multiple positive solutions for a quasilinear elliptic equation involving singular potential and critical sobolev exponent in this paper we are concerned with a class of quasilinear elliptic equation in which the potential is singular at the origin and the nonlinearity is p sublinear near the origin by the ekeland variational principle and mountain pass theorem we prove that there exist at least two positive solutions under suitable assumptions on the potential and the nonlinearity

multi mode trimming of imperfect thin rings using masses at pre selected locations abstract this paper presents a practical method for trimming the natural frequencies of an initially imperfect ring to simultaneously eliminate certain of the frequency splits present compared with previous work the novel feature of this method is that the trimming masses are positioned at pre selected locations on the ring the basis for the proposed method is the concept of equivalent imperfection mass which allows any imperfect ring to be considered as a perfect ring with equivalent imperfection masses attached by considering this trimming problem it is deduced that it is possible to trim n pairs of modes simultaneously by removing a minimum of 2 n trimming masses at particular locations around the ring by positioning the trimming masses at pre selected locations it is shown that a simple set of trimming masses can be calculated easily and from this set an infinite number of solution sets can be found methods for generating these sets are outlined for the trimming of both a single and a dual pair of modes in practice it is likely that the trimming masses will be spaced regularly for this special case it is found that it is not possible to trim all single and dual mode pairs with any arrangement of masses validation of the derived simple solution set and the proposed procedure to generate further sets is achieved by studying a number of theoretical examples

free vibrations of elliptical rings with circumferentially variable thickness abstract this paper deals with the in plane free vibration of rings with a nominally elliptical centreline results are presented for rings of constant axial length that have a rectangular cross section the radial thickness of which is constant or has a simple analytically defined circumferential variation additionally and for the first time the effects of small variations in in plane profile such as those arising in practical rings due to manufacturing tolerances are considered the problem is tackled using an approach in which the true middle surface is determined numerically from the outer and inner surface profiles which can be defined either by exact analytical expressions or in a more general way using fourier series the rayleigh ritz method is used to obtain the natural frequencies and mode shapes results are presented for a range of cases including some that have previously been studied by other authors and some that have not the effects on frequency splitting due to profile variations and the aspect ratio of the ellipse are emphasized results obtained using the developed numerical approach show excellent agreement with finite element predictions

sklyanin algebras and hilbert schemes of points we construct projective moduli spaces for torsion free sheaves on noncommutative projective planes these moduli spaces vary smoothly in the parameters describing the noncommutative plane and have good properties analogous to those of moduli spaces of sheaves over the usual commutative projective plane p2 the generic noncommutative plane corresponds to the sklyanin algebra s skl e constructed from an automorphism of infinite order on an elliptic curve e p2 in this case the fine moduli space of line bundles over s with first chern class zero and euler characteristic 1 n provides a symplectic variety that is a deformation of the hilbert scheme of n points on p2 e

multi mode trimming of imperfect rings abstract this paper proposes a method for trimming the natural frequencies of an imperfect ring to simultaneously eliminate certain of the frequency splits present initially the effect of the addition of a number of imperfection masses on a perfect ring is considered this is achieved by using a rayleigh ritz approach in which it is assumed that the mode shapes are identical to those of a perfect ring by considering the inverse the so called trimming problem it is deduced that it is possible to trim n pairs of modes simultaneously by removing a minimum of n trimming masses at particular locations around the ring to calculate the trimming mass locations it is necessary to solve n non linear algebraic equations once this has been achieved the magnitude of the trimming masses can be calculated easily for the special case of trimming a single pair of modes analytic solutions for the magnitude and position of the single required trimming mass are available to trim two pairs of modes it is shown that a simple analytic relationship exists between the angular positions of the two required trimming masses and that the magnitude of these masses can be obtained easily to trim more pairs of modes numerical techniques are required and for this purpose a numerical procedure is proposed validation of the derived analytic results and the proposed numerical procedure is achieved by studying a number of theoretical examples

solutions of the quasilinear elliptic problem with a critical sobolev hardy exponent and a hardy type term abstract in the present paper the quasilinear elliptic problem with a critical sobolev hardy exponent and a hardy type term is considered by means of a variational method the existence of nontrivial solutions to the problem is obtained

solutions of a quasilinear elliptic problem involving a critical sobolev exponent and multiple hardy type terms in the present paper a quasilinear elliptic problem with a critical sobolev exponent and multiple hardy type terms is considered by means of a variational method the existence of positive solutions of the problem is obtained

multiple positive solutions for a semilinear elliptic equation with critical sobolev exponent in this paper we study a class of semilinear elliptic equations with hardy potential and critical sobolev exponent by means of the ekeland variational principle and mountain pass theorem multiple positive solutions are obtained

noncommutative projective schemes an analogue of the concept of projective scheme is defined for noncommutative n graded algebras using the quotient category c of graded right a modules modulo its full subcategory of torsion modules we define proj a c a s where a is the object corresponding to the module aa and s is the autoequivalence defined by the shift of degrees the triples equivalent to proj a for a right noetherian graded algebra a are characterized in terms of a condition on extensions

elliptic problems with critical exponents and hardy potentials this paper is devoted to the existence of positive solutions of a dirichlet problem with critical exponent and a singular potential under various assumption on the domain omega which include some kinds of unbounded domains we prove the existence of ground states and of symmetric solutions c 2002 elsevier science usa all rights reserved

measurement and analysis of modulated doublet mode response in mock bladed disks the steady state response of spatially modulated doublet modes that occur in low count flexible bladed disks is investigated for the case in which the structure is driven by a harmonic travelling wave excitation source finite element simulation and modal testing of prototypical bladed disk structures demonstrate the presence of particular wavenumbers beyond the base number of nodal diameters which contaminate and distort the appearance of certain doublet modes the manner in which the natural frequency and wavenumber content of such modes shift and split as functions of the number of blades and their span angle is discussed in the light of a companion perturbation analysis for rotationally periodic structures resonance conditions are established and verified through simultaneous measurements made with a spin test stand using sensors that are placed in the rotating structure and stationary excitation frames of reference the travelling wave response components of a repeated frequency doublet mode are shown to propagate either in the same or opposite direction as the excitation source depending on whether certain algebraic relationships between the excitation order the base number of nodal diameters and the contamination wavenumbers are satisfied to the extent that such components can travel at different phase speeds and directions relative to one another the placement of sensors on the structure can be optimized to best measure the response amplitude conversely other placements can result in submaximal measurement of peak vibration amplitude over the structure

on the statistics of natural frequency splitting for rings with random mass imperfections abstract this paper considers the statistical distribution of natural frequency splits for an initially perfect ring with different types of random mass imperfection the analysis used to derive analytical expressions for the natural frequency splits is based on a rayleigh ritz approach in which it is assumed that the mode shapes of the imperfect rings are identical to those of a perfect ring the types of random mass imperfection investigated are i random harmonic variations in the mass per unit length around the circumference of the ring ii the attachment of random point masses at random locations on the ring and iii the attachment of random point masses at uniformly spaced positions on the ring for case i it is found that the frequency splits always have a rayleigh distribution for case ii an expression for the statistical distribution is deduced which tends to a rayleigh distribution as the number of attached masses increases for case iii it is found that the frequency split distribution is dependent upon the mode considered and the number of attached masses and that in some situations the frequency splits have a half gaussian i e non rayleigh distribution

the in plane vibration of thin rings with in plane profile variations part ii application to nominally circular rings abstract geometric profile variations always exist in nominally circular rings due to limitations in the manufacturing processes such profile variations are known to lead to frequency splitting between pairs of modes which are degenerate in a perfect ring in this paper the effects of circumferential profile variations on the in plane vibration characteristics of such rings are studied using a numerical method the inner and outer ring surfaces are described in a very general way by fourier series and the rayleigh ritz method is used to obtain the natural frequencies and mode shapes results are presented for a number of example cases which include single and multiple harmonic variations in profile the relationship between the patterns of frequency splitting and the harmonic content of the ring profile is investigated and the most important causes of frequency splitting are identified

in plane free vibration of a single crystal silicon ring in this paper the natural frequencies and the associated mode shapes of in plane free vibration of a single crystal silicon ring are analyzed it is found that the si 1 1 1 ring is two dimensionally isotropic in the 1 1 1 plane for elastic constants but three dimensionally anisotropic while the si 1 0 0 ring is fully anisotropic hamilton s principle is used to derive the equations of vibration which is a set of partial differential equations with coefficients being periodic in polar variable expressing the radial and tangential displacements in sinusoidal form with non predetermined amplitudes and through the integration with respect to the circumferential variable the original governing equations in partial differential form can be converted into the amplitude equations in ordinary differential form the exact expressions for frequencies and mode shapes are obtained it is found that for si 1 0 0 rings the frequencies of a pair of modes which are equal for an isotropic ring split due to the anisotropic effect only for the second in plane vibration mode the phenomena of frequency splitting and degenerate modes can be proved either based on the conservation of averaged mechanical energy or by the concept of crystallographic symmetry groups when the single crystal silicon is replaced by the polycrystalline silicon which is isotropic in elastic constants the derived equations for frequencies correctly predict the vanishing of the phenomenon of frequency splitting

a note on the sign changing solutions to elliptic problems with critical sobolev and hardy terms let rn be a bounded domain such that 0 n 7 2 2nn 2 we obtain existence of sign changing solutions for the dirichlet problem u u x 2 u 2 2u u on u 0 on for suitable positive numbers and

generic noncommutative surfaces abstract we study a class of noncommutative surfaces and their higher dimensional analogs which come from generic subalgebras of twisted homogeneous coordinate rings of projective space such rings provide answers to several open questions in noncommutative projective geometry specifically these rings r are the first known graded algebras over a field k which are noetherian but not strongly noetherian in other words r k b is not noetherian for some choice of commutative noetherian extension ring b this answers a question of artin small and zhang the rings r are also maximal orders but they do not satisfy all of the conditions of artin and zhang in particular they satisfy the 1 condition but not i for i 2 answering a question of stafford and zhang and a question of stafford and van den bergh finally we show that the noncommutative scheme r proj has finite global dimension

solutions for semilinear elliptic equations with critical exponents and hardy potential abstract in this paper we answer affirmatively an open problem cf theorem 4 in ferrero and gazzola j differential equations 177 2001 494 let 0 be an open bounded domain r n n 5 and assume that 0 n 2 2 2 n 2 n 2 then for all 0 there exists a nontrivial solution with critical level in the range 0 1 n s n 2 for the problem u u x 2 u u 2 2 u in u 0 on

cohomology of schematic algebras letrbe a positively graded noetheriank algebra and let proj r r grbe the quotient category of the category of gradedr modulesr grmodulo those of finite length ifris schematic cf f van oystaeyen and l willaert j pure appl algebra104 1995 109 122 and has finite global dimension we show that the cohomology groups ofr as defined in m artin and j j zhang adv in math 109 1994 228 287 are finite dimensional furthermore we introduce cech cohomology groups for schematic algebras and prove that they coincide with the usual graded cohomology groups these generalized cech cohomology groups are still manageable we illustrate this with an example

a new method for in plane vibration analysis of circular rings with widely distributed deviation abstract a new analytical method was developed to predict the in plane mode shapes and the natural frequencies of a ring with widely distributed deviation the laplace transform was used to find the exact solution of eigenvalue problem without assuming any trial functions and finite elements the widely distributed deviation was effectively formulated in the theory using gauss legendre quadrature the validity of the proposed method was examined through finite element analysis and modal test the effects of partial change of the density the stiffness and the thickness on the natural frequencies of the ring were investigated