anti symmetric multivariate exponential functions and corresponding fourier transforms we define and study symmetrized and antisymmetrized multivariate exponential functions they are defined as determinants and antideterminants of matrices whose entries are exponential functions of one variable these functions are eigenfunctions of the laplace operator on the corresponding fundamental domains satisfying certain boundary conditions to symmetric and antisymmetric multivariate exponential functions there correspond fourier transforms there are three types of such fourier transforms expansions into the corresponding fourier series integral fourier transforms and multivariate finite fourier transforms eigenfunctions of the integral fourier transforms are found

character generators for elements of finite order in simple lie groups a1 a2 a3 b2 and g2 elements of finite order in the compact simple lie groups su 2 su 3 su 4 sp 4 or o 5 and g 2 are considered we provide the characters of the elements on irreducible representations of lie groups by assigning appro priate numerical values to the variables on which the characters of representations of the lie group depend in this way we specialize generating functions for the characters of the representations of the lie groups to the generating functions for characters of the elements of finite order particular attention is paid to rational elements all of whose characters are integers they are listed and the generating functions for their characters are obtained in a simplified form from which the characters can be read gaussian elements are also studied in detail their characters are complex valued with integer real and imaginary parts

discrete and continuous exponential transforms of simple lie groups of rank 2 we develop and describe continuous and discrete transforms of class functions on compact simple lie group g as their expansions into series of uncommon special functions called here e functions in recognition of the fact that the functions generalize common exponential functions the rank of g is the number of variables in the e functions a uniform discretization of the decomposition problem is described on lattices of any density and symmetry admissible for the lie group g

voronoi and delaunay cells of root lattices classification of their faces and facets by coxeter dynkin diagrams the voronoi domains their duals delaunay domains and all their faces of any dimension are classified and described in terms of the weyl group action on a representative of each type of face the representative of a face type is specified by a decoration of the corresponding coxeter dynkin diagram the rules of domain description are uniform for root lattices of simple lie groups of all types an explicit description of the representatives of all faces is carried out for the domains of root lattices of the four classical series and for the five exceptional simple lie groups the coxeter dynkin diagrams required here are the diagrams extended by the highest short root each diagram is partitioned into two subdiagrams one describing completely a d face of the voronoi domain its complement completely describing the dual of the d face the applicability of the authors classification method to generalized kaleidoscopes is explained

stabilization of a class of slow fast control systems at non hyperbolic points in this document we deal with the local asymptotic stabilization problem of a class of slow fastsystems or singularly perturbed ordinary differential equations the systems studied here have the following properties 1 they have one fast and an arbitrary number of slow variables and 2 they have a non hyperbolic singularity at the origin of arbitrary degeneracy our goal is to stabilize such a point the presence of the aforementioned singularity complicates the analysis and the controller design in particular the classical theory of singular perturbations cannot be used we propose a novel design based on geometric desingularization which allows the stabilization of a non hyperbolic point of singularly perturbed control systems our results are exemplified on a didactic example and on an electric circuit

canards in r3 abstract we give a geometric analysis of canard solutions in three dimensional singularly perturbed systems with a folded two dimensional critical manifold by analysing the reduced flow we obtain singular canard solutions passing through a singularity on the fold curve we classify these singularities called canard points as folded saddles folded nodes and folded saddle nodes we prove the existence of canard solutions in the case of the folded saddle we show the existence of canards in the folded node case provided a generic non resonance condition is satisfied and in a subcase of the folded saddle node the proof is based on the blow up method

discrete and continuous sine transform generalized to semisimple lie groups of rank two a new continuous group transform together with its discretization over a lattice of any density and admissible symmetry is defined and described for the four compact semisimple lie groups of rank 2 in the case of rank 1 the discrete version is the sine transform properties of the expansion functions of the transform called s functions are studied digital data processing is our motivating application

computation of character decompositions of class functions on compact semisimple lie groups a new algorithm is described for splitting class functions of an arbitrary semisimple compact lie group k into sums of irreducible characters the method is based on the use of elements of finite order efo in k and is applicable to a number of problems including decompositions of tensor products and various symmetry classes of tensors as well as branching rules in group subgroup reductions the main feature is the construction of a decomposition matrix d computed once and for all for a given range of problems and for a given k which then reduces any particular splitting to a simple matrix multiplication determination of d requires selection of a suitable set s of conjugacy classes of efo representing a finite subgroup of a maximal torus t of k and the evaluation of weyl group orbit sums on s in fact the evaluation of d can be coupled with the evaluation of the orbit sums in such a way as to greatly enhance the efficiency of the latter the use of the method is illustrated by some extensive examples of tensor product decompositions in e6 modular arithmetic allows all computations to be performed exactly

a unified framework for input to state stability in systems with two time scales this paper develops a unified framework for studying robustness of the input to state stability iss property and presents new results on robustness of iss to slowly varying parameters to rapidly varying signals and to generalized singular perturbations the common feature in these problems is a time scale separation between slow and fast variables which permits the definition of a boundary layer system like in classical singular perturbation theory to address various robustness problems simultaneously the asymptotic behavior of the boundary layer is allowed to be complex and it generates an average for the derivative of the slow state variables the main results establish that if the boundary layer and averaged systems are iss then the iss bounds also hold for the actual system with an offset that converges to zero with the parameter that characterizes the separation of time scales the generality of the framework is illustrated by making connection to various classical two time scale problems and suggesting extensions

properties of continuous fourier extension of the discrete cosine transform and its multidimensional generalization a versatile method is described for the practical computation of the exact discrete fourier transforms dft both the direct and the inverse ones of a continuous function g given by its values gj at the points of a uniform grid fn generated by conjugacy classes of elements of finite adjoint order n in the fundamental region f of compact semisimple lie groups the present implementation of the method is for the groups su 2 when f is reduced to a one dimensional segment and for su 2 su 2 su 2 in multidimensional cases this simplest case turns out to be a version of the discrete cosine transform dct implementations abbreviated as dgt for discrete group transform based on simple lie groups of higher ranks are to be considered separately dct is often taken to be simply a specific type of the standard dft here we show that the dct is very different from the standard dft when the properties of the continuous extensions of the two inverse discrete transforms are studied the following propertie

alternating multivariate trigonometric functions and corresponding fourier transforms we define and study multivariate sine and cosine functions symmetric with respect to the alternating group an which is a subgroup of the permutation symmetric group sn these functions are eigenfunctions of the laplace operator they determine fourier type transforms there exist three types of such transforms expansions into corresponding sine fourier and cosine fourier series integral sine fourier and cosine fourier transforms and multivariate finite sine and cosine transforms in all these transforms alternating multivariate sine and cosine functions are used as a kernel

unstable periodic orbits and semiclassical quantisation the bohr sommerfeld quantisation condition has a meaningful extension to classically chaotic systems whose periodic unstable orbits are isolated it provides a semiclassical euler factorisation for the functional determinant of the quantal hamiltonian in contrast to the hadamard infinite product over the eigenvalues by which the exact determinant is defined

technical communique improvement of system order reduction via balancing using the method of singular perturbations in this paper we first present several alternative techniques for system order reduction based on system balancing by employing the method of singular perturbations the techniques obtained have the same robustness accuracy evaluated with respect to the h norm of the reduced order system as the two techniques existing in the control literature known as the direct truncation and balancing residualization methods the newly proposed techniques preserve the exact dc gain as the original system and produce very good to excellent accuracy at low and medium frequencies we also present a simple formula that fixes the dc gain for the direct truncation method in the second part of the paper we propose how to perform order reduction of a balanced system using theory of singular perturbations that can produce very good accuracy at high frequencies particularly for systems that have lightly damped highly oscillatory modes the result obtained for such kind of systems requires that the modes with higher hankel singular values should be neglected and those with smaller hankel singular values should be retained this is a counter intuitive result in the spirit of h theory but an intuitive result in the spirit of theory of singular perturbations the second part of the paper raises an important question about validity of the system robustness estimate based on the h norm for linear systems with lightly damped highly oscillatory modes

six types of e functions of the lie groups o 5 and g 2 new families of e functions are described in the context of the compact simple lie groups o 5 and g 2 these functions of two real variables generalize the common exponential functions and for each group only one family is currently found in the literature all the families are fully characterized their most important properties are described namely their continuous and discrete orthogonalities and decompositions of their products

orbit orbit branching rules between simple low rank algebras and equal rank subalgebras complete orbit orbit branching rules or equivalently reduction of weyl group orbits between each simple algebra of rank 3 or less and its equal rank subalgebras and between f4 and each of its equal rank subalgebras are given the generic case an an 1 u 1 the subjoining f4 b3 u 1 and e8 e6 a2 first and seventh highest weight labels nonzero are also treated branching rules between rank 4 and 5 simple algebras and e6 and their equal rank subalgebras are available in a depository

analysis of a slow fast system near a cusp singularity this paper studies a slow fast system whose principal characteristic is that the slow manifold is given by the critical set of the cusp catastrophe our analysis consists of two main parts first we recall a formal normal form suitable for systems as the one studied here afterwards taking advantage of this normal form we investigate the transition near the cusp singularity by means of the blow up technique our contribution relies heavily in the usage of normal form theory allowing us to refine previous results

local analysis near a folded saddle node singularity abstract folded saddle nodes occur generically in one parameter families of singularly perturbed systems with two slow variables we show that these folded singularities are the organizing centers for two main delay phenomena in singular perturbation problems canards and delayed hopf bifurcations we combine techniques from geometric singular perturbation theory the blow up technique and from delayed hopf bifurcation theory complex time path analysis to analyze the flow near such folded saddle nodes in particular we show the existence of canards as intersections of stable and unstable slow manifolds to derive these canard results we extend the singularly perturbed vector field into the complex domain and study it along elliptic paths this enables us to extend the invariant slow manifolds beyond points where normal hyperbolicity is lost furthermore we define a way in way out function describing the maximal delay expected for generic solutions passing through a folded saddle node singularity branch points associated with the change from a complex to a real eigenvalue structure in the variational equation along the critical slow manifold make our analysis significantly different from the classical delayed hopf bifurcation analysis where these eigenvalues are complex only

relaxation oscillation and canard explosion abstract we give a geometric analysis of relaxation oscillations and canard cycles in singularly perturbed planar vector fields the transition from small hopf type cycles to large relaxation cycles which occurs in an exponentially thin parameter interval is described as a perturbation of a family of singular cycles the results are obtained by means of two blow up transformations combined with standard tools of dynamical systems theory the efficient use of various charts is emphasized the results are applied to the van der pol equation

on discretization of tori of compact simple lie groups the discrete orthogonality of special function families called c and s functions which are derived from the characters of compact simple lie groups is described in hrivnak and patera 2009 j phys a math theor 42 385208 here the results of hrivnak and patera are extended to two additional recently discovered families of special functions called ss and sl functions the main result is an explicit description of their pairwise discrete orthogonality within each family when the functions are sampled on finite fragments fsm and flm of a lattice in any dimension n 2 and of any density controlled by m and of the symmetry of the weight lattice of any compact simple lie group with two different lengths of roots

discrete and continuous cosine transform generalized to lie groups su 3 and g 2 in the paper we complete the development and description of the four variants of two dimensional generalization of the cosine transform started in patera and zaratsyan j math phys 46 053514 2005 each variant is based on a compact semisimple lie group g of rank 2 here the groups are su 3 and g 2 the cosines are generalized as the corresponding c functions of the lie group a c function is the contribution to an irreducible character from one orbit of the appropriate weyl group an explicit description is provided for expansions of functions given on the fundamental region f of the two compact simple lie groups into series of c functions the fundamental region f is an equilateral triangle for su 3 and half of such a triangle for g 2 expansion coefficients are calculated using orthogonality of c functions on f discrete expansions are set up on a grid fm f the grid is defined group theoretically for all positive integers m it consists of points in f that represent conjugacy classes of ele

dynamic surface control via singular perturbation analysis this note contributes to the technical nature of dynamic surface control dsc based on a class of strict feedback nonlinear systems the dsc technique prevents the complexity problem in integrator backstepping control ibc yet the stability results obtained by existing dscs are conservative and consequently cannot fully reveal the technical nature of dsc by the exploitation of the singular perturbation theory several novel features of dsc are revealed as follows 1 semiglobal practical exponential stability can be guaranteed only by a suitably small filter parameter 2 steady state tracking accuracy does not rely on high control gains 3 the performance of ibc can be recovered via the decrease of the filter parameter these features are not only significant for understanding the technical nature of dsc but also useful for establishing an efficient strategy of parameter selection and system debugging simulation results have verified correctness of the theoretical result

affine reflection groups for tiling applications knot theory and dna we present in this paper some non conventional applications of affine weyl groups waff of rank 2 the symmetry group of the tiling lattice we first develop and present the tools for applications requiring tilings of a real euclidean plane r2 we then elucidate the equivalence of these tilings with 2d projections of knots the resulting mathematical structure provides a framework within which is encompassed recent work utilizing knot theory for modeling the structure and function of genetic molecules specifically the action of particular enzymes in altering the topology of dna in site specific recombination

brief a descriptor system approach to nonlinear singularly perturbed optimal control problem this paper considers the infinite horizon nonlinear quadratic optimal control problem for a singularly perturbed system which is nonlinear in both the slow and the fast variables the relationship between this problem and the analogous one for a descriptor system is investigated parameter independent controllers are constructed that solve the problem for the descriptor system and lead the full order system to the near optimal performance estimates on the closeness of the cost under near optimal controllers to the optimal one are obtained

weighted trace formula near a hyperbolic trajectory and complex orbits in this paper we consider a weighted trace formula for schrodinger operators more precisely let j and ej denote the eigenfunctions and eigenvalues of a schrodinger type operator h with a discrete spectrum let x be a coherent state centered at a point x of a hyperbolic closed orbit we show that as 0 the leading term of j ej e x j 2 can be expressed in terms of the analytic continuation on the upper and lower half planes of the positive and negative frequencies part of the result is also related to complex trajectories surrounding

a contraction theory approach to singularly perturbed systems in this technical note we revisit standard results on singular perturbations and multiple time scales using convergence analysis tools based on nonlinear contraction theory specifically assuming that the fast and slow subsystems are each partially contracting we obtain explicit bounds on the convergence rate of the trajectories to the slow manifold and on the asymptotic error between the trajectories of the singularly perturbed system and those of the reduced system as an application example we illustrate the design of a biomolecular insulation device

semiclassical resonances generated by a closed trajectory of hyperbolic type we determine all the resonances in certain rectangular regions of the complex plane of the schrodinger operator h2 v whenh 0 under the assumption that the set of trapped points of energy 0 for the classical flow form a closed trajectory and that the corresponding poincare map is hyperbolic

mixed mode oscillations with multiple time scales mixed mode oscillations mmos are trajectories of a dynamical system in which there is an alternation between oscillations of distinct large and small amplitudes mmos have been observed and studied for over thirty years in chemical physical and biological systems few attempts have been made thus far to classify different patterns of mmos in contrast to the classification of the related phenomena of bursting oscillations this paper gives a survey of different types of mmos concentrating its analysis on mmos whose small amplitude oscillations are produced by a local multiple time scale mechanism recent work gives substantially improved insight into the mathematical properties of these mechanisms in this survey we unify diverse observations about mmos and establish a systematic framework for studying their properties numerical methods for computing different types of invariant manifolds and their intersections are an important aspect of the analysis described in this paper

on singular perturbations due to fast actuators in hybrid control systems a stability result is given for hybrid control systems singularly perturbed by fast but continuous actuators if a hybrid control system has a compact set globally asymptotically stable when the actuator dynamics are omitted or equivalently are infinitely fast then the same compact set is semiglobally practically asymptotically stable in the finite speed of the actuator dynamics this result which generalizes classical results for differential equations justifies using a simplified plant model that ignores fast but continuous actuator dynamics even when using a hybrid feedback control algorithm

four families of weyl group orbit functions of b 3 and c 3 the properties of the four families of special functions of three real variables called here c s ss and sl functions are studied the ss and sl functions are considered in all details required for their exploitation in fourier expansions of digital data sampled on finite fragment of lattices of any density and of the 3d symmetry imposed by the weight lattices of b3 and c3 simple lie algebras groups the continuous interpolations which are induced by the discrete expansions are exemplified and compared for some model functions

singular perturbations and time scale methods in control theory survey 1976 1983 recent progress in the use of singular perturbation and two time scale methods of modeling and design for control systems is reviewed over 350 references are organized into major problem areas representative issues and results are discussed with a view to outlining research directions and indicating potential areas of application the survey is aimed at engineers and applied mathematicians interested in model order reduction separation of time scales and allied simplified methods of control system analysis and design the exposition does not assume prior knowledge of singular perturbation methods

the semi classical trace formula and propagation of wave packets abstract we study spectral and propagation properties of operators of the form s h n j 0 h j p j where j p j is a differential operator of order j on a manifold m asymptotically as h 0 the estimates are in terms of the flow t of the classical hamiltonian h x p n j 0 p j x p on t m where p j is the principal symbol of p j we present two sets of results i the semiclassical trace formula on the asymptotic behavior of eigenvalues and eigenfunctions of s h in terms of periodic trajectors of h ii associated to certain isotropic submanifolds t m we define families of functions h and prove that t exp iths is a family of the same kind associated to t