tools in the orbit space approach to the study of invariant functions rational parametrization of strata functions which are equivariant or invariant under the transformations of a compact linear group g acting in a euclidean space n can profitably be studied as functions defined in the orbit space of the group the orbit space is the union of a finite set of strata which are semialgebraic manifolds formed by the g orbits with the same orbit type in this paper we provide a simple recipe to obtain rational parametrizations of the strata our results can be easily exploited in many physical contexts where the study of equivariant or invariant functions is important for instance in the determination of patterns of spontaneous symmetry breaking in the analysis of phase spaces and structural phase transitions landau theory in equivariant bifurcation theory in crystal field theory and in most areas where use is made of symmetry adapted functions a physically significant example of utilization of the recipe is given related to spontaneous polarization in chiral biaxial liquid crystals where the advantages with respect to previous heuristic approaches are shown

kronecker products of representations of exceptional lie groups a technique is described for evaluating kronecker products of irreducible representations of an exceptional lie group g the method depends upon the natural embedding in g of a classical lie subgroup h the problem is reduced to that of finding the branching of one irreducible representation of g on restriction to h evaluating kronecker products in h and finally using the modification rules appropriate to g

nontrivial solutions of m point boundary value problems for singular second order differential equations with a sign changing nonlinear term this paper concerns the existence of nontrivial solutions for the following singular m point boundary value problem with a sign changing nonlinear term lu t h t f t u 0 0 is a sign changing continuous function and may be unbounded from below by applying the topological degree of a completely continuous field and the first eigenvalue and its corresponding eigenfunction of a special linear operator some new results on the existence of nontrivial solutions for the above singular m point boundary value problem are obtained an example is then given to demonstrate the application of the main results the work improves and generalizes the main results of g han y wu nontrivial solutions of singular two point boundary value problems with sign changing nonlinear terms j math anal appl 325 2007 1327 1338 j sun g zhang nontrivial solutions of singular superlinear sturm liouville problem j math anal appl 313 2006 518 536

existence theorems for a second order three point boundary value problem abstract letf 0 1 r 2 r be continuous ande c 0 1 let i 0 1 ai r all of theai s having the same sign i 1 2 m 2 0 x t f t x t x t e t t 0 1 e x 0 0 x 1 i 1 m 2 a i x i bc m we discuss existence theorems for the problem e bcm under some sign condition offabout the origin our analysis is based on a nonlinear alternative of leray schauder

inverse branching rules a method is given for inverting branching rules which is based on a technique for computing kronecker products due to king 1981 this approach simplifies the calculation of plethysms and branching rules of representations of exceptional groups it may also be applied to basic spin plethysms

symmetric positive solutions to singular system with multi point coupled boundary conditions in this paper we study the existence and multiplicity of symmetric positive solutions for a nonlinear system with multi point coupled boundary conditions the arguments are based upon a specially constructed cone and the fixed point index theorem in cones an example is then given to demonstrate the applicability of our results

numerical computation of cross coupled algebraic riccati equations related to h constrained lqg control problem abstract in this paper the numerical algorithm for solving the state and output feedback h constrained lqg control problem is investigated although the equations that have to be solved to design the controller consist of the nonlinear cross coupled algebraic riccati equations cares it is newly proven that both the uniqueness and the positive semidefiniteness of the iterative solutions can be guaranteed when disturbance attenuation level is sufficiently large the computational examples are given to demonstrate the efficiency of the proposed algorithm

lqg control with an h sup infinity performance bound a riccati equation approach an lqg linear quadratic gaussian control design problem involving a constraint on h sup infinity disturbance attenuation is considered the h sup infinity performance constraint is embedded within the optimization process by replacing the covariance lyapunov equation by a riccati equation whose solution leads to an upper bound on l sub 2 performance in contrast to the pair of separated riccati equations of standard lqg theory the h sup infinity constrained gains are given by a coupled system of three modified riccati equations the coupling illustrates the breakdown of the separation principle for the h sup infinity constrained problem both full and reduced order design problems are considered with an h sup infinity attenuation constraint involving both state and control variables an algorithm is developed for the full order design problem and illustrative numerical results are given

positive solutions for n order m point p laplacian operator singular boundary value problems in this paper we study the existence of positive solutions for the nonlinear n order m point p laplacian operator singular boundary value problems by using the method of defining operator by the reverse function of green function and the fixed point index theory the existence of positive solution and many positive solutions for nonlinear third order two point singular boundary value problem with p laplacian operator are obtained

symmetry breaking patterns for third rank totally antisymmetric representations of unitary groups some possible symmetry breaking patterns for unitary group gauge theoretic models based on higgs scalars in the third rank totally antisymmetric tensor representations of u n are studied the critical points are expressed in terms of a single parameter xi such that 1 n or 9 and remarks are made concerning a conjecture of michel 1979

existence of at least three solutions of a second order three point boundary value problem abstract we study existence of at least three solutions in the presence of two lower and two upper solutions of some second order nonlinear three point boundary value problem of the type x f t x x t i 0 1 x 0 0 x 1 x 0 1 0 1 the growth of f with respect to x is allowed to be quadratic we use some degree theory arguments to get the multiplicity result

orbit spaces of low dimensional representations of simple compact connected lie groups and extrema of a group invariant scalar potential orbit spaces of low dimensional representations of classical and exceptional lie groups are constructed and tabulated we observe that the orbit spaces of some single irreducible representations adjoints second rank symmetric and antisymmetric tensors of classical lie groups and the defining representations of f4 and e6 are warped polyhedrons with locally more protrudent boundaries corresponding to higher level little groups the orbit spaces of two irreducible representations have different shapes we observe that dimension and concavity of different strata are not sharply distinguished we explain that the observed orbit space structure implies that a physical system tends to retain as much symmetry as possible in a symmetry breaking process in appendix a we interpret our method of minimization in the orbit space in terms of conventional language and show how to find all the extrema in the representation space of a general group invariant scalar potential monotonic in the orbit space we also present the criterion to tell whether an extremum is a local minimum or maximum or an inflection point in appendix b we show that the minimization problem can always be reduced to a two dimensional one in the case of the most general higgs potential for a single irreducible representation and to a three dimensional one in the case of an even degree higgs potential for two irreducible representations we explain that the absolute minimum condition prompts the boundary conditions enough to determine the representation vector

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

solvability of a three point nonlinear boundary value problem for a second order ordinary differential equation in case f x u u ps x p x u pjx u with pk 0 l 4 r locally integrable the boundary value problem l lt 1 2 was studied by kiguradze and lomtatidze i the purpose of this paper is to obtain existence and uniqueness theorems for the boundary value problem 1 1 t 1 2 under natural conditions on f using degree theoretic arguments we note that if u is a solution of 1 1 1 2 then there exists at least one 5 e q l such that u c 0 accordingly the boundary value problem

positive solutions for n 1 1 three point boundary value problems with coefficient that changes sign abstract in this paper we establish existence results for positive solutions for the n 1 1 three point boundary value problems consisting of the equation u n a t f u t 0 t 0 1 with one of the following boundary value conditions u 0 u u 1 u u i 0 0 for i 1 2 n 2 and u n 2 0 u n 2 u n 2 1 u n 2 u i 0 0 for i 0 1 n 3 where 0 1 0 0 and a 0 1 r may change sign and r f 0 0 0 is a parameter our approach is based on the leray schauder degree theory this paper is motivated by eloe and henderson nonlinear anal 28 1997 1669 1680

positive solution of singular higher order m point boundary value problem with nonlinearity that changes sign this paper is about a class of singular higher order m point boundary value problems with nonlinearity that changes sign using the fixed point index theory we get a sufficient condition for the existence of positive solution to this problem

the existence of countably many positive solutions for some nonlinear nth order m point boundary value problems in this paper we consider the existence of countably many positive solutions for nth order m point boundary value problems consisting of the equation u n t a t f u t 0 email protected 0 1 with one of the following boundary value conditions u 0 email protected i 1m 2k iu x i u 0 u n 2 0 0 u 1 0 and u 0 0 u 0 u n 2 0 0 u 1 email protected i 1m 2k iu x i where n 2 k i 0 i 1 2 m 2 0 1 and has countably many singularities in 0 12 the associated green s function for the nth order m point boundary value problem is first given and we show that there exist countably many positive solutions using holder s inequality and krasnoselskii s fixed point theorem for operators on a cone

twin solutions of boundary value problems for ordinary differential equations and finite difference equations a new twin fixed point theorem is applied first to obtain the existence of at least two positive solutions for the right focal boundary value problem y f ity 0 0 t 1 y 0 y 1 0 it is applied later to obtain the existence of at least two positive solutions for the analogous discrete right focal boundary value problem 2u k g u k 0 k 0 n u 0 u n 1 0

existence and nonexistence of positive solutions for a class of nth order three point boundary value problems in banach spaces by using the fixed point principle in cone and the fixed point index theory for strict set contraction operator this paper investigates the existence nonexistence and multiplicity of positive solutions for a class of nonlinear three point boundary value problems of nth order differential equations in ordered banach spaces in addition an example is included to demonstrate the main results

multiple positive solutions of multi point boundary value problem for second order impulsive differential equations this paper is devoted to study the existence of multiple positive solutions for the second order multi point boundary value problem with impulse effects the arguments are based upon fixed point theorems in a cone an example is worked out to demonstrate the main results

positive solutions to a system of second order nonlocal boundary value problems abstract in this paper we study the existence of positive solutions to a system of second order nonlocal boundary value problems by using fixed point index theory in a cone our hypotheses imposed on nonlinearities are those which characterize systems of nonlocal boundary value problems and our boundary value conditions are expressed in terms of possibly nonlinear functions of riemann stieltjes integrals thus generalizing and unifying the boundary value conditions in the literature therefore our results cannot be routinely deduced from the ones for single nonlocal problem in the literature

symmetrised powers of rotation group representations the results given permit the unambiguous evaluation of all possible kronecker products of the irreducible representations tensor and spinor of on and son for n 2 nu and n 2 nu 1 a complete resolution of the second and third powers of the basic spinor representations of so2 nu and so2 nu 1 is given together with a prescription for analysing the fourth power of these representations detailed application is made to the enumeration of properties of so10 relevant to grand unified theories and sufficient information given to resolve the fourth power of any representation of so10

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

parisi s mean field solution for spin glasses as an analytic continuation in the replica number parisi s replica symmetry breaking solution for spin glasses is extended to finite replica number n the free energy fp n obtained this way as well as its first two derivatives with respect to n are shown to join the corresponding values in the sherrington kirkpatrick sk solution at a characteristic value ns t where stability breaks down in the latter the continuation composed of the sk branch fsk n for n or ns t and the parisi branch fp n for 0 or n or ns t fulfils the requirements of convexity monotonicity and stability for all n

on global existence of solutions to coupled matrix riccati equations in closed loop nash games presents comparison and global existence results for solutions of coupled matrix riccati differential equations appearing in closed loop nash games and in mixed h sub 2 h sub spl infin type problems convergence of solutions is established for the diagonal case solutions of the corresponding algebraic equations are discussed using numerical examples

positive solutions for a class of boundary value problem with integral boundary conditions in banach spaces this paper investigates the existence and multiplicity of positive solutions for a class of nonlinear boundary value problem of second order differential equations with integral boundary conditions in ordered banach spaces the arguments are based upon a specially constructed cone and the fixed point theory in a cone for strict set contraction operators the nonexistence of a positive solution is also studied

several sufficient conditions of solvability for a nonlinear higher order three point boundary value problem with all derivatives in this paper firstly some errors in the proof of our paper several sufficient conditions of solvability for a nonlinear higher order three point boundary value problem on time scales appl math comput 190 2007 566 575 are pointed and we make the corresponding correction when t r then the more general problem with all derivatives is considered under certain growth conditions on the nonlinearity several sufficient conditions for the existence and uniqueness of nontrivial solution are obtained by using leray schauder nonlinear alternative and banach fixed point theorem

efficient numerical procedures for solving closed loop stackelberg strategies with small singular perturbation parameter in this paper the computation of the linear closed loop stackelberg strategies with small singular perturbation parameter that characterizes singularly perturbed systems sps are studied the attention is focused on a new numerical algorithm for solving a set of cross coupled algebraic lyapunov and riccati equations calre it is proven that the new algorithm guarantees the local quadratic convergence a numerical example is solved to show a reduction of the average cpu time compared with the existing algorithm

absolute minima of the higgs potential for the 75 of su 5 the problem of minimising the higgs potential for the 75 dimensional representation of su 5 is considered the most general form of the potential is established in the absence of a cubic term it is shown that the possible symmetry breaking patterns include su 3 su 2 u 1 and a new counterexample to michel s conjecture 1979 is found

nonlinear triple point problems with change of sign in this paper we study the existence of at least one or two positive solutions to the second order triple point nonlinear boundary value problem y x h x f y x 0 x a b y a ay h y b by h where 0 a b 1 and h a b here h changes sign in h as an application we also give some examples to demonstrate our results

existence of solutions of boundary value problems with integral boundary conditions for second order impulsive integro differential equations in banach spaces this paper investigates the existence of solutions for a class of second order boundary value problems with integral boundary conditions of nonlinear impulsive integro differential equations in banach spaces the arguments are based upon the fixed point theorem of strict set contraction operators meanwhile an example is worked out to demonstrate the main results

branching rules for e8 so16 branching rules for e8 down arrow so16 are derived from those of e8 down arrow su2 e7 down arrow su8 and so16 down arrow u1 su8 by noting that both group chains share a common u1 su8 subgroup additional branching rules are deduced from kronecker products of e8 and so16 irreps in each case the rules distinguish unambiguously between conjugate irreps of so16 an alternative labelling scheme for the irreps of so2k based on its maximal u1 suk subgroup is outlined

global structure of positive solutions for nonlocal boundary value problems involving integral conditions abstract in this paper we consider the nonlinear eigenvalue problems u h t f u 0 0 t 1 u 0 0 u 1 0 1 u s d a s where 0 1 u s d a s is a stieltjes integral with a nondecreasing and a t is not a constant on 0 1 h c 0 1 0 and h t 0 on any subinterval of 0 1 f c 0 0 and f s 0 for s 0 and f 0 f 0 f 0 lim s 0 f s s f lim s f s s we investigate the global structure of positive solutions by using global bifurcation techniques

symmetric positive solutions of boundary value problems with integral boundary conditions under conditions weaker than those used by ma we establish various results on the existence and nonexistence of symmetric positive solutions to fourth order boundary value problems with integral boundary conditions the arguments are based upon a specially constructed cone and the fixed point theory in a cone our results improve and extend that obtained in h ma symmetric positive solutions for nonlocal boundary value problems of fourth order nonlinear anal 68 2008 645 651 we illustrate our results by one example which cannot be handled using the existing results

positive solutions for some second order four point boundary value problems abstract in this paper the second order four point boundary value problem x t h t f t x t 0 0 t 1 x 0 a x x 1 b x is studied where 0 1 0 a b 1 and h 0 1 0 f 0 1 0 0 are nonnegative continuous functions by the use of the property of the corresponding green s function fixed point index theory leray schauder degree and upper and lower solution method some existence nonexistence and multiplicity results of positive solutions are acquired

natural labelling schemes for simple roots and irreducible representations of exceptional lie algebras the simple root system of each exceptional simple lie algebra is explicitly constructed in a variety of forms each construction is based on the natural embedding in the exceptional algebra of a classical semi simple algebra of the same rank the procedure adopted leads to the discovery of a number of new chains of subalgebras illustrated by means of supplemented dynkin diagrams the various sets of simple roots are then used to determine natural labels for the irreducible representations of each of the exceptional algebras for each such labelling scheme the modification rules for dealing with non standard representation labels are tabulated the connection between the natural labels and dynkin labels is given in detail and a comparison is made with the labels of b g wybourne and m j bowick 1977

positive solutions for second order quasilinear multi point boundary value problems abstract in this paper a new fixed point theorem in a cone is applied to obtain the existence of at least one positive solution for second order quasilinear m point boundary value problem p x t a t f t x x 0 0 t 1 x 0 0 x 1 i 1 m 2 i x i where f is a nonnegative continuous function and p x x p 2 x p 1 a simple example is presented to illustrate the applications of the obtained results

positive solutions to a generalized second order three point boundary value problem abstract in this paper by using krasnoselskii fixed point theorem we study the existence of at least or two positive solutions to the nonlinear second order three point boundary value problem y t a t f y t 0 0 t t y 0 y y t y where 0 t 0 t 0 t t are given constants as an application we also give some examples to demonstrate our results

a sharper condition for the solvability of a three point second order boundary value problem letf 0 1 r2 rbe a function satisfying caratheodory s conditions ande t l1 0 1 let 0 1 r 1 1 be given this paper is concerned with the problem of existence of a solution for the three point boundary value problem formula omitted this problem was studied earlier by gupta ntouyas and tsamatos when 1 and when 1 with 1 in the general case this problem was studied by gupta as a multi point boundary value problem in this paper sharper existence conditions are obtained for the solvability of the above boundary value problem in the general case

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

a theorem on orbit structures strata of compact linear lie groups we present a comprehensive constructive proof of a theorem characterizing the tangent space to a stratum orbit structure of the euclidean space rn seat of an orthogonal representation of a compact group g the characterization is made in terms of gradients of a complete set integrity basis of g invariant polynomials in a recent paper m abud and g sartori phys lett b 104 147 1981 the theorem which may be considered a generalization of a theorem by michel c r acad sci ser a 272 433 1971 has been shown to be effective in the determination of the equations of the strata and in the determination of natural extrema of g invariant functions

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

positive solutions to a type of nonlinear three point boundary value problem with sign changing nonlinearities in this paper by using krasnoselskii s fixed point theorem in a cone we study the existence of positive solutions for the second order three point boundary value problem u t a t u t lf t u t 0 0

existence and multiplicity results for some three point boundary value problems abstract using the method of upper and lower solutions and some degree theory arguments we establish the existence of at least three solutions of some second order nonlinear differential equations of the type x f t x x t i 0 1 subject to nonlinear three point boundary conditions x 0 0 x 1 g x 0 1 the growth of f with respect to x is allowed to be quadratic

canonical forms of tensor representations and spontaneous symmetry breaking an algorithm for constructing canonical forms for any tensor representation of the classical compact lie groups is given this method is used to find a complete list of the symmetry breaking patterns produced by higgs fields in the third rank antisymmetric representations of u n su n and so n for n or 7 a simple canonical form is also given for kth rank symmetric tensor representations

triple solutions for second order three point boundary value problems abstract we establish the existence of at least three positive solutions to the second order three point boundary value problem u f t u 0 u 0 0 u u 1 where 0 lt f 0 1 0 0 is continuous we accomplish this by making growth assumptions on f which can apply to many more cases than the sublinear and superlinear ones discussed in recent works

upper and lower solutions method and a second order three point singular boundary value problem the singular boundary value problem u q t g t u u 0 t 0 1 h 0 1 c 0 1 u 0 0 u 1 cu h is studied in this paper the singularity may appear at t 0 and the function g may be superlinear at u and change sign the existence of solutions is obtained via an upper and lower solutions method

green s function and positive solutions of nth order m point boundary value problem abstract in this paper we study the expression and properties of green s function for nth order m point boundary value problem u n t a t f u 0 t 0 1 u 0 u 0 u n 2 0 0 u 1 i 1 m 2 i u i where 0 0 with i 1 m 2 i i n 1 1 furthermore we obtained the existence of positive solutions by meas of fixed point index theory

an exact line search method for solving generalized continuous time algebraic riccati equations we present a newton like method for solving algebraic riccati equations that uses an exact line search to improve the sometimes erratic convergence behavior of newton s method it avoids the problem of a disastrously large first step and accelerates convergence when newton steps are too small or too long the additional work to perform the line search is small relative to the work needed to calculate the newton step

positive solutions of nonlinear third order m point bvp for an increasing homeomorphism and homomorphism with sign changing nonlinearity in this paper several existence theorems of positive solutions are established for a nonlinear m point boundary value problem for the following third order differential equations f u a t f t u t 0 email protected 0 1 f u 0 email protected i 1m 2a email protected u x i u 0 0 u 1 email protected i 1m 2b iu x i where f email protected r is an increasing homeomorphism and homomorphism and f 0 0 the nonlinear term f may change sign as an application an example to demonstrate our results is given

several sufficient conditions of solvability for a nonlinear higher order three point boundary value problem on time scales abstract in this paper a nonlinear higher order three point boundary value problem on time scales is considered under certain growth conditions on the nonlinearity several sufficient conditions for the existence of nontrivial solution are obtained by using leray schauder nonlinear alternative as an application some examples to demonstrate our results are given

positive solutions of nonlinear three point boundary value problems abstract let a c 0 1 b c 0 1 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 to the nonlinear boundary value problem u t a t u t b t u t h t f u 0 t 0 1 u 0 0 u u 1 where 0 0 and f c 0 0 we show the existence of at least one positive solution if f is either superlinear or sublinear by applying the fixed point theorem in cones