the oscillation and asymptotically monotone solutions of second order quasilinear differential equations abstract we offer sufficient conditions for the oscillation of all solutions of the perturbed quasilinear differential equation a t y 1 y q t y p t y y as well as for the existence of a positive monotone solution of the damped differential equation a t y 1 y b t y 1 y h t y 0 where 0 examples that dwell upon the importance of our results are also included

oscillation and nonoscillation theorems for second order nonlinear difference equations abstract some new oscillation and nonoscillation theorems are obtained for second order nonlinear difference equations a p n a x n q n a x n f n x n b x n which generalize and improve all results proved by popenda in j math anal appl 123 1987 34 48

optimal reconstruction systems for erasures and for the q potential abstract in this paper we introduce the q potential as an extension of the benedetto fickus frame potential defined on general reconstruction systems and show that protocols are the minimizers of this potential under certain restrictions we extend recent results of b g bodmann on the structure of optimal protocols with respect to 1 and 2 lost packets where the worst normalized reconstruction error is computed with respect to a compatible unitarily invariant norm we finally describe necessary and sufficient spectral conditions that we call q fundamental inequalities for the existence of protocols with prescribed properties by relating this problem to klyachko s and fulton s theory on sums of hermitian operators

on the duality of fusion frames the fusion frames were considered recently by p g casazza g kutyniok and s li in connection with distributed processing and are related to the construction of global frames from local frames in this paper we give new results on the duality of fusion frames in hilbert spaces

asymptotic and oscillatory behavior of solutions of certain second order nonlinear difference equations mingshu peng department of applied mathematics beijing institute of technology beijing 100081 p r china qianli xu department of mathematics yiyang teachers college yiyang 413049 p r china lihong huang department of applied mathematics hunan university changsha 410082 p r china weigao ge

duality in reconstruction systems we consider the notion of finite dimensional reconstructions systems rs s which includes the fusion frames as projective rs s we study erasures some geometrical properties of these spaces the spectral picture of the set of all dual systems of a fixed rs the spectral picture of the set of rs operators for the projective systems with fixed weights and the structure of the minimizers of the joint potential in this setting we give several examples

generating relations for reducing matrices i ordinary representations auxiliary groups are constructed that make it possible to reduce the multiplicity problem and to derive consistent generating relations for the elements of reducing matrices three examples are worked out to illustrate the general scheme

coefficients of fractional parentage for u m p n q u m n u p q and u m n u m u n the outer product reduction coefficients orc which reduce the representation rep induced from the irreps of the permutation groups s f1 and s f2 into the irreps of s f1 f2 are shown to be the indirect coupling coefficients for the u m p n q u m n u p q irreducible basis the non standard orc for reducing the rep induced from the non standard irreps of s f13 s f1 s f3 and s f24 s f2 s f4 into that of s f s f12 s f34 with fij fi fj f f12 f34 are identified with the u f u f12 u f34 clebsch gordan coefficient for the special gel fand bases of u f12 and u f34 the u m p n q u m n u p q cfp as well as its special case the u m p u m u p cfp are identified with the s f s f12 s f34 outer product isoscalar factor the u m n u m u n cfp are obtained from the u m n u m u n cfp by simply changing all the partition labels for u n into their conjugates interchanging rows with columns and taking into account a phase change the cfp can be calculated from the orc numerical values of the one body cfp for systems with up to six particles are tabulated

symmetric and unitary group representations i duality theory an extension of the schur weyl duality connecting the representations of the symmetric and unitary groups is given the schur weyl basis is constructed using annihilation and creation operators three factorisation lemmas are derived their importance lies in the fact that they relate the phase freedoms within the racah wigner algebra of the symmetric groups and the unitary groups extensions of the regge symmetries are also given these are expressed in five duality relations

some 6 j symbols for symplectic and orthogonal groups by cerkaski s method the method devised by cerkaski 1987 has been used for the group sp 2n to find some 6 j symbols for which a single multiplicity index is required this approach possesses some advantages over that based on racah s techniques for so 3 the formulae for the 6 j symbols are comparatively simple and they automatically exhibit the symmetries of jucys 1973 elementary means are used to show that the eigenvalues of casimir s operator c2 for sp 2n are identical to those of c2 for o 2d if the authors take d n and replace the irreps of sp 2n by their transposed counterparts for o 2d where d is integral or half integral this enables the formulae for sp 2n to be used to generate expressions for some 6 j symbols for the orthogonal groups o 2d and so 2d illustrations are provided by so 2l 1 and o 4

a second order nonlinear difference equation oscillation and asymptotic behavior discrete analogues are investigated for well known results on oscillation growth and asymptotic behavior of solutions of y q t y 0 for q t 0 and for q t 0 the analogue of atkinson s oscillation criterion is shown to be true for 2yn 1 qn yn 0 but the analogue for atkinson s nonoscillation criterion is shown to be false

asymptotic and oscillatory behavior of solutions of general nonlinear difference equations of second order abstract the asymptotic and oscillatory behavior of solutions of some general second order nonlinear difference equations of the form a n h y n 1 y n p n y n q n 1 f y n 1 0 n z is studied oscillation criteria for their solutions when pn is of constant sign are established results are also presented for the damped forced equation a n h y n 1 y n p n y n q n 1 f y n 1 e n n z examples are inserted in the text for illustrative purposes

oscillation theorems for second order advanced functional difference equations abstract we consider the second order advanced functional difference equation a n n p n g n 0 where a n 0 s n 0 1 a s p n 0 p n 0 g n n 1 g n is a monotone increasing integer sequence we obtain some new oscillation criteria through an appropriate riccati equation

the oscillation of solutions of difference equations abstract using the solutions of a suitable inequality we shall discuss the asymptotic properties of the solutions of a generalized emden fowler difference equation in the absence of specified constant sign solutions of certain inequalities oscillatory behavior of all solutions of this equation are also deduced

on the existence of real clebsch gordan coefficients the question of the possible general existence of real clebsch gordan coefficients coupling coefficients for compact groups is considered criteria are established for a group to be susceptible to the classical rotation group approach in which a choice of standard irreducible matrix representations is made such that there is a fixed inner automorphism of the group carrying all standard representations into their complex conjugate in connection with a generalization of this approach the concept of quasi ambivalence of a group is shown to be relevant

calculation of primitive 6 j symbols the 6 j symbols of a group are independent of the subgroup chain chosen to define the basis states the authors present an improved algorithm for calculating the primitive 6 j symbols for a compact group with a faithful irrep by recursive building up using only the kronecker product rules and the general properties of 6 j previously one has sometimes needed to search for the useful equations by systematically trying all equations which involve unknown 6 j the authors show that the primitive 6 j may all be easily solved in terms of a subclass the core 6 j they discuss how the core 6 j can usually be solved proving that the method is complete for so3 the authors conjecture that the algorithm is complete for all groups

su mn su m su n isoscalar factors and s f1 f2 s f1 s f2 isoscalar factors a simple relation is found between the isoscalar factor isf of the unitary group and those of the permutation group i e the su mn su m su n isf is equal to the s f1 f2 s f1 s f2 isf since the values of s f1 f2 s f1 s f2 isf are independent of m and n one arrives at an important conclusion that the values of su mn su m su n isf are also independent of m and n therefore they can be calculated for all m and n by a single stroke instead of one m and one n at a time an eigenfunction metho for evaluating the su mn su m su n isf is given which can be easily translated into a computer program

3jm and 6j tables for some bases of su6 and su3 tables of 6j symbols for su6 and su3 and tables of 3jm factors for su6 contains implies su2 su3 su3 contains implies u1 su2 and su3 contains implies so3 are presented the tables are computer produced using a program that implements the building up principle in a general form these tables are useful for calculations in high energy nuclear and solid state physics some other tabulations contain errors and none uses all the symmetries available the n independence of sun results is discussed by using the various symmetric group unitary group duality relations

riccati type transformations for second order linear difference equations abstract oscillation and comparison theorems for a linear homogeneous second order difference equation are proved by employing various equivalent non linear equations obtained by means of transformations analogous to the riccati transformation for ordinary differential equations

coupling subduction and induction transformations for group representations the basis of a representation space of any group may be chosen according to its decomposition into representations of the group and a subgroup chain coupling recoupling subduction and resubduction factors are each shown to arise as a result of schur s lemmas and a particular group subgroup structure when induction and mackey s subgroup theorem are considered new transformation factors occur those of induction and reinduction key properties of these factors are given together with their relationship with double coset matrix elements

properties of finite dual fusion frames abstract a new notion of dual fusion frame has been recently introduced by the authors in this article that notion is further motivated and it is shown that it is suitable to deal with questions posed in a finite dimensional real or complex hilbert space reinforcing the idea that this concept of duality solves the question about an appropriate definition of dual fusion frames it is shown that for overcomplete fusion frames there always exist duals different from the canonical one conditions that assure the uniqueness of duals are given the relation of dual fusion frame systems with dual frames and dual projective reconstruction systems is established optimal dual fusion frames for the reconstruction in case of erasures of subspaces and optimal dual fusion frame systems for the reconstruction in case of erasures of local frame vectors are determined examples that illustrate the obtained results are exhibited

on the construction of wavefunctions in the six quark system a method for calculating the fractional parentage coefficients for many quark systems in the scheme ucst 12 contains implies ucs 6 ut 2 contains implies uc 3 us 2 ut 2 based on the complementarity of the permutation sn and unitary u n groups is developed the scalar factors of the clebsch gordan coefficients for the chain of groups u mn contains implies u m u n are shown to be independent of the ranks mn and m and n of the groups and to be determined by the young schemes associated with them tables of fractional parentage coefficients for low states of the six quark 6q system are presented

analysis of the outer product for the symmetric group expressions are derived to write the basis vectors for an irreducible representation of the symmetric group in terms of basis vectors for irreducible representations whose outer product yields

simultaneity and reality of u n and su n 3jm and 6j symbols a simple proof is given of the simultaneous existence of a real set of 3jm and 6j symbols for the unitary groups u n and su n including the case of mixed tensor representations we observe that simultaneity is incompatible with the conventional permutation symmetries for u n with n 3 the relevance of these results to the schur weyl duality of u n with the symmetric groups sl is discussed

scalar factors for non canonical subgroup restriction of unitary group a procedure has been outlined for obtaining the scalar factors reduced wigner coefficients of the unitary group u nm contains implies u n x u m this has been done at the permutation group level for sn down arrow sn x sn and the equality between the scalar factors of these two groups has been exploited it has been shown that the scalar factors can be uniquely expressed in terms of the inner and outer product coupling coefficients of the permutation group the ambiguity due to multiplicity is resolved at the level of the above coefficients

kronecker products for compact semisimple lie groups a review is given of the application of s function techniques to the evaluation of kronecker products of irreducible representations of compact semisimple lie groups explicit formulae are derived for all irreducible representations of all such groups recent developments involving composite young diagrams are brought to fruition and the vexed problem of so2k is dealt with completely new branching rules for the classical groups are given in an appendix these are exploited in the evaluation of kronecker products by means of a technique which is applied to both classical and exceptional groups a discussion is made of various modification rules which are needed to express the final results in standard form

focal boundary value problems for nonlinear difference equations ii on considere l unicite des solutions impliquant l existence des solutions de problemes aux valeurs limites de type focal pour l equation aux differences non lineaire d ordre n u t n f t u t u t n 1 ou f a r n r est continue et l equation u n 1 f t u 1 u n peut etre resolue pour u 1 comme fonction continue des variables t u 2 u u 1

algebraic formulas for some nontrivial un 6j symbols and umn um un 3jm symbols we give tables of algebraic formulas for some nontrivial 6j symbols and 3jm symbols of the unitary groups the tables demonstrate that the building up method can be used successfully to obtain the rank dependence of unitary group j and jm symbols to emphasize the rank dependent nature of this calculation we have employed the composite young tableaux notation or back to back notation to label the unitary group irreps in using this notation the transpose conjugate symmetry of the corresponding composite young diagram leads to a new symmetry of the unitary group 6j and 3jm symbols the transposition of the groups um and un gives rise to a further symmetry of the 3jm symbols of umn um un

bounded and monotone properties of solutions of second order quasilinear forced difference equations abstract the authors consider the quasilinear difference equation p n 1 y n 1 q n y n r n and obtain results on the asymptotic behavior of solutions of including sufficient conditions for all solutions to be bounded or unbounded some results on the existence and behavior of nonincreasing solutions of are also obtained examples are inserted to illustrate the results

g frames and g riesz bases g frames are generalized frames which include ordinary frames bounded invertible linear operators as well as many recent generalizations of frames e g bounded quasi projectors and frames of subspaces g frames are natural generalizations of frames and provide more choices on analyzing functions from frame expansion coefficients we give characterizations of g frames and prove that g frames share many useful properties with frames we also give a generalized version of riesz bases and orthonormal bases as an application we get atomic resolutions for bounded linear operators

optimal dual frames for erasures for any given frame for the purpose of encoding in a finite dimensional hilbert space we investigate its dual frames that are optimal for erasures for the purpose of decoding we show that in general the canonical dual is not necessarily optimal moreover optimal dual frames are not necessarily unique we present some sufficient conditions under which the canonical dual is the unique optimal dual frame for the erasure problem as an application we get that the canonical dual is the only optimal dual when either the frame is induced by a group representation or the frame is uniform tight

classification schemes for positive solutions of a second order nonlinear difference equation classification schemes for eventually positive solutions of a class of second order nonlinear difference equations are given in terms of their asymptotic magnitudes and necessary as well as sufficient conditions for the existence of these solutions are also provided

a duality consistent phase convention for complex conjugation in sun we have argued that a simple phase convention for all involutive mappings can be chosen so the algebra of inner product couplings in sun is identical to that for outer products in sl pure real phases with a single 1 entry in any row or column can represent the permutation matrix for transposition in order of the two component irreducible representations irreps in a binary coupling the matrix for association in sl with respect to the alternating group al the derome sharp matrix and the 1jm factor the latter two are required by complex conjugation in sun in our previous work we have proposed specific prescriptions for assigning the phase under transposition and association and have shown them to be consistent with duality in this work we propose a duality consistent prescription for assigning the phase of the derome sharp matrix and thus of the 1jm factor which is related to it by association