algebras of quotients of jordan algebras abstract we define a jordan analogue of lambek and utumi s associative algebra of quotients and we construct the maximal algebra of quotients for nondegenerate jordan algebras we apply those results to other classes of algebras of quotients appearing in the literature

positive solutions for higher order multi point boundary value problems with nonhomogeneous boundary conditions abstract we study nth order boundary value problems with a nonlinear term f t x subject to nonhomogeneous multi point boundary conditions criteria for the existence of positive solutions of such problems are established conditions are determined by the relationship between the behavior of f t x x near 0 and when compared with the smallest positive characteristic value of an associated linear integral operator this work improves and extends some recent results in the literature for the second order problems the results are illustrated with examples

a simplified algorithm for the second order sound fields an analytical description for the second order field of the sum difference frequency and second harmonic components is derived this treatment is based on the fact that an arbitrary axially symmetric beam can be expressed as the linear superposition of a set of gaussian beams j j wen and m a breazeale j acoust soc am 83 1752 1756 1988 so that the three dimensional integral for the second order components is typically expressed as a combination of a set of interaction terms of the gaussian beams correspondingly the evaluation of the field distribution is reduced to the summation of exponential integral functions from some examples the present approach provides the results for the second order field which are in good agreement with those obtained directly by numerical integration

non paraxial model for a parametric acoustic array this study is concerned with parametric radiation from an arbitrary axisymmetric planar source with a special focus on low frequency difference frequency fields as a model equation accounting for nonlinearity diffraction and dissipation the westervelt equation is used the difference frequency field patterns are calculated in the quasi linear approximation by the method of successive approximations a multi layer integral for calculation of the acoustic field is reduced to a three dimensional one by employing an approximate analytical description of the primary field with the use of a multi gaussian beam expansion this integral is subsequently reduced in the paraxial approximation to a one dimensional form which has previously been published in literature and which represents a means for fast calculations of secondary acoustic fields the three dimensional integral is calculated numerically and the numerical results predict nonzero amplitude of the low frequency field in the vicinity of the source which is an effect that cannot be correctly encompassed in the paraxial approximation

a multi point boundary value problem with two critical conditions abstract we apply a coincidence degree theorem due to mawhin to show the existence of at least one solution of the nonlinear second order multi point boundary value problem u t f t u t u t t 0 1 u 0 u i 1 n i u i u 1 where 0 1 0 i 1 i 1 n i 1 n i i 1 n i i 1 n 2 and f 0 1 r 2 r satisfying the caratheodory conditions in our setting both boundary conditions are responsible for resonance

positive solutions of nonlinear second order m point boundary value problem abstract in this paper by using krasnoselskii s fixed point theorem of cone expansion compression type and under suitable conditions we present the existence of single and multiple positive solutions to the nonlinear second order m point boundary value problem u t a t f u t 0 0 t 1 u 0 0 u 1 i 1 m 2 a i u i where is a positive parameter a i 0 for i 1 2 m 3 and a m 2 0 i satisfy 0 1 2 m 2 1 and i 1 m 2 a i i 1 we derive an explicit interval of such that for any in this interval the existence of at least one positive solution to the boundary value problem is guaranteed and the existence of at least two solutions for in an appropriate interval is also discussed

nontrivial solutions of singular nonlinear m point boundary value problems in this paper by using the method of topology degree some existence theorems of nontrivial solutions for singular nonlinear m point boundary value problems are established our nonlinearity may be singular in its dependent variable

second harmonic generation of practical bessel beams a fast gaussian expansion approach is used to investigate fundamental and second harmonic generation in practical bessel beams of finite aperture the analysis is based on the integral solutions of the kzk equation under the quasilinear approximation the influence of the medium s attenuation on the beam profile is considered analysis results show that the absorption parameter has a significant effect on the far field beam profile of the second harmonic under certain circumstances the second harmonic of a practical bessel beam still has the main properties of an ideal bessel beam of infinite aperture when it propagates within its depth of field

a generalized multi point boundary value problem for second order ordinary differential equations let f 0 1 x r 2 r be a function satisfying caratheodory s conditions and e t l 1 0 1 let x i t j 0 1 a i b j r i 1 2 m 2 j 1 2 n 2 0 x 1 x 2 x m 2 1 0 t 1 t 2 t n 2 1 be given this paper is concerned with the problem of existence of a solution for the generalized multi point boundary value problems x t f t x t x t e t 0

local orders in jordan algebras abstract making use of results on general algebras of quotients of jordan algebras we study a notion of local order based on the version for linear jordan algebras of the ideas of fountain and gould 14 as adapted to the jordan context by fernandez lopez and garcia rus in 7 in particular we characterize the set of lesieur croisot elements of a nondegenerate jordan algebra as those elements of the jordan algebra lying in the socle of its maximal algebra of quotients and apply this relationship to extend to quadratic jordan algebras the results of fernandez lopez and garcia rus on local orders in nondegenerate jordan algebras satisfying the descending chain condition on principal inner ideals and not containing ideals which are nonartinian quadratic factors

existence results for n point boundary value problem of second order ordinary differential equations abstract this study concerns the existence of positive solutions to the boundary value problem u a t f u 0 t 0 1 u 0 i 1 n 2 b i u i u 1 i 1 n 2 a i u i where i 0 1 with 0 1 2 n 2 1 a i b i 0 with 0 i 1 n 2 a i 1 and i 1 n 2 b i 1 by applying the krasnoselskii s fixed point theorem in banach spaces some sufficient conditions guaranteeing the existence of at least one positive solution or at least two positive solutions are established for the above general n point boundary value problem

nearfield of a large acoustic transducer part i linear radiation results are reported of an investigation of the nearfield of a large circular transducer 182 cm diameter and composed of 364 smaller elements operating in water at 11 16 khz and with parametric generated difference frequencies in the range 0 5 5 khz linear radiation is considered in this paper part i whereas parametric radiation is discussed in a related paper part ii amplitude and phase are measured at various distances and compared with calculations based on a simplified integral representation of the field obtained by naze tjo tta and tjo tta j acoust soc am 68 334 339 1980 as a solution of the parabolic approximation of the helmholtz equation comparison is also made with an exact on axis calculation the main results are 1 the transducer radiates as a piston with a uniform velocity distribution 2 the integral solution represents an adequate description of the observed sound field it provides a simple method to obtain general properties of the field as well as numerical results by computing a simple line integral 3 within the same approximation scheme formulas are presented for the case of axial symmetry from which the field of an amplitude or phase shaded source can be calculated from a simple line integral 4 from this it is further shown how the farfield amplitude and phase can be calculated readily from the nearfield measurements

symmetric solutions of a multi point boundary value problem abstract we apply the fixed point theorem of avery and peterson to the nonlinear second order multi point boundary value problem u t a t f t u t u t t 0 1 u 0 i 1 n i u i u 1 t u t t 0 1 where 0 1 2 n 1 2 i 0 for i 1 n with i 1 n i 1 n 2 we show that under the appropriate growth conditions on the inhomogeneous term symmetric about t 1 2 the problem has triple symmetric solutions

nonlinear effects in the farfield of a directive sound source nonlinear propagation of a periodic sound beam in a dissipative fluid is considered using fourier series expansion and numerical methods to solve the governing equation of motion in the parabolic approximation the nearfield was considered in a previous paper aanonsen et al j acoust soc am 75 749 768 1984 the analysis is now extended to the farfield numerical and asymptotic results are derived and used to explain the development of the fundamental and harmonic components from the nearfield into the farfield a discussion is also given of some earlier models for the farfield of directional waves emphasis is put on the importance of imposing the proper matching conditions between the nearfield solution and the spherical solution in the farfield in order to obtain a good approximation propagation and saturation curves are calculated as well as beam patterns for various harmonic components the results are compared with available experimental observations nonlinear effects although generated

a gaussian finite element method for description of sound diffraction a new method for solving diffraction problems is presented it is based on the use of gaussian diffraction theory comparison of the gaussian beam expansion and fourier series expansion reveals that the gaussian expansion is a more general and more powerful technique the new method combines the gaussian beam superposition technique wen and breazeale j acoust soc am 83 1752 1756 1988 and the numerical finite element solution to the parabolic equation huang j acoust soc am 84 1405 1413 1988 the new method is capable of solving for the sound field even in an inhomogeneous medium or in the presence of reflection as shown by computer modeling the source may be a gaussian source or a distributed source calculated results are compared with experimental results in a laboratory water tank

second order boundary value problems with nonhomogeneous boundary conditions ii sufficient conditions are given for the existence of solutions of the following nonlinear boundary value problem with nonhomogeneous multi point boundary condition u f t u u 0 t 0 1 u 0 i 1maiu ti 1 u 1 i 1mbiu ti 2 we prove that the whole plane r2 is divided by a continuous decreasing curve into two disjoint connected regions e and n such that the above problem has at least one solution for 1 2 has at least two solutions for 1 2 e and has no solution for 1 2 n we also find explicit subregions of e where the above problem has at least two solutions and two positive solutions respectively

generalized approximations and rapid convergence of solutions of m point boundary value problems abstract the upper and lower solutions method and the generalized quasilinearization technique for second order nonlinear m point boundary value problem of the type x f t x x t 0 1 x 0 x 0 0 x 1 i 1 m 2 i x i i 0 1 is developed a monotone sequence of solutions of linear problems converging uniformly and quadratically to a solution of the problem is obtained in the c 1 norm

nonlinear effects in focused sound fields focused finite amplitude sound fields are investigated with numerical solutions of the khokhlov zabolotskaya kuznetsov kzk equation the numerical solution is based on the algorithm developed by aanonsen et al j acoust soc am 75 749 768 1984 who used a fourier series expansion of the sound pressure to reduce the kzk equation to a set of coupled parabolic equations the basic algorithm has been modified by introducing a coordinate system that follows the convergent geometry of focused sound fields in this way more efficient numerical evaluation of the detailed field structure within the focal region is achieved arbitrary axisymmetric sources can be modeled here circular sources having linear focusing gains of order 50 will be considered the calculated time waveforms propagation curves and beam patterns illustrate clearly the combined effects of nonlinearity diffraction and absorption on finite amplitude sound that passes through a focal region among the new results are power curves

supplementary notes on the gaussian beam expansion the letter provides alternatively a simple way of computing the fresnel field integral a further extension to the gaussian beam expansion the zeroth order bessel function of the first kind is expanded into an approximate sum of gaussian functions the field integral is then expressible in terms of these simple functions the approach is useful in treatment of the field radiation problem for a large and important group of piston sources in acoustics as examples the calculation results for the uniform and the simply supported piston sources are presented in a good agreement with those evaluated by numerical integration

development of a simple and accurate approximation method for the gaussian beam expansion technique the calculation of the sound field can be greatly simplified by using the gaussian beam expansion technique the source distribution function is expressed as the superposition of a small number of gaussian functions and the expansion coefficients could be obtained by minimizing an object function in the spatial or k space domain in this paper a fast algorithm is developed to determine the gaussian function coefficients for a more accurate approximation two stage procedures are employed in the proposed method firstly two real coefficients are estimated by a simple search approach and then the least mean square lms algorithm is adopted for determining the optimal expansion coefficients finally the presented method is evaluated in the case of calculation of sound fields radiated from a piston and a rectangular planar source simulation results show that compared with the previous approaches the developed scheme is simple to implement with high accuracy

some extensions of the gaussian beam expansion radiation fields of the rectangular and the elliptical transducer a straightforward extension of gaussian beam expansion is presented for calculation of the fresnel field integral j j wen and m a breazeale j acoust soc am 83 1752 1756 1988 the source distribution function is expanded into the superposition of a series of two dimensional gaussian functions the corresponding radiation field is expressed as the superposition of these two dimensional gaussian beams and is then reduced to the computation of these simple functions this treatment overcomes the limit that the shape of source is of circular axial symmetry the numerical examples are presented for the field of the uniform elliptical and the rectangular piston transducers and agree well with the results given by complicated computation

multi point boundary value problems of second order differential equations i abstract we study the nonlinear boundary value problem with nonhomogeneous multi point boundary condition u f t u u 0 t 0 1 u 0 i 1 m a i u t i 1 u 1 i 1 m b i u t i 2 sufficient conditions are found for the existence of solutions of the problem based on the existence of lower and upper solutions with certain relations using one of the results under some assumptions we obtain explicit ranges of values of 1 and 2 with which the problem has a solution has a positive solution and has no solution respectively furthermore we prove that the whole plane for 1 and 2 can be divided into two disjoint connected regions e and n such that the problem has a solution for 1 2 e and has no solution for 1 2 n

the maximal symmetric ring of quotients abstract the maximal symmetric ring of quotients q r as defined by utumi is a symmetric version of the maximal ring of quotients of r for the most part we study this ring when r k g is a group algebra for example we show that if g is a free product of groups and if r k g is a domain then q r is usually equal to r on the other hand there are certainly groups for which q r is properly larger than r and we construct a number of such examples

rings of quotients of incidence algebras and path algebras we compute the maximal right left symmetric rings of quotients of finite dimensional incidence and graph algebras we show that these rings of quotients are morita equivalent to incidence algebras and path algebras respectively with respect to simpler well determined partially ordered sets and finite quivers respectively

notes on the gaussian beam expansion a powerful gaussian beam expansion technique j j wen and m a breazeale j acoust soc am 83 1752 1756 1988 is widely used in calculating the fresnel field integral a basic ideal of this treatment is to expand the source distribution function into the superposition of a series of gaussian functions the evaluation of the radiation field is then greatly simplified as a supplement this paper presents a simple approach to determine the expansion coefficients for the sound source the results for the ultrasonic field radiated by piston transducers are typically given in a fairly good agreement with those calculated by numerical integration and breazeale s the limits of the present method are also discussed

computing the symmetric ring of quotients abstract we discuss and compute the symmetric martindale ring of quotients for various classes of prime rings in particular we consider free algebras and group algebras

a simple calculation approach for the second harmonic sound field generated by an arbitrary axial symmetric source in a linear approximation an arbitrary axial symmetric source can be expressed as the linear superposition of a set of gaussian sources and the corresponding radiated sound field can be represented as the superposition of this set of gaussian beams the analysis is extended to the second harmonic generation due to nonlinear effects under a quasilinear approximation then the second harmonic sound field could be considered as a sum of self and cross interaction terms produced by a series of gaussian beams therefore the calculation of the second harmonic sound field of an axial symmetric source can be reduced to a simple linear combination of a set of exponential integral functions instead of a complicated triple integral in order to verify this calculation approach the second harmonic generation of a focused and a simple piston source is examined and the result is in good agreement with that in earlier paper by complicated computation

positive solutions for second order m point boundary value problems by using the schauder fixed point theorem and analysis method we establish the existence of solutions for the m point boundary value problem u t a t f u 0 u 0 0 u 1 i 1m 2kiu i b where b ki 0 i 1 2 m 2 0 b

existence of positive solutions to n point nonhomogeneous boundary value problem in this paper we are concerned with the existence of positive solutions to a n point nonhomogeneous boundary value problem by using the krasnoselskii s fixed point theorem in banach spaces some sufficient conditions guaranteeing the existence of positive solution is established for the n point nonhomogeneous boundary value problem

algebras of quotients of nonsingular jordan algebras abstract we define a jordan analogue of johnson s associative algebra of quotients and study it under a suitable condition of nonsingularity of the jordan algebra which we call strong nonsingularity in particular we prove the existence and describe the maximal algebras of quotients of prime strongly nonsingular jordan algebras

associative systems of left quotients abstract in this paper we construct the maximal left quotient system of every pair of modules in some generalized matrix rings these rings can be seen as 3 graded algebras or as superalgebras we show the relation among the three different notions of left quotients

solvability of an m point boundary value problem for second order ordinary differential equations let 0 1 r2 r be a function satisfying caratheodory s conditions and e t l1 0 1 let i 0 1 ai r all of the ai s having the same sign i 1 2 m 2 0 1 2 m 2 1 be given this paper is concerned with the problem of existence of a solution for the m point boundary value problem x t t x t x t e t t 0 1 e formula conditions for the existence of a solution for the above boundary value problem are given using the leray schauder continuation theorem

multiple positive solutions for second order m point boundary value problems abstract in this paper the second order m point boundary value problem u t f t u 0 0 t 1 u 0 0 u 1 i 1 m 2 k i u i 0 is studied where k i 0 i 1 2 m 2 0 1 2 m 2 1 we give at first the associated green s function and its properties for the m point boundary value problem and growth conditions are imposed on f which yield the existence of at least three positive solutions by using the five functionals fixed point theorem and the properties

a simplified algorithm for second order sound beams with arbitrary source distribution and geometry l an earlier investigation j acoust soc am 108 2759 2764 2000 of the simplified algorithm for the second order sound fields with circular axial symmetry is extended to a more general case of the sound sources with an arbitrary distribution this extension is based on the fact that the source function is expanded into the sum of a series of two dimensional gaussian ones the two and five dimensional integral expressions for the primary and second order fields are respectively reduced in terms of gaussian functions and simple one dimensional integrals numerical calculations are then greatly simplified

positive solutions for nonhomogeneous m point boundary value problems let a c 0 1 b c 0 1 0 let d r and d 0 let 1 t be the unique solution of the linear boundary value problem u t a t u t b t u t 0 t 0 1 u 0 0 u 1 1 we study the existence of positive solutions for the m point boundary value problem u a t u t b t u h t f u 0 u 0 0 u 1 i 1m 2 iu ei d where i 0 1 and i 0 for i 1 m 2 are given constants satisfying i 1m 2 ix 1 d

positive solutions of an m point boundary value problem with sign changing nonlinearities let a i 0 for i 1 m 2 let x i satisfy 0 x 0 x 1 x 2 x m 2 x m 1 1 and m 2 i 1a email protected i 1 we study the existence of positive solutions to the boundary value problem u f t u 0 0

monotone positive solutions of second order multi point boundary value problems abstract in this paper by using guo krasnosel skii fixed point theorem and schauder s fixed point theorem we establish sufficient conditions for the existence and nonexistence of monotone increasing positive solution for a class of second order differential equation with multi point nonhomogeneous boundary conditions