azumaya algebras with involutions res 2wr zbr s cores br r 1 one could ask whether this sequence is exact indeed if r and s are fields and char r 2 the exactness of the above sequence is proved in cl corollary 4 6 p 4731 let a be an azumaya s algebra whose class a belongs to kernel cores br s br r for a to belong to res br r it is necessary and sufficient that there exists an azumaya s algebra b brauer equivalent to a admitting involutions of first and second kinds which commute results of saltman assert lo that there exist algebras bi i 1 2 brauer equivalent to a which support involutions of the first and second kind respectively we construct section 4 a homomorphism h ker cores br s br r pic rjcores pit s whose kernel consists precisely of classes of azumaya s algebras which support commuting involutions of both kinds thus h is a measure of the obstruction to the sequence 1 being exact in particular if pit r cores pit s 1 is exact

an orientation selective orthogonal lapped transform a novel critically sampled orientation selective orthogonal lapped transform called the lapped hartley transform lht is derived in a first step overlapping basis functions are generated by modulating basis functions of a 2 d block hartley transform by a cosine wave to achieve invertibility and orthogonality an iterative filter is applied as prefilter in the analysis and as postfilter in the synthesis operation respectively alternatively filtering can be restricted to analysis or synthesis ending up with a biorthogonal transform lht pr lht po a statistical analysis based on a 4000 image data base shows that the lht and lht po have better redundancy removal properties than other block or lapped transforms finally image compression and noise removal examples are given showing the advantages of the lht especially in images containing oriented textures

cyclic galois extensions and normal bases a kummer theory is presented which does not need roots of unity in the ground ring for r commutative with p i e r we study the group of cyclic galois extensions of fixed degree pn in detail our theory is well suited for dealing with cyclic pn extensions of a number field k which are unramified outside p we then consider the group gal 6f p i cpn of all such extensions and its subgroup nb k k p1 cpn of extensions with integral normal basis outside p for the size of the latter we get a simple asymptotic formula n oo and the discrepancy between the two groups is in some way measured by the defect a in leopoldt s conjecture introduction this work begins with a general study of galois extensions of a commutative ring r with finite abelian galois group g part i the results are applicable in a number theoretical setting part ii and give theorems concerning the existence of p integral normal bases in cpn extensions and zp extensions of an arbitrary number field k here p is an odd prime cpn is the cyclic group of order pn and p integral means integral outside places over p let us start by describing the general situation let r be a commutative ring with 1 and g a finite abelian group we study the finite abelian group gal r g which consists of the isomorphism classes of g galois ring extensions of r such extensions were defined by chase harrison and rosenberg 4 the group structure on gal r g is due to harrison 13 and goes back to hasse 14 for the case r k a field in that case the basic idea is to admit some nonfields galois algebras in order to form gal k g the principal link between general galois theory of rings and number theory is the study of ramification let l be a g galois field extension of the number field k x a set of finite places of k and r the ring of x integers in k then the integral closure s of r in l is a g galois extension of r if and only if l k is unramified at all finite places outside x see 4 remark 1 5d in particular 6 l is galois over k iff l k is unramified except at infinity received by the editors june 13 1989 1980 mathematics subject classification 1985 revision primary 1 lr33 1 lr27 13b05

lattice structure for generalized support multidimensional linear phase perfect reconstruction filter bank multidimensional linear phase perfect reconstruction filter bank mdlpprfb can be designed and implemented via lattice structure the lattice structure for the mdlpprfb with filter support n m has been published by muramatsu where m is the decimation matrix is a positive integer diagonal matrix and n n denotes the set of integer vectors in the fundamental parallelepiped of the matrix n obviously if is chosen to be other positive diagonal matrices instead of only positive integer ones the corresponding lattice structure would provide more choices of filter banks offering better trade off between filter support and filter performance we call such resulted filter bank as generalized support mdlpprfb gsmdlpprfb the lattice structure for gsmdlpprfb however cannot be designed by simply generalizing the process that muramatsu employed furthermore the related theories to assist the design also become different from those used by muramatsu such issues will be addressed in this paper to guide the design of gsmdlpprfb the necessary and sufficient conditions are established for a generalized support multidimensional filter bank to be linear phase to determine the cases we can find a gsmdlpprfb the necessary conditions about the existence of it are proposed to be related with filter support and symmetry polarity i e the number of symmetric filters ns and antisymmetric filters na based on a process different from the one muramatsu used that combines several polyphase matrices to construct the starting block one of the core building blocks of lattice structure the lattice structure for gsmdlpprfb is developed and shown to be minimal additionally the result in this paper includes muramatsu s as a special case

the contourlet transform an efficient directional multiresolution image representation the limitations of commonly used separable extensions of one dimensional transforms such as the fourier and wavelet transforms in capturing the geometry of image edges are well known in this paper we pursue a true two dimensional transform that can capture the intrinsic geometrical structure that is key in visual information the main challenge in exploring geometry in images comes from the discrete nature of the data thus unlike other approaches such as curvelets that first develop a transform in the continuous domain and then discretize for sampled data our approach starts with a discrete domain construction and then studies its convergence to an expansion in the continuous domain specifically we construct a discrete domain multiresolution and multidirection expansion using nonseparable filter banks in much the same way that wavelets were derived from filter banks this construction results in a flexible multiresolution local and directional image expansion using contour segments and thus it is named the contourlet transform the discrete contourlet transform has a fast iterated filter bank algorithm that requires an order n operations for n pixel images furthermore we establish a precise link between the developed filter bank and the associated continuous domain contourlet expansion via a directional multiresolution analysis framework we show that with parabolic scaling and sufficient directional vanishing moments contourlets achieve the optimal approximation rate for piecewise smooth functions with discontinuities along twice continuously differentiable curves finally we show some numerical experiments demonstrating the potential of contourlets in several image processing applications

the hodge conjecture for self products of certain k3 surfaces abstract we use a result of van geemen 2008 vg4 to determine the endomorphism algebra of the kuga satake variety of a k3 surface with real multiplication this is applied to prove the hodge conjecture for self products of double covers of p 2 which are ramified along six lines

directional lapped orthogonal transform theory and design this paper proposes a directional design method of 2 d nonseparable linear phase paraunitary filter banks the proposed method is based on a lattice structure consisting of the 2 d separable dct block and nonseparable support extension processes because of the nonseparability the bases are allowed to be directional with the critically fixed subsampling overlapping orthogonal symmetric real valued and compact support properties first a novel vanishing moment vm condition is introduced as a suitable directional constraint where the moment is referred to as the trend vm the condition forces wavelet filters i e high pass and bandpass filters to annihilate trend surface components second some theoretical properties of tvms are discussed for general 2 d paraunitary systems and then the properties are applied to the lattice parameters in order to verify the significance several design examples are shown the trend surface annihilation properties are numerically confirmed and the denoising capability is evaluated for images through shrinkage it is shown that our proposed transforms yield perceptually preferable results

corestriction for algebras with group action abstract we define a corestriction map for equivariant brauer groups in the sense of frohlich and wall which contain as a special case the brauer clifford groups introduced by turull we show that this corestriction map has similar properties as the corestriction map in group cohomology especially galois cohomology in particular composing corestriction and restriction associated to a subgroup h g amounts to powering with the index g h

a development of symmetric extension method for subband image coding describes the development of a technique for subband image coding called the symmetric extension method which utilizes the nature of a symmetrically extended image to achieve high quality coding there are however some restrictions on the usable subband coding system the development is done based on the property of a symmetrically extended signal in the analysis process to remove the restriction the behavior of the sequence after two processes filtering and decimation is examined and formulated the development extends the application area of the technique

complete controllability of finite level quantum systems complete controllability is a fundamental issue in the field of control of quantum systems not least because of its implications for dynamical realizability of the kinematical bounds on the optimization of observables in this paper we investigate the question of complete controllability for finite level quantum systems subject to a single control field for which the interaction is of dipole form sufficient criteria for complete controllability of a wide range of finite level quantum systems are established and the question of limits of complete controllability is addressed finally the results are applied to give a classification of complete controllability for four level systems

on the residue fields of henselian valued stable fields abstract let k v be a henselian valued field satisfying the following conditions for a given prime number p i central division k algebras of finite p primary dimensions have schur indices equal to their exponents ii the value group v k properly includes its subgroup p v k the paper shows that if k is the residue field of k v and r is an intermediate field of the maximal p extension k p k then the natural homomorphism br k br r of brauer groups maps surjectively the p component br k p on br r p it proves that br k p is divisible if p 2 or k is a nonreal field and that br k 2 is of order 2 when k is formally real we also obtain that r embeds as a k subalgebra in a central division k algebra if and only if the degree r k divides the index of

on two channel filter banks with directional vanishing moments the contourlet transform was proposed to address the limited directional resolution of the separable wavelet transform one way to guarantee good approximation behavior is to let the directional filters in the contourlet filter bank have sharp frequency response this requires filters with large support size we seek to isolate the key filter property that ensures good approximation in this direction we propose filters with directional vanishing moments dvm these filters we show annihilate information along a given direction we study two channel filter banks with dvm filters we provide conditions under which the design of dvm filter banks is possible a complete characterization of the product filter is thus obtained we propose a design framework that avoids 2 d factorization using the mapping technique the filters designed when used in the contourlet transform exhibit nonlinear approximation comparable to the conventional filters while being shorter and therefore providing better visual quality with less ringing artifacts furthermore experiments show that the proposed filters outperform the conventional ones in image approximation and denoising

boundary operation of 2 d nonseparable linear phase paraunitary filter banks this paper proposes a boundary operation technique of 2 d nonseparable linear phase paraunitary filter banks ns lppufbs for size limitation the proposed technique is based on a lattice structure consisting of the 2 d separable block discrete cosine transform and nonseparable support extension processes the bases are allowed to be anisotropic with the fixed critically subsampling overlapping orthogonal symmetric real valued and compact support properties first the blockwise implementation is developed so that the basis images can be locally controlled the local control of basis images is shown to maintain orthogonality this property leads a basis termination bt technique as a boundary operation the technique overcomes the drawback of ns lppufbs that the popular symmetric extension method is invalid through some experimental results of diagonal texture coding the significance of the bt is verified

a design method of multidimensional linear phase paraunitary filter banks with a lattice structure a lattice structure of multidimensional md linear phase paraunitary filter banks lppufbs is proposed which makes it possible to design such systems in a systematic manner our proposed structure can produce md lppufbs whose filters all have the region of support spl nscr m spl xi where m and spl xi are the decimation and positive integer diagonal matrices respectively and spl nscr n denotes the set of integer vectors in the fundamental parallelepiped of a matrix n it is shown that if spl nscr m is reflection invariant with respect to some center then the reflection invariance of spl nscr m spl xi is guaranteed this fact is important in constructing md linear phase filter banks because the reflection invariance is necessary for any linear phase filter since our proposed system structurally restricts both the paraunitary and linear phase properties an unconstrained optimization process can be used to design md lppufbs our proposed structure is developed for both an even and an odd number of channels and includes the conventional 1 d system as a special case it is also shown to be minimal and the no dc leakage condition is presented some design examples show the significance of our proposed structure for both the rectangular and nonrectangular decimation cases

the gradient flow for control of closed quantum systems the gradient flow is commonly used to numerically solve the problem of maximizing quantum observables for finite dimensional closed quantum systems in this note we analyze the asymptotic behavior of the gradient flow we show that under a regularity assumption on the controls the flow almost always converges to a solution of the maximization problem

implicit lyapunov control of finite dimensional schrodinger equations an implicit lyapunov based approach is proposed for generating trajectories of a finite dimensional controlled quantum system the main difficulty comes from the fact that we consider the degenerate case where the linearized control system around the target state is not controllable the controlled lyapunov function is defined by an implicit equation and its existence is shown by a fix point theorem the convergence analysis is done using lasalle invariance principle closed loop simulations illustrate the performance of such feedback laws for the open loop control of a test case considered by chemists

symmetric extension methods for m channel linear phase perfect reconstruction filter banks the symmetric extension method has been shown to he an efficient way for subband processing of finite length sequences this paper presents an extension of this method to general linear phase perfect reconstruction filter banks we derive constraints on the length and symmetry polarity of the permissible filter banks and propose a new design algorithm in the algorithm different symmetric sequences are formulated in a unified form based on the circular symmetry framework the length constraints in symmetrically extending the input sequence and windowing the subband sequences are investigated the effect of shifting the input sequence is included when the algorithm is applied to equal length filter banks we explicitly show that symmetric extension methods can always be constructed to replace the circular convolution approach

analysis of lyapunov method for control of quantum states the natural trajectory tracking problem is studied for generic quantum states represented by density operators a control design based on the hilbert schmidt distance as a lyapunov function is considered the control dynamics is redefined on an extended space where the lasalle invariance principle can be correctly applied even for non stationary target states lasalle s invariance principle is used to derive a general characterization of the invariant set which is shown to always contain the critical points of the lyapunov function critical point analysis of the latter is used to show that for generic states it is a morse function with n isolated critical points including one global minimum one global maximum and n 2 saddles it is also shown however that the actual dynamics of the system is not a gradient flow and therefore a full eigenvalue analysis of the linearized dynamics about the critical points of the dynamical system is necessary to ascertain stability of the critical points this analysis shows that a generic target state is locally asymptotically stable if the linearized system is controllable and the invariant set is regular and in fact convergence to the target state trajectory in this case is almost global in that the stable manifolds of all other critical points form a subset of measure zero of the state space on the other hand if either of these sufficient conditions is not satisfied the target state ceases to be asymptotically stable a center manifold emerges around the target state and the control design ceases to be effective

induced orthogonal representations of galois groups abstract we prove a result showing the connection between induced orthogonal representations and corestrictions of group extensions derived from their clifford groups by the means of the corestriction map we then obtain new obstructions to the p embedding problems given by the group extensions of the modular p group and to one 2 embedding problem given by a group extension of the dihedral group

isotropic trialitarian algebraic groups abstract by modifying a construction from knus et al we construct all isotropic algebraic groups of type 3 d 4 and 6 d 4 over an arbitrary field of characteristic 2 we also provide a nice isomorphism criterion for such groups the results of this paper extend the main results of allison using entirely different methods to fields of nonzero characteristic and algebraic groups

quantum state filter with disturbance and noise a quantum state filter qsf is proposed in this paper to estimate a low rank quantum density matrix from informationally incomplete and contaminated measurements there exist sparse disturbances on the quantum density matrix and gaussian noise in the measurements a proximal jacobian variant of the alternating direction method of multipliers pj admm is proposed to design the qsf the closed form solutions to three resulting subproblems are given and the iterative qsf is developed the proposed qsf is proved to be convergent and its superiority is demonstrated in the numerical illustrations compared with different state of the art methods

directional lapped transforms for image coding in this paper we present the design of directional lapped transforms for image coding a lapped transform which can be implemented by a prefilter followed by a discrete cosine transform dct can be factorized into elementary operators the corresponding directional lapped transform is generated by applying each elementary operator along a given direction the proposed directional lapped transforms are not only nonredundant and perfectly reconstructed but they can also provide a basis along an arbitrary direction these properties along with the advantages of lapped transforms make the proposed transforms appealing for image coding a block based directional transform scheme is also presented and integrated into hd phtoto one of the state of the art image coding systems to verify the effectiveness of the proposed transforms

control landscapes are almost always trap free a geometric assessment a proof is presented that almost all closed finite dimensional quantum systems have trap free i e free from local optima landscapes for a large and physically general class of circumstances which includes qubit evolutions in quantum computing this result offers an explanation for why gradient based methods succeed so frequently in quantum control the role of singular controls is analyzed using geometric tools in the case of the control of the propagator and thus in the case of observables as well singular controls have been implicated as a source of landscape traps the conditions under which singular controls can introduce traps and thus interrupt the progress of a control optimization are discussed and a geometrical characterization of the issue is presented it is shown that a control being singular is not sufficient to cause control optimization progress to halt and sufficient conditions for a trap free landscape are presented it is further shown that the local surjectivity full rank assumption of landscape analysis can be refined to the condition that the end point map is transverse to each of the level sets of the fidelity function this mild condition is shown to be sufficient for a quantum system s landscape to be trap free the control landscape is shown to be trap free for all but a null set of hamiltonians using a geometric technique based on the parametric transversality theorem numerical evidence confirming this analysis is also presented this new result is the analogue of the work of altifini wherein it was shown that controllability holds for all but a null set of quantum systems in the dipole approximation these collective results indicate that the availability of adequate control resources remains the most physically relevant issue for achieving high fidelity control performance while also avoiding landscape traps

lyapunov control of bilinear schrodinger equations a lyapunov based approach for trajectory tracking of the schrodinger equation is proposed in the finite dimensional case convergence is precisely analyzed connection between the controllability of the linearized system around the reference trajectory and asymptotic tracking is studied when the linearized system is controllable such a feedback ensures almost global asymptotic convergence when the linearized system is not controllable the stability of the closed loop system is not asymptotic to overcome such lack of convergence we propose when the reference trajectory is an eigenstate a modification based on adiabatic invariance simulations illustrate the simplicity and also the interest for trajectory generation