solvability of linear local and nonlocal robin problems over c ω abstract let r n be a bounded lipschitz domain we first consider an elliptic boundary value problem with general robin boundary conditions the boundary conditions can be either local or nonlocal depending on the conditions imposed on the elliptic operator we prove that this boundary value problem is uniquely solvable and moreover we show that such weak solution is holder continuous on we also prove that a realization of the associated differential operator with generalized local or nonlocal robin boundary conditions generates an analytic c 0 semigroup of angle 2 over c we conclude by applying the elliptic regularity theory to solve the corresponding cauchy problem over c

global regularity for a class of quasi linear local and nonlocal elliptic equations on extension domains abstract we consider the solvability of quasi linear elliptic equations with either local or nonlocal neumann robin or wentzell boundary conditions defined in the generalized sense on a bounded w 1 p extension domain whose boundary is an upper d set for an appropriate d 0 then we extend the fine regularity theory for weak solutions of the elliptic equations with the above boundary conditions known for bounded lipschitz domains to bounded w 1 p extension domains whose boundaries are upper d sets by showing that such weak solutions are globally holder continuous consequently we generalize substantially the class of bounded domains where weak solutions of boundary value problems of type neumann robin or wentzell may be uniformly continuous up to the boundary

quasi linear boundary value problems with generalized nonlocal boundary conditions abstract we investigate a quasi linear boundary value problem of the form div u p 2 u 0 involving a general boundary map and mixed neumann boundary conditions on a bounded lipschitz domain we show existence uniqueness and holder continuity of the weak solution of this mixed boundary value problem and obtain maximum principles for this class of mixed equations as a consequence we obtain uniform continuity up to the boundary to solutions associated with a class of electrical models described by maxwell s equations with nonlocal boundary conditions an extension to boundary value problems with generalized nonlocal robin boundary conditions is also achieved

stochastic differential equations with reflecting boundary conditions on resout des equations differentielles stochastiques a conditions aux limites reflechissantes par une approche directe basee sur le probleme de skorokhod

a machine learning method for extracting symbolic knowledge from recurrent neural networks neural networks do not readily provide an explanation of the knowledge stored in their weights as part of their information processing until recently neural networks were considered to be black boxes with the knowledge stored in their weights not readily accessible since then research has resulted in a number of algorithms for extracting knowledge in symbolic form from trained neural networks this article addresses the extraction of knowledge in symbolic form from recurrent neural networks trained to behave like deterministic finite state automata dfas to date methods used to extract knowledge from such networks have relied on the hypothesis that networks states tend to cluster and that clusters of network states correspond to dfa states the computational complexity of such a cluster analysis has led to heuristics that either limit the number of clusters that may form during training or limit the exploration of the space of hidden recurrent state neurons these limitations while necessary may lead to decreased fidelity in which the extracted knowledge may not model the true behavior of a trained network perhaps not even for the training set the method proposed here uses a polynomial time symbolic learning algorithm to infer dfas solely from the observation of a trained network s input output behavior thus this method has the potential to increase the fidelity of the extracted knowledge

brownian motion in a wedge with variable reflection existence and uniqueness existence and uniqueness in law of reflecting brownian motion in a wedge is proved the direction of reflection along the sides of the wedge varies in a reasonable fashion except perhaps at the corner

on the stochastic regularity of distorted brownian motions we systematically develop general tools to apply fukushima s absolute continuity condition these tools comprise methods to obtain a hunt process on a locally compact separa ble metric state space whose transition function has a density w r t the reference measure and methods to estimate drift potentials comfortably we then apply our results to distorted brownian motions and construct weak solutions to singular stochastic differential equations i e equations with possibly unbounded and discontinuous drift and reflect ion terms which may be the sum of countably many local times the solutions can start from any point of the explicitly specified state space we consider different kind of weights like muckenhoupt a2 weights and weights with moderate growth at singularities as well as different kind of multiple boundary conditions our approach leads in particular to the construction and explicit identification of countably skew reflected and normally reflected brownian motions with singu lar drift in bounded and unbounded multi dimensional domains

a skorohod type lemma and a decomposition of reflected brownian motion we consider two dimensional reflected brownian motions in sharp thorns pointed downward with horizontal vectors of reflection we present a decomposition of the process into a brownian motion and a process which has bounded variation away from the tip of the thorn the construction is based on a new skorohod type lemma

a quasi linear neumann problem of ambrosetti prodi type on extension domains abstract let r n be a bounded domain with the extension property whose boundary is an upper d set with respect to a measure for n 2 and d n p n we investigate the solvability of the ambrosetti prodi problem for the p laplace operator p with neumann boundary conditions using a priori estimates regularity theory a sub supersolution method and the leray schauder degree theory we obtain a necessary condition for the non existence of solutions in the weak sense the existence of at least one minimal solution and the existence of at least two distinct solutions moreover we establish global holder continuity for weak solutions of the neumann problem of ambrosetti prodi type on a large class of bad domain extending the solvability of this type of elliptic problem for the first time to a wide class of non smooth domains

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

brownian motion in a wedge with oblique reflection this work is concerned with the existence and uniqueness of a strong markov process that has continuous sample paths and the following additional properties i the state space is an infinite two dimensional wedge and the process behaves in the interior of the wedge like an ordinary brownian motion ii the process reflects instantaneously at the boundary of the wedge the angle of reflection being constant along each side iii the amount of time that the process spends at the comer of the wedge is zero i e the set of times for which the process is at the comer has lebesgue measure zero hereafter let be the angle of the wedge 0 2 let 1 and 2 be the angles of reflection on the two sides of the wedge measured from the inward normals the positive angles being toward the corner 1 2 and set 1 2 the question of existence and uniqueness is recast as a submartingale problem in the style used by stroock and varadhan diffusion processes with boundary conditions comm pure appl math 24 1971 pp 147 225 for diffusions on smooth domains with smooth boundary conditions it is shown that no solution exists if 2 in this case there is a unique continuous strong markov process satisfying i ii above it reaches the corner of the wedge almost surely and it remains there if 2 however then there is a unique continuous strong markov process statisfying i iii it is shown that starting away from the corner this process does not reach the corner of the wedge if 0 and does reach the corner if 0 2 the general theory of multi dimensional diffusions does not apply to the above problem because in general the boundary of the state space is not smooth and there is a discontinuity in the direction of reflection at the corner for some values of the process arises from diffusion approximations to storage systems and queueing networks i the state space is an infinite two dimensional wedge and the process behaves in the interior of the wedge like an ordinary brownian motion ii the process reflects instantaneously at the boundary of the wedge and the angle of reflection being constant along each side iii the amount of time that the process spends at the corner of the wedge is zero i e the set of times for which the process is at the corner has lebesgue measure zero

on the complexity of computing and learning with multiplicative neural networks in a great variety of neuron models neural inputs are combined using the summing operation we introduce the concept of multiplicative neural networks that contain units that multiply their inputs instead of summing them and thus allow inputs to interact nonlinearly the class of multiplicative neural networks comprises such widely known and well studied network types as higher order networks and product unit networks we investigate the complexity of computing and learning for multiplicative neural networks in particular we derive upper and lower bounds on the vapnik chervonenkis vc dimension and the pseudodimension for various types of networks with multiplicative units as the most general case we consider feedforward networks consisting of product and sigmoidal units showing that their pseudo dimension is bounded from above by a polynomial with the same order of magnitude as the currently best known bound for purely sigmoidal networks moreover we show that this bound holds even when the unit type product or sigmoidal may be learned crucial for these results are calculations of solution set components bounds for new network classes as to lower bounds we construct product unit networks of fixed depth with superlinear vc dimension for sigmoidal networks of higher order we establish polynomial bounds that in contrast to previous results do not involve any restriction of the network order we further consider various classes of higher order units also known as sigma pi units that are characterized by connectivity constraints in terms of these we derive some asymptotically tight bounds multiplication plays an important role in both neural modeling of biological behavior and computing and learning with artificial neural networks we briefly survey research in biology and in applications where multiplication is considered an essential computational element the results we present here provide new tools for assessing the impact of multiplication on the computational power and the learning capabilities of neural networks

skorokhod decomposition of reflected diffusions on bounded lipschitz domains with singular non reflection part let overline g subset mathbb r d be a compact set with interior g let l1 g dx 0 dx a e on g and m dx let a aij be symmetric and globally uniformly strictly elliptic on g let be such that mathcal e r f g frac 1 2 sum i j 1 d int g a ij partial i f partial j g dm f g in c infty overline g is closable in l2 g m with closure r d r the latter is fulfilled if satisfies the hamza type condition or i l1loc g dx 1 i d conservative non symmetric diffusion processes xt related to the extension of a generalized dirichlet form mathcal e r f g sum i 1 d int g rho 1 overline b i partial i f g dm f g in d mathcal e r b where rho 1 overline b 1 overline b d in l 2 g mathbb r d m satisfies sum i 1 d int g overline b i partial i f dx 0 quad rm for all f in c infty overline g are constructed and analyzed if g is a bounded lipschitz domain h1 1 g and aij d r a skorokhod decomposition for xt is given this happens through a local time that is uniquely associated to the smooth measure 1 tr 0 d where tr denotes the trace and the surface measure on g

an algebraic framework to represent finite state machines in single layer recurrent neural networks in this paper we present an algebraic framework to represent finite state machines fsms in single layer recurrent neural networks slrnns which unifies and generalizes some of the previous proposals this framework is based on the formulation of both the state transition function and the output function of an fsm as a linear system of equations and it permits an analytical explanation of the representational capabilities of first order and higher order slrnns the framework can be used to insert symbolic knowledge in rnns prior to learning from examples and to keep this knowledge while training the network this approach is valid for a wide range of activation functions whenever some stability conditions are met the framework has already been used in practice in a hybrid method for grammatical inference reported elsewhere sanfeliu and alquezar 1994

symmetric orthogonal filters and wavelets with linear phase moments in this paper we study symmetric orthogonal filters with linear phase moments which are of interest in wavelet analysis and its applications we investigate relations and connections among the linear phase moments sum rules and symmetry of an orthogonal filter as one of the results we show that if a real valued orthogonal filter a is symmetric about a point then a has sum rules of order m if and only if it has linear phase moments of order 2m these connections among the linear phase moments sum rules and symmetry help us to reduce the computational complexity of constructing symmetric real valued orthogonal filters and to understand better symmetric complex valued orthogonal filters with linear phase moments to illustrate the results in the paper we provide many examples of univariate symmetric orthogonal filters with linear phase moments in particular we obtain an example of symmetric real valued 4 orthogonal filters whose associated orthogonal 4 refinable function lies in c 2 r

regularity and well posedness of some quasi linear elliptic and parabolic problems with nonlinear general wentzell boundary conditions on nonsmooth domains abstract we show that on a bounded domain r n with lipschitz continuous boundary if 2 n n 2 p n 1 then weak solutions of the quasi linear elliptic equation p u u p 2 u 1 x u f in with the general wentzell boundary conditions p u u p 2 u u p 2 u 2 x u g weakly on are globally holder continuous on here p and p denote the p laplace operator and the p laplace beltrami operator on the boundary respectively the functions 1 x for x and 2 x for x are continuous on r and satisfy a certain growth condition we also obtain that a realization of the operator p u 1 x u in c with the boundary conditions p u 1 x u p u u p 2 u 2 x u 0 on generates a strongly continuous semigroup of nonlinear operators on c

on the spectrum of markov semigroups via sample path large deviations the essential spectral radius of a sub markovian process is defined as the infimum of the spectral radiuses of all local perturbations of the process when the family of rescaled processes satisfies sample path large deviation principle the spectral radius and the essential spectral radius are expressed in terms of the rate function the paper is motivated by applications to reflected diffusions and jump markov processes describing stochastic networks for which the sample path large deviation principle has been established and the rate function has been identified while essential spectral radius has not been calculated

oblique derivative boundary conditions and the image method the image method of solving boundary value problems is extended to oblique derivative boundary conditions on plane boundaries for a class of equations including the wave equation the heat equation and laplace s equation this generalizes the recent result of gilbarg and trudinger elliptic partial differential equations of second order springer verlag berlin 1977 for laplace s equation the usefulness of the method is illustrated by solving with it both the wave equation and the heat equation in a half space with an initial point source and an oblique derivative boundary condition in both cases a surface wave is found for the wave equation there is a region near the boundary where the surface wave is the first arrival for the heat equation the surface wave is the largest wave in this region for small values of the time

fuzzy finite state automata can be deterministically encoded into recurrent neural networks there has been an increased interest in combining fuzzy systems with neural networks because fuzzy neural systems merge the advantages of both paradigms on the one hand parameters in fuzzy systems have clear physical meanings and rule based and linguistic information can be incorporated into adaptive fuzzy systems in a systematic way on the other hand there exist powerful algorithms for training various neural network models however most of the proposed combined architectures are only able to process static input output relationships they are not able to process temporal input sequences of arbitrary length fuzzy finite state automats ffas can model dynamical processes whose current state depends on the current input and previous states unlike in the case of deterministic finite state automats dfas ffas are not in one particular state rather each state is occupied to some degree defined by a membership function based on previous work on encoding dfas in discrete time second order recurrent neural networks we propose an algorithm that constructs an augmented recurrent neural network that encodes a ffa and recognizes a given fuzzy regular language with arbitrary accuracy we then empirically verify the encoding methodology by correct string recognition of randomly generated ffas in particular we examine how the networks performance varies as a function of synaptic weight strengths

brownian motion reflected on brownian motion we study brownian motion reflected on an independent brownian path we prove results on the joint distribution of both processes and the support of the parabolic measure in the space time domain bounded by a brownian path we show that there exist two different natural local times for a brownian path reflected on a brownian path

construction and decomposition of reflecting diffusions on lipschitz domains with holder cusps we consider a d dimensional euclidean domain d whose boundary is lipschitz continuous but admits locally finite number of outward or inward holder cusp points using a method of stampacchia and moser for pde we first construct a conservative diffusion process on the euclidean closure of d possessing a strong feller resolvent and associated with a second order uniformly elliptic differential operator of divergence form with measurable coefficients a ij the sample path of the constructed diffusion can be uniquely decomposed as a sum of a martingale additive functional and an additive functional locally of zero energy the second additive functional will be proved to be of bounded variation with a skorohod type expression whenever a ij is weakly differentiable and the holder exponent at each outward cusp boundary point is greater than 1 2 regardless the dimension d

efficient simulation of finite automata by neural nets let k m denote the smallest number with the property that every m state finite automaton can be built as a neural net using k m or fewer neurons a counting argument shows that k m is at least o m log m 1 3 and a construction shows that k m is at most o m 3 4 the counting argument and the construction allow neural nets with arbitrarily complex local structure and thus may require neurons that themselves amount to complicated networks mild and in practical situations almost necessary constraints on the local structure of the network give again by a counting argument and a construction lower and upper bounds for k m that are both linear in m

a stochastic differential game in the orthrant we study a zero sum stochastic differential game in the nonnegative orthrant the state of the system is governed by controlled reflecting diffusions in the nonnegative orthrant we consider discounted and average payoff evaluation criteria we prove the existence of values and optimal strategies for both payoff criteria

stable encoding of large finite state automata in recurrent neural networks with sigmoid discriminants we propose an algorithm for encoding deterministic finite state automata dfas in second order recurrent neural networks with sigmoidal discriminant function and we prove that the languages accepted by the constructed network and the dfa are identical the desired finite state network dynamics is achieved by programming a small subset of all weights a worst case analysis reveals a relationship between the weight strength and the maximum allowed network size which guarantees finite state behavior of the constructed network we illustrate the method by encoding random dfas with 10 100 and 1000 states while the theory predicts that the weight strength scales with the dfa size we find empirically the weight strength to be almost constant for all the random dfas these results can be explained by noting that the generated dfas represent average cases we empirically demonstrate the existence of extreme dfas for which the weight strength scales with dfa size

reflected brownian motion with skew symmetric data in a polyhedral domain this paper is concerned with the characterization and invariant measures of certain reflected brownian motions rbm s in polyhedral domains the kind of rbm studied here behaves like d dimensional brownian motion with constant drift in the interior of a simple polyhedron and is instantaneously reflected at the boundary in directions that depend on the face that is hit under the assumption that the directions of reflection satisfy a certain skew symmetry condition first introduced in harrison williams 9 it is shown that such an rbm can be characterized in terms of a family of submartingales and that it reaches non smooth parts of the boundary with probability zero in 9 a purely analytic problem associated with such an rbm was solved here the exponential form solution obtained in 9 is shown to be the density of an invariant measure for the rbm furthermore if the density is integrable over the polyhedral state space then it yields the unique stationary distribution for the rbm in the proofs of these results a key role is played by a dual process for the rbm and by results in 9 for reflected brownian motions on smooth approximating domains

reflecting brownian motion in a cusp let c be the cusp x y x 0 x y x where 1 set c 1 x y x 0 y x and c 2 x y x 0 y x we study the existence and uniqueness in law of reflecting brownian motion in c the angle of reflection at c j 0 relative to the inward unit normal is a constant j 2 2 and is positive iff the direction of reflection has a negative first component in all sufficiently small neighborhoods of 0 when 1 2 0 existence and uniqueness in law hold when 1 2 0 existence fails we also obtain results for a large class of asymmetric cusps we make essential use of results of warschawski on the differentiability at the boundary of conformal maps

on the robin problem in fractal domains research partially supported by national science foundation nsf grants dms 0244737 and dms 0303310

generalized interpolating refinable function vectors interpolating scalar refinable functions with compact support are of interest in several applications such as sampling theory signal processing computer graphics and numerical algorithms in this paper we shall generalize the notion of interpolating scalar refinable functions to compactly supported interpolating d refinable function vectors with any multiplicity r and dilation factor d more precisely we are interested in a d refinable function vector f f 1 f r t such that f is an rx1 column vector of compactly supported continuous functions with the following interpolation property f mr k email protected email protected 1 m email protected z m 0 r 1 1 r where d 0 1 and d k 0 for k 0 now for any function f email protected c the function f can be interpolated and approximated by email protected email protected email protected email protected email protected zf 1r k f k email protected email protected z f k f 1r k f r 1r k f k since f is interpolating email protected k r f k r for all email protected z that is email protected agrees with f on r 1z moreover for r 2 or d 2 such interpolating refinable function vectors can have the additional orthogonality property r email protected d k k for all k k z and 1

induction of finite state languages using second order recurrent networks second order recurrent networks that recognize simple finite state languages over 0 1 are induced from positive and negative examples using the complete gradient of the recurrent network and sufficient training examples to constrain the definition of the language to be induced solutions are obtained that correctly recognize strings of arbitrary length

recurrent neural networks with small weights implement definite memory machines recent experimental studies indicate that recurrent neural networks initialized with small weights are inherently biased toward definite memory machines tino cernansky benuskova 2002a 2002b this article establishes a theoretical counterpart transition function of recurrent network with small weights and squashing activation function is a contraction we prove that recurrent networks with contractive transition function can be approximated arbitrarily well on input sequences of unbounded length by a definite memory machine conversely every definite memory machine can be simulated by a recurrent network with contractive transition function hence initialization with small weights induces an architectural bias into learning with recurrent neural networks this bias might have benefits from the point of view of statistical learning theory it emphasizes one possible region of the weight space where generalization ability can be formally proved it is well known that standard recurrent neural networks are not distribution independent learnable in the probably approximately correct pac sense if arbitrary precision and inputs are considered we prove that recurrent networks with contractive transition function with a fixed contraction parameter fulfill the so called distribution independent uniform convergence of empirical distances property and hence unlike general recurrent networks are distribution independent pac learnable

constructing deterministic finite state automata in recurrent neural networks recurrent neural networks that are trained to behave like deterministic finite state automata dfas can show deteriorating performance when tested on long strings this deteriorating performance can be attributed to the instability of the internal representation of the learned dfa states the use of a sigmoidel discriminant function together with the recurrent structure contribute to this instability we prove that a simple algorithm can construct second order recurrent neural networks with a sparse interconnection topology and sigmoidal discriminant function such that the internal dfa state representations are stable that is the constructed network correctly classifies strings of arbitrary length the algorithm is based on encoding strengths of weights directly into the neural network we derive a relationship between the weight strength and the number of dfa states for robust string classification for a dfa with n state and m input alphabet symbols the constructive algorithm generates a programmed neural network with o n neurons and o mn weights we compare our algorithm to other methods proposed in the literature

a family of nonseparable scaling functions and compactly supported tight framelets abstract given integers b and d with d 1 and b 1 we construct even nonseparable compactly supported refinable functions with dilation factor b that generate multiresolution analyses on l 2 r d these refinable functions are nonseparable in the sense that they cannot be expressed as the product of two functions defined on lower dimensions we use these scaling functions and a slight generalization of a theorem of lai and stockler to construct smooth compactly supported tight framelets both the refinable functions and the framelets they generate can be made as smooth as desired estimates for the supports of these refinable functions and framelets are given

reflected brownian motion on an orthant we consider a k dimensional diffusion process z whose state space is the nonnegative orthant on the interior of the orthant z behaves like a kdimensional brownian motion with arbitrary covariance matrix and drift vector at each of the k 1 dimensional hyperplanes that form the boundary of the orthant z reflects instantaneously in a direction that is constant over that hyperplane there is no extant theory of multidimensional diffusion that applies to this process because the boundary of its state space is not smooth we adopt an approach that requires a restriction on the directions of reflection but reiman has shown that this restriction is met by all diffusions z arising as heavy traffic limits in open k station queuing networks our process z is defined as a path to path mapping of k dimensional brownian motion from this construction it follows that z is a continuous markov process and a semimartingale using the latter property we obtain a change of variable formula from which one can develop a complete analytical theory for the process z

comments on constructive learning of recurrent neural networks limitations of recurrent cascade correlation and a simple solution giles et al 1995 have proven that fahlman s recurrent cascade correlation rcc architecture is not capable of realizing finite state automata that have state cycles of length more than two under a constant input signal this paper extends the conclusions of giles et al by showing that there exists a corollary to their original proof which identifies a large second class of automata that is also unrepresentable by rcc

brownian motion in a wedge with variable skew reflection does planar brownian motion confined to a wedge by skew reflection on the sides approach the vertex of the wedge this question has been answered by varadhan and williams in the case where the direction of reflection is constant on each of the sides but here we address the question when the direction reflected is allowed to vary a necessary condition and a sufficient condition are obtained for the vertex to be reached the conditions are of a geometric nature and the gap between them is quite small

on the semimartingale representation of reflecting brownian motion in a cusp letc be the symmetric cusp x y 2 x y x x 0 where 1 in this paper we decide whether or not reflecting brownian motion inc has a semimartingale representation here the reflecting brownian motion has directions of reflection that make constant angles with the unit inward normals to the boundary our results carry through for a wide class of asymmetric cusps too

an algorithm for matrix extension and wavelet construction this paper gives a practical method of extending an n x r matrix p z r n with laurent polynomial entries in one complex variable z to a square matrix also with laurent polynomial entries if p z has orthonormal columns when z is restricted to the torus t it can be extended to a paraunitary matrix if p z has rank r for each z t it can be extended to a matrix with nonvanishing determinant on t the method is easily implemented in the computer it is applied to the construction of compactly supported wavelets and prewavelets from multiresolutions generated by several univariate scaling functions with an arbitrary dilation parameter

a multidimensional process involving local time on presente une approche probabiliste a un probleme pose par mckean 1967 pour la construction d un systeme de n particules sur r avec un generateur formel

reflected brownian motion in a wedge semimartingale property in this paper the object of study is reflected brownian motion in a two dimensional wedge with constant direction of reflection on each side of the wedge the basic question considered here is when is this process a semimartingale it is first shown that a related process defined by specifying the corner of the wedge to be an absorbing state rather than an instantaneous one is a semimartingale conditions for the existence and uniqueness of the process for which the corner is an instantaneous state were given by vardhan and williams brownian motion in a wedge with oblique reflection comm pure appl math to appear under these conditions it is shown that starting away from the corner the process is a semimartingale if and only if there is a convex combination of the directions of reflection that points into the wedge this equivalence is also shown to hold starting from the corner except in one unresolved case for which the wedge angle exceeds and the directions of reflection are exactly opposed

matrix splitting with symmetry and dyadic framelet filter banks over algebraic number fields abstract algebraic number fields are of particular interest and play an important role in both mathematics and engineering since an algebraic number field can be viewed as a finite dimensional linear space over the rational number field q algorithms using algebraic number fields can be efficiently implemented involving only integer arithmetics in this paper we properly formulate the matrix splitting problem over any general subfield of it c including an algebraic number field as a special case and provide a simple necessary and sufficient condition for a 2 2 matrix of laurent polynomials with symmetry to be able to be factorized by a 2 2 matrix of laurent polynomials with certain symmetry structure we propose an effective algorithm on how to obtain the factorization matrix step by step as an application we obtain a satisfactory algorithm for constructing dyadic framelet filter banks with the perfect reconstruction property and with symmetry over algebraic number fields several examples are provided to illustrate the algorithms proposed in this paper