complexity of matrix problems in representation theory the classification problem is called wild if it contains the problem of classifying pairs of matrices up to simultaneous similarity we show in an explicit form that the last problem contains all classification matrix problems given by quivers or posets then we prove that this problem does not contain but is contained in the problem of classifying three valent tensors hence every wild classification problem given by a quiver or poset has the same complexity moreover a solution of one of them implies a solution of each of the remaining problems the problem of classifying three valent tensors is more complicated

reduced storage vq via secondary quantization this paper introduces methods for reducing the table storage required for encoding and decoding with unstructured vector quantization uvq or tree structured vector quantization tsvq specifically a low storage secondary quantizer is used to compress the code vectors and test vectors of the primary quantizer the relative advantages of uniform and nonuniform secondary quantization are investigated a linde buzo gray lbg like algorithm that optimizes the primary uvq codebook for a given secondary codebook and another that jointly optimizes both primary and secondary codebooks are presented in comparison to conventional methods it is found that significant storage reduction is possible typically a factor of two to three with little loss of signal to noise ratio snr moreover when reducing dimension is considered as another method of reducing storage it is found that the best strategy is a combination of both the method of secondary quantization is also applied to tsvq to reduce the table storage required for both encoding and decoding it is shown that by exploiting the correlation among the test vectors in the tree both encoder and decoder storage can be significantly reduced with little loss of snr by a factor of about four or two relative to the conventional method of storing test vectors or test hyperplanes

advances in residual vector quantization a review advances in residual vector quantization rvq are surveyed definitions of joint encoder optimality and joint decoder optimality are discussed design techniques for rvqs with large numbers of stages and generally different encoder and decoder codebooks are elaborated and extended fixed rate rvqs and variable rate rvqs that employ entropy coding are examined predictive and finite state rvqs designed and integrated into neural network based source coding structures are revisited successive approximation rvqs that achieve embedded and refinable coding are reviewed a new type of successive approximation rvq that varies the instantaneous block rate by using different numbers of stages on different blocks is introduced and applied to image waveforms and a scalar version of the new residual quantizer is applied to image subbands in an embedded wavelet transform coding system

variable length constrained storage tree structured vector quantization constrained storage vector quantization csvq introduced by chan and gersho 1990 1991 allows for the stagewise design of balanced tree structured residual vector quantization codebooks with low encoding and storage complexities on the other hand it has been established by makhoul et al 1985 riskin et al 1991 and by mahesh et al see ieee trans inform theory vol 41 p 917 30 1995 that variable length tree structured vector quantizer vltsvq yields better coding performance than a balanced tree structured vector quantizer and may even outperform a full search vector quantizer due to the nonuniform distribution of rate among the subsets of its input space the variable length constrained storage tree structured vector quantization vlcs tsvq algorithm presented in this paper utilizes the codebook sharing by multiple vector sources concept as in csvq to greedily grow an unbalanced tree structured residual vector quantizer with constrained storage it is demonstrated by simulations on test sets from various synthetic one dimensional 1 d sources and real world images that the performance of vlcs tsvq whose codebook storage complexity varies linearly with rate can come very close to the performance of greedy growth vltsvq of riskin et al and mahesh et al the dramatically reduced size of the overall codebook allows the transmission of the code vector probabilities as side information for source adaptive entropy coding

linear systems theory revisited this paper investigates and clarifies how different definitions of reachability observability controllability reconstructability and minimality that appear in the control literature may be equivalent or different depending on the type of linear system the differences are caused by 1 whether or not the linear system has state dimensions that vary with time 2 bounds on the time axis of the linear system 3 whether or not the initial state is non zero and 4 whether or not the system is time invariant also 5 time reversibility of systems plays a role discrete time linear strictly proper systems are considered a recently published result is used to argue that all the results carry over to continuous time out of the investigation two types of definitions emerge one type applies naturally to systems with constant dimensions while the other applies naturally to systems with variable dimensions this paper reveals that time varying state dimensions that are allowed to be zero are necessary to obtain equivalence between minimality and weak reachability together with observability at the systems level besides their theoretical significance the results of this paper are of practical importance for model reduction and control of time varying discrete time linear systems because they result in minimal realizations with smaller dimensions that are also computed more easily

constrained storage vector quantization with a universal codebook many image compression techniques require the quantization of multiple vector sources with significantly different distributions with vector quantization vq these sources are optimally quantized using separate codebooks which may collectively require an enormous memory space since storage is limited in most applications a convenient way to gracefully trade between performance and storage is needed earlier work addressed this problem by clustering the multiple sources into a small number of source groups where each group shares a codebook we propose a new solution based on a size limited universal codebook that can be viewed as the union of overlapping source codebooks this framework allows each source codebook to consist of any desired subset of the universal code vectors and provides greater design flexibility which improves the storage constrained performance a key feature of this approach is that no two sources need be encoded at the same rate an additional advantage of the proposed method is its close relation to universal adaptive finite state and classified quantization necessary conditions for optimality of the universal codebook and the extracted source codebooks are derived an iterative design algorithm is introduced to obtain a solution satisfying these conditions possible applications of the proposed technique are enumerated and its effectiveness is illustrated for coding of images using finite state vector quantization multistage vector quantization and tree structured vector quantization

a characterization of finite auslander reiten quivers a we give necessary and sufficient conditions for a finite k modulated translation quiver to be an auslander reiten quiver in terms of certain homology groups associated to the quiver section 3 b we show that whether a finite k modulated translation quiver is an auslander reiten quiver depends only on the underlying valued translation quiver and the characteristic of k and give necessary and sufficient conditions only in terms of the valued quiver and char k we also show that any valued quiver with valuations 1 n or n l n q 3 admits a modulation over any prime field sections 5 6 7 c to a valued translation quiver r we associate an unvalued translation quiver r i e all valuations are 1 and show that a finite k modulated translation quiver is an auslander reiten quiver if and only if the associated unvalued translation quiver with the trivial k modulation k prime field of k is an auslander reiten quiver section 8

constrained controllability of perturbed discrete time systems in this paper we study the global null controllability with bounded controls of perturbed linear discrete time systems in rn more explicitly a globally null controllable autonomous discrete time system is perturbed by suitably small terms v k and b k to obtain a system of the form x k 1 a v k x k b b k u k and sufficient conditions are given to ensure the global null controllability of the resulting perturbed system

on the constrained controllability of linear time varying discrete systems in this paper the authors study the constrained complete controllability to the origin of various classes of time varying discrete time systems under an assumption of a kalman type uniform controllability they will give sufficient conditions for complete controllability for both stable and unstable systems the admissible controllers considered assume values in the unit balls of the sequence spaces l sup p

almost periodic parameters for the second moment stability of linear stochastic systems the technical note presents conditions to assure the uniform second moment stability for a class of linear time varying stochastic systems the system parameters are assumed to be almost periodic a concept that is weaker than the periodic one under the existence of the long run average cost associated with the stochastic system we apply the almost periodicity to prove the desired stability result an application illustrates the usefulness of the approach by implementing an almost periodic state feedback strategy to control the velocity of a dc motor device

socle reductions and socle projective modules a useful and simple idea in determining the representation type of an artinian ring a is to relate its module category mod a with a module category of another ring a whose representation type is known or it is easier to study this was successfully used by several authors let us mention the coxeter functors connection 9 10 3 38 stable equivalences 2 231 the connection between radical square zero rings and hereditary rings 11 221 connections between representations of algebras and representations of partially ordered sets 19 20 24 253 or more generally with representations of bows 29 and the covering technique 131 an idea close to the one above is to find some subcategory 9 of mod a whose representations type is known and to look at the factor category mod a w of mod a modulo the two sided ideal i in mod a generated by maps in sk if mod a sy zmod a 8 and the representation type of a is known then we get a lot of information about mod a 2 22 23 on the other hand there are several situations when mod is equivalent to a nice subcategory 55 in some mod r and there are constructive methods for studying indecomposable objects in v it follows from 6 s 31 32 34 361 that the matrix problems technique and the differentiation procedure 19 201 lead to this scheme in connection with these ideas the notion of a right peak ring r is introduced and socle projective modules mod r over r are studied in 31 32f it is shown there that several factor categories of mod a where a is an artinian ring are of the form mod r where r is a right peak ring in particular if a t y where f is a division pi ring and t is a

hamiltonian structures and stability for rigid bodies with flexible attachments the dynamics of a rigid body with flexible attachments is studied a general framework for problems of this type is established in the context of poisson manifolds and reduction a simple model for a rigid body with an attached linear extensible shear beam is worked out for illustration second the energy casimir method for proving nonlinear stability is recalled and specific stability criteria for our model example are worked out the poisson structure and stability results take into account vibrations of the string rotations of the rigid body their coupling at the point of attachment and centrifugal and coriolis forces

constrained relative motions in rotational mechanics the dynamical effects of imposing constraints on the relative motions of component parts in a rotating mechanical system or structure are explored it is noted that various simplifying assumptions in modeling the dynamics of elastic beams imply strain constraints i e that the structure being modeled is rigid in certain directions in a number of cases such assumptions predict features in both the equilibrium and dynamic behavior which are qualitatively different from what is seen if the assumptions are relaxed it is argued that many pitfalls may be avoided by adopting so called geometrically exact models and examples from the recent literature are cited to demonstrate the consequences of not doing this these remarks are brought into focus by a detailed discussion of the nonlinear nonlocal model of a shear free inextensible beam attached to a rotating rigid body here it is shown that linearization of the equations of motion about certain relative equilibrium configurations leads to a partial differential equation such spatially localized models are not obtained in general however and this leaves open questions regarding the way in which the geometry of a complex structure influences computational requirements and the possibility of exploiting parallelism in performing simulations a general treatment of linearization about implicit solutions to equilibrium equations is presented and it is shown that this approach avoids unintended imposition of constraints on relative motions in the models finally the example of a rotating kinematic chain shows how constraining the relative motions in a rotating mechanical system may destabilize uniformly rotating states

optimal pruning with applications to tree structured source coding and modeling an algorithm introduced by l breiman et al 1984 in the context of classification and regression trees is reinterpreted and extended to cover a variety of applications in source coding and modeling in which trees are involved these include variable rate and minimum entropy tree structured vector quantization minimum expected cost decision trees variable order markov modeling optimum bit allocation and computer graphics and image processing using quadtrees a concentration on the first of these and a detailed analysis of variable rate tree structured vector quantization are provided it is found that variable rate tree structured vector quantization outperforms not only the fixed rate variety but also full search vector quantization the successive approximation character of variable rate tree structured vector quantization permits it to degrade gracefully if the rate is reduced at the encoder this has applications to the problem of buffer overflow

lagrange asymptotic stability of weak detectable markov jump linear systems with bounded long run average cost in this note we study the stability of markov jump linear systems with additive noise we show in a rather direct manner that the system is mean square lagrange asymptotic stable if and only if the long run average cost is bounded and the system is weak detectable generalizing previous results employing observability notions in control applications this means that for detectable systems closed loop controls incurring in bounded long run average cost are ensured to be stabilizing a numerical example is included

partial coxeter functors and right pure semisimple hereditary rings an important problem in the representation theory of artinian rings is to obtain a charcterization of rings of finite representation type see 2 7 16 201 we recall that a ring r is of finite representation type if r is both left and right artinian and there is only a finite number of isomorphism classes of indecomposable finitely generated r modules the main tools for studying the representation theory of hereditary artinian rings are partial coxeter functors and coxeter functors see 4 8 10 141 they are successfuly applied in the investigation of hereditary rings of finite representation type 8 9 12 141 in the present paper we study partial coxeter functors for hereditary artinian rings and we use them as a tool for studying right pure semisimple hereditary rings which are close to rings of finite representation type we recall that for any ring r the right pure global dimension r p gl dim r and the left pure global dimension l p gl dim r are defined 19 21 261 there is an interesting problem how to characterize rings r with r p gl dim r 0 which are exactly those right artinian rings r all of whose right r modules are algebraically compact 19 20 261 or equivalently all of whose right r modules are direct sums of modules of finite length a ring r with r p gl dim r 0 is called right pure semisimple 25 271 it is well known that any ring of finite representation type is both left and right pure semisimple 12 20 23 251 and the converse implication also holds true 2 15 201 however it is still an open question if a right pure semisimple ring r is of finite representation type or equivalently if r p gl dim r 0 implies l p gl dim r 0 see 29 a positive solution of this problem is given by auslander 3 for artin algebras he also discusses the problem for arbitrary left artinian rings see also 1181 we recall that a 195 0021 8693 81 070195 24 02 00 0

almost split sequences in subcategories let r be a commutative artin ring and let i be an r algebra which is a finitely generated r module in 5 auslander and reiten introduced the notion of an almost split sequence in mod a the category of finitely generated modules over a and the existence and uniqueness of such short exact sequences were established there the theory of almost split sequences developed further and similar results for certain subcategories of mod i were proved by bautista and martinez 8 and roggenkamp 9 the main purpose of this paper is to develop a more general theory for subcategories of mod i having almost split sequences which has the previous examples as special cases notions closely related to that of almost split sequences are those of minimal left and right almost split morphisms these notions were introduced in 61 for mod i and further developed in 7 for subcategories of mod ii this paper is based upon 7 and we therefore use the same notations and conventions as used there we now give some definitions and then proceed to describe the content of the paper section by section let c be a subcategory of mod ii then a morphism g b c in c is said to be a right almost split morphism in c if i g is not a splittable epimorphism and ii whenever there is a nonsplittable epimorphism h c c in c there exists an h c b such that gh h dually a morphismfi a f b in c is said to be left almost split if i f is not a splittable monomorphism and ii whenever there is a nonsplittable monomorphism h a a there exists an h b a such that h f h we say that c has right almost split morphisms if for all indecomposable objects

schurian sp representation finite right peak pi rings and their indecomposable socle projective modules such that soc r is an essential right ideal in r and it is isomorphic to a direct sum of finitely many copies of p called a right peak of r here f f are local rings f f is a division ring imj is an fi fj bimodule and the multiplication in r is given by fi f bimodule maps cii imj im jm satisfying the natural associativity conditions we denote by p p p the right indecomposable row ideals of r

an algebraic approach to bounded controllability of linear systems in this paper we present an algebraic approach to the proof that a linear system with matrices a b is null controllable using bounded inputs if and only if it is null controllable with unbounde

variable rate tree structured vector quantizers in general growth algorithms for optimal tree structured vector quantizers do not exist in this paper we show that if the source satisfies certain conditions namely that of diminishing marginal returns optimal growth algorithms do exist we present such an algorithm and compare its performance with that of other tree growth algorithms even for sources that do not meet the necessary conditions for the growth algorithm to be optimal such as for speech with unknown statistics it is seen by simulation that the algorithm outperforms other known growth algorithms for sources that do not satisfy the required conditions the algorithm presented here can also be used to grow the initial tree for the pruning process the performance of such pruned trees is superior to that of trees pruned from full trees of the same rate

average cost and stability of time varying linear systems in this technical note the stability for time varying discrete time stochastic linear systems with possibly unbounded trajectories is associated to the existence of the long run average cost criteria under controllability and observability the stochastic system is stable in the sense that its state tends asymptotically to the origin in the mean further conditions related to a lower bound on the long run average cost or with periodic time varying systems provides uniform second moment stability

canonical matrices for linear matrix problems abstract we consider a large class of matrix problems which includes the problem of classifying arbitrary systems of linear mappings for every matrix problem from this class we construct belitskii s algorithm for reducing a matrix to a canonical form which is the generalization of the jordan normal form and study the set c mn of indecomposable canonical m n matrices considering c mn as a subset in the affine space of m n matrices we prove that either c mn consists of a finite number of points and straight lines for every m n or c mn contains a 2 dimensional plane for a certain m n

multistage storage and entropy constrained tree structured vector quantization this paper proposes a new algorithm for the design of a tree structured vector quantizer that operates under storage and entropy constraints hence it is called the storage and entropy constrained tree structured vector quantizer sectsvq the algorithm uses the tree growing approach and designs the tree one stage layer at a time the constraints on the rate and storage size i e the number of nodes or codewords at each stage are specified prior to the design procedure while growing the tree at each stage the algorithm optimally allocates the rate and the number of nodes available for the current stage to the nodes of the previous stage using the dynamic programming technique the nodes of the current stage are then determined based on the allocations in addition to being useful as a tree structured vq sectsvq is particularly suited for application in progressive transmission moreover the optimal allocation technique used in the design can be effectively applied to other optimal resource allocation problems the sectsvq algorithm is implemented for various sources and is shown to compare favorably with other vqs

constrained storage quantization of multiple vector sources by codebook sharing a codebook sharing technique called constrained storage vector quantization csvq is introduced this technique offers a convenient and optimal way of trading off performance against storage the technique can be used in conjunction with tree structured vector quantization vq and other structured vq techniques that alleviate the search complexity obstacle the effectiveness of csvq is illustrated for coding transform coefficients of audio signals with multistage vq

a matrix problem over a central quadratic skew field extension abstract let f g f u be a central quadratic skew field extension such that the generator u is central in g and w g f g a natural g g bimodule we deal with the matrix problem on finding a canonical form for rectangular matrices over w with help of left elementary transformations of their rows and right elementary transformations of columns over g we solve this problem reducing it in the separable resp inseparable case to the semilinear resp pseudolinear pencil problem

partial stabilisation of semilinear systems using bounded controls this paper studies a partial stabilisation problem for distributed semilinear systems this consists of steering only a part of the system s state to its equilibrium constrained controls are proposed to force the desired part of the state to approach the equilibrium necessary and sufficient conditions for weak and strong partial stabilisation are given applications are presented

partial asymptotic stabilization of nonlinear distributed parameter systems the paper is devoted to stability and stabilization of a class of evolution equations arising from mathematical modeling of hybrid mechanical systems with flexible parts a sufficient condition is obtained for partial strong asymptotic stability of nonlinear infinite dimensional dynamic systems in banach spaces this result is applied to deriving a control law that stabilizes a part of the variables describing a rotating rigid body endowed with a number of elastic beams results of numerical simulations are presented

a canonical form for matrices under consimilarity abstract square complex matrices a b are said to be consimilar if a sb s 1 for some nonsingular matrix s consimilarity is an equivalence relation that is a natural matrix generalization of rotation of scalars in the complex plane we survey the known forms to which a given complex matrix may be reduced by unitary consimilarity and describe a canonical form to which it may be reduced by a general consimilarity we derive a useful criterion for two matrices to be consimilar and show that every matrix is consimilar to its own conjugate transpose and adjoint to a real matrix and to a hermitian matrix

moment stability of nonlinear discrete stochastic systems with time delays based on h representation technique in this paper mean square stability of nonlinear discrete stochastic systems with time delays is investigated we obtain the second order moment equations of discrete stochastic systems by virtue of the h representation technique by the comparison principle and the established moment equations some moment stability theorems for nonlinear discrete stochastic systems with time delays are established a numerical example is presented to illustrate the efficiency of our results at the end of the paper

constrained controllability of discrete time systems the constrained controllability of the discrete time system xk 1 a k xk b k u k is considered where the control uk is termed admissible if it satisfies specified magnitude constraints constrained controllability is concerned with the existence of an admissible control which steers the state x to a given target set from a specified initial state conditions for checking constrained controllability to a given target set from a specified initial state are presented these conditions involve solving finite dimensional optimization problems and can be checked via numerical computation in addition conditions for checking global constrained controllability to a given target set are presented a system is globally constrained controllable if for every initial state there exists an admissible control that steers the system to the target if a given discrete time system is constrained controllable it may be desirable to obtain an admissible control that steers the system to the target from a specified initial s

three equipped posets and their representations and corepresentations inseparable case abstract we define three equipped posets i e partially ordered sets equipped with three kinds of binary relations and their representations and corepresentations over an inseparable cubic field extension f g in characteristic 3 these representations and corepresentations lead to some matrix problems of mixed type over the pair of fields f g through these problems of linear algebra we completely describe in evident matrix form the indecomposables for some critical three equipped posets reducing the task for the rest of the critical sets to some matrix problems containing the pseudolinear pencil problem as a subproblem

variable branch tree structured vector quantization tree structured vector quantizers tsvq and their variants have recently been proposed all trees used are fixed m ary tree structured such that the training samples in each node must be artificially divided into a fixed number of clusters this paper proposes a variable branch tree structured vector quantizer vbtsvq based on a genetic algorithm which searches for the number of child nodes of each splitting node for optimal coding in vbtsvq moreover one disadvantage of tsvq is that the searched codeword usually differs from the full searched codeword briefly the searched codeword in tsvq sometimes is not the closest codeword to the input vector this paper proposes the multiclassification encoding method to select many classified components to represent each cluster and the codeword encoded in the vbtsvq is usually the same as that of the full search vbtsvq outperforms other tsvqs in the experiments presented here

representations of generalized equipped posets and posets with automorphisms over galois field extensions abstract generalized equipped posets and their representations over a galois field extension k l are introduced and studied an equipment of a poset p is a pair s consisting of a finite group and a collection s x y x y of its non empty subsets such that x y y z x z for any chain x y z in p equipped posets in the old sense of our former papers correspond to the case 2 with some additional restriction since the category of representations of a strictly equipped poset p s is isomorphic to the category of representations of its evolvent q which is a poset with automorphisms we study in fact representations of posets with automorphisms over the pair k l in the case where gal l k we define the complexification functor c rep k l q rep l q and the realification functor r rep l q rep k l q and show that they induce reciprocal bijections between the isoclasses of indecomposables of the category rep k l q and the orbits of isoclasses of indecomposables of the category rep l q in this way the problem on classification of indecomposables of equipped posets and posets with automorphisms over k l actually is reduced to that one for representations of ordinary posets over l

detectability and stabilizability of time varying discrete time linear systems the concepts of detectability and stabilizability are explored for time varying systems we study duality invariance under feedback an extended version of the lemma of lyapunov existence of stabilizing feedback laws linear quadratic filtering and control and the existence of approximate canonical forms

on the kronecker problem and related problems of linear algebra abstract we consider some classification problems of linear algebra related closely to the classical kronecker problem on pairs of linear maps between two finite dimensional vector spaces as shown by djokovic and sergeichuk the kronecker s solution is extended to the cases of pairs of semilinear maps and more generally pseudolinear bundles respectively our objective is to deal with the semilinear case of the kronecker problem especially with its applications it is given a new short solution both to this case and to its contragredient variant the biquadratic matrix problem is investigated and reduced in the homogeneous case in characteristic 2 to the semilinear kronecker problem the integer matrix sequence n and transformation of polynomials are introduced and studied to get a simplified canonical form of indecomposables for the mentioned homogeneous problem some applications to the representation theory of posets with additional structures are presented