spectral properties of the transition operator associated to a multivariate refinement equation given a finitely supported sequence a on z s and an s s dilation matrix m the transition operator ta is the linear operator defined by tavoau a p b2z s aomabuvobu where a2 z s and v lies in 0oz s u the linear space of all finitely supported sequences on z s in this paper we investigate the spectral properties of the transition operator ta and apply these properties to the study of the approximation and smoothness properties of the normalized solution of the refinement equation a p a2z s aoau omau 1999

representation of surfaces using spinor operators we use the concept of spinor operator to study some aspects of the classical theory of surfaces in three dimensions we show that a spinor operator representing a surface satisfies a pair of dirac like equations and that this spinor operator representation is a generalization of the weierstrass representation of minimal surfaces we also obtain spinor operator representations of a surface directly from its parameterization in terms of isothermal parameters without the need of solving the pair of dirac like equations we apply these results to minimal surfaces and their deformations and also show how they can be written in terms of quaternions and complex numbers we use the concept of spinor operator to study some aspects of the classical theory of surfaces in three dimensions we show that a spinor operator representing a surface satisfies a pair of dirac like equations and that this spinor operator representation is a generalization of the weierstrass representation of minimal surfaces we also obtain spinor operator representations of a surface directly from its parameterization in terms of isothermal parameters without the need of solving the pair of dirac like equations we apply these results to minimal surfaces and their deformations and also show how they can be written in terms of quaternions and complex numbers

ergodic properties of equilibrium states possible ergodic properties of gibbs states are discussed by constructing a number of examples in particular existence of gibbs states which are mixing but not extremal is shown

constant mean curvature surfaces via an integrable dynamical system it is shown that the equation which describes constant mean curvature surfaces via the generalized weierstrass enneper induction has hamiltonian form its simplest finite dimensional reduction is the integrable hamiltonian system with two degrees of freedom this finite dimensional system admits action and classes of equivalence of its trajectories are in one to one correspondence with different helicoidal constant mean curvature surfaces thus the interpretation of well known delaunay and do carmo dajczer surfaces via an integrable finite dimensional hamiltonian system is established

nonseparable multidimensional perfect reconstruction filter banks and wavelet bases for r sup n new results on multidimensional filter banks and their connection to multidimensional nonseparable wavelets are presented among the topics discussed are sampling in multiple dimensions multidimensional perfect reconstruction filter banks the two channel case in multiple dimensions the synthesis of multidimensional filter banks and the design of compactly supported wavelets

how should one define a weak crystal we compare two proposals for the study of positional long range order one in terms of the spectrum of the translation operator the other in terms of the fourier spectrum we point out that only the first one allows for the consideration of molecular as opposed to atomic weakly periodic structures we illustrate this point on the thue morse system

from frame like wavelets to wavelet frames keeping approximation properties and symmetry abstract for a given symmetric refinable mask obeying the sum rule of order n an explicit method is suggested for the construction of mutually symmetric almost frame like wavelet system providing approximation order n a transformation based on the lifting scheme is described that allows to improve almost frame like wavelets up to dual wavelet frames and preserve other properties a direct method for the construction of dual wavelet frames providing approximation order n and mutual symmetry properties is also discussed for an abelian symmetry group h a technique providing the h symmetry property for each wavelet function is given for the above three methods

tight wavelet frames for subdivision in this paper we construct multivariate tight wavelet frame decompositions for scalar and vector subdivision schemes with nonnegative masks the constructed frame generators have one vanishing moment and are obtained by factorizing certain positive semi definite matrices the construction is local and allows us to obtain framelets even in the vicinity of irregular vertices constructing tight frames instead of wavelet bases we avoid extra computations of the dual masks in addition the frame decomposition algorithm is stable as the discrete frame transform is an isometry on 2 if the data are properly normalized

triangular 3 subdivision schemes the regular case the paper deals with the investigation of triangular 3 subdivision schemes in the stationary shift invariant setting in section 2 we collect the available theory on refinable functions subdivision surfaces with emphasis on their sobolev and holder smoothness families of interpolatory and approximating 3 subdivision schemes are investigated in section 3 some dual 3 subdivision schemes which are related to vector valued refinable functions are also analyzed for this purpose we have developed matlab routines for numerically investigating properties of vector subdivision schemes

split invariance principles for stationary processes the results of komlos major and tusnady give optimal wiener approximation of partial sums of i i d random variables and provide an extremely powerful tool in probability and statistical inference recently wu ann probab 35 2007 2294 2320 obtained wiener approximation of a class of dependent stationary processes with finite pth moments 2 0 and liu and lin stochastic process appl 119 2009 249 280 removed the logarithmic factor reaching the komlos major tusnady bound o n 1 p no similar results exist for p 4 and in fact no existing method for dependent approximation yields an a s rate better than o n 1 4 in this paper we show that allowing a second wiener component in the approximation we can get rates near to o n 1 p for arbitrary p 2 this extends the scope of applications of the results essentially as we illustrate it by proving new limit theorems for increments of stochastic processes and statistical tests for short term epidemic changes in stationary processes our method works under a general weak dependence condition covering wide classes of linear and nonlinear time series models and classical dynamical systems

hilbert space representations of m dependent processes a representation of one dependent processes is given in terms of hilbert spaces vectors and bounded linear operators on hilbert spaces this generalizes a construction of one dependent processes that are not two block factors we show that all one dependent processes admit a representation we prove that if there is in the hilbert space a closed convex cone that is invariant under certain operators and that is spanned by a finite number of linearly independent vectors then the corresponding process is a two block factor of an independent process apparently the difference between two block factors and non two block factors is determined by the geometry of invariant cones the dimension of the smallest hilbert space that represents a process is a measure for the complexity of the structure of the process for two valued one dependent processes if there is a cylinder with measure equal to zero then this process can be represented by a hilbert space with dimension smaller than or equal to the length of this cylinder in the two valued case a cylinder with measure equal to zero whose length is minimal and less than or equal to 7 is symmetric we generalize the concept of hilbert space representation to m dependent processes and it turns out that all m dependent processes admit a representation several theorems can be generalized to m dependent processes

tail triviality for sums of stationary random variables we study tail sigma fields and loss of memory associated with sums of stationary integer valued random variables an application concerns convergence in distribution of interarrival times in zero one sequences

bi frames with 4 fold axial symmetry for quadrilateral surface multiresolution processing when bivariate filter banks and wavelets are used for surface multiresolution processing it is required that the decomposition and reconstruction algorithms for regular vertices derived from them have high symmetry this symmetry requirement makes it possible to design the corresponding multiresolution algorithms for extraordinary vertices recently lifting scheme based biorthogonal bivariate wavelets with high symmetry have been constructed for surface multiresolution processing if biorthogonal wavelets have certain smoothness then the analysis or synthesis scaling function or both have big supports in general in particular when the synthesis low pass filter is a commonly used scheme such as loop s scheme or catmull clark s scheme the corresponding analysis low pass filter has a big support and the corresponding analysis scaling function and wavelets have poor smoothness big supports of scaling functions or in other words big templates of multiresolution algorithms are undesirable for surface processing on the other hand a frame provides flexibility for the construction of basis systems this paper concerns the construction of wavelet or affine bi frames with high symmetry in this paper we study the construction of wavelet bi frames with 4 fold symmetry for quadrilateral surface multiresolution processing with both the dyadic and 2 refinements considered the constructed bi frames have 4 framelets or frame generators for the dyadic refinement and 2 framelets for the 2 refinement namely with either the dyadic or 2 refinement a frame system constructed in this paper has only one more generator than a wavelet system the constructed bi frames have better smoothness and smaller supports than biorthogonal wavelets furthermore all the frame algorithms considered in this paper are given by templates so that one can easily implement them

wavelet bi frames with uniform symmetry for curve multiresolution processing this paper is about the construction of univariate wavelet bi frames with each framelet being symmetric as bivariate filter banks are used for surface multiresolution processing it is required that the corresponding decomposition and reconstruction algorithms have high symmetry so that it is possible to design the corresponding multiresolution algorithms for extraordinary vertices for open surfaces special multiresolution algorithms are designed to process boundary vertices when the multiresolution algorithms derived from univariate wavelet bi frames are used as the boundary algorithms it is desired that not only the scaling functions but also all framelets be symmetric in addition the algorithms for curve surface multiresolution processing should be given by templates so that they can be easily implemented in this paper first by appropriately associating the lowpass and highpass outputs to the nodes of z we show that both biorthogonal wavelet multiresolution algorithms and bi frame multiresolution algorithms can be represented by templates then using the idea of the lifting scheme we provide frame algorithms given by several iterative steps with each step represented by a symmetric template finally with the given templates of algorithms we obtain the corresponding filter banks and construct bi frames based on their smoothness and vanishing moments two types of symmetric bi frames are studied in this paper in order to provide a clearer picture on the template based procedure for bi frame construction in this paper we also consider the template based construction of biorthogonal wavelets the approach of the template based bi frame construction introduced in this paper can be extended easily to the construction of bivariate bi frames with high symmetry for surface multiresolution processing

quincunx fundamental refinable functions and quincunx biorthogonal wavelets we analyze the approximation and smoothness properties of quincunx fundamental refinable functions in particular we provide a general way for the construction of quincunx interpolatory refinement masks associated with the quincunx lattice in r2 their corresponding quincunx fundamental refinable functions attain the optimal approximation order and smoothness order in addition these examples are minimally supported with symmetry for two special families of such quincunx interpolatory masks we prove that their symbols are nonnegative finally a general way of constructing quincunx biorthogonal wavelets is presented several examples of quincunx interpolatory masks and quincunx biorthogonal wavelets are explicitly computed

mixing properties for random walk in random scenery consider the lattice z d d geq 1 together with a stochastic black white coloring of its points and on it a random walk that is independent of the coloring a local scenery perceived at a given time is a pattern of colors seen by the walker in a finite box around his current position under weak assumptions on the probability distributions governing walk and coloring we prove asymptotic independence of local sceneries perceived at times 0 and n in the limit as n rightarrow infty and at times 0 and t k in the limit as k rightarrow infty where t k is the random k th hitting time of a black point an immediate corollary of the latter result is the convergence in distribution of the interarrival times between successive black hits i e of t k 1 t k as k rightarrow infty the limit distribution is expressed in terms of the distribution of the first hitting time t 1 the proof uses coupling arguments and ergodic theory

extremal two correlations of two valued stationary one dependent processes the maximal value of the two correlation for two valued stationary one dependent processes with fixed probability of a single symbol is determined we show that the process attaining this bound is unique except when 1 2 when there are exactly two different processes the analogous problem for minimal two correlation is discussed and partial results are obtained

compactly supported tight affine spline frames in l 2 r d the theory of fiberization is applied to yield compactly supported tight affine frames wavelets in l 2 r d from box splines the wavelets obtained are smooth piecewise polynomials on a simple mesh furthermore they exhibit a wealth of symmetries and have a relatively small support the number of mother wavelets however increases with the increase of the required smoothness two bivariate constructions of potential practical value are highlighted in both the wavelets are derived from four direction mesh box splines that are refinable with respect to the dilation matrix 1 1 1 1

asymptotic expansions for potential functions of i i d random fields for sums of finite range potential functions of an iid random field we derive the validity of formal expansions of length two under standard conditions formal expansions are valid if and only if the characteristic functions of the sum converge to zero for all nonzero frequency parameters if this convergence fails the distribution of the sum can be approximated by a mixture of lattice distributions the result applies to m dependent random fields generated by independent random variables

triangular 7 and quadrilateral 5 subdivision schemes regular case abstract this paper is devoted to the study of triangular 7 and quadrilateral 5 surface subdivisions both approximation and interpolatory subdivision schemes are considered with illustrative examples of both scalar valued and matrix valued 7 and 5 subdivision masks that satisfy the sum rule of sufficiently high orders in particular optimal sobolev smoothness is determined and holder smoothness estimates are presented

z2 gradings of clifford algebras and multivector structures let c v g be the real clifford algebra associated with the real vector space v e ndowed with a nondegenerate metric g i n this paper we study the class of z2 gradings of c v g which are somehow compatible with the multivector structure of the grassmann algebra over v a complete characterization for such z2 gradings is obtained by classifying all the even subalgebras coming from them an expression relating such subalgebras to the usual even part of c v g is also obtained finally we employ this framework to define spinor spaces and to parametrize all the possible signature changes on c v g by z2 gradings of this algebra

close packed dimers on the line diffraction versus dynamical spectrum the translation action of d on a translation bounded measure leads to an interesting class of dynamical systems with a rather rich spectral theory in general the diffraction spectrum of which is the carrier of the diffraction measure lives on a subset of the dynamical spectrum it is known that under some mild assumptions a pure point diffraction spectrum implies a pure point dynamical spectrum the opposite implication always being true for other systems the diffraction spectrum can be a proper subset of the dynamical spectrum as was pointed out for the thue morse sequence with singular continuous diffraction by van enter and mi kisz j stat phys 66 1147 1153 1992 here we construct a random system of close packed dimers on the line that have some underlying long range periodic order as well and display the same type of phenomenon for a system with absolutely continuous spectrum an interpretation in terms of atomic versus molecular spectrum suggests a way to come to a more general correspondence between these two types of spectra

an algebraic construction of a class of one dependent processes a special class of stationary one dependent two valued stochastic processes is defined we associate to each member of this class two parameter values whereby different members receive different parameter values for any given values of the parameters we show how to determine whether 1 a process exists having the given parameter values and if so 2 this process can be obtained as a two block factor from an independent process this determines a two parameter subfamily of the class of stationary one dependent two valued stochastic processes which are not two block factors of independent processes

nonseparable two and three dimensional wavelets we present two and three dimensional nonseparable wavelets they are obtained from discrete time bases by iterating filter banks we consider three sampling lattices quincunx separable by two in two dimensions and fco the design methods are based either on cascade structures or on the mcclellan transformation in the quincunx case we give a few design examples in particular the first example of an orthonormal 2 d wavelet basis with symmetries is constructed

approximation properties of multivariate wavelets wavelets are generated from refinable functions by using multiresolution analysis in this paper we investigate the approximation properties of multivariate refinable functions we give a characterization for the approximation order provided by a refinable function in terms of the order of the sum rules satisfied by the refinement mask we connect the approximation properties of a refinable function with the spectral properties of the corresponding subdivision and transition operators finally we demonstrate that a refinable function in w k 1 1 r s provides approximation order k

symmetry property and construction of wavelets with a general dilation matrix in this note we are interested in the symmetry property of a refinable function with a general dilation matrix we investigate the symmetry group of a mask so that its associated refinable function with a general dilation matrix has a certain kind of symmetry given two dilation matrices which produce the same lattice we demonstrate that if a mask has a certain kind of symmetry then its associated refinable functions with respect to the two dilation matrices are the same therefore the two corresponding derived wavelet systems are essentially the same finally we illustrate that for any dilation matrix orthogonal masks as well as interpolatory masks having nonnegative symbols can be easily constructed with any preassigned order of sum rules by employing a linear transform without solving any equations the method in this note on constructing masks with certain desirable properties is simple painless and general examples of quincunx wavelets and wavelets with respect to the checkerboard lattice are presented to illustrate the general theory

continuity of eigenfunctions of uniquely ergodic dynamical systems and intensity of bragg peaks we study uniquely ergodic dynamical systems over locally compact sigma compact abelian groups we characterize uniform convergence in wiener wintner type ergodic theorems in terms of continuity of the limit our results generalize and unify earlier results of robinson and assani respectively