generalized invariant subspaces and parameter insensitive disturbance rejection problems with static output feedback in the geometric approach some generalized invariant subspaces for uncertain linear systems are investigated and then various parameter insensitive disturbance rejection problems with static output feedback are formulated and necessary and sufficient conditions for the problems to be solvable are presented further some sufficient conditions which are useful to check the solvability conditions are presented using the notions of generalized invariant subspaces

disturbance localization with dead beat control for linear periodic discrete time systems in this paper the classical notion of a controlled invariant subspace together with its main properties is extended to the case of linear periodic discrete time systems and used for deriving necessary and sufficient solvability conditions for the disturbance localization problem and a synthesis procedure for the solution moreover the notions of outer and inner controllable subspaces are introduced and studied for the same class of system thus allowing the derivation of necessary and sufficient solvability conditions for the disturbance localization problem with output or state dead beat control and to give synthesis procedures for the solutions

a unified approach to some recurrence sequences via faa di bruno s formula abstract the well known faa di bruno formula for higher derivatives of a composite function plays an important role in combinatorics in this paper we obtain a unified approach to some recurrence sequences for the complete bell polynomials by using faa di bruno s formula furthermore we obtain an inverse relation of lah numbers of two types and by using this relation we obtain some new and interesting combinatorial identities

solvability conditions for disturbance decoupling problems with static measurement feedback two well known disturbance decoupling problems are considered a checkable necessary and sufficient condition is derived under which the disturbance decoupling problem with static measurement feedback ddpkm is solvable for a class of multi input and multi output linear time invariant systems which have a left invertible transfer matrix function from the control input to the controlled output the set of all solutions that solve the ddpkm is constructed and parametrized explicitly this set is then used to solve the disturbance decoupling problem with static measurement feedback and with internal stability ddpkms

disturbance decoupling for linear time invariant systems a matrix pencil approach we give a new systematic analysis of disturbance decoupling problems for standard linear time invariant systems based on the theory of matrix pencils this approach is based on the computation of condensed forms under orthogonal equivalence transformations from these forms which can be computed in a numerically stable way we obtain new necessary and sufficient conditions that are numerically verifiable and furthermore we immediately obtain numerically stable algorithms to compute the desired compensators we present a numerical example that demonstrates the properties of the new approach

the sum of like powers of the zeros of the riemann zeta function on analyse une methode de calcul de la somme r p r ou r est un entier superieur a 1 la somme etant prise sur tous les zeros complexes de la fonction zeta de riemann

on the eigenvalues of the electrostatic integral operator abstract a fundamental result in scattering and potential theory in r 3 states that the spectrum of the electrostatic integral operator lies in the interval 1 1 in the case of a sphere and a prolate spheroid it is known that the spectrum of the operator lies in the interval 1 0 a fundamental question arises whether the spectrum of the operator always lies in this interval or whether there exists a smooth surface for which the electrostatic integral operator has a positive eigenvalue in this paper this question is answered a surface is produced whereby the underlying integral operator has a positive eigenvalue

special coordinate basis for multivariable linear systems finite and infinite zero structure squaring down and decoupling abstract a special coordinate basis for multivariable linear systems is introduced here several fundamental properties of linear systems regarding controllability stabiliza bility observability detectability invariant zeros decoupling zeros infinite zero structure effect of feedback on zero structure squaring down diagonal and triangular decoupling etc can be directly displayed in terms of the special coordinate basis moreover connections between the special coordinate basis and the important invariant subspaces of the geometric theory of linear systems are established

a new approach to synthesis problems in linear system theory we consider a rational nonlinear equation which is involved in the general synthesis design problem of linear multivariable systems the complete solution of this equation subject to causality stability and minimality constraints is provided formal power series and in particular the theory of partial realizations prove of central importance for the design problem

on the convergence of interior reflective newton methods for nonlinear minimization subject to bounds we consider a new algorithm an interior reflective newton approach for the problem of minimizing a smooth nonlinear function of many variables subject to upper and or lower bounds on some of the variables this approach generatesstrictly feasible iterates by using a new affine scaling transformation and following piecewise linear paths reflection paths the interior reflective approach does not require identification of an activity set in this paper we establish that the interior reflective newton approach is globally and quadratically convergent moreover we develop a specific example of interior reflective newton methods which can be used for large scale and sparse problems

on the magnetostatic integral operator for ellipsoids the magnetostatic integral operator in r3is of considerable interest in both scattering and potential theory as well as for its scalar pendant the electrostatic integral operator its eigenvalues and its eigenfunctions play a fundamental role in the theory of permanent magnetization while the eigenvalues of both operators are closely related explicit expressions for the eigenfunctions of the magnetostatic operator are known only in the case when the underlying surface is a sphere for the cases of ellipsoids and spheroids we present explicit expressions for the eigenvalues in terms of lame functions and associated legendre functions and expressions for eigenfunctions in terms of surface ellipsoidal and surface spheroidal harmonics respectively as an application the permanent magnetization of ellipsoids is discussed

divided differences and a general explicit formula for sequences of mansour mulay shattuck divided differences is a very important tool often used both in theory and in practice for the approximation of functions in this paper we employ divided differences with repetitions to find a general explicit formula for the sequences satisfying mansour mulay shattuck s two term recurrence with arbitrary b i i 0

the algebraic regulator problem from the state space point of view abstract we study the algebraic aspects of the regulator problem using some new ideas in the state space geometric approach to feedback design problems for linear multi variable systems necessary and sufficient conditions are given for the solvability of a general version of this problem requiring output stability internal stability and disturbance decoupling as well an algorithm is given by which these conditions can be verified from the system parameters

crandall s computation of the incomplete gamma function and the hurwitz zeta function with applications to dirichlet l series this paper extends tools developed by crandall 2012 16 to provide robust high precision methods for computation of the incomplete gamma function and the lerch transcendent we then apply these to the corresponding computation of the hurwitz zeta function and so of dirichlet l series and character polylogarithms

electric field and potential calculation for a bioelectric current dipole in an ellipsoid computer based modelling in ellipsoidal geometry can be useful in electro encephalography and gastrography due to the resemblance of the human brain and stomach to an ellipsoid both theoretically and computationally the bioelectric current dipole model is important for the study of electrical activity in these two organs computing the electric potential and electric field e due to a current dipole located in an ellipsoid requires truncated series expansions involving ellipsoidal harmonics which are the products of lame functions a theoretical model for this has appeared in the literature only for expansions of order 2 in however this may be insufficient in forward electrogastrography while an analogous situation is encountered in some geodetic problems where similar expansions are required in this paper we propose a generalized model for computing and e numerically using harmonic expansions of arbitrary order and degree the implementation of such a procedure involves finding roots of lame polynomials of degree 5 or higher using an optimization technique for solving nonlinear systems of equations this process can allow one to make a knowledgeable decision concerning the optimal expansion size required for the physical problem under investigation without compromising the overall accuracy of the computation

brief fixed poles of disturbance rejection by dynamic measurement feedback a geometric approach this paper deals with the problem of disturbance rejection by dynamic measurement feedback it aims at proposing new geometric characterizations for the fixed poles of this problem we also propose solutions to the problem for which all the poles except the fixed poles of the problem are freely placed these fixed poles are also characterized as some particular invariant zeros as a corollary we propose a necessary and sufficient condition for the existence of internally stable solutions which generalizes the sufficient condition proposed by basile and marro

the fixed poles of the disturbance decoupling problem and almost stability subspace mathcal v star_ b g rm ker c this paper is concerned with the fixed poles of the disturbance decoupling problem for linear time invariant systems by state feedback and measurement feedback algebraic characterizations for these fixed poles are given using the reducing subspace technique in numerical linear algebra these algebraic characterizations lead that the fixed poles can be directly computed by numerically reliable algorithms as an additional application of the reducing subspace technique we also develop a numerically stable method for computing the almost stability subspace mathcal v star b g rm ker c which is an extension of the method given by van dooren for computing the invariant subspaces in geometric control theory

two chain rules for divided differences and faa di bruno s formula in this paper we derive two formulas for divided differences of a function of a function both formulas lead to other divided difference formulas such as reciprocal and quotient rules the two formulas can also be used to derive faa di bruno s formula and other formulas for higher derivatives of composite functions we also derive a divided difference version of faa di bruno s determinant formula

disturbance localization and pole shifting by dynamic compensation this paper concerns the problem of disturbance localization and pole shifting by dynamic compensation in multivariable linear systems with inaccessible state vector a necessary and sufficient condition for the solvability of the problem is given two synthesis procedures of the compensator are presented one is based on an observer and the other is based on a dual observer the observer designed by the proposed procedure acts as as unknown input observer for a linear function an improved procedure of designing a reduced order unknown input observer is also given the result may be applicable to reducing the compensator order

magnetoencephalography in ellipsoidal geometry an exact analytic solution for the forward problem in the theory of biomagnetics of the human brain is known only for the 1d case of a sphere and the 2d case of a spheroid where the excitation field is due to an electric dipole within the corresponding homogeneous conductor in the present work the corresponding problem for the more realistic ellipsoidal brain model is solved and the leading quadrupole approximation for the exterior magnetic field is obtained in a form that exhibits the anisotropic character of the ellipsoidal geometry the results are obtained in a straightforward manner through the evaluation of the interior electric potential and a subsequent calculation of the surface integral over the ellipsoid using lame functions and ellipsoidal harmonics the basic formulas are expressed in terms of the standard elliptic integrals that enter the expressions for the exterior lame functions the laborious task of reducing the results to the spherical geometry is also included

a sum property of the eigenvalues of the electrostatic integral operator abstract the double layer potential integral operator and the electrostatic integral operator in r 3 occur while using integral equation methods in scattering and potential theory if the underlying surface is either a sphere a spheroid or a triaxial ellipsoid explicit expressions of the eigenvalues and eigenfunctions of order n n 0 are known the main result of this paper is the following the sum of all eigenvalues of order n n 0 each counted with respect to its multiplicity is 1 this is trivial for the sphere in the case of spheroids a proof is given for n n 0 for the triaxial ellipsoid this sum property is verified for n 0 1 2 3 whereas for n 3 some numerical results are provided

ddp and model following problem revisited abstract for linear multivariable systems with two input two output plant and compensator the paper solves the disturbance decoupling perfect tracking problems with stability results are obtained for the not necessarily robust as well as for the robust problem

regulator synthesis using c a b pairs the concept of a c a b pair of subspaces was introduced in an earlier paper by the author to study problems of compensator design here it is shown that the same concept plays a basic role in the theory of tracking and regulation from a few fundamental properties the main results of regulator theory are derived in a relatively simple and straightforward manner another new feature is that we are able to allow the presence of an unknown input thereby unifying the fields of tracking and regulation and disturbance decoupling all results are fully constructive and an example is given to illustrate the theory

electroencephalography in ellipsoidal geometry the human brain is shaped in the form of an ellipsoid with average semiaxes equal to 6 6 5 and 9 cm this is a genuine 3 d shape that reflects the anisotropic characteristics of the brain as a conductive body the direct electroencephalography problem in such anisotropic geometry is studied in the present work the results which are obtained through successively solving an interior and an exterior boundary value problem are expressed in terms of elliptic integrals and ellipsoidal harmonics both in jacobian as well as in cartesian form reduction of our results to spheroidal as well as to spherical geometry is included in contrast to the spherical case where the boundary does not appear in the solution the boundary of the realistic conductive brain enters explicitly in the relative expressions for the electric field moreover the results in all three geometrical models reveal that to some extend the strength of the electric source is more important than its location

combinatorial formulae for the derivatives of lame functions lame functions play a central role in the theory of ellipsoidal harmonics and have many varied applications in mathematical physics in this article a generalized approach to the computation of lame function derivatives of arbitrary order is derived and it is demonstrated how these derivatives can be expressed recursively in terms of the original lame functions by making use of combinatorial formulae discovered by di bruno girard and waring

brief solutions to disturbance decoupling problem with constant measurement feedback for linear systems we study in this paper the problem of disturbance decoupling with constant i e static measurement feedback ddpcm for linear systems for a class of systems which have a left invertible transfer function from the control input to the controlled output or a right invertible transfer function from the disturbance input to the measurement output we obtain a complete characterization of all solutions to the ddpcm for a system that does not satisfy the above invertibility condition we use the special co ordinate basis to obtain a reduced order system then a complete characterization of all possible solutions to the ddpcm for the given system can be explicitly obtained if the obtained reduced order system itself satisfies the invertibility condition the main advantage of these solutions is that the solutions are given in a set of linear equations this resolves the well known difficulty in solving non linear equations associated with the ddpcm

the stieltjes function definition and properties definition et proprietes d une fonction liee au developpement en serie de laurent de la fonction zeta de riemann au voisinage du point singulier s 1 fonction de stieltjes

disturbance decoupling by measurement feedback with stability or pole placement in this paper we solve the disturbance decoupling problem by measurement feedback and requiring stability or pole placement on the closed loop system the problem is attacked using the geometric approach through the concepts of a bmod mathcal b invariant and controllability subspaces and their duals a mid k invariant and complementary observability subspaces the solution of this problem has an interesting structure consisting of a feedback processor which decomposes into i a disturbance decoupling loop ii a disturbance input stabilization or pole placement loop and iii a controlled output stabilization or pole placement loop

computation and theory of extended mordell tornheim witten sums we consider some fundamental generalized mordell tornheim witten mtw zeta function values along with their derivatives and explore connections with multiple zeta values mzvs to achieve this we make use of symbolic integration high precision numerical integration and some interesting combinatorics and special function theory our original motivation was to represent unresolved constructs such as eulerian log gamma integrals we are able to resolve all such integrals in terms of a mtw basis we also present for a substantial subset of mtw values explicit closed form expressions in the process we signicantly extend methods for high precision numerical computation of polylogarithms and their derivatives with respect to order

supratransmission in a disordered nonlinear periodic structure abstract we study the interaction among dispersion nonlinearity and disorder effects in the context of wave transmission through a discrete periodic structure subjected to continuous harmonic excitation in its stop band we consider a damped nonlinear periodic structure of finite length with disorder disorder is introduced throughout the structure by small changes in the stiffness parameters drawn from a uniform statistical distribution dispersion effects forbid wave transmission within the stop band of the linear periodic structure however nonlinearity leads to supratransmission phenomenon by which enhanced wave transmission occurs within the stop band of the periodic structure when forced at an amplitude exceeding a certain threshold the frequency components of the transmitted waves lie within the pass band of the linear structure where disorder is known to cause anderson localization there is therefore a competition between dispersion nonlinearity and disorder in the context of supratransmission we show that supratransmission persists in the presence of disorder the influence of disorder decreases in general as the forcing frequency moves away from the pass band edge reminiscent of dispersion effects subsuming disorder effects in linear periodic structures we compute the dependence of the supratransmission force threshold on nonlinearity and strength of coupling between units we observe that nonlinear forces are confined to the driven unit for weakly coupled systems this observation together with the truncation of higher order nonlinear terms permits us to develop closed form expressions for the supratransmission force threshold in sum in the frequency range studied here disorder does not influence the supratransmission force threshold in the ensemble average sense but it does reduce the average transmitted wave energy

multidimensional bell polynomials of higher order we develop some extensions of the classical bell polynomials previously obtained by introducing a further class of these polynomials called multidimensional bell polynomials of higher order they arise considering the derivatives of functions f in several variables f i i 1 2 m where f i are composite functions of different orders i e f i t i 1 i 2 i r i t i 1 2 m we show that these new polynomials are always expressible in terms of the ordinary bell polynomials by means of suitable recurrence relations or formal multinomial expansions moreover we give a recurrence relation for their computation

the riemann zeta function and its derivatives formulas for higher derivatives of the riemann zeta function are developed from ramanujan s theory of the constant of series by using the euler maclaurin summation methods formulas for zeta n s zeta n 1 s and zeta n 0 are obtained additional formulas involving the stieltjes constants are also derived analytical expression for error bounds is given in each case the formulas permit accurate derivative evaluation and the error bounds are shown to be realistic a table of zeta prime s is presented to 20 significant figures for s 20 0 1 20 for rational arguments zeta 1 k zeta prime 1 k are given for k 10 1 10 the first ten zeros of zeta prime s are also tabulated because the stieltjes constants appear in many formulas the constants were evaluated freshly for this work formulas for the gamma n are derived with new error bounds and a tabulation of the constants is given from n 0 to 100

a numerical solution for the simultaneous disturbance decoupling and row by row decoupling problem the numerical computation of the simultaneous disturbance decoupling and row by row decoupling problem is considered based on a condensed form for the system matrices the solvability of the simultaneous disturbance decoupling and row by row decoupling problem is reduced to the verification of an integer equality and the nonsingularity of a lower dimensional constant matrix the condensed form we use is computed using only orthogonal transformations which can be implemented in a numerically stable way hence it leads directly to an effective numerical method for solving the disturbance decoupling and row by row decoupling problem using existing tools such as lapack and matlab our result complements existing geometric and structural approaches which are of a theoretical nature and are not suitable for numerical computation

a spectral property of the electrostatic integral operator for an ellipsoid the eigenfunctions of the electrostatic integral operator and its adjoint are governed by the lame polynomials it is shown that the arithmetic mean of the eigenvalues corresponding to the maximum number of linearly independent eigenfunctions of equal order is the same for the ellipsoid as for the sphere

formulas for higher derivatives of the riemann zeta function the functional equation for c s is used to obtain formulas for all derivatives t k s a closed form evaluation of t k 0 is given and numerical values are computed to

a multivariate faa di bruno formula with applications a multivariate faa di bruno formula for computing arbitrary partial derivatives of a function composition is presented it is shown by way of a general identity how such derivatives can also be expressed in the form of an infinite series applications to stochastic processes and multivariate cumulants are then delineated

some extensions of faa di bruno s formula with divided differences the well known formula of faa di bruno s for higher derivatives of a composite function has played an important role in combinatorics in this paper we generalize the divided difference form of faa di bruno s formula and give an explicit formula for the n th divided difference of a multicomposite function more generally we establish the relationships of the bell polynomials with respect to multicomposite functions applying these to multicomposite functions we obtain some extensions of faa di bruno s formula

new spectral results for the electrostatic integral operator abstract in this paper two new fundamental results are established concerning the spectra of both the double layer potential and the electrostatic integral operators in r 3 while it is known that 0 lies in the spectrum of each operator its specific nature is not well understood for some geometries e g a sphere and a prolate spheroid it is known that 0 lies in their continuous spectra in this paper however specific examples are produced where 0 lies in the point spectrum of each operator thus it is established that the spectral nature of 0 depends upon the geometry of the underlying surface the second new result given is an important generalization of a recent paper concerning the possible refinement of a classical result of plemelj

finite size effects on instabilities of discrete breathers abstract when the two arcs of the continuous phonon spectrum of the floquet matrix of a discrete breather overlap on the unit circle the breather solution in the infinite lattice might be stable while the corresponding solutions in finite systems appear to be unstable more precisely when the model parameters vary the breather in the finite system exhibits a large number of collisions between the floquet eigenvalues belonging to the phonon spectrum these collisions correspond to complex cascades of instability thresholds followed near after by re entrant stability thresholds we interpret this complex structure on the basis of the band analysis of the matrix of the second variation of the action then we can predict that in the limit of an infinite system the number of instability and stability thresholds in the cascade diverges but simultaneously the maximum amplitude of the instabilities vanishes so that the breather in the infinite system recovers its linear stability as long as all its other localized modes remain stable this is the situation which is required in cretegny et al physica d 119 1998 73 87 for having inelastic phonon scattering with two channels we also analyze the size effects when a floquet eigenvalue associated with a localized mode collides with the floquet continuous phonon spectrum with different krein signature in contrast to the previous case the infinite system is unstable after the collision

an extension of the bell polynomials abstract after recalling the most important properties and applications of the bell polynomials we introduce an extension of this special class of functions more precisely we consider the case of multicomposite functions and we show connections with the ordinary bell polynomials

nonlinear resonances and energy transfer in finite granular chains in the present work we test experimentally and compute numerically the stability and dynamics of harmonically driven monoatomic granular chains composed of an increasing number of particles n n 1 50 in particular we investigate the inherent effects of dissipation and finite size on the evolution of bifurcation instabilities in the statically compressed case the findings of the study suggest that the nonlinear bifurcation phenomena which arise due to finite size can be useful for efficient energy transfer away from the drive frequency in transmitted waves

the structure of robust disturbance rejection control this paper deals with the problem of disturbance rejection under state feedback in linear multivariable control systems this problem is ill posed under arbitrary state space parameter perturbations however it is shown that with respect to arbitrary perturbations of the plant transfer function coefficients a robust solvability condition and possibly a robust synthesis can be obtained these results are significant because the class of arbitrary state space parameter variations is too rich for practical interest whereas the class of arbitrary variations in transfer functions more closely represents the realistic problem of engineering concern

some identities involving exponential functions and stirling numbers and applications guo and qi 2013 posed a problem asking to determine the coefficients a k i 1 for 1 i k such that 1 1 e t k 1 i 1 k a k i 1 1 e t 1 i 1 the authors answer this question alternatively by faa di bruno s formula unify the eight identities due to guo and qi to two identities involving two parameters and apply them to obtain an explicit expression for the apostol bernoulli numbers and the fubini numbers respectively