grading refinements in the contractions of lie algebras and their invariants all mutually inequivalent toroidal gradings of the lie algebra are constructed each of them provides several different graded contractions to other eight dimensional lie algebras these can be either continuous or discrete the continuous ones are related to generalized wigner inonu contractions the behaviour of casimir operators under the graded contractions is studied and the casimir and generalized casimir operators of the contracted lie algebras are presented

contractions of lie algebras and separation of variables the wigner inonu contraction from the rotation group o 3 to the euclidean group e 2 is used to relate the separation of variables in the laplace beltrami operators on two corresponding homogeneous spaces different realizations of the contraction take the two separable coordinate systems on the sphere to the four on the plane

general charge conjugation operators in simple lie groups a description of particular elements charge conjugation operators found in any compact simple lie group k is presented such elements ri transform a physical state weight vector of a basis of a representation space into others with opposite charge ith component of the weight sometime changing also the sign of the state it is demonstrated that exploitation of these elements and the finite subgroup n of k generated by them offer new powerful methods for computing with representations of the lie group their application to construction of bases in representation spaces is considered in detail it represents a completely new direction to the problem

robustness in the presence of mixed parametric uncertainty and unmodeled dynamics continuing the development of the structured singular value approach to robust control design the authors investigate the problem of computing mu in the case of mixed real parametric and complex uncertainty the problem is shown to be equivalent to a smooth constrained finite dimensional optimization problem in view of the fact that the functional to be maximized may have several local extrema an upper bound on mu whose computation is numerically tractable is established this leads to a sufficient condition of robust stability and performance a historical perspective on the development of the mu theory is included

generalized mixed μ bounds for real and complex multiple block uncertainty with internal matrix structure new absolute stability results that unify and extend existing structured singular value bounds for mixed uncertainty are developed using frequency domain arguments these bounds generalize prior upper bounds for mixed by permitting the treatment of non diagonal real uncertain blocks as well as accounting for internal matrix structure in the uncertainty several specializations to well known absolute stability criteria from the classical literature are given which allow for a systematic comparison of these results in addressing the problem of robust stability for real parameter uncertainty

graded contractions of representations of special linear lie algebras with respect to their maximal parabolic subalgebras parabolic gradings of the classical simple lie algebras sl n c n or 3 are described for all maximal parabolic subalgebras all contractions are found which leave a maximal parabolic subalgebra intact and which preserve a parabolic grading parabolic contractions of lie algebras contractions of the irreducible representations of sl n c for each parabolic contraction of the lie algebra are the main results of the article

the fundamental invariants of inhomogeneous classical groups the casimir operators of the following groups are explicitly constructed rn gl n r rn sl n r rn o p q cn u p q r2n sp n r cn gl n c cn sl n c cn o n c c2n sp n c hn u 2n hn su 2n hn o 2n hn sp p q and r r2n sp n r the method is based on a particular fiber bundle structure of the generic orbits generated by the co adjoint representation of a semidirect product

on the identification of a lie algebra given by its structure constants i direct decompositions levi decompositions and nilradicals abstract methods are given for identifying a lie algebra l given by its structure constants the identification involves a transformation to a canonical basis in which the structure becomes obvious thus decomposable lie algebras are already decomposed into direct sums of indecomposable ones an indecomposable lie algebra that is not simple or solvable has its radical exhibited a solvable algebra has its nilradical displayed the methods all lead to simple algorithms that have been implemented as computer programs involving some symbolic manipulations

graded contractions of sl 3 c the wigner inonu contraction is generalized to include gradings other than z2 and the formalism is applied to obtain all toroidal contractions of sl 3 c examples of contractions based on nontoroidal gradings are also given

graded contractions and kinematical groups of space time g graded contractions of the complex lie algebra b2 and of its real forms are considered it is found that a particular g z2 z2 is a proper choice of the grading group preserved when graded contractions leading to lie algebras of the kinematical groups of space time are studied

solvable lie algebras with abelian nilradicals a procedure is presented for classifying solvable lie algebras with abelian nilradicals theorems on the structure of such algebras are proven and their centres are constructed many examples are analysed

the octomorphic criterion for real parameter uncertainty real m bounds without circles and d n scales abstract in this paper we introduce new bounds for robust stability analysis with real parameter uncertainty the approach is based on an absolute stability criterion that excludes the nyquist plot from a paraboloidal region containing the point 1 j0 transformation of this criterion to the case of norm bounded uncertainty leads to a stability criterion in terms of the octomorphic or figure eight shaped region the requirement that the nyquist plot lie inside the octomorphic region thus yields a bound on the allowable real parameter uncertainty this stability criterion is distinct from recent bounds for real which involve frequency dependent scales having a frequency dependent off axis circle interpretation since the octomorphic region includes both upper and lower halves it is able to encompass the entire nyquist plot without using frequency dependent scales

generalized casimir operators of solvable lie algebras with abelian nilradicals a solvable complex lie algebra l of dimension n with an abelian nilradical of dimension r is shown to have precisely 2r n generalized casimir invariants we always have r or n 2 they are constructed as invariants of the coadjoint representation of l and depend only on variables dual to elements of the nilradical their form in general involves logarithms of these variables in addition to rational and irrational functions they give rise to genuine casimir operators whenever they happen to be polynomials

extensions of mixed µ bounds to monotonic and odd monotonic nonlinearities using absolute stability theory in this paper we make explicit connections between classical absolute stability theory and modern mixed analysis and synthesis specifically using the parameter dependent lyapunov function framework of haddad and bernstein and the frequency dependent off axis circle interpretation of how and hall we extend previous work on absolute stability theory for monotonic and odd monotonic nonlinearities to provide tight approximations for constant real parameter uncertainty an immediate application of this framework is the generalization and reformulation of mixed analysis and synthesis in terms of lyapunov functions and riccati equations this observation is exploited to provide robust reduced order controller synthesis while avoiding the standard d n k iteration and curve fitting procedures

contractions of lie algebras generalized inonu wigner contractions versus graded contractions recently a new notion of contraction has been introduced the purely algebraically defined graded contractions that fall into two disjoint classes continuous and discrete our main result is that all continuous graded contractions can be realized by generalized inonu wigner contractions furthermore we give a complete characterization of the discrete graded contractions via certain higher order identities inherent to the grading group

time domain and frequency domain conditions for strict positive realness the author states various time domain and frequency domain conditions pertaining to multivariate strict positive real systems and establishes the relationship between these conditions their relationship is clarified by means of a three tier diagram each tier represents a set of equivalent conditions successively stronger than the lower tier the main theorem in the present study can be considered as the positive realness lemma for strictly positive real systems

parameter dependent lyapunov functions and the popov criterion in robust analysis and synthesis many practical applications of robust feedback control involve constant real parameter uncertainty whereas small gain or norm bounding techniques guarantee robust stability against complex frequency dependent uncertainty thus entailing undue conservatism since conventional lyapunov bounding techniques guarantee stability with respect to time varying perturbations they possess a similar drawback in this paper we develop a framework for parameter dependent lyapunov functions a less conservative refinement of fixed lyapunov functions an immediate application of this framework is a reinterpretation of the classical popov criterion as a parameter dependent lyapunov function this result is then used for robust controller synthesis with full order and reduced order controllers

discrete and continuous graded contractions of representations of lie algebras simultaneous grading of lie algebras and their representation spaces is used to develop a new theory of grading preserving contractions of representations of all lie algebras admitting the chosen grading the theory is completely different from the traditional ways of contracting representations the graded contractions fall naturally into two classes discrete and continuous ones related respectively to 2 cocycles and coboundaries of the grading group

discrete and continuous graded contractions of lie algebras and superalgebras grading preserving contractions of lie algebras and superalgebras of any type over the complex number field are defined and studied such contractions fall naturally into two classes the wigner inonu like continuous contractions and new discrete contractions a general method is described for any abelian grading semigroup and any lie algebra or superalgebra admitting such a grading all contractions preserving z2 z3 and z2 z2 gradings are found examples of these gradings and contractions for the simple lie algebra a2 affine kac moody algebra a 1 and the simple superalgebra osp 2 1 are shown

real m bounds based on fixed shapes in the nyquist plane parabolas hyperbolas cissoids nephroids and octomorphs in this paper we introduce new bounds for the real structured singular value the approach is based on absolute stability criteria with plant dependent multipliers that exclude the nyquist plot from fixed plane curve shapes containing the critical point jo unlike half plane and circle based bounds the critical feature of the fixed curve bounds is their ability to differentiate between the real and imaginary components of the uncertainty since the plant dependent multipliers have the same functional form at all frequencies the resulting graphical interpretation of the absolute stability criteria are frequency independent in contrast to the frequency dependent off axis circles that arise in standard real bounds