on the general linear group and hochschild homology our main result here is a rational computation of the homology of the adjoint action of the infinite general linear group of an arbitrary ring before stating the result we establish some notation and conventions rings are associative and with unit if a is a ring then gl a uk 0glk a is its infinite general linear group an a bimodule is an abelian group b which is both a left a module and a right a module and satisfies alb a2 al ba2 for ai ea be b for example b a if b is an a bimodule then m b uk omk b is the infinite additive group of matrices with entries in b conjugation defines an action the adjoint action of gl a on m b note that an a q bimodule is just an a bimodule which is also a rational vector space if b is an a q bimodule then hj a c q b denotes the hochschild homology of a q with coefficients in b our main result it appears in slightly more detailed form as theorem v 3 is

disturbance rejection of left invertible systems abstract a new approach is presented for the study of the disturbance rejection of left invertible systems via proportional state feedback reducing the problem to that of solving a linear algebraic system of equations on the basis of this system of equations the necessary and sufficient conditions for the existence of a solution to the disturbance rejection problem are established in a form of simple algebraic criteria thus reducing the computational effort over known results and the general analytical expressions of all solutions of the controller matrices are derived these solutions are parameterized only by an arbitrary matrix

controllability observability and duality in singular systems the concepts of controllability and observability for systems of the form e dot x ax bu y cx e singular are considered a theory is presented which unifies the three main approaches to this topic already existing in the literature the development includes a generalization of the duality theorem from state space theory

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

disturbance decoupling of linear time varying singular systems the problem of disturbance decoupling by state feedback is defined for a linear time varying singular system it is required that the closed loop system has a unique impulse free solution and its output is not affected by disturbances an algorithm namely disturbance decoupling algorithm is proposed it is proved that the feasibility of the disturbance decoupling algorithm is invariant under any regular feedback control law based on the disturbance algorithm a constructive method is provided to design a disturbance decoupling feedback control law sufficient conditions for the solvability of the disturbance decoupling problem are derived it is proved that one of the sufficient conditions is also necessary provided that other conditions are satisfied

disturbance decoupling problems by measurements feedback a characterization of all solutions and fixed modes this paper presents a new approach to disturbance decoupling problems by measurement feedback ddpm the novelty in this approach is that it yields a characterization of the sets of all solutions to ddpm and ddpm with internal stability and that the set of fixed modes of ddpm with respect to dynamic output feedback is easily identified the central object used in deriving these results is a minimal polynomial basis of a rational vector space which conveniently represents the problem data

disturbance rejection a polynomial approach a detailed analysis of the disturbance rejection problem in single input single output linear systems is presented the analysis is based on an external polynomial model and employs the algebra of polynomials all solutions are expressed in a parametric form and solvability conditions are given for a number of constraints including those of internal stability and arbitrary pole placement measurable and unmeasurable disturbances are distinguished and the effect of output feedback is investigated a simple design procedure is then proposed which provides a stable dynamical solution and which can be extended so as to minimize its order the procedure consists of solving two linear polynomial equations and yields the compensator in a compact form suitable for direct realization

conductivity exponent and backbone dimension in 2 d percolation abstract we present high statistics simulations for 2 d percolation clusters in the bus bar geometry at the critical point for site and for bond percolation we measured their backbone sizes and electrical conductivities for all sets of measurements we find large corrections to scaling most of which do not seem to be described by single powers using single power terms for the corrections to scaling of the backbone masses we would obtain fractal dimensions which are different for site and bond percolation while the correct result is d b 1 6432 0 0008 for both for the conductivity the corrections to scaling are strongly non monotonic for bond percolation the exponent t t is measured as 0 9826 0 0008 in disagreement with the alexander orbach and other conjectures

topological proof for the alexander orbach conjecture this paper advocates an unconventional analytical approach to studying the fractal geometry of percolation at the threshold which is based on the most general methods of the differential topology our particular interest concentrates on the alexander orbach ao conjecture j phys france lett 43 l625 1982 which assigns the universal mean field value 4 3 to the spectral fractal dimension stackrel ifmmode tilde else fi d at the percolation threshold for all embedding euclidean dimensions n greater than one i e ng 2 using the topological arguments we show that the ao conjecture might be improved for the relatively low embedding dimensions 2l nl 5 for which the analytical result stackrel ifmmode tilde else fi d 1 327 ifmmode pm else textpm fi 0 001 is proposed instead of the original ao estimate 4 3 meanwhile we assume that the exact value stackrel ifmmode tilde else fi d 4 3 holds for all ng 6 as it follows directly from the well known mean field theory the improved value of stackrel ifmmode tilde else fi d ensuremath approx 1 327 for 2l nl 5 is obtained from an analysis of the basic topological properties of the percolating fractal sets at the threshold of percolation we show that these properties could be investigated fruitfully with the introduction of the concept of the fractal manifold which might serve as an effective instrument when considering the topology of the fractal objects our results indicate that the proposed value of stackrel ifmmode tilde else fi d ensuremath approx 1 327 for the spectral dimension at the percolation threshold has the fundamental topological origin related to the most general features of the fractal geometry of percolation at criticality we argue that the constraint 2l nl 5 on the topological dimension n of the embedding euclidean space is the direct consequence of the famous whitney theorem which establishes the embedding properties of manifolds from the viewpoint of their dimensionality a simple topological condition that identifies the threshold of percolation is obtained for 2l nl 5 the particular topological restrictions implied throughout the present study are discussed and the important issue of contractibility of the fractal manifolds is pointed out

general linear and functor cohomology over finite fields in recent years there has been considerable success in computing extgroups of modular representations associated to the general linear group by relating this problem to one of computing ext groups in functor categories f l s f s in this paper we extend our ability to make such ext group calculations by establishing several fundamental results throughout this paper we work over fields of positive characteristic p the reader familiar with the representation theory of algebraic objects will recognize the importance of an understanding of ext groups for example the existence of nonzero ext groups of positive degree is equivalent to the existence of objects which are not direct sums of simple objects indeed a knowledge of ext groups provides considerable knowledge of compound objects in the study of modular representation theory of finite chevalley groups such as gln fq ext groups play an even more central role it has been shown in cps that a knowledge of certain ext groups is sufficient to prove lusztig s conjecture concerning the dimension and characters of irreducible representations we consider two different categories of functors the category f fq of all functors from finite dimensional fq vector spaces to fq vector spaces where fq is the finite field of cardinality q and the category p fq of strict polynomial functors of finite degree as defined in f s the category p fq presents several advantages over the category f fq from the point of view of computing extgroups these are the accessibility of injectives and projectives the existence of a base change and an even easier access to ext groups of tensor products this explains the usefulness of our comparison in theorem 3 10 of ext groups in the category p fq with ext groups in the category f fq weaker forms of this theorem have been known to us since 1995 and to s betley independently

measuring complexity nonextensivity and chaos in the dna sequence of the major histocompatibility complex abstract we analyze 4 mb sequences of the major histocompatibility complex mhc which is a dna segment on chromosome 6 with high gene density controlling many immunological functions and associated with many diseases the analysis is based on modern theoretical and mathematical tools of complexity theory such as nonlinear time series analysis and tsallis non extensive statistics the results revealed that the dna complexity and self organization can be related to fractional dynamical nonlinear processes with low dimensional deterministic chaotic and non extensive statistical character which generate the dna sequences under the extremization of tsallis q entropy principle while it still remains an open question as to whether the dna walk is a fractional brownian motion fbm a static anomalous diffusion process or a non gaussian dynamical fractional anomalous diffusion process the results of this study testify for the latter providing also a possible explanation for the previously observed long range power law correlations of nucleotides as well as the long range correlation properties of coding and non coding sequences present in dna sequences

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

anomalous diffusion on a fractal mesh we present exact analytical results for properties of anomalous diffusion on a fractal mesh the fractal mesh structure is a direct product of two fractal sets one belonging to a main branch of backbones the other to the side branches of fingers both fractal sets are constructed on the entire infinite y and x axes we suggest a special algorithm in order to construct such sets out of standard cantor sets embedded in the unit interval the transport properties of the fractal mesh are studied in particular subdiffusion along the backbones is obtained with the dispersion relation x 2 t t where the transport exponent 1 has been observed as well when the environment is controlled by means of a memory kernel

precise calculation of the dynamical exponent of two dimensional percolation we report numerical data obtained on the special purpose computer percola for the exponent tilde t of the electrical conductivity of 2d percolation the extrapolation yields tilde t 0 9745 pm 0 0015 and a correction to the scaling exponent 1 2 0 2

a case study for the open question disturbance decoupling problem for singular systems by output feedback we study the disturbance decoupling problem for singular systems ex spl dot ax bu gd y cx z hx with left invertibility we give necessary and sufficient conditions for the existence of a solution to the disturbance decoupling problem with or without stability by output feedback that also makes the resulting closed loop system regular and or of index at most one all results are proved based on a condensed form that can be computed using orthogonal transformations i e can be computed in a numerically stable way

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

noninteracting control of descriptor systems involving disturbances for m input p output descriptor systems involving disturbances it is proven that if the problem of disturbance rejection is solvable via static state feedback and the input output transfer function matrix is right invertible there always exists a static state feedback control law yielding simultaneously to disturbance rejection a triangular input output relation the structural properties of the closed loop system stability pole assignment etc are extensively studied

on disturbance decoupling in descriptor systems the disturbance decoupling problem for a linear multivariable control system is to determine when and how a state feedback may be chosen so that some class of disturbances has no effect on the system output the development of necessary and sufficient conditions for the solvability of this problem for state space systems by wonham and his co workers has profoundly influenced geometric control theory particularly by means of the concept of an a b invariant subspace this paper develops necessary and sufficient conditions for solvability of this problem for linear systems described by a mixture of algebraic and differential equations although the argument presented has the same structure as in the state space case and incorporates a generalisation of a b invariant subspaces to descriptor systems there are some additional issues demanding particularly careful attention is the ensuring of existence and uniqueness of classical solutions of the underlying differential equations which cannot be tak

cluster structure near the percolation threshold derives exact relations that allow one to describe unambiguously and quantitatively the structure of clusters near the percolation threshold pc in particular the author proves the relations p dpij dp lambda ij where p is the bond density pij is the pair connectedness function and lambda ij is the average number of cutting bonds between i and j from this relation it follows that the average number of cutting bonds between two points separated by a distance of the order of the connectedness length xi diverges as mod p pc mod 1 the remaining multiply connected bonds in the percolating backbone which lump together in blobs diverge with a dimensionality dependent exponent he also shows that in the cell renormalisation group of reynolds et al 1978 1980 the thermal eigenvalue is simply related to the average number of cutting bonds in the spanning cluster he discusses a percolation model in which the blobs can be controlled by varying a parameter and study the influence on the critical exponents

observer design for generalized state space systems with unknown inputs abstract the following four major aspects for the observer design problem for generalized state space systems with unknown inputs are resolved necessary and sufficient conditions for the problem to have a solution the order of the minimal observer the properties of the closed loop system separation principle and general analytical expressions of the minimal order observer matrices all above results are first in the field

disturbance rejection of left invertible generalized state space systems a new approach is presented for the study of the disturbance rejection of left invertible generalized state space systems via pure proportional state feedback reducing the problem to that of solving a linear algebraic system of equations on the basis of this system of equations the necessary and sufficient conditions for the problem to have a solution are established in a form of simple algebraic criteria thus reducing the computational effort over known results and the general analytical expressions of the controller matrices are derived the structural properties of the closed loop system feedback invariant poles disturbance rejection with simultaneous elimination of the infinite poles are studied

analytical solutions for diffusion on fractal objects a generalization of the euclidean diffusion equation is proposed for diffusion on fractals on the basis of a scaling argument a renormalization group theory for the green s function and numerical tests conjectures on the applicability to natural fractals such as porous media are presented

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

frequency domain conditions for the solvability of the multivariable disturbance decoupling problem given a plant description in terms of its transfer matrices necessary and sufficient conditions are derived in the frequency domain for the solvability of the multivariable disturbance decoupling problem with independent controlled and measured outputs