 The effect of the Kerr nonlinearity on linear non-diffractive Bessel beams is investigated analytically and numerically using the nonlinear Schr\\"odinger equation. The nonlinearity is shown to primarily affect the central parts of the Bessel beam, giving rise to radial compression or decompression depending on whether the nonlinearity is focusing or defocusing, respectively. The dynamical properties of Gaussian-truncated Bessel beams are also analysed in the presence of a Kerr nonlinearity. It is found that although a condition for width balance in the root-mean-square sense exists, the beam profile becomes strongly deformed during propagation and may exhibit the phenomena of global and partial collapse. 2. Generating Generalized Bessel Equations by Virtue of Bose Operator Algebra and Entangled State Representations FAN Hong-Yi; WANG Yong 2006-01-01 With the help of Bose operator identities and entangled state representation and based on our previous work [Phys. Lett. A 325 (2004) 188] we derive some new generalized Bessel equations which also have Bessel function as their solution. It means that for these intricate higher-order differential equations, we can get Bessel function solutions without using the expatiatory power-series expansion method. 3. Using Bessel functions to find fixed solutions to wave equations%利用Bessel函数求解波动方程的定解问题 金启胜 2014-01-01 分离变量法是求解波动方程定解问题的一种重要方法。分离变量法的重点在于求特征值及其对应的特征函数。Bessel函数是应用很广泛的一种特征函数，运用Bessel函数的有关性质可以很方便地求解波动方程的定解问题。%The method of separation of variables is an important method for finding solutions to heat conduction equations. The method is used to find the characteristic values of characteristic functions. Bessel functions are characteristic functions,which can be widely used to find fixed solutions to wave equations. 4. Large Degree Asymptotics of Generalized Bessel Polynomials López, J. L.; Temme, Nico 2011-01-01 Asymptotic expansions are given for large values of$n$of the generalized Bessel polynomials$Y_n^\\mu(z)$. The analysis is based on integrals that follow from the generating functions of the polynomials. A new simple expansion is given that is valid outside a compact neighborhood of the origin in the$z-$plane. New forms of expansions in terms of elementary functions valid in sectors not containing the turning points$z=\\pm i/n$are derived, and a new expansion in terms of modified Bessel fu... 5. Apertured paraxial Bessel beams. Umul, Yusuf Z 2010-03-01 The paraxial Bessel beam is obtained by applying an approximation in the wavenumbers. The scattering of the beams by a circular aperture in an absorbing screen is investigated. The scattered fields are expressed in terms of the Fresnel integrals by evaluating the Kirchhoff diffraction integral in the paraxial approximation. The results are examined numerically. PMID:20208927 6. Eigenvalue asymptotics for Dirac-Bessel operators Hryniv, Rostyslav O.; Mykytyuk, Yaroslav V. 2016-06-01 In this paper, we establish the eigenvalue asymptotics for non-self-adjoint Dirac-Bessel operators on (0, 1) with arbitrary real angular momenta and square integrable potentials, which gives the first step for solution of the related inverse problem. The approach is based on a careful examination of the corresponding characteristic functions and their zero distribution. 7. Large degree asymptotics of generalized Bessel polynomials López, J.L.; Temme, N.M. 2011-01-01 Asymptotic expansions are given for large values of$n$of the generalized Bessel polynomials$Y_n^\\mu(z)$. The analysis is based on integrals that follow from the generating functions of the polynomials. A new simple expansion is given that is valid outside a compact neighborhood of the origin in t 8. Bessel potential spaces in Beurling's distributions Sohn, Byung Keun 2009-01-01 We introduce the generalized Bessel potential spaces in the Beurling’s distributions. We give the topological characterizations of the generalized Bessel potential spaces and consider multiplication and convolution operations in the generalized Bessel potential spaces. 9. PSICE Model for a Coaxial Cable in High Frequency Domain Submitted to a Longitudinal Temperature Gradient Using Kelvin-Bessel Asymptotic Functions Hatem Mokhtari 2012-08-01 Full Text Available This paper investigates on the propagation in a coaxial cable with the assumption that the distributed parameters are variable with the longitudinal distance from the source due to a linear temperature variation. This work addresses the problem of propagation in borehole conditions where the temperature varies by approximately 3°C/100m. In addition to the temperature variation along the coax we have considered the high frequency domain where the skin effect is severe and leading to very high losses. Unlike the uniform temperature distribution study this case is very complex and that is the reason why we have provided a model for PSPICE simulator to represent the cascaded cells because the analytic solution is extremely tedious to solve. The idea is thus to replace the coax cable by cascaded elements whose resistance and impedance are calculated via Kelvin-Bessel functions. Because the number of cells is huge we have written a C language program to generate automatically the PSPICE .CIR file. Each cell has its own resistance and inductance according to its temperature. The capacity is assumed to be constant along the coax for each cell. Besides the PSPICE program, and for comparison, we have developed a recursive method for computing the attenuation due to the whole cascaded cells. The comparison between PSPICE and the recursive method has shown results in very good agreement. 10. Finite extensions of Bessel sequences Bakić, Damir; Berić, Tomislav 2015-01-01 The paper studies finite extensions of Bessel sequences in infinite-dimensional Hilbert spaces. We provide a characterization of Bessel sequences that can be extended to frames by adding finitely many vectors. We also characterize frames that can be converted to Parseval frames by finite-dimensional perturbations. Finally, some results on excesses of frames and near-Riesz bases are derived. 11. Entangled Bessel beams McLaren, Melanie; Leach, Jonathan; Roux, Filippus S; Padgett, Miles J; Forbes, Andrew 2012-01-01 Orbital angular momentum (OAM) entanglement is investigated in the Bessel-Gauss (BG) basis. Having a readily adjustable radial scale, BG modes provide a more favourable basis for OAM entanglement over Laguerre-Gaussian (LG) modes. The OAM bandwidth in terms of BG modes can be increased by selection of particular radial modes and leads to a flattening of the spectrum. The flattening of the spectrum allows for higher entanglement. We demonstrate increased entanglement in terms of BG modes by performing a Bell-type experiment and violating the appropriate Clauser Horne Shimony Holt (CHSH) inequality. In addition, we reconstruct the quantum state of BG modes entangled in high-dimensions. 12. Filamentation with nonlinear Bessel vortices. Jukna, V; Milián, C; Xie, C; Itina, T; Dudley, J; Courvoisier, F; Couairon, A 2014-10-20 We present a new type of ring-shaped filaments featured by stationary nonlinear high-order Bessel solutions to the laser beam propagation equation. Two different regimes are identified by direct numerical simulations of the nonlinear propagation of axicon focused Gaussian beams carrying helicity in a Kerr medium with multiphoton absorption: the stable nonlinear propagation regime corresponds to a slow beam reshaping into one of the stationary nonlinear high-order Bessel solutions, called nonlinear Bessel vortices. The region of existence of nonlinear Bessel vortices is found semi-analytically. The influence of the Kerr nonlinearity and nonlinear losses on the beam shape is presented. Direct numerical simulations highlight the role of attractors played by nonlinear Bessel vortices in the stable propagation regime. Large input powers or small cone angles lead to the unstable propagation regime where nonlinear Bessel vortices break up into an helical multiple filament pattern or a more irregular structure. Nonlinear Bessel vortices are shown to be sufficiently intense to generate a ring-shaped filamentary ionized channel in the medium which is foreseen as opening the way to novel applications in laser material processing of transparent dielectrics. PMID:25401574 13. Single Bessel tractor-beam tweezers Mitri, F G 2014-01-01 The tractor behavior of a zero-order Bessel acoustic beam acting on a fluid sphere, and emanating from a finite circular aperture (as opposed to waves of infinite extent) is demonstrated theoretically. Conditions for an attractive force acting in opposite direction of the radiating waves, determined by the choice of the beam's half-cone angle, the size of the radiator, and its distance from a fluid sphere, are established and discussed. Numerical predictions for the radiation force function, which is the radiation force per unit energy density and cross-sectional surface, are provided using a partial-wave expansion method stemming from the acoustic scattering. The results suggest a simple and reliable analysis for the design of Bessel beam acoustical tweezers and tractor beam devices. 14. The probability distributions of the first hitting times of Bessel processes Hamana, Yuji; Matsumoto, Hiroyuki 2011-01-01 We consider the first hitting times of the Bessel processes. We give explicit expressions for the distribution functions and for the densities by means of the zeros of the Bessel functions. The results extend the classical ones and cover all the cases. 15. Propagation and self-healing ability of a Bessel-Gaussian beam modulated by Bessel gratings Qiao, Chunhong; Feng, Xiaoxing; Chu, Xiuxiang 2016-04-01 A new type of Bessel-like beam which can be generated by using Bessel gratings to modulate the amplitude and phase of a Bessel beam is proposed. In analogy to study a Bessel beam in free space, the intensity evolution and self-healing property of the Bessel-like beam have been studied. Meanwhile, based on the Fresnel diffraction integral, the propagation of the Bessel-like beam in free space has also been investigated. Results show that the Bessel-like beam and the Bessel-Gaussian-like beams have some special and interesting properties. 16. Depth-controlled Bessel beams Müller, Angelina; Wallrabe, Ulrike 2016-01-01 We present a ring aperture with independently switchable segments for the three-dimensional control of quasi propagation invariant beams. We demonstrate that our liquid crystal design concept preserves coherence and generates the Bessel beam structure. 17. Harmonic Analysis Associated with the Generalized q-Bessel Operator Ahmed Abouelaz; Radouan Daher; El Mehdi Loualid 2016-01-01 In this article, we give a new harmonic analysis associated with the generalized q-Bessel operator. We introduce the generalized$q$-Bessel transform, the generalized q-Bessel translation and the generalized$q$-Bessel convolution product. 18. Review of nondiffracting Bessel beams Lapointe, Michael R. 1991-01-01 The theory of nondiffracting beam propagation and experimental evidence for nearly-nondiffractive Bessel beam propagation are reviewed. The experimental results are reinterpreted using simple optics formulas, which show that the observed propagation distances are characteristic of the optical systems used to generate the beams and do not depend upon the initial beam profiles. A set of simple experiments are described which support this interpretation. It is concluded that nondiffracting Bessel beam propagation has not yet been experimentally demonstrated. 19. Heat kernel analysis for Bessel operators on symmetric cones Möllers, Jan 2014-01-01 heat kernel is explicitly given in terms of a multivariable$I$-Bessel function on$Ω$. Its corresponding heat kernel transform defines a continuous linear operator between$L^p$-spaces. The unitary image of the$L^2$-space under the heat kernel transform is characterized as a weighted Bergmann space...... on the complexification$G_{\\mathbb C}/K_{\\mathbb C}$of$Ω$, the weight being expressed explicitly in terms of a multivariable$K$-Bessel function on$Ω$. Even in the special case of the symmetric cone$Ω=\\mathbb{R}_+$these results seem to be new.... 20. Studies of Transverse Momentum Dependent Parton Distributions and Bessel Weighting Aghasyan, M; De Sanctis, E; Gamberg, L; Mirazita, M; Musch, B; Prokudin, A; Rossi, P 2014-01-01 In this paper we present a new technique for analysis of transverse momentum dependent parton distribution functions, based on the Bessel weighting formalism. The procedure is applied to studies of the double longitudinal spin asymmetry in semi-inclusive deep inelastic scattering using a new dedicated Monte Carlo generator which includes quark intrinsic transverse momentum within the generalized parton model. Using a fully differential cross section for the process, the effect of four momentum conservation is analyzed using various input models for transverse momentum distributions and fragmentation functions. We observe a few percent systematic offset of the Bessel-weighted asymmetry obtained from Monte Carlo extraction compared to input model calculations, which is due to the limitations imposed by the energy and momentum conservation at the given energy/Q2. We find that the Bessel weighting technique provides a powerful and reliable tool to study the Fourier transform of TMDs with controlled systematics du... 1. Selfhealing of asymmetric Bessel-like modes Israelsen, Stine Møller; Rishøj, Lars Søgaard; Rottwitt, Karsten 2014-01-01 We numerically investigate asymmetric Bessel-like modes in an aircladding fiber. The selfhealing ability of asymmetric Bessel-like modes is demonstrated and quantified including the angular dependency of this ability.......We numerically investigate asymmetric Bessel-like modes in an aircladding fiber. The selfhealing ability of asymmetric Bessel-like modes is demonstrated and quantified including the angular dependency of this ability.... 2. Expansion of Bessel and g -Bessel sequences to dual frames and dual g -frames Golsa Kavian; Mohammad Sadegh Asgari 2014-01-01 In this paper we study the duality of Bessel and$g-Bessel sequences‎ ‎in Hilbert spaces‎. ‎We show that a Bessel sequence is an inner summand of a frame and‎ ‎the sum of any Bessel sequence with Bessel bound less than one with a Parseval frame is‎ ‎a frame‎. ‎Next we develop this results to the g-frame situation. 3. Maximal inequalities for bessel processes Graversen SE 1998-01-01 Full Text Available It is proved that the uniform law of large numbers (over a random parameter set for the -dimensional ( Bessel process started at 0 is valid: for all stopping times for . The rate obtained (on the right-hand side is shown to be the best possible. The following inequality is gained as a consequence: for all stopping times for , where the constant satisfies as . This answers a question raised in [4]. The method of proof relies upon representing the Bessel process as a time changed geometric Brownian motion. The main emphasis of the paper is on the method of proof and on the simplicity of solution. 4. Ablation of metal thin films using femtosecond laser Bessel vortex beams Sahin, Ramazan; Ersoy, Tansu; Akturk, Selcuk 2015-01-01 Femtosecond lasers can provide submicron ablation resolution, making them suitable and attractive for various micro/nanofabrication applications. Laser beam shaping lends further advantages and increases the versatility of these sources. In this work, we report on the use of femtosecond laser pulses with first-order Bessel function (Bessel vortex) beam profiles in ablation of metal thin films. The diffraction-free nature of Bessel beams provides significant convenience regarding alignment and repeatability. Ablation profiles with Bessel vortex beams generally consist of single or multiple concentric rings, determined by pulse fluence on target. We investigate single-pulse ablation behavior with two laser wavelengths (1,030 and 515 nm) and three different Bessel beam cone angles. For each case, we measure inner and outer ring diameters and compare our results with theoretical calculations. 5. Transformation of phase dislocations under acousto-optic interaction of optical and acoustical Bessel beams Belyi, V. N.; Khilo, P. A.; Kazak, N. S.; Khilo, N. A. 2016-07-01 The generation of wavefront phase dislocations of vortex Bessel light beams under acousto-optic (AO) diffraction in uniaxial crystals has been investigated. For the first time the process of AO interaction is studied with participation of Bessel acoustic beams instead of plane waves. A mathematical description of AO interaction is provided, which supposes the satisfaction of two types of phase-matching condition. The acousto-optic processes of transferring optical singularities onto the wavefront of BLBs are investigated and the generation of high-order optical vortices is considered at the interaction of optical and acoustical Bessel beams. The change of Bessel function order or phase dislocation order is explained as a result of the spin–orbital interaction under acousto-optic diffraction of vortex Bessel beams. 6. Generalized weighted Besov spaces on the Bessel hypergroup Miloud Assal 2006-01-01 Full Text Available In this paper we study generalized weighted Besov type spaces on the Bessel-Kingman hypergroup. We give different characterizations of these spaces in terms of generalized convolution with a kind of smooth functions and by means of generalized translation operators. Also a discrete norm is given to obtain more general properties on these spaces. 7. On the product and ratio of Bessel random variables Saralees Nadarajah 2005-01-01 Full Text Available The distributions of products and ratios of random variables are of interest in many areas of the sciences. In this paper, the exact distributions of the product |XY| and the ratio |X/Y| are derived when X and Y are independent Bessel function random variables. An application of the results is provided by tabulating the associated percentage points. 8. Sobre la inversión de los potenciales de Bessel-Riesz Sobre la inversión de los potenciales de Bessel-Riesz Ruben Alejandro Cerutti 2010-02-01 Full Text Available En este trabajo se obtiene la inversión de un operador del tipo convolución usando técnicas de integrales hipersingulares. El operador de Bessel-Riesz de una función ϕ perteneciente a S , el espacio de funciones de prueba de Schwartz, es definido por la convolución con las funciones generalizadas (fórmula expresables en términos de la función de Bessel de primera especie (formula es también una combinación lineal infinita del núcleo ultrahiperbólico de Riesz de diferentes ordenes. Este hecho nos permite invertir los potenciales de Bessel-Riesz de un modo análogo a lo αhecho en el caso de los potenciales ultrahiperbólicos de Bessel (cf. [01] y los potenciales causales de Riesz (cf. [2].In this paper the inversion of a convolution type operator is obtained by using hypersingular integral technics. The Bessel-Riesz operator of a function ϕ belonging to S , the space of test functions of Schwartz, is definied by the convolution with the generalized functions (formula expressible in terms of the Bessel function of first kind (formula . γ is also an infinite linear combination of the ultrahyperbolic Riesz kernel of differents orders. This fact allows us to invert the Bessel-Riesz potential in an analogue manner of the ultrahyperbolic Bessel potentials (cf. [01] and causal Riesz potentials (cf. [2]. 9. A relativistic study of Bessel beams We present a fully relativistic analysis of Bessel beams revealing some noteworthy features that are not explicit in the standard description. It is shown that there is a reference frame in which the field takes a particularly simple form, the wave appearing to rotate in circles. The concepts of polarization and angular momentum for Bessel beams are also reanalysed 10. A relativistic study of Bessel beams Hacyan, S; Jauregui, R [Instituto de Fisica, Universidad Nacional Autonoma de Mexico, Apdo. Postal 20-364, Mexico DF 01000 (Mexico) 2006-04-14 We present a fully relativistic analysis of Bessel beams revealing some noteworthy features that are not explicit in the standard description. It is shown that there is a reference frame in which the field takes a particularly simple form, the wave appearing to rotate in circles. The concepts of polarization and angular momentum for Bessel beams are also reanalysed. 11. A relativistic study of Bessel beams Hacyan, S 2006-01-01 We present a fully relativistic analysis of Bessel beams revealing some noteworthy features that are not explicit in the standard description. It is shown that there is a reference frame in which the field takes a particularly simple form, the wave appearing to rotate in circles. The concepts of polarization and angular momentum for Bessel beams is also reanalyzed. 12. The Bessel polynomials and their differential operators Differential operators associated with the ordinary and the generalized Bessel polynomials are defined. In each case the commutator bracket is constructed and shows that the differential operators associated with the Bessel polynomials and their generalized form are not commutative. Some applications of these operators to linear differential equations are also discussed. (author). 4 refs 13. Superluminal Behaviour of Modified Bessel Waves WANG Zhi-Yong; XIONG Cai-Dong 2006-01-01 @@ Much experimental evidence of superluminal phenomena has been available by electromagnetic wave propagation experiments, with the results showing that the phase time describes the barrier traversal time. Based on the extrapolated phase time approach and numerical methods, we show that, in contrast to the ordinary Bessel waves of real argument, the group velocities of modified Bessel waves are superluminal. We obtain the following results. 14. Fourier-Bessel heat kernel estimates Malecki, Jacek; Serafin, Grzegorz; Zorawik, Tomasz 2015-01-01 We provide sharp two-sided estimates of the Fourier-Bessel heat kernel and we give sharp two-sided estimates of the transition probability density for the Bessel process in (0,1) killed at 1 and killed or reflected at 0. 15. On the Diamond Bessel Heat Kernel Wanchak Satsanit 2011-01-01 We study the heat equation in n dimensional by Diamond Bessel operator. We find the solution by method of convolution and Fourier transform in distribution theory and also obtain an interesting kernel related to the spectrum and the kernel which is called Bessel heat kernel. 16. On a novel iterative method to compute polynomial approximations to Bessel functions of the first kind and its connection to the solution of fractional diffusion/diffusion-wave problems Yuste, Santos Bravo; Abad, Enrique, E-mail: santos@unex.es [Departamento de Fisica, Universidad de Extremadura, E-06071 Badajoz (Spain) 2011-02-18 We present an iterative method to obtain approximations to Bessel functions of the first kind J{sub p}(x) (p > -1) via the repeated application of an integral operator to an initial seed function f{sub 0}(x). The class of seed functions f{sub 0}(x) leading to sets of increasingly accurate approximations f{sub n}(x) is considerably large and includes any polynomial. When the operator is applied once to a polynomial of degree s, it yields a polynomial of degree s + 2, and so the iteration of this operator generates sets of increasingly better polynomial approximations of increasing degree. We focus on the set of polynomial approximations generated from the seed function f{sub 0}(x) = 1. This set of polynomials is useful not only for the computation of J{sub p}(x) but also from a physical point of view, as it describes the long-time decay modes of certain fractional diffusion and diffusion-wave problems. 17. Harmonic Analysis Associated with the Generalized q-Bessel Operator Ahmed Abouelaz 2016-01-01 Full Text Available In this article, we give a new harmonic analysis associated with the generalized q-Bessel operator. We introduce the generalizedq$-Bessel transform, the generalized q-Bessel translation and the generalized$q$-Bessel convolution product. 18. Bessel multipliers in Hilbert$C^\\ast$--modules Khosravi, Amir; Mirzaee Azandaryani, Morteza 2015-01-01 In this paper we introduce Bessel multipliers, g-Bessel multipliers and Bessel fusion multipliers in Hilbert$C^\\ast$--modules and we show that they share many useful properties with their corresponding notions in Hilbert and Banach spaces. We show that various properties of multipliers are closely related to their symbols and Bessel sequences, especially we consider multipliers when their Bessel sequences are modular Riesz bases and we see that in this case multipliers can be ... 19. Detection of Bessel beams with digital axicons Trichili, Abderrahmen; Ismail, Yaseera; Roux, Filippus; McLaren, Melanie; Zghal, Mourad; Forbes, Andrew 2014-01-01 We propose a simple method for the detection of Bessel beams with arbitrary radial and azimuthal indices, and then demonstrate it in an all-digital setup with a spatial light modulator. We confirm that the fidelity of the detection method is very high, with modal cross-talk below 5%, even for high orbital angular momentum carrying fields with long propagation ranges. To illustrate the versatility of the approach we use it to observe the modal spectrum changes during the self-reconstruction process of Bessel beams after encountering an obstruction, as well as to characterize modal distortions of Bessel beams propagating through atmospheric turbulence. 20. Materials processing with superposed Bessel beams Yu, Xiaoming; Trallero-Herrero, Carlos A.; Lei, Shuting 2016-01-01 We report experimental results of femtosecond laser processing on the surface of glass and metal thin film using superposed Bessel beams. These beams are generated by a combination of a spatial light modulator (SLM) and an axicon with >50% efficiency, and they possess the long depth-of-focus (propagation-invariant) property as found in ordinary Bessel beams. Through micromachining experiments using femtosecond laser pulses, we show that multiple craters can be fabricated on glass with single-shot exposure, and the 1+(-1) superposed beam can reduce collateral damage caused by the rings in zero-order Bessel beams in the scribing of metal thin film. 1. Causal (anticausal Bessel derivative and the ultrahyperbolic Bessel operator Manuel Aguirre Téllez 1997-11-01 Full Text Available Let B^α_C and B^α_A be ultrahyperbolic Bessel operator causal (anticausal of the order α defined by B^α_C f = G_α ( P + i0, m, n ∗ f , B^α f = G_α ( P −i0, m, n ∗ f and let D^α_C and D^α_A be generalized causal (anticausal Bessel derivative of order α defined by D^α_C f = G_{−α} ( P − i0, m, n ∗ f , D^α_A f =G_{−α} ( P + i0, m, n ∗ f . In this note we give a sense to several relations of type: B^α_C (B^β_ A f + B^α_A (B^β_C f , D^ α_C (D^β_ A f + D^α_A (D^β_C f , . . . 2. Bessel bridges decomposition with varying dimension. Applications to finance Faraud, Gabriel 2012-01-01 We consider a class of stochastic processes containing the classical and well-studied class of Squared Bessel processes. Our model, however, allows the dimension be a function of the time. We first give some classical results in a larger context where a time-varying drift term can be added. Then in the non-drifted case we extend many results already proven in the case of classical Bessel processes to our context. Our deepest result is a decomposition of the Bridge process associated to this generalized squared Bessel process, much similar to the much celebrated result of J. Pitman and M. Yor. On a more practical point of view, we give a methodology to compute the Laplace transform of additive functionals of our process and the associated bridge. This permits in particular to get directly access to the joint distribution of the value at t of the process and its integral. We finally give some financial applications to illustrate the panel of applications of our results. 3. Catenary nanostructures as compact Bessel beam generators Li, Xiong; Pu, Mingbo; Zhao, Zeyu; Ma, Xiaoliang; Jin, Jinjin; Wang, Yanqin; Gao, Ping; Luo, Xiangang 2016-02-01 Non-diffracting Bessel beams, including zero-order and high-order Bessel Beams which carry orbital angular momentum (OAM), enable a variety of important applications in optical micromanipulation, sub-diffraction imaging, high speed photonics/quantum communication, etc. The commonly used ways to create Bessel beams, including an axicon or a digital hologram written to a spatial light modulator (SLM), have great challenges to operate at the nanoscale. Here we theoretically design and experimentally demonstrate one kind of planar Bessel beam generators based on metasurfaces with analytical structures perforated in ultra-thin metallic screens. Continuous phase modulation between 0 to 2π is realized with a single element. In addition, due to the dispersionless phase shift stemming from spin-orbit interaction, the proposed device can work in a wide wavelength range. The results may find applications in future optical communication, nanofabrication and super-resolution imaging, etc. 4. Catenary nanostructures as compact Bessel beam generators. Li, Xiong; Pu, Mingbo; Zhao, Zeyu; Ma, Xiaoliang; Jin, Jinjin; Wang, Yanqin; Gao, Ping; Luo, Xiangang 2016-01-01 Non-diffracting Bessel beams, including zero-order and high-order Bessel Beams which carry orbital angular momentum (OAM), enable a variety of important applications in optical micromanipulation, sub-diffraction imaging, high speed photonics/quantum communication, etc. The commonly used ways to create Bessel beams, including an axicon or a digital hologram written to a spatial light modulator (SLM), have great challenges to operate at the nanoscale. Here we theoretically design and experimentally demonstrate one kind of planar Bessel beam generators based on metasurfaces with analytical structures perforated in ultra-thin metallic screens. Continuous phase modulation between 0 to 2π is realized with a single element. In addition, due to the dispersionless phase shift stemming from spin-orbit interaction, the proposed device can work in a wide wavelength range. The results may find applications in future optical communication, nanofabrication and super-resolution imaging, etc. PMID:26843142 5. Bessel's Differential Equation and Its Hyers-Ulam Stability Kim Byungbae 2007-01-01 Full Text Available We solve the inhomogeneous Bessel differential equation and apply this result to obtain a partial solution to the Hyers-Ulam stability problem for the Bessel differential equation. 6. Bessel's Differential Equation and Its Hyers-Ulam Stability Soon-Mo Jung 2007-12-01 Full Text Available We solve the inhomogeneous Bessel differential equation and apply this result to obtain a partial solution to the Hyers-Ulam stability problem for the Bessel differential equation. 7. Propagation of Bessel beam for ground-to-space applications Ituen, Iniabasi; Birch, Philip; Young, Rupert; Chatwin, Chris 2015-01-01 We model the propagation of Gaussian and Bessel beams from ground through 22km altitude of atmospheric turbulence. We observe the Bessel beam has better performance based on RMS intensity error and the captured beam power 8. On Relations between Bessel Potential Spaces and Riesz Potential Spaces Kurokawa, Takahide 2000-01-01 We present a relation between the Bessel potential spaces and the Riesz potential spaces. The ideas of the proof are to characterize each potential spaces and to give a correspondence between individual Bessel potentials and Riesz potentials. 9. Identities involving Bessel polynomials arising from linear differential equations Kim, Taekyun; Kim, Dae San 2016-01-01 In this paper, we study linear di?erential equations arising from Bessel polynomials and their applications. From these linear differential equations, we give some new and explicit identities for Bessel polynomials. 10. Multipliers of pg-Bessel sequences in Banach spaces Abdollahpour, M. R.; Najati, A.; Gavruta, P. 2015-01-01 In this paper, we introduce (p,q)g-Bessel multipliers in Banach spaces and we show that under some conditions a (p,q)g-Bessel multiplier is invertible. Also, we show the continuous dependency of (p,q)g-Bessel multipliers on their parameters. 11. A Determinant Expression for the Generalized Bessel Polynomials Sheng-liang Yang; Sai-nan Zheng 2013-01-01 Using the exponential Riordan arrays, we show that a variation of the generalized Bessel polynomial sequence is of Sheffer type, and we obtain a determinant formula for the generalized Bessel polynomials. As a result, the Bessel polynomial is represented as determinant the entries of which involve Catalan numbers. 12. Bessel beams with spatial oscillating polarization Fu, Shiyao; Zhang, Shikun; Gao, Chunqing 2016-08-01 Bessel beams are widely used in optical metrology mainly because of their large Rayleigh range (focal length). Radial/azimuthal polarization of such beams is of interest in the fields of material processing, plasma absorption or communication. In this paper an experimental set-up is presented, which generates a Bessel-type vector beam with a spatial polarization, oscillating along the optical axis, when propagating in free space. A first holographic axicon (HA) HA1 produces a normal, linearly polarized Bessel beam, which by a second HA2 is converted into the spatial oscillating polarized beam. The theory is briefly discussed, the set-up and the experimental results are presented in detail. 13. Bessel beams with spatial oscillating polarization. Fu, Shiyao; Zhang, Shikun; Gao, Chunqing 2016-01-01 Bessel beams are widely used in optical metrology mainly because of their large Rayleigh range (focal length). Radial/azimuthal polarization of such beams is of interest in the fields of material processing, plasma absorption or communication. In this paper an experimental set-up is presented, which generates a Bessel-type vector beam with a spatial polarization, oscillating along the optical axis, when propagating in free space. A first holographic axicon (HA) HA1 produces a normal, linearly polarized Bessel beam, which by a second HA2 is converted into the spatial oscillating polarized beam. The theory is briefly discussed, the set-up and the experimental results are presented in detail. PMID:27488174 14. Microwave bessel beams generation using guided modes Salem, Mohamed 2011-06-01 A novel method is devised for Bessel beams generation in the microwave regime. The beam is decomposed in terms of a number of guided transverse electric modes of a metallic waveguide. Modal expansion coefficients are computed from the modal power orthogonality relation. Excitation is achieved by means of a number of inserted coaxial loop antennas, whose currents are calculated from the excitation coefficients of the guided modes. The efficiency of the method is evaluated and its feasibility is discussed. Obtained results can be utilized to practically realize microwave Bessel beam launchers. © 2006 IEEE. 15. Quantum-mechanical properties of Bessel beams Bessel beams are studied within the general framework of quantum optics. The two modes of the electromagnetic field are quantized and the basic dynamical operators are identified. As we show explicitly, the operators that are usually associated with linear momentum, orbital angular momentum, and spin do not satisfy the algebra of the translation and rotation group. Nevertheless, we identify some components of these operators that represent observable quantities in an appropriate basis, thus characterizing the quantum numbers of Bessel photons. Some physical consequences of these results are discussed 16. The Logvinenko-Sereda Theorem for the Fourier-Bessel transform Ghobber, Saifallah 2012-01-01 The aim of this paper is to establish an analogue of Logvinenko-Sereda's theorem for the Fourier-Bessel transform (or Hankel transform)$\\ff_\\alpha$of order$\\alpha>-1/2$. Roughly speaking, if we denote by$PW_\\alpha(b)$the Paley-Wiener space of$L^2$-functions with Fourier-Bessel transform supported in$[0,b]$, then we show that the restriction map$f\\to f|_\\Omega$is essentially invertible on$PW_\\alpha(b)$if and only if$\\Omega$is sufficiently dense. Moreover, we give an estimate of the norm of the inverse map. As a side result we prove a Bernstein type inequality for the Fourier-Bessel transform. 17. Burkholder inequalities for submartingales, Bessel processes and conformal martingales Bañuelos, Rodrigo 2011-01-01 The motivation for this paper comes from the following question on comparison of norms of conformal martingales$X$,$Y$in$\\R^d$,$d\\geq 2$. Suppose that$Y$is differentially subordinate to$X$. For$01$is not an integer, which has further interesting applications to stopped Bessel processes and to the behavior of smooth functions on Euclidean domains. The inequality for conformal martingales, which has its roots on the study of the$L^p$norms of the Beurling-Ahlfors singular integral operator \\cite{BW}, extends a recent result of Borichev, Janakiraman and Volberg \\cite{BJV2}. 18. Elliptic integral evaluations of Bessel moments Bailey, David H.; Borwein, Jonathan M.; Broadhurst, David; Glasser, M.L. 2008-01-06 We record what is known about the closed forms for variousBessel function moments arising in quantum field theory, condensed mattertheory and other parts of mathematical physics. More generally, wedevelop formulae for integrals of products of six or fewer Besselfunctions. In consequence, we are able to discover and prove closed formsfor c(n,k) := Int_0 inf tk K_0 n(t) dt, with integers n = 1, 2, 3, 4 andk greater than or equal to 0, obtaining new results for the even momentsc3,2k and c4,2k . We also derive new closed forms for the odd momentss(n,2k+1) := Int_0 inf t(2k+1) I_0(t) K_0n(t) dt,with n = 3, 4 and fort(n,2k+1) := Int_0 inf t(2k+1) I_02(t) K_0(n-2) dt, with n = 5, relatingthe latter to Green functions on hexagonal, diamond and cubic lattices.We conjecture the values of s(5,2k+1), make substantial progress on theevaluation of c(5,2k+1), s(6,2k+1) and t(6,2k+1) and report more limitedprogress regarding c(5,2k), c(6,2k+1) and c(6,2k). In the process, weobtain 8 conjectural evaluations, each of which has been checked to 1200decimal places. One of these lies deep in 4-dimensional quantum fieldtheory and two are probably provable by delicate combinatorics. Thereremains a hard core of five conjectures whose proofs would be mostinstructive, to mathematicians and physicists alike. 19. 求解一类 Riccati - Bessel 方程边值问题的新方法%A New Method of Solving the Boundary Value Problem of a Class of Riccati - Bessel Equation 王强; 李顺初; 蒲俊 2015-01-01 This paper solved a boundary value problem of Riccati - Bessel equation;and the similar kernel function and similar structure of the solution were obtained. By further analysis and solving this class of boundary value problem,the guiding functions were firstly constructed by using two linearly independent solutions of Riccati- Bessel equation,and then the similar kernel function was assembled by the guiding functions and coefficient of right boundary value condition. The solution to the boundary value problem was assembled by similar kernel func-tion and coefficient of left boundary value condition. Therefore a new idea is put forward for solving this class of boundary value problem of Riccati - Bessel equation:similar structure.%针对 Riccati - Bessel 方程一类边值问题进行求解，获得了解式的相似核函数和相似结构，通过进一步分析，发现求解该类边值问题可先利用 Riccati - Bessel 方程的两个线性无关解构造引解函数，再结合右边值条件的系数组装得到相似核函数；通过相似核函数和左边值条件的系数组装就可以得到 Riccati - Bessel 方程边值问题的解，由此提出了解决该类 Riccati - Bessel 方程边值问题的一种新思路———相似构造。 20. On the$q$-Bessel Fourier transform Dhaouadi, Lazhar 2013-01-01 In this work, we are interested by the$q$-Bessel Fourier transform with a new approach. Many important results of this$q$-integral transform are proved with a new constructive demonstrations and we establish in particular the associated$q$-Fourier-Neumen expansion which involves the$q$-little Jacobi polynomials. 1. Bessel potential space on the Laguerre hypergroup Ahmed Taieb 2011-01-01 Full Text Available Abstract In this article, we define the fractional differentiation Dδ of order δ, δ > 0, induced by the Laguerre operator L and associated with respect to the Haar measure dmα. We obtain a characterization of the Bessel potential space using Dδ and different equivalent norms. 2. Point vortex equilibria related to Bessel polynomials O'Neil, Kevin A. 2016-05-01 The method of polynomials is used to construct two families of stationary point vortex configurations. The vortices are placed at the reciprocals of the zeroes of Bessel polynomials. Configurations that translate uniformly, and configurations that are completely stationary, are obtained in this way. 3. Programs for high-speed Fourier, Mellin and Fourier-Bessel transforms Ikhabisimov, D. K.; Debabov, A. S.; Kolosov, B. I.; Usikov, D. A. 1979-01-01 Several FORTRAN program modules for performing one-dimensional and two-dimensional discrete Fourier transforms, Mellin, and Fourier-Bessel transforms are described along with programs that realize the algebra of high speed Fourier transforms on a computer. The programs can perform numerical harmonic analysis of functions, synthesize complex optical filters on a computer, and model holographic image processing methods. 4. Bessel filters applied in biomedical image processing Mesa Lopez, Juan Pablo; Castañeda Saldarriaga, Diego Leon 2014-06-01 A magnetic resonance is an image obtained by means of an imaging test that uses magnets and radio waves to create body images, however, in some images it's difficult to recognize organs or foreign agents present in the body. With these Bessel filters the objective is to significantly increase the resolution of magnetic resonance images taken to make them much clearer in order to detect anomalies and diagnose the illness. As it's known, Bessel filters appear to solve the Schrödinger equation for a particle enclosed in a cylinder and affect the image distorting the colors and contours of it, therein lies the effectiveness of these filters, since the clear outline shows more defined and easy to recognize abnormalities inside the body. 5. Quantum Mechanical Properties of Bessel Beams Jauregui, R.; Hacyan, S. 2004-01-01 Bessel beams are studied within the general framework of quantum optics. The two modes of the electromagnetic field are quantized and the basic dynamical operators are identified. The algebra of these operators is analyzed in detail; it is shown that the operators that are usually associated to linear momentum, orbital angular momentum and spin do not satisfy the algebra of the translation and rotation group. In particular, what seems to be the spin is more similar to the helicity. Some physi... 6. Ladder operators and recursion relations for the associated Bessel polynomials Introducing the associated Bessel polynomials in terms of two non-negative integers, and under an integrability condition we simultaneously factorize their corresponding differential equation into a product of the ladder operators by four different ways as shape invariance symmetry equations. This procedure gives four different pairs of recursion relations on the associated Bessel polynomials. In spite of description of Bessel and Laguerre polynomials in terms of each other, we show that the associated Bessel differential equation is factorized in four different ways whereas for Laguerre one we have three different ways 7. Ladder operators and recursion relations for the associated Bessel polynomials Fakhri, H.; Chenaghlou, A. 2006-10-01 Introducing the associated Bessel polynomials in terms of two non-negative integers, and under an integrability condition we simultaneously factorize their corresponding differential equation into a product of the ladder operators by four different ways as shape invariance symmetry equations. This procedure gives four different pairs of recursion relations on the associated Bessel polynomials. In spite of description of Bessel and Laguerre polynomials in terms of each other, we show that the associated Bessel differential equation is factorized in four different ways whereas for Laguerre one we have three different ways. 8. Ladder operators and recursion relations for the associated Bessel polynomials Fakhri, H. [Institute for Studies in Theoretical Physics and Mathematics (IPM), PO Box 19395-5531, Tehran (Iran, Islamic Republic of) and Department of Theoretical Physics and Astrophysics, Physics Faculty, Tabriz University, PO Box 51666-16471, Tabriz (Iran, Islamic Republic of)]. E-mail: hfakhri@ipm.ir; Chenaghlou, A. [Institute for Studies in Theoretical Physics and Mathematics (IPM), PO Box 19395-5531, Tehran (Iran, Islamic Republic of) and Physics Department, Faculty of Science, Sahand University of Technology, PO Box 51335-1996, Tabriz (Iran, Islamic Republic of)]. E-mail: a.chenaghlou@sut.ac.ir 2006-10-30 Introducing the associated Bessel polynomials in terms of two non-negative integers, and under an integrability condition we simultaneously factorize their corresponding differential equation into a product of the ladder operators by four different ways as shape invariance symmetry equations. This procedure gives four different pairs of recursion relations on the associated Bessel polynomials. In spite of description of Bessel and Laguerre polynomials in terms of each other, we show that the associated Bessel differential equation is factorized in four different ways whereas for Laguerre one we have three different ways. 9. BESSEL FILTER AND CHAOS: THREE-IN-ONE ACTION Tamaševicius, Arunas; Mykolaitis, Gytis; Bumeliene, Skaidra; 2006-01-01 Low-pass active Bessel filters are proposed to be used in a chaotic oscillator. The Bessel unit plays the role of three-in-one: the delay line, the amplifier, and the filter. Results of Spice simulations and hardware experiments are presented.......Low-pass active Bessel filters are proposed to be used in a chaotic oscillator. The Bessel unit plays the role of three-in-one: the delay line, the amplifier, and the filter. Results of Spice simulations and hardware experiments are presented.... 10. Geometric Bessel models for GSp_4 and multiplicity one Lysenko, Sergey 2004-01-01 I this paper, which is a sequel to math.AG/0310361, we study Bessel models of representations of GSp_4 over a local non archimedian field in the framework of the geometric Langlands program. The Bessel module over the nonramified Hecke algebra of GSp_4 admits a geometric counterpart, the Bessel category of perverse sheaves on some ind-algebraic stack. We use it to prove a geometric version of the multiplicity one for Bessel models. It implies a geometric Casselman-Shalika type formula for the... 11. Acoustic scattering of a Bessel vortex beam by a rigid fixed spheroid Mitri, F. G. 2015-12-01 Partial-wave series representation of the acoustic scattering field of high-order Bessel vortex beams by rigid oblate and prolate spheroids using the modal matching method is developed. The method, which is applicable to slightly elongated objects at low-to-moderate frequencies, requires solving a system of linear equations which depends on the partial-wave index n and the order of the Bessel vortex beam m using truncated partial-wave series expansions (PWSEs), and satisfying the Neumann boundary condition for a rigid immovable surface in the least-squares sense. This original semi-analytical approach developed for Bessel vortex beams is demonstrated for finite oblate and prolate spheroids, where the mathematical functions describing the spheroidal geometry are written in a form involving single angular (polar) integrals that are numerically computed. The transverse (θ = π / 2) and 3D scattering directivity patterns are evaluated in the far-field for both prolate and oblate spheroids, with particular emphasis on the aspect ratio (i.e., the ratio of the major axis over the minor axis of the spheroid) not exceeding 3:1, the half-cone angle β and order m of the Bessel vortex beam, as well as the dimensionless size parameter kr0. Periodic oscillations in the magnitude plots of the far-field scattering form function are observed, which result from the interference of the reflected waves with the circumferential (Franz') waves circumnavigating the surface of the spheroid in the surrounding fluid. Moreover, the 3D directivity patterns illustrate the far-field scattering from the spheroid, that vanishes in the forward (θ = 0) and backward (θ = π) directions. Particular applications in underwater acoustics and scattering, acoustic levitation and the detection of submerged elongated objects using Bessel vortex waves to name a few, would benefit from the results of the present investigation. 12. Optical trapping with Bessel beams generated from semiconductor lasers In this paper, we study generation of Bessel beams from semiconductor lasers with high beam propagation parameter M2 and their utilization for optical trapping and manipulation of microscopic particles including living cells. The demonstrated optical tweezing with diodegenerated Bessel beams paves the way to replace their vibronic-generated counterparts for a range of applications towards novel lab-on-a-chip configurations 13. Higher-order chaotic oscillator using active bessel filter Lindberg, Erik; Mykolaitis, Gytis; Bumelien, Skaidra; 2010-01-01 A higher-order oscillator, including a nonlinear unit and an 8th-order low-pass active Bessel filter is described. The Bessel unit plays the role of "three-in-one": a delay line, an amplifier and a filter. Results of hardware experiments and numerical simulation are presented. Depending... 14. Bessel type inequalities in Hilbert C*-modules S. S. Dragomir; M. Khosravi; Moslehian, M. S. 2009-01-01 Regarding the generalizations of the Bessel inequality in Hilbert spaces which are due to Bombiari and Boas--Bellman, we obtain a version of the Bessel inequality and some generalizations of this inequality in the framework of Hilbert$C^*$-modules. 15. Orthogonal fast spherical Bessel transform on uniform grid Serov, Vladislav V 2015-01-01 We propose an algorithm for the orthogonal fast discrete spherical Bessel transform on an uniform grid. Our approach is based upon the spherical Bessel transform factorization into the two subsequent orthogonal transforms, namely the fast Fourier transform and the orthogonal transform founded on the derivatives of the discrete Legendre orthogonal polynomials. 16. Efficient Generation of Truncated Bessel Beams using Cylindrical Waveguides Ilchenko, Vladimir S.; Mohageg, Makan; Savchenkov, Anatoliy A.; Matsko, Andrey B.; Maleki, Lute 2007-01-01 In this paper we address efficient conversion between a Gaussian beam (a truncated plane wave) and a truncated Bessel beam of agiven order, using cylindrical optical waveguides and whispering gallery mode resonators. Utilizing a generator based on waveguides combined with whispering gallery mode resonators, we have realized Bessel beams of the order of 200 with a conversion efficiency exceeding 10 %. 17. Electromagnetic modified Bessel-Gauss beams and waves. Seshadri, S R 2008-01-01 The transverse magnetic (TM) modified Bessel-Gauss beams and their full-wave generalizations are treated. Attention is paid to the spreading properties on propagation of the null in the radiation intensity pattern for the azimuthal mode numbers m=0 and 1. The rate of spreading of the null in the propagation direction is significantly less for the TM modified Bessel-Gauss waves than those for the corresponding TM Bessel-Gauss waves. The total power transported by the waves is determined and compared with that of the corresponding paraxial beam to estimate the quality of the paraxial beam approximation of the wave. The dependence of the quality of the paraxial beam approximation on the azimuthal mode number, the beam shape parameter, and the ratio of the beam waist to the wavelength has a regular pattern for the TM Bessel-Gauss wave and not for the TM modified Bessel-Gauss wave. PMID:18157205 18. Nanolithography using Bessel Beams of Extreme Ultraviolet Wavelength. Fan, Daniel; Wang, Li; Ekinci, Yasin 2016-01-01 Bessel beams are nondiffracting light beams with large depth-of-focus and self-healing properties, making them suitable as a serial beam writing tool over surfaces with arbitrary topography. This property breaks the inherent resolution vs. depth-of-focus tradeoff of photolithography. One approach for their formation is to use circularly symmetric diffraction gratings. Such a ring grating was designed and fabricated for the extreme ultraviolet (EUV) wavelength of 13.5 nm, a candidate wavelength for future industrial lithography. Exposure of the aerial images showed that a Bessel beam with an approximately 1 mm long z-invariant central core of 223 nm diameter had been achieved, in good agreement with theory. Arbitrary patterns were written using the Bessel spot, demonstrating possible future application of Bessel beams for serial beam writing. Lithographic marks of ~30 nm size were also observed using a high resolution Bessel beam. PMID:27501749 19. Generation and application of bessel beams in electron microscopy. Grillo, Vincenzo; Harris, Jérémie; Gazzadi, Gian Carlo; Balboni, Roberto; Mafakheri, Erfan; Dennis, Mark R; Frabboni, Stefano; Boyd, Robert W; Karimi, Ebrahim 2016-07-01 We report a systematic treatment of the holographic generation of electron Bessel beams, with a view to applications in electron microscopy. We describe in detail the theory underlying hologram patterning, as well as the actual electron-optical configuration used experimentally. We show that by optimizing our nanofabrication recipe, electron Bessel beams can be generated with relative efficiencies reaching 37±3%. We also demonstrate by tuning various hologram parameters that electron Bessel beams can be produced with many visible rings, making them ideal for interferometric applications, or in more highly localized forms with fewer rings, more suitable for imaging. We describe the settings required to tune beam localization in this way, and explore beam and hologram configurations that allow the convergences and topological charges of electron Bessel beams to be controlled. We also characterize the phase structure of the Bessel beams generated with our technique, using a simulation procedure that accounts for imperfections in the hologram manufacturing process. PMID:27203186 20. Optical forces in higher order Bessel beam Šiler, Martin; Jákl, Petr; Chvátal, Lukáš; Zemánek, Pavel Bellingham : SPIE, 2012, 86970R:1-6. ISBN 978-0-8194-9481-8. [CPS 2012. Czech-Polish-Slovak Optical Conference on Wave and Quantum Aspects of Contemporary Optics /18./. Ostravice (CZ), 03.09.2012-07.09.2012] R&D Projects: GA MŠk LH12018; GA MŠk ED0017/01/01; GA ČR GPP205/12/P868 Institutional support: RVO:68081731 Keywords : Bessel beam * optical vortex * optical forces * generalized Lorenz-Mie theory * size-effect Subject RIV: BH - Optics, Masers, Lasers 1. Time-reversal and the Bessel equation Alfinito, Eleonora; Vitiello, Giuseppe 2015-07-01 The system of two damped/amplified oscillator equations is of widespread interest in the study of many physical problems and phenomena, from inflationary models of the Universe to thermal field theories, in condensed matter physics as well in high energy physics, and also in neuroscience. In this report we review the equivalence, in a suitable parametrization, between such a system of equations and the Bessel equations. In this connection, we discuss the breakdown of loop-antiloop symmetry, its relation with time-reversal symmetry and the mechanism of group contraction. Euclidean algebras such as e(2) and e(3) are also discussed in relation with Virasoro-like algebra. 2. Quantum Mechanical Properties of Bessel Beams Jauregui, R 2004-01-01 Bessel beams are studied within the general framework of quantum optics. The two modes of the electromagnetic field are quantized and the basic dynamical operators are identified. The algebra of these operators is analyzed in detail; it is shown that the operators that are usually associated to linear momentum, orbital angular momentum and spin do not satisfy the algebra of the translation and rotation group. In particular, what seems to be the spin is more similar to the helicity. Some physical consequences of these results are examined. 3. Microparticle trapping in an ultrasonic Bessel beam Choe, Youngki; Kim, Jonathan W; Shung, K. Kirk; Kim, Eun Sok 2011-01-01 This paper describes an acoustic trap consisting of a multi-foci Fresnel lens on 127 μm thick lead zirconate titanate sheet. The multi-foci Fresnel lens was designed to have similar working mechanism to an Axicon lens and generates an acoustic Bessel beam, and has negative axial radiation force capable of trapping one or more microparticle(s). The fabricated acoustic tweezers trapped lipid particles ranging in diameter from 50 to 200 μm and microspheres ranging in diameter from 70 to 90 μm at... 4. Second-harmonic generation with Bessel beams Shatrovoy, Oleg We present the results of a numerical simulation tool for modeling the second-harmonic generation (SHG) interaction experienced by a diffracting beam. This code is used to study the simultaneous frequency and spatial profile conversion of a truncated Bessel beam that closely resembles a higher-order mode (HOM) of an optical fiber. SHG with Bessel beams has been investigated in the past and was determined have limited value because it is less efficient than SHG with a Gaussian beam in the undepleted pump regime. This thesis considers, for the first time to the best of our knowledge, whether most of the power from a Bessel-like beam could be converted into a second-harmonic beam (full depletion), as is the case with a Gaussian beam. We study this problem because using HOMs for fiber lasers and amplifiers allows reduced optical intensities, which mitigates nonlinearities, and is one possible way to increase the available output powers of fiber laser systems. The chief disadvantage of using HOM fiber amplifiers is the spatial profile of the output, but this can be transformed as part of the SHG interaction, most notably to a quasi-Gaussian profile when the phase mismatch meets the noncollinear criteria. We predict, based on numerical simulation, that noncollinear SHG (NC-SHG) can simultaneously perform highly efficient (90%) wavelength conversion from 1064 nm to 532 nm, as well as concurrent mode transformation from a truncated Bessel beam to a Gaussian-like beam (94% overlap with a Gaussian) at modest input powers (250 W, peak power or continuous-wave operation). These simulated results reveal two attractive features -- the feasibility of efficiently converting HOMs of fibers into Gaussian-like beams, and the ability to simultaneously perform frequency conversion. Combining the high powers that are possible with HOM fiber amplifiers with access to non-traditional wavelengths may offer significant advantages over the state of the art for many important applications 5. 无衍射Bessel及Bessel-Gauss光的理论与实验%The Theory and Experiment of Diffraction-Free Bessel and Bessel-Gauss Beams 邱振兴; 吴逢铁; 郭东栋 2008-01-01 理论分析Bessel光束和Bessel-Gauss光束的相互联系及区别,数值模拟理想Bessel光束和Bessel-Gauss光束在任意平面的径向光强分布,以及光腰半径和谐振腔腔长对输出Bessel-Gauss光束的影响.模拟结果表明,相对于Bessel光,Bessel-Gauss光衰减较快,实验结果也验证了理论分析.同时,光腰半径越大,光强的径向分布衰减越慢;而腔长越大,其主模式的振幅分布衰减越慢. 6. Bessel Series in the Space H1(D)%H1(D)空间的Bessel级数 木乐华 2001-01-01 An identity concerning the partial sums of Bessel series and power series for H1(D) functions is given.Based on it,many of precise extimates about the deviation of the partial sums of Bessel series can be obtained.%本文给出关于H1(D)空间中函数的Bessel级数的部分和用幂级数的部分和表示的一个恒等式.基于它，可以得到Bessel级数部分和偏差的诸多精确估计. 7. Multifunctional diffractive optical elements for the generation of higher order Bessel-like-beams Vijayakumar, A.; Bhattacharya, Shanti 2015-01-01 Higher Order Bessel Beams (HOBBs) have many useful applications in optical trapping experiments. The generation of HOBBs is achieved by illuminating an axicon by a Laguerre-Gaussian beam generated by a spiral phase plate. It can also be generated by a Holographic Optical Element (HOE) containing the functions of the Spiral Phase Plate (SPP) and an axicon. However the HOBB's large focal depth reduces the intensity at each plane. In this paper, we propose a multifunctional Diffractive Optical Element (DOE) containing the functions of a SPP, axicon and a Fresnel Zone Lens (FZL) to generate higher efficiency higher order Bessel-like-beams with a reduced focal depth. The functions of a SPP and a FZL were combined by shifting the location of zones of FZL in a spiral fashion. The resulting element is combined with an axicon by modulo-2π phase addition technique. The final composite element contains the functions of SPP, FZL and axicon. The elements were designed with different topological charges and fabricated using electron beam direct writing. The elements were tested and the generation of a higher order Bessel-like-beams is confirmed. Besides, the elements also generated high quality donut beams at two planes equidistant from the focal plane of the FZL. 8. Confined nanoparticle measurement using Bessel Beam Microscopy Chakraborty, Chumki; Snoeyink, Craig 2015-11-01 With the advent of Lab-on-chip technologies, study of near surface phenomenon has gained a lot of importance due to their huge impact on bulk fluid properties. Such studies demand imaging techniques with utmost precision to capture the intricate details of the interface. But, resolution for most of the optical imaging systems is limited due to the light spreading effects of diffraction. This diffraction limited resolution, can be improved by the use of Bessel Beam microscopy. Bessel beam imaging technique when combined with a TIRF (Total Internal Reflection Fluorescence) system can be used for high resolution particle tracking experiments, to reveal detailed information about near surface particle positions and motions with their velocity profile and distribution. With the experimental set up combining these two powerful tools, we plan to present our particle tracking velocimetry results in the interface regime of confined nanoparticles in a binary fluid mixture. Such a study can contribute towards a better understanding of near surface fluid-particle interfaces. 9. Bessel and Grüss Type Inequalities in Inner Product Modules over Banach -Algebras Dragomir SS 2011-01-01 Full Text Available We give an analogue of the Bessel inequality and we state a simple formulation of the Grüss type inequality in inner product -modules, which is a refinement of it. We obtain some further generalization of the Grüss type inequalities in inner product modules over proper -algebras and unital Banach -algebras for -seminorms and positive linear functionals. 10. Behavior of obliquely incident vector Bessel beams at planar interfaces Salem, Mohamed 2013-01-01 We investigate the behavior of full-vector electromagnetic Bessel beams obliquely incident at an interface between two electrically different media. We employ a Fourier transform domain representation of Bessel beams to determine their behavior upon reflection and transmission. This transform, which is geometric in nature, consists of elliptical support curves with complex weighting associated with them. The behavior of the scattered field at an interface is highly complex, owing to its full-vector nature; nevertheless, this behavior has a straightforward representation in the transform domain geometry. The analysis shows that the reflected field forms a different vector Bessel beam, but in general, the transmitted field cannot be represented as a Bessel beam. Nevertheless, using this approach, we demonstrate a method to propagate a Bessel beam in the refractive medium by launching a non- Bessel beam at the interface. Several interesting phenomena related to the behavior of Bessel beams are illustrated, such as polarized reflection at Brewster\\'s angle incidence, and the Goos-Hänchen and Imbert-Federov shifts in the case of total reflection. © 2013 Optical Society of America. 11. Generation of spatial Bessel beams using holographic metasurface. Cai, Ben Geng; Li, Yun Bo; Jiang, Wei Xiang; Cheng, Qiang; Cui, Tie Jun 2015-03-23 We propose to use backward radiations of leaky waves supported by a holographic metasurface to produce spatial Bessel beams in the microwave frequency regime. The holographic metasurface consists of a grounded dielectric slab and a series of metal patches. By changing the size of metal patches, the surface-impedance distribution of the holographic metasurface can be modulated, and hence the radiation properties of the leaky waves can be designed to realize Bessel beams. Both numerical simulations and experiments verify the features of spatial Bessel beams, which may be useful in imaging applications or wireless power transmissions with the dynamic focal-depth controls. PMID:25837097 12. Jacobi-Bessel analysis of reflector antennas with elliptical apertures Rahmat-Samii, Yahya 1987-01-01 Although many reflector antennas possess circular projected apertures, there are recent satellite and ground antenna applications for which it is desirable to employ reflectors with elliptical apertures. Here a modification of the Jacobi-Bessel expansion is presented for the diffraction analysis of reflectors with elliptical apertures. A comparative study is also performed between this modified Jacobi-Bessel algorithm and the one which uses the Jacobi-Bessel expansion over a circumscribing circular region. Numerical results are presented for offset reflectors with elliptical and circular apertures and the improved convergence properties of the modified algorithm are highlighted. 13. Surface multipole solitons on photorefractive media with Bessel optical lattices Hong, Woo-Pyo 2015-03-01 We find the existence conditions for new surface crescent, dipole, tripole, and quadrupole solitons formed at the interface of a focusing photorefractive medium and a medium imprinted with a Bessel optical lattice. We demonstrate by using numerical simulations that the crescent and the dipole solitons show oscillatory behaviors in their amplitude and shape while the tripole and the quadrupole solitons maintain a remarkable rigidity during propagation. Based on a linear stability analysis, we classify the stability region of the tripole and the quadrupole surface solitons in terms of the Bessel optical lattice strength and the Bessel index. 14. Axisymmetric scattering of an acoustical Bessel beam by a rigid fixed spheroid Mitri, F G 2015-01-01 Based on the partial-wave series expansion (PWSE) method in spherical coordinates, a formal analytical solution for the acoustic scattering of a zeroth-order Bessel acoustic beam centered on a rigid fixed (oblate or prolate) spheroid is provided. The unknown scattering coefficients of the spheroid are determined by solving a system of linear equations derived for the Neumann boundary condition. Numerical results for the modulus of the backscattered pressure (\\theta = \\pi) in the near-field and the backscattering form function in the far-field for both prolate and oblate spheroids are presented and discussed, with particular emphasis on the aspect ratio (i.e., the ratio of the major axis over the minor axis of the spheroid), the half-cone angle of the Bessel beam \\beta, and the dimensionless frequency. The plots display periodic oscillations (versus the dimensionless frequency) due to the interference of specularly reflected waves in the backscattering direction with circumferential Franz' waves circumnavigati... 15. Millijoule femtosecond micro-Bessel beams for ultra-high aspect ratio machining. Mitra, Sambit; Chanal, Margaux; Clady, Raphaël; Mouskeftaras, Alexandros; Grojo, David 2015-08-20 We report on a functional experimental design for Bessel beam generation capable of handling high-energy ultrashort pulses (up to 1.2 mJ per pulse of 50 fs duration). This allows us to deliver intensities exceeding the breakdown threshold for air or any dielectric along controlled micro-filaments with lengths exceeding 4 mm. It represents an unprecedented upscaling in comparison to recent femtosecond Bessel beam micromachining experiments. We produce void microchannels through glass substrates to demonstrate that aspect ratios exceeding 1200∶1 can be achieved by using single high-intensity pulses. This demonstration must lead to new methodologies for deep-drilling and high-speed cutting applications. PMID:26368773 16. On the validity of localized approximations for Bessel beams: All N-Bessel beams are identically equal to zero Gouesbet, Gérard 2016-06-01 Localized approximation procedures are efficient ways to evaluate beam shape coefficients of a laser beam. They are particularly useful when other methods are ineffective or inefficient. Several papers in the literature have reported the use of such procedures to evaluate the beam shape coefficients of Bessel beams. Relying on the concept of N-beams, it is demonstrated that care must be taken when constructing a localized approximation for a Bessel beam, namely a localized Bessel beam is satisfactorily close enough to the intended beam only when the axicon angle is small enough. 17. Riesz's and Bessel's Operators in Bilateral Grand Lebesgue Spaces Ostrovsky, E; Sirota, L 2009-01-01 In this paper we obtain the non - asymptotic estimations for Riesz's and Bessel's potential integral operators in the so - called Bilateral Grand Lebesgue Spaces. We also give examples to show the sharpness of these inequalities. 18. Extensions of Bessel sequences to dual pairs of frames Christensen, Ole; Kim, Hong Oh; Kim, Rae Young 2013-01-01 Tight frames in Hilbert spaces have been studied intensively for the past years. In this paper we demonstrate that it often is an advantage to use pairs of dual frames rather than tight frames. We show that in any separable Hilbert space, any pairs of Bessel sequences can be extended to a pair of...... dual frames. If the given Bessel sequences are Gabor systems in L2(R), the extension can be chosen to have Gabor structure as well. We also show that if the generators of the given Gabor Bessel sequences are compactly supported, we can choose the generators of the added Gabor systems to be compactly...... supported as well. This is a significant improvement compared to the extension of a Bessel sequence to a tight frame, where the added generator only can be compactly supported in some special cases. We also analyze the wavelet case, and find sufficient conditions under which a pair of wavelet systems can be... 19. Long-distance Bessel beam propagation through Kolmogorov turbulence. Birch, Philip; Ituen, Iniabasi; Young, Rupert; Chatwin, Chris 2015-11-01 Free-space optical communication has the potential to transmit information with both high speed and security. However, since it is unguided it suffers from losses due to atmospheric turbulence and diffraction. To overcome the diffraction limits the long-distance propagation of Bessel beams is considered and compared against Gaussian beam properties. Bessel beams are shown to have a number of benefits over Gaussian beams when propagating through atmospheric turbulence. PMID:26560921 20. Second-Harmonic Generation of Bessel Beams in Lossy Media 丁德胜; 许坚毅; 王耀俊 2002-01-01 We present a further analysis for the second-harmonic generation of Bessel beams in lossy media. The emphasis is put on the effect of absorption to the radial pattern of the second-harmonic beam. It is shown that within the absorption length of the second harmonic, the Bessel second-harmonic beam approaches limited diffraction in the radial direction and behaves as in the case of lossless media. 1. GENERALIZED BROWNIAN SHEET IMAGES AND BESSEL-RIESZ CAPACITY 2007-01-01 Let (W) be a two-parameter Rd-valued generalized Brownian sheet. The author obtains an explicit Bessel-Riesz capacity estimate for the images of a two-dimensional set under (W). He also presents the connections between the Lebesgue measure of the image of (W) and Bessel-Riesz capacity. His conclusions also solve a problem proposed by J.P.Kahane. 2. Light scattering of a non-diffracting zero-order Bessel beam by uniaxial anisotropic bispheres Based on the generalized multi-particle Mie theory and the Fourier transformation approach, light scattering of two interacting homogeneous uniaxial anisotropic spheres with parallel primary optical axes illuminated by a zero-order Bessel beam (ZOBB) is investigated. The size and configuration of the particles are arbitrary. The expansion expressions of the ZOBB are given in terms of the spherical vector wave functions (SVWFs) and the expansion coefficients are derived. Utilizing the vector addition theorem of the SVWFs, the interactive scattering coefficients are derived through the continuous boundary conditions on which the interaction of the bispheres is considered. The effects of the conical angle, beam centre position, sphere separation distance, and anisotropic parameters on the far-region field distributions are numerically analyzed in detail. Some results are compared with those results for a Gaussian beam incidence. Selected results of bispheres consisting of typical medium such as TiO2, SiO2, Silicon, water are exhibited. This investigation could provide an effective test for further research on the scattering characteristic of an aggregate of anisotropic spheres by a high-order Bessel vortex beam and radiation forces, which are important in optical tweezers and particle manipulation applications. - Highlights: • Scattering of a zero-order Bessel beam by uniaxial anisotropic bispheres is studied. • The parallel primary optical axes of the anisotropic spheres are considered. • The accuracy of the theory and codes is verified. • Scattering performances of uniaxial anisotropic bispheres are numerically discussed. • Different properties of multiple scattering by a zero-order Bessel beam are shown 3. Axisymmetric scattering of an acoustical Bessel beam by a rigid fixed spheroid. Mitri, Farid G 2015-10-01 Based on the partial-wave series expansion (PWSE) method in spherical coordinates, a formal analytical solution for the acoustic scattering of a zeroth-order Bessel acoustic beam centered on a rigid fixed (oblate or prolate) spheroid is provided. The unknown scattering coefficients of the spheroid are determined by solving a system of linear equations derived for the Neumann boundary condition. Numerical results for the modulus of the backscattered pressure (θ = π) in the near field and the backscattering form function in the far field for both prolate and oblate spheroids are presented and discussed, with particular emphasis on the aspect ratio (i.e., the ratio of the major axis over the minor axis of the spheroid), the half-cone angle of the Bessel beam, and the dimensionless frequency. The plots display periodic oscillations (versus the dimensionless frequency) because of the interference of specularly reflected waves in the backscattering direction with circumferential Franz' waves circumnavigating the surface of the spheroid in the surrounding fluid. Moreover, the 3-D directivity patterns illustrate the near- and far-field axisymmetric scattering. Investigations in underwater acoustics, particle levitation, scattering, and the detection of submerged elongated objects and other related applications utilizing Bessel waves would benefit from the results of the present study. PMID:26470043 4. Combinatorial proofs of inverse relations and log-concavity for Bessel numbers Han, Hyuk; Seo, Seunghyun 2004-01-01 Let the Bessel number of the second kind B(n,k) be the number of set partitions of [n] into k blocks of size one or two, and let the Bessel number of the first kind b(n,k) be a certain coefficient in n-th Bessel polynomial. In this paper, we show that Bessel numbers satisfy two properties of Stirling numbers: The two kinds of Bessel numbers are related by inverse formulas, and both Bessel numbers of the first kind and the second kind form log-concave sequences. By constructing sign-reversing ... 5. Non-linear optical measurements using a scanned, Bessel beam Collier, Bradley B.; Awasthi, Samir; Lieu, Deborah K.; Chan, James W. 2015-03-01 Oftentimes cells are removed from the body for disease diagnosis or cellular research. This typically requires fluorescent labeling followed by sorting with a flow cytometer; however, possible disruption of cellular function or even cell death due to the presence of the label can occur. This may be acceptable for ex vivo applications, but as cells are more frequently moving from the lab to the body, label-free methods of cell sorting are needed to eliminate these issues. This is especially true of the growing field of stem cell research where specialized cells are needed for treatments. Because differentiation processes are not completely efficient, cells must be sorted to eliminate any unwanted cells (i.e. un-differentiated or differentiated into an unwanted cell type). In order to perform label-free measurements, non-linear optics (NLO) have been increasingly utilized for single cell analysis because of their ability to not disrupt cellular function. An optical system was developed for the measurement of NLO in a microfluidic channel similar to a flow cytometer. In order to improve the excitation efficiency of NLO, a scanned Bessel beam was utilized to create a light-sheet across the channel. The system was tested by monitoring twophoton fluorescence from polystyrene microbeads of different sizes. Fluorescence intensity obtained from light-sheet measurements were significantly greater than measurements made using a static Gaussian beam. In addition, the increase in intensity from larger sized beads was more evident for the light-sheet system. 6. Sum-type Kaiser-Bessel windows for apodized antenna arrays Anukhin, I. P.; Lukin, V. V.; Ponomarenko, N. N.; Zelensky, A. A.; Saramaki, Tapio; Zbaida, K. 1997-01-01 A class of sum-type Kaiser-Bessel windows is introduced. These windows are optimized for improving the performance of apodized array patterns. Compared with the basic Kaiser-Bessel windows, the proposed windows are shown provide several benefits. 7. Quaternion Bessel-Fourier moments and their invariant descriptors for object reconstruction and recognition Shao, Zhuhong; Shu, Huazhong; Wu, Jiasong; Chen, Beijing; Coatrieux, Jean-Louis 2014-01-01 In this paper, the quaternion Bessel-Fourier moments are introduced. The significance of phase information in quaternion Bessel-Fourier moments is investigated and an accurate estimation method for rotation angle is described. Furthermore, a new set of invariant descriptors based on the magnitude and the phase information of quaternion Bessel-Fourier moments is derived. Experimental results show that quaternion Bessel-Fourier moments lead to better performance for color image reconstruction t... 8. A short note on the propagation of a Bessel beam in conducting media Mugnai, D. 2009-01-01 Recently, the use of Bessel beams in evaluating the possibility of using them for a new generation of GPR (ground penetrating radar) systems has been considered. Therefore, an analysis of the propagation of Bessel beam in conducting media is worthwhile. We present here an analysis of this type. Specifically, for normal incidence we analyze the propagation of a Bessel beam coming from a perfect dielectric and impinging on a conducting medium, i.e. the propagation of a Bessel beam generated by ... 9. Propagation of Bessel beams from a dielectric to a conducting medium Mugnai, D. 2010-01-01 Recently, the use of Bessel beams in evaluating the possibility of using them for a new generation of GPR (ground penetrating radar) systems has been considered. Therefore, an analysis of the propagation of Bessel beam in conducting media is worthwhile. We present here an analysis of this type. Specifically, for normal incidence we analyze the propagation of a Bessel beam coming from a perfect dielectric and impinging on a conducting medium, i.e. the propagation of a Bessel beam generated by ... 10. RIEMANN-HILBERT CHARACTERIZATION FOR MAIN BESSEL POLYNOMIALS WITH VARYING LARGE NEGATIVE PARAMETERS 段萍; 杜金元 2014-01-01 In this article, we establish the Bessel polynomials with varying large negative parameters and discuss their orthogonality based on the generalized Bessel polynomials. By using the Riemann-Hilbert boundary value problem on the positive real axis, we get the Riemann-Hilbert characterization of the main Bessel polynomials with varying large negative parameters. 11. Change of the size of vector Bessel beam rings under reflection Novitsky, Andrey V.; Novitsky, Denis V. 2008-01-01 We theoretically predict the change of the size of Bessel beam rings under reflection. Considered electromagnetic Bessel beam is the superposition of phase shifted TE and TM polarized Bessel beams. Reflection from a semi-infinite medium and from a slab are studied. The sets of parameters maximizing the effect are discussed. 12. High-Order Bessel-Gaussian Beam and its Propagation Properties 陆璇辉; 陈许敏; 张蕾; 薛大建 2003-01-01 A high-order Bessel-Gaussian mode is introduced to describe hollow beams. The results for high-order BesselGaussian beams propagating through lens focusing system and free space are derived in terms of Collins integral formula. The diffraction patterns and profile for high-order Bessel-Gaussian beams propagating through the above-mentioned optical systems are illustrated. 13. The Solutions of the -Dimensional Bessel Diamond Operator and the Fourier–Bessel Transform of their Convolution Hüseyin Yildirim; M Zeki Sarikaya; Sermin Öztürk 2004-11-01 In this article, the operator$\\Diamond^k_B$is introduced and named as the Bessel diamond operator iterated times and is defined by $$\\Diamond^k_B=[(B_{x_1}+B_{x_2}+\\cdots +B_{x_p})^2-(B_{x_{p+1}}+\\cdots +B_{x_{p+q}})^2]^k,$$ where$p + q = n, B_{x_i} = \\frac{^2}{ x^2_i}+\\frac{2_i}{x_i}\\frac{}{ x_i}$, where$2_i = 2_i + 1, _i > - \\frac{1}{2} [8], x_i > 0, i = 1, 2,\\ldots ,n, k$is a non-negative integer and is the dimension of$\\mathbb{R}^+_n$. In this work we study the elementary solution of the Bessel diamond operator and the elementary solution of the operator$\\Diamond^k_B$is called the Bessel diamond kernel of Riesz. Then, we study the Fourier–Bessel transform of the elementary solution and also the Fourier–Bessel transform of their convolution. 14. Generation of high-order Bessel-Gauss beams%高阶Bessel-Gauss光束的产生方法 靳李丽; 朱艳英; 魏勇; 沈军峰; 窦红星; 李云涛 2012-01-01 Higher-order Bessel - Gauss beams shows "diffraction-free" characteristics under certain conditions. It is a kind of hollow beam that has broad application prospect. In this paper, we summarize and classify the generation methods. They can be divided into two categories-active way and passive way. The present generation methods-Resonator method, Geometrical optics method, Optical holographic method, computer generated hologram, Nonlinear optical method are described and analyzed. Finally we introduce the advantages and disadvantages of each method.%高阶贝塞尔-高斯(Bessel-Gauss)光束在一定条件下呈现“无衍射”特性,是一种具有广阔应用前景的空心光束.本文首先对高阶Bessel-Gauss光束的产生方法进行了分析和归类,将其产生方式分为主动式和被动式两大类.其次对获得高阶Bessel-Gauss光束的谐振腔法、几何光学法、光学全息法、计算全息法、非线性光学法等实验方法进行了阐述.最后总结了各种方法产生高阶Bessel-Gauss光束的优缺点. 15. Bessel beam fluorescence lifetime tomography of live embryos (Conference Presentation) Xu, Dongli; Peng, Leilei 2016-03-01 Optical tomography allows isotropic 3D imaging of embryos. Scanning-laser optical tomography (SLOT) has superior light collecting efficiency than wide-field optical tomography, making it ideal for fluorescence imaging of live embryos. We previously reported an imaging system that combines SLOT with a novel Fourier-multiplexed fluorescence lifetime imaging (FmFLIM) technique named FmFLIM-SLOT. FmFLIM-SLOT performs multiplexed FLIM-FRET readout of multiple FRET sensors in live embryos. Here we report a recent effort on improving the spatial resolution of the FmFLIM-SLOT system in order to image complex biochemical processes in live embryos at the cellular level. Optical tomography has to compromise between resolution and the depth of view. In SLOT, the commonly-used focused Gaussian beam diverges quickly from the focal plane, making it impossible to achieve high resolution imaging in a large volume specimen. We thus introduce Bessel beam laser-scanning tomography, which illuminates the sample with a spatial-light-modulator-generated Bessel beam that has an extended focal depth. The Bessel beam is scanned across the whole specimen. Fluorescence projection images are acquired at equal angular intervals as the sample rotates. Reconstruction artifacts due to annular-rings of the Bessel beam are removed by a modified 3D filtered back projection algorithm. Furthermore, in combination of Fourier-multiplexing fluorescence lifetime imaging (FmFLIM) method, the Bessel FmFLIM-SLOT system is capable of perform 3D lifetime imaging of live embryos at cellular resolution. The system is applied to in-vivo imaging of transgenic Zebrafish embryos. Results prove that Bessel FmFLIM-SLOT is a promising imaging method in development biology research. 16. Optical orbital angular momentum of evanescent Bessel waves. Yang, Zhenshan 2015-05-18 We show that the orbital angular momentum (OAM) of evanescent light is drastically different from that of traveling light. Specifically, the paraxial contribution (typically the most significant part in a traveling wave) to the OAM vanishes in an evanescent Bessel wave when averaged over the azimuthal angle. Moreover, the OAM per unit energy for the evanescent Bessel field is reduced by a factor of (1+κ2/k2) from the standard result for the corresponding traveling field, where k and κ are the wave number and the evanescent decay rate, respectively. PMID:26074524 17. Practical realization of a microwave Bessel beam launcher Manzhura, Oksana 2011-08-01 An experimental setup is realized to practically generate Bessel beams in the microwave regime. The setup, which consists of a series of circular loop antennas inserted coaxially inside a circular metallic waveguide, excites the waveguide\\'s transverse-electric modes such that their superposition forms a Bessel beam at the open-end of the waveguide. The excitation currents are calculated from the needed excitation coefficients of each guided mode, which, in turn, are calculated from the modal decomposition of the beam. The efficiency of the setup is evaluated and the obtained experimental results are compared to the theoretical estimates. © 2011 IEEE. 18. Jacobi-Bessel Analysis Of Antennas With Elliptical Apertures. Rahmat-Samii, Y. 1989-01-01 Coordinate transformation improves convergence pattern analysis of elliptical-aperture antennas. Modified version of Jacobi-Bessel expansion for vector diffraction analysis of reflector antennas uses coordinate transformation to improve convergence with elliptical apertures. Expansion converges rapidly for antennas with circular apertures, but less rapidly for elliptical apertures. Difference in convergence behavior between circular and elliptical Jacobi-Bessel algorithms indicated by highest values of indices m, n, and p required to achieve same accuracy in computed radiation pattern of offset paraboloidal antenna with elliptical aperture. 19. Elliptic boundary value problems for Bessel operators, with applications to anti-de Sitter spacetimes Gannot, Oran 2015-01-01 This paper considers boundary value problems for a class of singular elliptic operators which appear naturally in the study of anti-de Sitter spacetimes. These problems involve a singular Bessel operator acting in the normal direction. After formulating a Lopatinskii condition, elliptic estimates are established for functions supported near the boundary. A global Fredholm property follows from additional hypotheses in the interior. The results of this paper provide a rigorous framework for the study of quasinormal modes on anti-de Sitter black holes for the full range of boundary conditions considered in the physics literature. 20. Generation of arbitrary order Bessel beams via 3D printed axicons at the terahertz frequency range. Wei, Xuli; Liu, Changming; Niu, Liting; Zhang, Zhongqi; Wang, Kejia; Yang, Zhengang; Liu, Jinsong 2015-12-20 We present the generation of arbitrary order Bessel beams at 0.3 THz through the implementation of suitably designed axicons based on 3D printing technology. The helical axicons, which possess thickness gradients in both radial and azimuthal directions, can convert the incident Gaussian beam into a high-order Bessel beam with spiral phase structure. The evolution of the generated Bessel beams are characterized experimentally with a three-dimensional field scanner. Moreover, the topological charges carried by the high-order Bessel beams are determined by the fork-like interferograms. This 3D-printing-based Bessel beam generation technique is useful not only for THz imaging systems with zero-order Bessel beams but also for future orbital-angular-momentum-based THz free-space communication with higher-order Bessel beams. PMID:26837031 1. Bessel process, Schramm-Loewner evolution, and Dyson model Katori, Makoto 2011-01-01 Bessel process is defined as the radial part of the Brownian motion (BM) in the$D$-dimensional space, and is considered as a one-parameter family of one-dimensional diffusion processes indexed by$D$, BES$^{(D)}$. It is well-known that$D_{\\rm c}=2$is the critical dimension. Bessel flow is a notion such that we regard BES$^{(D)}$with a fixed$D$as a one-parameter family of initial value. There is another critical dimension$\\bar{D}_{\\rm c}=3/2$and, in the intermediate values of$D$,$\\bar{D}_{\\rm c} < D < D_{\\rm c}$, behavior of Bessel flow is highly nontrivial. The dimension D=3 is special, since in addition to the aspect that BES$^{(3)}$is a radial part of the three-dimensional BM, it has another aspect as a conditional BM to stay positive. Two topics in probability theory and statistical mechanics, the Schramm-Loewner evolution (SLE) and the Dyson model (Dyson's BM model with parameter$\\beta=2$), are discussed. The SLE$^{(D)}$is introduced as a 'complexification' of Bessel flow on the upper-h... 2. Asymmetric skew Bessel processes and their applications to finance M. Decamps; M.J. Goovaerts; W. Schoutens 2006-01-01 In this paper, we extend the Harrison and Shepp's construction of the skew Brownian motion (1981) and we obtain a diffusion similar to the two-dimensional Bessel process with speed and scale densities discontinuous at one point. Natural generalizations to multi-dimensional and fractional order Besse 3. Two-photon flow cytometer with laser scanning Bessel beams Wang, Yongdong; Ding, Yu; Ray, Supriyo; Paez, Aurelio; Xiao, Chuan; Li, Chunqiang 2016-03-01 Flow cytometry is an important technique in biomedical discovery for cell counting, cell sorting and biomarker detection. In vivo flow cytometers, based on one-photon or two-photon excited fluorescence, have been developed for more than a decade. One drawback of laser beam scanning two-photon flow cytometer is that the two-photon excitation volume is fairly small due to the short Rayleigh range of a focused Gaussian beam. Hence, the sampling volume is much smaller than one-photon flow cytometry, which makes it challenging to count or detect rare circulating cells in vivo. Bessel beams have narrow intensity profiles with an effective spot size (FWHM) as small as several wavelengths, making them comparable to Gaussian beams. More significantly, the theoretical depth of field (propagation distance without diffraction) can be infinite, making it an ideal solution as a light source for scanning beam flow cytometry. The trade-off of using Bessel beams rather than a Gaussian beam is the fact that Bessel beams have small concentric side rings that contribute to background noise. Two-photon excitation can reduce this noise, as the excitation efficiency is proportional to intensity squared. Therefore, we developed a two-photon flow cytometer using scanned Bessel beams to form a light sheet that intersects the micro fluidic channel. 4. Fractional Solutions of Bessel Equation with N-Method Erdal Bas; Resat Yilmazer; Etibar Panakhov 2013-01-01 This paper deals with the design fractional solution of Bessel equation. We obtain explicit solutions of the equation with the help of fractional calculus techniques. Using the N-fractional calculus operator N ν method, we derive the fractional solutions of the equation. 5. On the Operator ⨁Bk Related to Bessel Heat Equation Wanchak Satsanit 2010-01-01 Full Text Available We study the equation (∂/∂tu(x,t=c2⊕Bku(x,t with the initial condition u(x,0=f(x for x∈Rn+. The operator ⊕Bk is the operator iterated k-times and is defined by ⊕Bk=((∑i=1pBxi4-(∑j=p+1p+qBxi4k, where p+q=n is the dimension of the Rn+, Bxi=∂2/∂xi2+(2vi/xi(∂/∂xi, 2vi=2αi+1, αi>-1/2, i=1,2,3,…,n, and k is a nonnegative integer, u(x,t is an unknown function for (x,t=(x1,x2,…,xn,t∈Rn+×(0,∞, f(x is a given generalized function, and c is a positive constant. We obtain the solution of such equation, which is related to the spectrum and the kernel, which is so called Bessel heat kernel. Moreover, such Bessel heat kernel has interesting properties and also related to the kernel of an extension of the heat equation. 6. Two-color above threshold ionization of atoms and ions in XUV Bessel beams and combined with intense laser light Seipt, D; Surzhykov, A; Fritzsche, S 2016-01-01 The two-color above-threshold ionization (ATI) of atoms and ions is investigated for a vortex Bessel beam in the presence of a strong near-infrared (NIR) light field. While the photoionization is caused by the photons from the weak but extreme ultra-violet (XUV) vortex Bessel beam, the energy and angular distribution of the photoelectrons and their sideband structure are affected by the plane-wave NIR field. We here explore the energy spectra and angular emission of the photoelectrons in such two-color fields as a function of the size and location of the target (atoms) with regard to the beam axis. In addition, analogue to the circular dichroism in typical two-color ATI experiments with circularly polarized light, we define and discuss seven different dichroism signals for such vortex Bessel beams that arise from the various combinations of the orbital and spin angular momenta of the two light fields. For localized targets, it is found that these dichroism signals strongly depend on the size and position of t... 7. Acoustic beam interaction with a rigid sphere: The case of a first-order non-diffracting Bessel trigonometric beam Mitri, F. G. 2011-11-01 Mathematical expressions for the acoustic scattering, instantaneous (linear), and time-averaged (nonlinear) forces resulting from the interaction of a new type of Bessel beam, termed here a first-order non-diffracting Bessel trigonometric beam (FOBTB) with a sphere, are derived. The beam is termed "trigonometric" because of the dependence of its phase on the cosine function. The FOBTB is regarded as a superposition of two equi-amplitude first-order Bessel vortex (helicoidal) beams having a unit positive and negative order (known also as topological charge), respectively. The FOBTB is non-diffracting, possesses an axial null, a geometric phase, and has an azimuthal phase that depends on cos( ϕ± ϕ0), where ϕ0 is an initial arbitrary phase angle. Beam rotation around its wave propagation axis can be achieved by varying ϕ0. The 3D directivity patterns are computed, and the resulting modifications of the scattering are illustrated for a rigid sphere centered on the beam's axis and immersed in water. Moreover, the backward and forward acoustic scattering by a sphere vanish for all frequencies. The present paper will shed light on the novel scattering properties of an acoustical FOBTB by a sphere that may be useful in particle manipulation and entrapment, non-destructive/medical imaging, and may be extended to other potentially useful applications in optics and electromagnetism. 8. Multiple scattering of a zero-order Bessel beam with arbitrary incidence by an aggregate of uniaxial anisotropic spheres Li, Z. J.; Wu, Z. S.; Qu, T.; Shang, Q. C.; Bai, L. 2016-01-01 Based on the generalized multiparticle Mie theory, multiple scattering of an aggregate of uniaxial anisotropic spheres illuminated by a zero-order Bessel beam (ZOBB) with arbitrary propagation direction is investigated. The particle size and configuration are arbitrary. The arbitrary incident Bessel beam is expanded in terms of spherical vector wave functions (SVWFs). Utilizing the vector addition theorem of SVWFs, interactive and total scattering coefficients are derived through the continuous boundary conditions on which the interaction of the particles is considered. The accuracy of the theory and codes are verified by comparing results with those obtained for arbitrary plane wave incidence by CST simulation, and for ZOBB incidence by a numerical method. The effects of angle of incidence, pseudo-polarization angle, half-conical angle, beam center position, and permittivity tensor elements on the radar cross sections (RCSs) of several types of collective uniaxial anisotropic spheres, such as a linear chain, a 4×4×4 cube-shaped array, and other periodical structures consisting of massive spheres, are numerically analyzed. Selected results on the properties of typical particles such as TiO2, SiO2, or other particle lattices are calculated. This investigation could provide an effective test for further research on the scattering characteristics of an aggregate of anisotropic spheres by a high-order Bessel vortex beam. The results have important application in optical tweezers and particle manipulation. 9. Light scattering of a non-diffracting zero-order Bessel beam by uniaxial anisotropic bispheres Li, Z. J.; Wu, Z. S.; Qu, T.; Li, H. Y.; Bai, L.; Gong, L. 2015-09-01 Based on the generalized multi-particle Mie theory and the Fourier transformation approach, light scattering of two interacting homogeneous uniaxial anisotropic spheres with parallel primary optical axes illuminated by a zero-order Bessel beam (ZOBB) is investigated. The size and configuration of the particles are arbitrary. The expansion expressions of the ZOBB are given in terms of the spherical vector wave functions (SVWFs) and the expansion coefficients are derived. Utilizing the vector addition theorem of the SVWFs, the interactive scattering coefficients are derived through the continuous boundary conditions on which the interaction of the bispheres is considered. The effects of the conical angle, beam centre position, sphere separation distance, and anisotropic parameters on the far-region field distributions are numerically analyzed in detail. Some results are compared with those results for a Gaussian beam incidence. Selected results of bispheres consisting of typical medium such as TiO2, SiO2, Silicon, water are exhibited. This investigation could provide an effective test for further research on the scattering characteristic of an aggregate of anisotropic spheres by a high-order Bessel vortex beam and radiation forces, which are important in optical tweezers and particle manipulation applications. 10. Elliptic Bessel processes and elliptic Dyson models realized as temporally inhomogeneous processes Katori, Makoto 2016-01-01 The Bessel process with parameter$D>1$and the Dyson model of interacting Brownian motions with coupling constant$\\beta >0$are extended to the processes, in which the drift term and the interaction terms are given by the logarithmic derivatives of Jacobi's theta functions. They are called the elliptic Bessel process, eBES$^{(D)}$, and the elliptic Dyson model, eDYS$^{(\\beta)}$, respectively. Both are realized on the circumference of a circle$[0, 2 \\pi r)$with radius$r >0$as temporally inhomogeneous processes defined in a finite time interval$[0, t_*), t_* < \\infty$. Transformations of them to Schr\\"odinger-type equations with time-dependent potentials lead us to proving that eBES$^{(D)}$and eDYS$^{(\\beta)}$can be constructed as the time-dependent Girsanov transformations of Brownian motions. In the special cases where$D=3$and$\\beta=2$, observables of the processes are defined and the processes are represented for them using the Brownian paths winding round a circle and pinned at time$t_*\$. We...