superinduction for pattern groups abstract it is well known that the representation theory of the finite group of unipotent upper triangular matrices u n over a finite field is a wild problem by instead considering approximately irreducible representations supercharacters one obtains a rich combinatorial theory analogous to that of the symmetric group where we replace partition combinatorics with set partitions this paper studies diaconis isaacs concept of superinduction in pattern groups while superinduction shares many desirable properties with usual induction it no longer takes characters to characters we begin by finding sufficient conditions guaranteeing that superinduction is in fact induction it turns out for two natural embeddings of u m in u n superinduction is induction we conclude with an explicit combinatorial algorithm for computing this induction analogous to the pieri formulas for the symmetric group

counting characters of upper triangular groups abstract working over a finite field of order q let u n q be the group of upper triangular n n matrices with all diagonal entries equal to 1 it is known that all complex irreducible characters of u n q have degrees that are powers of q and that all powers q e occur for 0 e n where the upper bound n is defined by the formulas 2 m m m 1 and 2 m 1 m 2 it has been conjectured that the number of irreducible characters having degree q e is some polynomial in q with integer coefficients depending on n and e in this paper we construct explicit polynomials for the cases where e is one of n n 1 and 1 also a list of polynomials is presented that appear to give the correct character counts for n 9 several related results concerning pattern groups are also included

supercharacters of unipotent groups defined by involutions abstract we construct supercharacter theories of finite unipotent groups in the orthogonal symplectic and unitary types our method utilizes group actions in a manner analogous to that of diaconis and isaacs in their construction of supercharacters of algebra groups the resulting supercharacter theories agree with those of andre and neto in the case of the unipotent orthogonal and symplectic groups and generalize to a large collection of subgroups in the unitary group case we describe the supercharacters and superclasses in terms of labeled set partitions and calculate the supercharacter table

an approximation theory of solutions to operator riccati equations for h infty control as in the finite dimensional case the appropriate state feedback for the infinite dimensional h infty disturbance attenuation problem may be calculated by solving a riccati equation this operator riccati equation can rarely be solved exactly we approximate the original infinite dimensional system by a sequence of finite dimensional systems and consider the corresponding finite dimensional disturbance attenuation problems we make the same assumptions required in approximations for the classical linear quadratic regulator problem and show that the sequence of solutions to the corresponding finite dimensional riccati equations converge strongly to the solution to the infinite dimensional riccati equation furthermore the corresponding finite dimensional feedback operators yield performance arbitrarily close to that obtained with the infinite dimensional solution

comparison techniques for random walk on finite groups we develop techniques for bounding the rate of convergence of a symmetric random walk on a finite group to the uniform distribution the techniques gives bounds on the second largest and other eigenvalues in terms of the eigenvalues of a comparison chain with known eigenvalues the techniques yield sharp rates for a host of previously intractable problems on the symmetric group

nonzero coefficients in restrictions and tensor products of supercharacters of un q abstract the standard supercharacter theory of the finite unipotent upper triangular matrices u n q gives rise to beautiful combinatorics based on set partitions as with the representation theory of the symmetric group embeddings of u m q u n q for m n lead to branching rules diaconis and isaacs established that the restriction of a supercharacter of u n q is a nonnegative integer linear combination of supercharacters of u m q in fact it is polynomial in q in a first step towards understanding the combinatorics of coefficients in the branching rules of the supercharacters of u n q this paper characterizes when a given coefficient is nonzero in the restriction of a supercharacter and the tensor product of two supercharacters these conditions are given uniformly in terms of complete matchings in bipartite graphs

sensor choice for minimum error variance estimation a kalman filter is optimal in that the variance of the error is minimized by the estimator it is shown here in an infinite dimensional context that the solution to an operator riccati equation minimizes the steady state error variance this extends a result previously known for lumped parameter systems to distributed parameter systems it is shown then that minimizing the trace of the riccati operator is a reasonable criterion for choosing sensor locations it is then shown that multiple inaccurate sensors that is those with large noise variance can provide as good an estimate as a single highly accurate but probably more expensive sensor optimal sensor location is then combined with estimator design a framework for calculation of the best sensor locations using approximations is established and sensor location as well as choice is investigated with three examples simulations indicate that the sensor locations do affect the quality of the estimation and that multiple low quality sensors can lead to better estimation than a single high quality sensor

supercharacter formulas for pattern groups c andre and n yan introduced the idea of a supercharacter theory to give a tractable substitute for character theory in wild groups such as the unipotent uppertriangular group u n f q in this theory superclasses are certain unions of conjugacy classes and supercharacters are a set of characters which are constant on superclasses this paper gives a character formula for a supercharacter evaluated at a superclass for pattern groups and more generally for algebra groups

the finite primitive permutation groups containing an abelian regular subgroup a complete classification is given of finite primitive permutation groups which contain an abelian regular subgroup this solves a long standing open problem in permutation group theory initiated by w burnside in 1900

counting irreducible representations of large degree of the upper triangular groups abstract let u n q be the upper triangular group of degree n over the finite field f q with q elements in this paper we present constructions of large degree complex irreducible representations of u n q where n 7 and then determine the number of irreducible representations of largest second largest and third largest degrees

supercharacters and superclasses for algebra groups we study certain sums of irreducible characters and compatible unions of conjugacy classes in finite algebra groups these groups generalize the unimodular upper triangular groups over a finite field and the supercharacter theory we develop extends results of carlos andre and ning yan that were originally proved in the upper triangular case this theory sometimes allows explicit computations in situations where it would be impractical to work with the full character table we discuss connections with the kirillov orbit method and with gelfand pairs and we give conditions for a supercharacter or a superclass to be an ordinary irreducible character or conjugacy class respectively we also show that products of supercharacters are positive integer combinations of supercharacters

a spillover phenomenon in the optimal location of actuators in this paper we are interested in finding the optimal location and shape of the actuators in a stabilization problem namely we consider the one dimensional wave equation damped by an internal feedback supported on a subdomain omega of given length the criterion we want to optimize represents the rate of decay of the total energy of the system it theoretically involves all the eigenmodes of the operator from an engineering point of view it seems more realistic to consider only a finite number of modes say the n first ones in that context we are able to prove existence and uniqueness of an optimal domain omega n it is the better possible location for the actuators we characterize this optimal domain and we point out the following strange phenomenon at least for small lengths the optimal domain omega n which is the better one for the n first modes is actually the worse one for the n 1th mode this looks like the well known spillover phenomenon in control theory at last we will give some possible extension and open problems in higher dimension

finite permutation groups with a regular dihedral subgroup and edge transitive dihedrants abstract we study finite permutation groups that contain a regular dihedral subgroup called d groups in particular a classification is given of bi quasiprimitive d groups it is then applied to classify edge transitive dihedrants of order 2 p m and valency coprime to p where p is an odd prime and m is a positive integer

characters of the sylow p subgroups of the chevalley groups d4 pn abstract let u q be a sylow p subgroup of the chevalley groups d 4 q where q is a power of a prime p we describe a construction of all complex irreducible characters of u q and obtain a classification of these irreducible characters via the root subgroups which are contained in the center of these characters furthermore we show that the multiplicities of the degrees of these irreducible characters are given by polynomials in q 1 with nonnegative integer coefficients

h sub infinity control by state feedback and fast algorithms for the computation of optimal h sub infinity norms the suboptimality of some parameter for h sub infinity optimization by dynamic state feedback is characterized in terms of the solvability of riccati inequalities this is done without restricting the finite zero structure of the plant if there are no system zeros on the imaginary axis the h sub infinity problem can be treated in a complete and satisfactory way explicit characterizations optimum to be achieved are provided and a closed formula for the optimal value is derived in terms of the h sub infinity norm of some fixed transfer matrix if the optimum is not attained any sequence of controllers of bounded size which is constructed to approach the infimal norm must necessarily be high gain a globally and quadratically convergent algorithm to compute the optimal value is proposed this algorithm is generalized to the h sub infinity optimization problem by measurement feedback

h optimal actuator location in many control applications there is freedom to place the actuators the actuator location should be chosen to optimize certain performance objectives in this paper h performance with state feedback is considered that is both the controller and the actuator locations are chosen to minimize the effect of disturbances on the output a framework for calculating h optimal actuator locations is developed conditions for well posedness of the h optimal actuator location problem are presented many optimal actuator problems involve systems modelled by partial differential equations and conditions under which approximations yield reliable results are given a derivative free optimization algorithm to calculate h optimal actuator locations is described the results are illustrated using several examples

h control by state feedback an iterative algorithm and characterization of high gain occurrence we present an iterative algorithm to compute the achievable h norm by state feedbak for a standard regular problem and give a characterization when high gain feedback is necessary to approach the optimal value if there is no need for high gain components to approximate the optimal value we can reduce the problem to the computation of the h norm of a certain stable transfer matrix

irreducible characters of large degree of sylow p subgroups of classical groups abstract if g is a sylow p subgroup of a classical group defined over a finite field of characteristic p 2 and g is the associated lie algebra we show that the irreducible characters of g with sufficiently large degree p f can be parameterized by the orbits of g on the dual space of g of size p 2 f we use this correspondence to count the number of irreducible characters of g of maximum degree

disturbance attenuation and h optimization a design method based on the algebraic riccati equation this note presents a method for designing a state feedback control law to reduce the effect of disturbances on the output of a given linear system the problem under consideration involves attenuating the disturbances to a prespecified level the construction of the state feedback control law requires the solution of a certain algebraic riccati equation by applying the procedure with successively smaller values of the prespecifled disturbance level the result also provides an approach to the h infty optimization problem for the state feedback case

optimal sensor actuator placement for active vibration control using explicit solution of algebraic riccati equation abstract this paper deals with an optimal placement problem of sensors and actuators for active vibration control of flexible structures for undamped structures with collocated rate sensors and actuators two solutions of generalized algebraic riccati equations generalized control algebraic riccati equation gcare generalized filtering algebraic riccati equation gfare are obtained explicitly employing these explicit solutions we can obtain a stabilizing h controller based on the normalized coprime factorization approach without solving any algebraic riccati equations numerically generally in a optimal sensor actuator placement problem with a model based control law lqg or h the feedback controller needs to be obtained for all candidates of the optimal placement which may be derived with some numerical optimization techniques by solving algebraic riccati equations numerically therefore the amount of computation required to determine the optimal sensor actuator placement and the controller increases rapidly for large scale structures which have many pairs of sensor actuators the h controller in this paper can be obtained just by addition and multiplication of several matrices furthermore a closed loop property on h norm is automatically bounded for all candidates of the optimal placement hence we can formulate the optimal sensor actuator placement problem to optimize other closed loop properties norm of the closed loop system with less computational requirement than the model based method mentioned above the gradient of the h2norm of the closed loop system which is necessary for a descent based optimization technique is derived using this sensitivity formula we obtain the optimal placements of two pairs of sensors and actuators which minimize the h2norm of the closed loop system for a simply supported beam by the quasi newton method the simulation results show the effectiveness of the proposed design method

convex analysis and global optimization of joint actuator location and control problems it is shown that the optimal value of the continuous time linear quadratic problem regarded as a function of the system model and index parameters exhibits properties convexity concavity and monoticity specially suitable for optimization purposes based on this fact a procedure for the global solution determination of eventually nonconvex problems involving the above mentioned function is proposed such problems embody some known designs such as filtering under noise uncertainty or precision constraints and optimal actuator sensor location the last problem is deeply analyzed and two practical applications namely satellite attitude control and large flexible system control are included

linear quadratic optimal actuator location in control of vibrations diffusion and many other problems governed by partial differential equations there is freedom in the choice of actuator location the actuator location should be chosen to optimize performance objectives in this paper we consider linear quadratic performance two types of cost are considered the choice depends on whether the response to the worst initial condition is to be minimized or whether the initial condition is regarded as random in practice approximations are used in controller design and thus in selection of the actuator locations the optimal cost and location of the approximating sequence should converge to the exact optimal cost and location in this work conditions for this convergence are given in the case of linear quadratic control examples are provided to illustrate that convergence may fail when these conditions are not satisfied

on primitive permutation groups with small suborbits and their orbital graphs abstract in this paper we study finite primitive permutation groups with a small suborbit based on the classification result of quirin math z 122 1971 267 and wang comm algebra 20 1992 889 we first produce a precise list of primitive permutation groups with a suborbit of length 4 in particular we show that there exist no examples of such groups with the point stabiliser of order 2 4 3 6 clarifying an uncertain question since 1970s then we analyse the orbital graphs of primitive permutation groups with a suborbit of length 3 or of length 4 we obtain a complete classification of vertex primitive arc transitive graphs of valency 3 and valency 4 and we prove that there exist no vertex primitive half arc transitive graphs of valency less than 10 finally we construct vertex primitive half arc transitive graphs of valency 2 k for infinitely many integers k with 14 as the smallest valency

simple groups probabilistic methods and a conjecture of kantor and lubotzky abstract we prove that a randomly chosen involution and a randomly chosen additional element of a finite simple groupggenerategwith probability 1 as g this confirms a conjecture of kantor and lubotzky applications and related results are derived for example we show that except for the suzuki groups and finitely many possible other exceptions all finite simple groups can be generated by two elements one of which has order 3 we also obtain sharp estimates on the probabilityp g of generating a finite simple group of exceptional lie typegby two randomly chosen elements this complements analogous estimates of babai and kantor for alternating and classical groups denoting bym g the minimal index of a proper subgroup of a finite simple groupg we conclude in particular thatm g 1 p g is bounded between two positive absolute constants

supercharacters symmetric functions in noncommuting variables and related hopf algebras we identify two seemingly disparate structures supercharacters a useful way of doing fourier analysis on the group of unipotent uppertriangular matrices with coefficients in a finite field and the ring of symmetric functions in noncommuting variables each is a hopf algebra and the two are isomorphic as such this allows developments in each to be transferred the identification suggests a rich class of examples for the emerging field of combinatorial hopf algebras

on symmetries of cayley graphs and the graphs underlying regular maps abstract by definition cayley graphs are vertex transitive and graphs underlying regular or orientably regular maps on surfaces are arc transitive this paper addresses questions about how large the automorphism groups of such graphs can be in particular it is shown how to construct 3 valent cayley graphs that are 5 arc transitive in answer to a question by cai heng li and cayley graphs of valency 3 t 1 that are 7 arc transitive for all t 0 the same approach can be taken in considering the graphs underlying regular or orientably regular maps leading to classifications of all such maps having a 1 4 or 5 arc regular 3 valent underlying graph in answer to questions by cheryl praeger and sanming zhou

super characters of finite unipotent groups of types b n c n and d n abstract we define and study super characters over the complex field of the classical finite unipotent groups of types b n c n and d n under the assumption that the prime is sufficiently large we extend the known results for the unitriangular group proved by the first author in the papers c a m andre basic characters of the unitriangular group j algebra 175 1995 287 319 and c a m andre basic characters of the unitriangular group for arbitrary primes proc amer math soc 130 7 2002 1943 1954 in particular we prove that every irreducible complex character occurs as a constituent of a unique super character we also give a combinatorial description of all the irreducible characters of maximum degree

restrictions of rainbow supercharacters abstract the maximal subgroup of unipotent upper triangular matrices of the finite general linear groups is a fundamental family of p groups their representation theory is well known to be wild but there is a standard supercharacter theory replacing irreducible representations by super representations that gives us some control over its representation theory while this theory has a beautiful underlying combinatorics built on set partitions the structure constants of restricted super representations remain mysterious this paper proposes a new approach to solving the restriction problem by constructing natural intermediate modules that help factor the computation of the structure constants we illustrate the technique by solving the problem completely in the case of rainbow supercharacters and some generalizations along the way we introduce a new q analogue of the binomial coefficients that depend on an underlying poset

a supercharacter theory for involutive algebra groups if j is a finite dimensional nilpotent algebra over a finite field k the algebra group p 1 j admits a standard supercharacter theory as defined in 16 if j is endowed with an involution then naturally defines a group automorphism of p 1 j and we may consider the fixed point subgroup cp x p x x 1 assuming that k has odd characteristic p we use the standard supercharacter theory for p to construct a supercharacter theory for cp in particular we obtain a supercharacter theory for the sylow p subgroups of the finite classical groups of lie type and thus extend in a uniform way the construction given by andre and neto in 7 8 for the special case of the symplectic and orthogonal groups

on nonconvergence of adjoint semigroups for control systems with delays it is shown that the adjoints of a spline based approximation scheme for delay equations do not converge strongly

closed form h infinity optimal control for a class of infinite dimensional systems h infinity optimal control and estimation are addressed for a class of systems governed by partial differential equations with bounded input and output operators diffusion equations are an important example in this class explicit formulas for the optimal state feedback controller as well as the optimal state estimator are given unlike traditional methods for h infinity synthesis no iteration is needed to obtain the optimal solution moreover the optimal performance for both the state feedback and state estimation problems are explicitly calculated this is shown to be useful for problems of h infinity optimal actuator and sensor location furthermore the results can be used in testing and bench marking of general purpose algorithms for h infinity synthesis the results also apply to finite dimensional systems

computing in unitriangular matrices over finite fields abstract let g n p be the sylow p subgroup of sl n p formed by the upper unitriangular matrices the aim of this paper is to describe algorithms for the computation of the number of conjugacy classes the conjugacy vector of g n p the character rational or real of the elements of g n p the cardinality of the centralizer of each matrix of g n p the conjugacy vector of the normal subset n corresponding to a pivot disposition and the character inert or ramification of each entry of any matrix of g n p for p 2 by using these algorithms we have proved that kirillov s conjecture every matrix of g n 2 is conjugate to its inverse holds for n 12 but for n 13 there exists a unique pair of inverse conjugacy classes not conjugate a representative pair of these conjugacy classes is given in j algebra 202 1998 704 for n 14 we give the complete list of the canonical matrices of the 22 counterexamples to kirillov s conjecture for n 14 we have proved that a and a 5 are conjugate and for n 25 we have found a matrix a g 25 2 such that a and a 5 are not conjugate in addition for n 32 we have found a matrix a g 32 2 such that a and a 1 are conjugate but a and a 5 are not conjugate so isaacs conjecture every real matrix in g n 2 m is actually rational is not true for p 3 isaacs and kereguezian give in ibid a matrix a g 20 3 such that a and a 4 are not conjugate in this paper for every odd prime p we obtain a matrix a g n p with n 6 p 1 such that a and a 1 p are not conjugate consequently there exist irreducible characters of g 6p 1 p that are not q p valued where p is a primitive p th root of unity besides for p 3 and n 13 we have computed the conjugacy vector of g n 3 and verified that every matrix a g n 3 is conjugate to a 4 thus for all n 13 the character values of g n 3 always lie in q 3

state space solutions to standard h sub 2 and h sub infinity control problems simple state space formulas are derived for all controllers solving the following standard h sub infinity problem for a given number gamma 0 find all controllers such that the h sub infinity norm of the closed loop transfer function is strictly less than gamma it is known that a controller exists if and only if the unique stabilizing solutions to two algebraic riccati equations are positive definite and the spectral radius of their product is less than gamma sup 2 under these conditions a parameterization of all controllers solving the problem is given as a linear fractional transformation lft on a contractive stable free parameter the state dimension of the coefficient matrix for the lft constructed using the two riccati solutions equals that of the plant and has a separation structure reminiscent of classical lqg i e h sub 2 theory this paper is intended to be of tutorial value so a standard h sub 2 solution is developed in parallel

on finite permutation groups with a transitive cyclic subgroup abstract in this paper we study the structure of finite permutation groups with a transitive cyclic subgroup in particular we extend the classification due to w feit and g a jones of the primitive groups containing such subgroups to permutation groups that are i quasiprimitive ii almost simple and iii innately transitive we also analyse the actions of normal subgroups of arbitrary groups with transitive cyclic subgroups we give an application to circulant homogeneous factorisations of complete graphs obtaining new structural information and giving a new proof of their possible parameters

optimal actuator sensor location for active noise regulator and tracking control problems in this paper we investigate the problem of finding the optimal location of sensors and actuators to achieve reduction of the noise field in an acoustic cavity we offer two control strategies the first is based on linear quadratic tracking where the offending noise is tracked and the second considers the formulation of the harmonic control strategy as a periodic static output feedback control problem the first method which is based on full state information is suitable for optimal location of actuators while the second strategy can extend the results to finding optimal location of sensors as well as actuators for both methods we consider the optimization of an appropriate quadratic performance criterion with respect to the location of the actuators and or the sensors numerical examples are presented to compare the effectiveness of each control strategy and also the effect of optimal placement of actuators and sensors

iterative character constructions for algebra groups abstract we construct a family of orthogonal characters of an algebra group which decompose the supercharacters defined by diaconis and isaacs 2008 6 like supercharacters these characters are given by nonnegative integer linear combinations of kirillov functions and are induced from linear supercharacters of certain algebra subgroups we derive a formula for these characters and give a condition for their irreducibility generalizing a theorem of otto 2010 20 we also show that each such character has the same number of kirillov functions and irreducible characters as constituents in proving these results we observe as an application how a recent computation by evseev 2010 7 implies that every irreducible character of the unitriangular group ut n q of unipotent n n upper triangular matrices over a finite field with q elements is a kirillov function if and only if n 12 as a further application we discuss some more general conditions showing that kirillov functions are characters and describe some results related to counting the irreducible constituents of supercharacters

a supercharacter theory for the sylow p subgroups of the finite symplectic and orthogonal groups abstract we define the superclasses for a classical finite unipotent group u of type b n q c n q or d n q and show that together with the supercharacters defined in c a m andre a m neto supercharacters of the sylow p subgroups of the finite symplectic and orthogonal groups pacific j math 239 2 2009 201 230 they form a supercharacter theory in the sense of p diaconis i m isaacs supercharacters and superclasses for algebra groups trans amer math soc 360 5 2008 2359 2392 in particular we prove that the supercharacters take a constant value on each superclass and evaluate this value as a consequence we obtain a factorization of any superclass as a product of elementary superclasses in addition we also define the space of superclass functions and prove that it is spanned by the supercharacters as a consequence we re obtain the decomposition of the regular character as an orthogonal linear combination of supercharacters finally we define the supercharacter table of u and prove various orthogonality relations for supercharacters similar to the well known orthogonality relations for irreducible characters

classical groups probabilistic methods and the 2 3 generation problem we study the probability that randomly chosen elements of prescribed type in a finite simple classical group g generate g in particular we prove a conjecture of kantor and lubotzky in this area the probabilistic approach is then used to determine the finite simple classical quotients of the modular group psl2 z up to finitely many exceptions

finite edge transitive cayley graphs and rotary cayley maps this paper aims to develop a theory for studying cayley graphs especially for those with a high degree of symmetry the theory consists of analysing several types of basic cayley graphs normal bi normal and core free and analysing several operations of cayley graphs core quotient normal quotient and imprimitive quotient it provides methods for constructing and characterising various combinatorial objects such as half transitive graphs orientable and non orientable regular cayley maps vertex transitive non cayley graphs and permutation groups containing certain regular subgroups in particular a characterisation is given of locally primitive holomorph cayley graphs and a classification is given of rotary cayley maps of simple groups also a complete classification is given of primitive permutation groups that contain a regular dihedral subgroup

computing the positive stabilizing solution to algebraic riccati equations with an indefinite quadratic term via a recursive method an iterative algorithm to solve algebraic riccati equations with an indefinite quadratic term is proposed the global convergence and local quadratic rate of convergence of the algorithm are guaranteed and a proof is given numerical examples are also provided to demonstrate the superior effectiveness of the proposed algorithm when compared with methods based on finding stable invariant subspaces of hamiltonian matrices a game theoretic interpretation of the algorithm is also provided