on conditional fault tolerance of hierarchical cubic networks abstract this paper considers the conditional fault tolerance h super connectivity h and h super edge connectivity h of the hierarchical cubic network h c n n an attractive alternative network to the hypercube and shows h h c n n h h c n n 2 h n 1 h for any h with 0 h n 1 the results imply that at least 2 h n 1 h vertices or edges have to be removed from h c n n to make it disconnected with no vertices of degree less than h and generalize some known results

resonance modes in a one dimensional medium with two purely resistive boundaries calculation methods orthogonality and completeness studying the problem of wave propagation in media with resistive boundaries can be made by searching for resonance modes or free oscillations regimes in the present article a simple case is investigated which allows one to enlighten the respective interest of different classical methods some of them being rather delicate this case is the one dimensional propagation in a homogeneous medium having two purely resistive terminations the calculation of the green function being done without any approximation using three methods the first one is the straightforward use of the closed form solution in the frequency domain and the residue calculus then the method of separation of variables space and time leads to a solution depending on the initial conditions the question of the orthogonality and completeness of the complex valued resonance modes is investigated leading to the expression of a particular scalar product the last method is the expansion in biorthogonal modes in the frequency domain the modes having eigenfrequencies depending on the frequency results of the three methods generalize or and correct some results already existing in the literature and exhibit the particular difficulty of the treatment of the constant mode

variational formulation of interior cavity frequencies for spheroidal bodies this paper presents interior cavity frequencies for spheroidal bodies in cases of neumann and dirichlet homogenous boundary conditions the helmholtz wave equation is expressed in terms of prolate spheroidal coordinates from which a variational formulation is derived for the determination of the interior cavity frequencies the search of stationarity is performed by means of a rayleigh ritz type expansion of trial functions the trial functions are expressed as a double summation of basis functions in radial and angular coordinates a variable transformation is applied to the variational form in order to have homogenous boundary conditions which are essential for establishing a complete function space of the associated boundary conditions the present analysis is verified by comparisons of interior frequencies of a spherical body of prolate spheroidal bodies and of the interior characteristic functions with the radial and angular functions obtained from prolate spheroidal differential equations interior frequencies are tabulated for various aspect ratios of spheroidal bodies in accordance with harmonic orders in azimuthal directions for both neumann and dirichlet conditions

a combined integro modal approach for predicting acoustic properties of irregular shaped cavities an integro modal approach is presented in this paper for computing the acoustic properties of irregular shaped cavities the method consists of discretizing the whole cavity into a series of subcavities whose acoustic pressure is decomposed either over a modal basis of regular subcavities or over that of the bounding cavities in the case of irregular shaped boundaries an integral formulation is then established to ensure continuity of both pressure and velocity between adjacent subcavities using a membrane with zero mass and stiffness to some extent the method provides a combined approach retaining the advantages of both the acoustoelastic method aem and the green s function method gfm numerical and experimental results are presented demonstrating the efficiency and accuracy of the suggested technique comparisons with other existing methods are also made it is shown that good accuracy on the computation of cavity modes can be obtained using a very limited number of subcavities

determination of invariant zeros and zero directions of the system s a b c e mita 1977 presented a simple method for the invariant zeros of a linear multivariable system by constructing its maximal unobservable subspace mus however the results of this method have been restricted to a class of controllable and right invertible systems satisfying the diagonal decoupling condition and described by the triple a b c here a general method based on the theory of decoupling extends mita s results for all classes of systems described by the 4 tuples a b c e with no special requirements the mus of all classes of such systems is simply constructed and its order is given in terms of the system order n and the infinite zeros or the minimal observability indices of the system in addition the invariant zeros and zero directions are found from the eigenvalues and eigenvectors of a reduced order matrix having the dimension of the mus obtained directly from the state space parameters algorithms are presented for the invariant zeros and zero directions of invertible decouplable o

on conditional diagnosability of the folded hypercubes the n dimensional folded hypercube fq n a variation of the hypercube proposed by ahmed et al a el amawy s latifi properties and performance of folded hypercubes ieee transactions on parallel and distributed systems 2 3 1991 31 42 is an n 1 regular n 1 connected graph conditional diagnosability a new measure of diagnosability introduced by lai et al pao lien lai jimmy j m tan chien ping chuang lih hsing hsu conditional diagnosability measures for large multiprocessor systems ieee transactions on computers 54 2 2005 165 175 can better measure the diagnosability of regular interconnection networks this paper determines that under pmc model the conditional diagnosability of fq n t c fq n is 4n 3 when n 5 or n 8 t c fq 3 3 t c fq 4 7

conditional diagnosability of arrangement graphs under the pmc model abstract processor fault diagnosis has played an important role in measuring the reliability of a multiprocessor system and the diagnosabilities of many well known multiprocessor systems have been investigated the conditional diagnosability has been widely accepted as a measure of diagnosability by assuming an additional condition that any fault set can t contain all the neighbors of any node in a multiprocessor system this paper considers the conditional diagnosability of an n k arrangement graph a n k a flexible interconnection network model for multiprocessor systems under the classical pmc diagnostic model and determines that the conditional diagnosability of a n k k 2 n k 2 is 4 k 4 n k 3 which is about four times its traditional diagnosability

acoustic analysis of a rectangular cavity with general impedance boundary conditions a fourier series method is proposed for the acoustic analysis of a rectangular cavity with impedance boundary conditions arbitrarily specified on any of the walls the sound pressure is expressed as the combination of a three dimensional fourier cosine series and six supplementary two dimensional expansions introduced to ensure accelerate the uniform and absolute convergence rate of the series representation in the cavity including the boundary surfaces the expansion coefficients are determined using the rayleigh ritz method since the pressure field is constructed adequately smooth throughout the entire solution domain the rayleigh ritz solution is mathematically equivalent to what is obtained from a strong formulation based on directly solving the governing equations and the boundary conditions to unify the treatments of arbitrary nonuniform impedance boundary conditions the impedance distribution function on each specified surface is invariantly expressed as a double fourier series expansion so that all the relevant integrals can be calculated analytically the modal parameters for the acoustic cavity can be simultaneously obtained from solving a standard matrix eigenvalue problem instead of iteratively solving a nonlinear transcendental equation as in the existing methods several numerical examples are presented to demonstrate the effectiveness and reliability of the current method for various impedance boundary conditions including nonuniform impedance distributions

eigenmode analysis of arbitrarily shaped two dimensional cavities by the method of point matching in the proposed method as in the boundary element method bem a field problem is solved on its boundary along which nodes are distributed the distinct feature of the proposed method is related to the fact that no interpolation functions between the nodes are required so that the basic collocation method is employed to satisfy a given boundary condition this approach reduces a large amount of numerical calculation induced due to the interpolation functions unlike the method of auxiliary sources mas in the proposed method only the regular functions that have finite functional values everywhere including the origins are employed as basis functions as a result the proposed method does not need the auxiliary surface the construction of which is a troublesome work because its form dimension shifting etc strongly influence the accuracy of the mas the eigenvalues calculated by the proposed method rapidly converge to the exact values thanks to the simplicity of the method moreover the method gives mode shapes successfully without using the interpolation functions between the nodes the efficiency and accuracy of the method are verified through several applications

acoustical cavity excitation a tutorial exposition is given of cavity excitations and it is shown that relatively simple analytical solutions for the excitation of acoustic fields in cavities with lossy walls can be obtained for a broad class of such cavities the standard analytic procedure whereby one expresses the acoustic field as a sum over the eigenmodes of the cavity is reviewed and its limitations are pointed out an alternate procedure is to seek to make use of green s functions depending on a single coordinate and to not express such green s functions as sums over eigenfunctions instead one expresses the green s function as a closed form function that has a discontinuous slope at the value of the coordinate corresponding to the source location although the class of cavities for which this approach is possible is limited it is sufficiently broad to be of intrinsic interest and the results may be helpful to those seeking benchmark solutions and analytic insight displayed examples are for cavities whose shapes are spheres and cylinders with interior monopole dipole line and distributed sources the formulation yields simple expressions for the total power absorbed by the cavity walls

relating diagnosability strong diagnosability and conditional diagnosability of strong networks an interconnection network s diagnosability is an important measure of its self diagnostic capability based on the classical notion of diagnosability strong diagnosability and conditional diagnosability were proposed later to better reflect the networks self diagnostic capability under more realistic assumptions in this paper we study a class of interconnection networks called strong networks which are n regular n 1 connected and with cn number no more than n 3 we build a relationship among the three diagnosability measures for strong networks under both pmc and rm mm ast models given a strong network g with diagnosability t we prove that g is strongly t diagnosable if and only if g s conditional diagnosability is greater than t a simple check can show that almost all well known regular interconnection networks are strong networks the significance of this paper s result is that it reveals an important relationship between strong and conditional diagnosabilities and the proof of strong diagnosability for many interconnection networks under rm mm ast or pmc model is not necessary if their conditional diagnosability can be shown to be strictly larger than their diagnosability

an approach to conditional diagnosability analysis under the pmc model and its application to torus networks a general technique is proposed for determining the conditional diagnosability of interconnection networks under the pmc model several graph invariants are involved in the approach such as the length of the shortest cycle the minimum number of neighbors p resp p over all p vertex subsets resp cycles and a variant of connectivity called the r super connectivity an n dimensional torus network is defined as a cartesian product of n cycles ck1 ckn where ckj is a cycle of length kj for 1 j n the proposed technique is applied to the two or higher dimensional torus networks and their conditional diagnosabilities are established completely the conditional diagnosability of every torus network g is equal to 4 g 1 excluding the three small ones c3 c3 c3 c4 and c4 c4 in addition p g as well as 4 g is derived for 2 p 4 and the r super connectivity is also derived for 1 r 3

diagnosable evaluation of dcc linear congruential graphs under the pmc diagnostic model a dcc disjoint consecutive cycles linear congruential graph g f n consists of n nodes and is generated by a set of linear functions f with special properties it was proved that g f n is a 2t regular graph and has connectivity 2t where t f and 1 3 and 2

conditional diagnosability of the round matching composition networks the conditional diagnosability of many well known networks has been explored in this paper we analyze the conditional diagnosability of a family of networks called the round matching composition networks which are a class of networks composed of r r 4 components of the same order linked by r perfect matchings applying the result we determine the conditional diagnosability of the k ary n cubes and the recursive circulant graphs under the pmc model and the mm model respectively

the g good neighbor conditional diagnosability of hypercube under pmc model abstract processor fault diagnosis plays an important role in multiprocessor systems for reliable computing and the diagnosability of many well known networks has been explored for example hypercubes crossed cubes mobius cubes and twisted cubes of dimension n all have diagnosability n the conditional diagnosability of n dimensional hypercube q n is proved to be 4 n 2 1 under the pmc model in this paper we study the g good neighbor conditional diagnosability of q n under the pmc model and show that it is 2 g n g 2 g 1 for 0 g n 3 the g good neighbor conditional diagnosability of q n is several times larger than the classical diagnosability

on the conditional diagnosability of matching composition networks the conditional diagnosability of interconnection networks has been studied by using a number of ad hoc methods recently gathering various ad hoc methods developed in the last decade a unified approach was developed and this approach was used to find the conditional diagnosability of many interconnection networks in this paper we study the conditional diagnosability of matching composition networks including those that are not triangle free

a kind of conditional vertex connectivity of cayley graphs generated by 2 trees let g be a graph then t v g is called an r k vertex cut if g t is disconnected and each vertex in v g t has at least k neighbors in g t the size of a smallest r k vertex cut is the r k vertex connectivity of g and is denoted by k k g in this paper we determine the numbers k 1 and k 2 for cayley graphs generated by 2 trees including the popular alternating group graphs

new explicit solutions in acoustics of closed spaces on the basis of divergent series a new approach to the acoustics of closed spaces is developed that involves solutions for polygonal shapes in explicit form it is shown that exact solutions can be constructed for polygonal geometries where all the interior angles are equal to n n is an integer it is stated that the set of such polygons consists of the rectangle known result and three types of triangles some new explicit formulas are obtained for the eigenfrequencies of the triangles it is demonstrated that the proposed technique also permits an exact representation of the impulse response function for the geometries described

on conditional fault tolerance and diagnosability of hierarchical cubic networks fault tolerance is especially important for interconnection networks since the growing size of networks increases their vulnerability to component failures a classical measure for the fault tolerance of a network in the case of vertex failures is its connectivity given a network based on a graph g and a positive integer h the r h connectivity of g is the minimum cardinality of a set of vertices in g if any whose deletion disconnects g and the minimum degree of every connected component is at least h this paper investigates the r h connectivity h 1 2 of the hierarchical cubic network hcn n n 2 and shows that 1 hcn n 2 n 2 hcn n 4 n 4 respectively furthermore the paper establishes the conditional diagnosability of hcn n under the pmc diagnostic model

the conditional fault diagnosability of n k star graphs abstract the growing size of the multiprocessor system increases its vulnerability to component failures it is crucial to locate and replace the faulty processors to maintain a system s high reliability the fault diagnosis is the process of identifying faulty processors in a system through testing this paper shows that the largest connected component of the survival graph contains almost all the remaining vertices when a lot of faulty vertices occur in the n k star graph s n k n 5 k 3 n k 2 based on this fault resiliency it establishes the conditional fault diagnosability of s n k under the comparison model

applications of waveguide type base functions for the eigenproblems of two dimensional cavities an acoustic modal analysis technique useful for two dimensional cavities of general shapes is developed as an extension of the technique developed earlier by kim and kang j acoust soc am 99 2938 2946 1996 for the modal analysis of convex membranes both neumann and dirichlet problems can be solved with the present choice of cosinusoidally varying waveguide type functions instead of sinusoidally varying waveguide type functions adopted earlier j acoust soc am 99 2938 2946 1996 a cavity subdivision technique for general nonconvex cavities overcomes the limitation of the earlier technique useful only for convex cavities successful applications of the present technique are demonstrated in some practical two dimensional cavity problems the present technique is potentially a very efficient technique in handling higher eigenmodes

on g extra conditional diagnosability of hierarchical cubic networks abstract connectivity and diagnosability are important parameters in measuring the fault tolerance and reliability of interconnection networks given a graph g and a non negative integer g the g extra connectivity of g denoted by g g is the minimum cardinality of a set of vertices of g if it exists whose deletion disconnects g and every remaining component has more than g vertices the g extra conditional diagnosability t g g of a graph g is the maximum value of t such that g is g extra conditional t diagnosable the n dimensional hierarchical cubic network h c n n was proposed by ghose and desai 12 as a hypercube based topology while preserving its attractive features for n 5 we first determine g h c n n for 0 g n 1 then establish t g h c n n under the pmc model for 0 g n 1 and t g h c n n under the mm model for 0 g n 1 3 respectively

acoustic eigenfrequencies in a spheroidal cavity with a concentric penetrable sphere the acoustic eigenfrequencies fnsm in a spheroidal cavity containing a concentric penetrable sphere are determined analytically for both dirichlet and neumann conditions in the spheroidal boundary two different methods are used for the evaluation in the first the pressure field is expressed in terms of both spherical and spheroidal wave functions connected with one another by well known expansion formulas in the second a shape perturbation method this field is expressed in terms of spherical wave functions only while the equation of the spheroidal boundary is given in spherical coordinates the analytical determination of the eigenfrequencies is possible when the solution is specialized to small values of h d 2r2 h 1 with d the interfocal distance of the spheroidal boundary and 2r2 the length of its rotation axis in this case exact closed form expressions are obtained for the expansion coefficients gnsm 2 and gnsm 4 in the resulting relation fnsm h fns 0 1 h2gnsm 2 h4gnsm 4 o h6

classification of proper bases of rational vector spaces minimal macmillian degree bases abstract the algebraic structure of the get of all proper rational vectors contained in a given rational vector space x1d4af s is shown to be that of a noothorian pr s motiulo m ropf pr a is the ring of proper rational functions the proper submodules mt of m form an ascending chain of submodules partially ordered by an invariant of mt defined as the valuation at s ofmt tho various bases of mt are examined and classified according to their property of column reduceness at s the concept of a prime column reduced at basis of mt is introduced it is shown that tho prime bases of mt can be further classified by their macmillan degrees and the existence of minimal macmillan degree bases for mt is established a prime and minimal macmillan degree basis ofmt extends forney s concept of a minimal polynomial basis of s for the rpr mt module the macmillan degrees of the columns of such bases form a set of invariants mt for which are defined as the x1d4af s generalized invariant dynamicol indices o