threshold circuits of small majority depth we investigate the complexity of computations with constant depth threshold circuits such circuits are composed of gates that determine if the sum of their inputs is greater than a certain threshold when restricted to polynomial size these circuits compute exactly the functions in the class tc sp0 these circuits are usually studied by measuring their efficiency in terms of their total depth using this point of view the best division and iterated multiplication circuits have depth three and four respectively in this thesis we propose a different approach since threshold gates are much more powerful than and or gates we allow the explicit use of and or gates and consider the main measure of complexity to be the majority depth of the circuit i e the maximum number of threshold gates on any path in the circuit using this approach we obtain division and iterated multiplication circuits of total depth four and five but of majority depth two and three the technique used is called chinese remaindering we present this technique as a general tool for computing functions with integer values and use it to obtain depth four threshold circuits of majority depth two for other arithmetic problems such as the logarithm and power series approximation we also consider the iterated multiplication problem for integers modulo q and for finite fields the notion of majority depth naturally leads to a hierarchy of subclasses of tc sp0 we investigate this hierarchy and show that it is closely related to the usual depth hierarchy

constant depth reducibility the purpose of this paper is to study reducibilities that can be computed by combinational logic networks of polynomial size and constant depth containing and s or s and not s with no bound placed on the fan in of and gates and or gates two such reducibilities are defined and reductions and equivalences among several common problems such as parity sorting integer multiplication graph connectivity bipartite matching and network flow are given certain problems are shown to be complete with respect to these reducibilities in the complexity classes deterministic logarithmic space nondeterministic logarithmic space and deterministic polynomial time new upper bounds on the size depth unbounded fan in circuit complexity of symmetric boolean functions are established

ensemble manifold rank preserving for acceleration based human activity recognition with the rapid development of mobile devices and pervasive computing technologies acceleration based human activity recognition a difficult yet essential problem in mobile apps has received intensive attention recently different acceleration signals for representing different activities or even a same activity have different attributes which causes troubles in normalizing the signals we thus cannot directly compare these signals with each other because they are embedded in a nonmetric space therefore we present a nonmetric scheme that retains discriminative and robust frequency domain information by developing a novel ensemble manifold rank preserving emrp algorithm emrp simultaneously considers three aspects 1 it encodes the local geometry using the ranking order information of intraclass samples distributed on local patches 2 it keeps the discriminative information by maximizing the margin between samples of different classes and 3 it finds the optimal linear combination of the alignment matrices to approximate the intrinsic manifold lied in the data experiments are conducted on the south china university of technology naturalistic 3 d acceleration based activity dataset and the naturalistic mobile devices based human activity dataset to demonstrate the robustness and effectiveness of the new nonmetric scheme for acceleration based human activity recognition

bounded depth polynomial size circuits for symmetric functions abstract let f f 1 f 2 be a family of symmetric boolean functions where f n has n boolean variables for each n 1 let f n be the minimum number of variables of f n that each have to be set to constant values so that the resulting function is a constant function we show that the growth rate of f n completely determines whether or not the family f is good that is can be realized by a family of constant depth polynomial size circuits with unbounded fan in furthermore if f n log n k for some k then the family f is good however if f n n for some 0 then the family is not good

a novel characterization of the complexity class θkp based on counting and comparison abstract the complexity class 2 p which is the class of languages recognizable by deterministic turing machines in polynomial time with at most logarithmic many calls to an np oracle received extensive attention in the literature its complete problems can be characterized by different specific tasks such as deciding whether the optimum solution of an np problem is unique or whether it is in some sense odd e g whether its size is an odd number in this paper we introduce a new characterization of this class and its generalization k p to the k th level of the polynomial hierarchy we show that problems in k p are also those whose solution involves deciding for two given sets a and b of instances of two k 1 p complete or k 1 p complete problems whether the number of yes instances in a is greater than those in b moreover based on this new characterization we provide a novel sufficient condition for k p hardness we also define the general problem comp valid k which is proven here k 1 p complete comp valid k is the problem of deciding given two sets a and b of quantified boolean formulas with at most k alternating quantifiers whether the number of valid formulas in a is greater than those in b notably the problem comp sat of deciding whether a set contains more satisfiable boolean formulas than another set which is a particular case of comp valid 1 demonstrates itself as a very intuitive 2 p complete problem nonetheless to our knowledge it eluded its formal definition to date in fact given its strict adherence to the count and compare semantics here introduced comp valid k is among the most suitable tools to prove k p hardness of problems involving the counting and comparison of the number of yes instances in two sets we support this by showing that the 2 p hardness of the max voting scheme over mcp nets is easily obtained via the new characterization of k p introduced in this paper

adaptive learning in complex reproducing kernel hilbert spaces employing wirtinger s subgradients this paper presents a wide framework for non linear online supervised learning tasks in the context of complex valued signal processing the complex input data are mapped into a complex reproducing kernel hilbert space rkhs where the learning phase is taking place both pure complex kernels and real kernels via the complexification trick can be employed moreover any convex continuous and not necessarily differentiable function can be used to measure the loss between the output of the specific system and the desired response the only requirement is the subgradient of the adopted loss function to be available in an analytic form in order to derive analytically the subgradients the principles of the recently developed wirtinger s calculus in complex rkhs are exploited furthermore both linear and widely linear in rkhs estimation filters are considered to cope with the problem of increasing memory requirements which is present in almost all online schemes in rkhs the sparsification scheme based on projection onto closed balls has been adopted we demonstrate the effectiveness of the proposed framework in a non linear channel identification task a non linear channel equalization problem and a quadrature phase shift keying equalization scheme using both circular and non circular synthetic signal sources

exact analysis of dodgson elections lewis carroll s 1876 voting system is complete for parallel access to np in 1876 lewis carroll proposed a voting system in which the winner is the candidate who with the fewest changes in voters preferences becomes a condorcet winner a candidate who beats all other candidates in pairwise majority rule elections bartholdi tovey and trick provided a lower bound np hardness on the computational complexity of determining the election winner in carroll s system we provide a stronger lower bound and an upper bound that matches our lower bound in particular determining the winner in carroll s system is complete for parallel access to np that is it is complete for theta 2 p for which it becomes the most natural complete problem known it follows that determining the winner in carroll s elections is not np complete unless the polynomial hierarchy collapses

reproducing kernel hilbert space approach for the online update of radial bases in neuro adaptive control classical work in model reference adaptive control for uncertain nonlinear dynamical systems with a radial basis function rbf neural network adaptive element does not guarantee that the network weights stay bounded in a compact neighborhood of the ideal weights when the system signals are not persistently exciting pe recent work has shown however that an adaptive controller using specifically recorded data concurrently with instantaneous data guarantees boundedness without pe signals however the work assumes fixed rbf network centers which requires domain knowledge of the uncertainty motivated by reproducing kernel hilbert space theory we propose an online algorithm for updating the rbf centers to remove the assumption in addition to proving boundedness of the resulting neuro adaptive controller a connection is made between pe signals and kernel methods simulation results show improved performance

new set of symmetries of the integrable equations lie algebra and non isospectral evolution equations ii akns system the authors define directly a new set of symmetries for the akns system and prove that they constitute an infinite dimensional lie algebra with the old symmetries they also point out the relation between the new symmetry and the non isospectral problem they use the reduction technique and point out that the nls mkdv sg and sinh g hierarchies have two sets of symmetries which constitute an infinite dimensional lie algebra these results are extensions of those of chen et al 1982

on sparse hard sets for counting classes abstract in this paper we study one word decreasing self reducible sets which are introduced by lozano and toran 1991 these are the usual self reducible sets with the peculiarity that the self reducibility machine makes at most one query and this is lexicographically smaller than the input we show first that for all counting classes defined by a predicate on the number of accepting paths there exist complete sets which are one word decreasing self reducible using this fact we can prove that for any class k chosen from pp np c p mod2p mod3 p it holds that 1 if there is a sparse pbtt hard set for k then k p and 2 if there is a sparse snbtt hard set for k then k np co np this generalizes the result of ogiwara and watanabe 1991 to the mentioned complexity classes

widely linear estimation with complex data mean square estimation of complex and normal data is not linear as in the real case but widely linear the purpose of the paper is to calculate the optimum widely linear mean square estimate and to present its main properties the advantage with respect to the linear procedure is especially analyzed

enumerative counting is hard abstract an n variable boolean formula may have anywhere from 0 to 2n satisfying assignments can a polynomial time machine given such a formula reduce this exponential number of possibilities to a small number of possibilities we call such a machine an enumerator and prove that if there is a good polynomial time enumerator for p i e one where for every boolean formula f the small set has at most o f 1 e numbers then p np p p and probabilistic polynomial time equals polynomial time furthermore we show that p polynomial time turing reduces to enumerating p

efficient threshold circuits for power series abstract we show that functions with convergent real power series can be well approximated by two classes of polynomial size small weight threshold circuits depth three circuits with threshold gates on all levels and depth four circuits with threshold gates on the first two levels and and or gates on the last two this is done without restricting the input to a fixed closed subinterval of the interval of convergence of the series we also point out that rational functions and the logarithm of x in base b can be well approximated by the same classes of circuits when both x and b are given as input

extension of wirtinger s calculus to reproducing kernel hilbert spaces and the complex kernel lms over the last decade kernel methods for nonlinear processing have successfully been used in the machine learning community the primary mathematical tool employed in these methods is the notion of the reproducing kernel hilbert space rkhs however so far the emphasis has been on batch techniques it is only recently that online techniques have been considered in the context of adaptive signal processing tasks moreover these efforts have only been focussed on real valued data sequences to the best of our knowledge no adaptive kernel based strategy has been developed so far for complex valued signals furthermore although the real reproducing kernels are used in an increasing number of machine learning problems complex kernels have not yet been used in spite of their potential interest in applications that deal with complex signals with communications being a typical example in this paper we present a general framework to attack the problem of adaptive filtering of complex signals using either real reproducing kernels taking advantage of a technique called complexification of real rkhss or complex reproducing kernels highlighting the use of the complex gaussian kernel in order to derive gradients of operators that need to be defined on the associated complex rkhss we employ the powerful tool of wirtinger s calculus which has recently attracted attention in the signal processing community wirtinger s calculus simplifies computations and offers an elegant tool for treating complex signals to this end in this paper the notion of wirtinger s calculus is extended for the first time to include complex rkhss and use it to derive several realizations of the complex kernel least mean square cklms algorithm experiments verify that the cklms offers significant performance improvements over several linear and nonlinear algorithms when dealing with nonlinearities

study of the convergence behavior of the complex kernel least mean square algorithm the complex kernel least mean square cklms algorithm is recently derived and allows for online kernel adaptive learning for complex data kernel adaptive methods can be used in finding solutions for neural network and machine learning applications the derivation of cklms involved the development of a modified wirtinger calculus for hilbert spaces to obtain the cost function gradient we analyze the convergence of the cklms with different kernel forms for complex data the expressions obtained enable us to generate theory predicted mean square error curves considering the circularity of the complex input signals and their effect on nonlinear learning simulations are used for verifying the analysis results

lie algebraic structures of 1 1 dimensional lax integrable systems an approach of constructing isospectral flows kl nonisospectral flows k and their implicit representations of a general lax integrable system is proposed by introducing product function matrices it is shown that the two sets of flows and of related symmetries both constitute infinite dimensional lie algebras with respect to the commutator given in this paper algebraic properties for some well known integrable systems such as the akns system the generalized harry dym system and the n wave interaction system are obtained as particular examples

an information theoretic approach of designing sparse kernel adaptive filters this paper discusses an information theoretic approach of designing sparse kernel adaptive filters to determine useful data to be learned and remove redundant ones a subjective information measure called surprise is introduced surprise captures the amount of information a datum contains which is transferable to a learning system based on this concept we propose a systematic sparsification scheme which can drastically reduce the time and space complexity without harming the performance of kernel adaptive filters nonlinear regression short term chaotic time series prediction and long term time series forecasting examples are presented

the complexity of iterated multiplication abstract for a monoid g the iterated multiplication problem is the computation of the product of n elements from g by refining known completeness arguments we show that as g varies over a natural series of important groups and monoids the iterated multiplication problems are complete for most natural low level complexity classes the completeness is with respect to first order projections low level reductions that do not obscure the algebraic nature of these problems

kernel recursive least squares tracker for time varying regression in this paper we introduce a kernel recursive least squares krls algorithm that is able to track nonlinear time varying relationships in data to this purpose we first derive the standard krls equations from a bayesian perspective including a sensible approach to pruning and then take advantage of this framework to incorporate forgetting in a consistent way thus enabling the algorithm to perform tracking in nonstationary scenarios the resulting method is the first kernel adaptive filtering algorithm that includes a forgetting factor in a principled and numerically stable manner in addition to its tracking ability it has a number of appealing properties it is online requires a fixed amount of memory and computation per time step incorporates regularization in a natural manner and provides confidence intervals along with each prediction we include experimental results that support the theory as well as illustrate the efficiency of the proposed algorithm

stochastic behavior analysis of the gaussian kernel least mean square algorithm the kernel least mean square klms algorithm is a popular algorithm in nonlinear adaptive filtering due to its simplicity and robustness in kernel adaptive filters the statistics of the input to the linear filter depends on the parameters of the kernel employed moreover practical implementations require a finite nonlinearity model order a gaussian klms has two design parameters the step size and the gaussian kernel bandwidth thus its design requires analytical models for the algorithm behavior as a function of these two parameters this paper studies the steady state behavior and the transient behavior of the gaussian klms algorithm for gaussian inputs and a finite order nonlinearity model in particular we derive recursive expressions for the mean weight error vector and the mean square error the model predictions show excellent agreement with monte carlo simulations in transient and steady state this allows the explicit analytical determination of stability limits and gives opportunity to choose the algorithm parameters a priori in order to achieve prescribed convergence speed and quality of the estimate design examples are presented which validate the theoretical analysis and illustrates its application

lie algebras and hamiltonian structures of multi component ablowitz kaup newell segur hierarchy isospectral and non isospectral hierarchies of multi component ablowitz kaup newell segur akns are obtained from a matrix spectral problem then by means of the zero curvature representations of the isospectral flows km and non isospectral flows n we construct the symmetries and their algebraic structures for isospectral multi component akns hierarchies demonstrate the recursive operator l is a strong and hereditary symmetry for the isospectral hierarchy we also derive that there are implectic operator and symplectic operator j such that l j and discuss the multi hamiltonian structures and the liouville integrability of the isospectral hierarchies

quantum and classical complexity classes separations collapses and closure properties we study the complexity of quantum complexity classes such as eqp bqp and nqp quantum analogs of p bpp and np respectively using classical complexity classes such as zpp wpp and c p the contributions of this paper are threefold first via oracle constructions we show that no relativizable proof technique can improve the best known classical upper bound for bqp bqp awpp journal of computer and system sciences 59 2 1999 240 to bqp wpp and the best known classical lower bound for eqp p eqp to zpp eqp second we prove that there are oracles a and b such that relative to a corp is immune to nqp and relative to b bqp is immune to pc p extending a result of de graaf and valiant technical report quant ph 0211179 quantum physics 2002 we construct a relativized world where eqp is immune to modpkp third motivated by the fact that counting classes e g lwpp awpp etc are the best known classical upper bounds on quantum complexity classes we study properties of these counting classes we prove that wpp is closed under polynomial time truth table reductions while we construct an oracle relative to which wpp is not closed under polynomial time turing reductions the latter result implies that proving the equality of the similar appearing classes lwpp and wpp would require nonrelativizable proof techniques we also prove that both awpp and app are closed under tup reductions we use closure properties of wpp and awpp to prove interesting consequences in terms of the complexity of the polynomial hierarchy of the following hypotheses nqp bqp and eqp nqp

two integrable hamiltonian hierarchies in sl 2 r and so 3 r with three potentials by introducing two specific matrix spectral problems associated with sl 2 r and so 3 r matrix lie algebras we generate two integrable hamiltonian hierarchies with three potentials the computation and analysis on their hamiltonian structures by means of the trace identity show that the resulting hierarchies are liouville integrable namely that each hierarchy consists of commuting hamiltonian soliton equations

quantized kernel least mean square algorithm in this paper we propose a quantization approach as an alternative of sparsification to curb the growth of the radial basis function structure in kernel adaptive filtering the basic idea behind this method is to quantize and hence compress the input or feature space different from sparsification the new approach uses the redundant data to update the coefficient of the closest center in particular a quantized kernel least mean square qklms algorithm is developed which is based on a simple online vector quantization method the analytical study of the mean square convergence has been carried out the energy conservation relation for qklms is established and on this basis we arrive at a sufficient condition for mean square convergence and a lower and upper bound on the theoretical value of the steady state excess mean square error static function estimation and short term chaotic time series prediction examples are presented to demonstrate the excellent performance

a trace identity and its applications to the theory of discrete integrable systems a general theory for studying discrete integrable systems is developed based on a trace identity that the author previously proposed a scheme for generating hierarchies of discrete integrable systems is presented under broad assumptions the resulting hierarchies are shown to consist of liouville integrable hamiltonian systems

depth size tradeoffs for neural computation the tradeoffs between the depth i e the time for parallel computation and the size i e the number of threshold gates in neural networks are studied the authors focus the study on the neural computations of symmetric boolean functions and some arithmetic functions it is shown that a significant reduction in the size is possible for symmetric functions and some arithmetic functions at the expense of a small constant increase in depth in the process several neural networks which have the minimum size among all the known constructions have been developed results on implementing symmetric functions can be used to improve results about arbitrary boolean functions in particular it is shown that any boolean function can be computed in a depth 3 neural network with o 2 sup n sup 2 threshold gates it is also proven that at least omega 2 sup n sup 3 threshold gates are required

integrable systems associated with the schrodinger spectral problem in the plane associated with the schrodinger spectral problem in the plane the nonlinear integrable systems are investigated the commutative properties of equations of the flow and their symmetries and mastersymmetries are derived in spite of the noncanonicality of the related extended recursion operator by applying the inverse scattering transform the systems can be solved and then the correspondences between commuting resp noncommuting flows and isospectral resp nonisospectral deformations of the spectral problem are found

some observations on the connection between counting and recursion abstract based on valiant s class p of all functions counting the number of accepting computations of nondeterministic polynomial time turing machines the polynomial time hierarchy of counting functions is introduced the class phcf of all functions of this hierarchy and some of its subclasses are characterized by recursion theoretic means it turns out that from the recursion theoretic point of view phcf is an analogue to kalmar s class e of elementary functions to the class pspace of polynomial space computable functions as well as to the class p of polynomial time computable functions

the complexity of kemeny elections kemeny proposed a voting scheme which is distinguished by the fact that it is the unique voting scheme that is neutral consistent and condorcet bartholdi tovey and trick showed that determining the winner in kemeny s system is np hard we provide a stronger lower bound and an upper bound matching the lower bound namely we show that determining the winner in kemeny s system is complete for p np the class of sets solvable via parallel access to np

model checking for fragments of the interval temporal logic hs at the low levels of the polynomial time hierarchy abstract some temporal properties of reactive systems such as actions with duration and temporal aggregations which are inherently interval based can not be properly expressed by the standard point based temporal logics ltl ctl and ctl as they give a state by state account of system evolution conversely interval temporal logics which feature intervals instead of points as their primitive entities naturally express them we study the model checking mc problem for halpern and shoham s modal logic of time intervals hs interpreted on kripke structures under the homogeneity assumption hs is the best known interval based temporal logic which has one modality for each of the 13 ordering relations between pairs of intervals allen s relations apart from equality we focus on mc for some hs fragments featuring modalities for a subset of allen s relations meet met by started by and finished by showing that it is in p np additionally we provide some complexity lower bounds to the problem

symmetries and hierarchies of equations for the 2 1 dimensional n wave interaction abstract infinite sets of symmetries for the 2 1 dimensional n wave interaction are derived and two hierarchies of equations are given the lax pairs for these hierarchies are derived by an algebraic approach consequently two hierarchies are proved to be respectively the isospectral and non isospectral flows thus the inverse scattering transform of the multidimensional akns spectral problem can be applied and the corresponding time evolution of scattering data are obtained

two remarks on the power of counting the relationship between the polynomial hierarchy and valiant s class p is at present unknown we show that some low portions of the polynomial hierarchy namely deterministic polynomial algorithms using an np oracle at most a logarithmic number of times can be simulated by one p computation we also show that the class of problems solvable by polynomial time nondeterministic turing machines which accept whenever there is an odd number of accepting computations is idempotent that is closed under usage of oracles from the same class

hierarchies of nonlinear integrable equations and their symmetries in 2 1 dimensions abstract for a given nonlinear integrable equation in 2 1 dimensions an approach is described to construct the hierarchies of equations and relevant lie algebraic properties the commutability and noncommutability of equations of the flow their symmetries and mastersymmetries are then derived as direct results of these algebraic properties the details for the modified kadomtsev petviashvilli equation are shown as an example and the main results for the 2 1 dimensional caudrey dodd gibbon katera sawada equation are given

a sequential learning algorithm for complex valued self regulating resource allocation network csran this paper presents a sequential learning algorithm for a complex valued resource allocation network with a self regulating scheme referred to as complex valued self regulating resource allocation network csran the self regulating scheme in csran decides what to learn when to learn and how to learn based on the information present in the training samples csran is a complex valued radial basis function network with a sech activation function in the hidden layer the network parameters are updated using a complex valued extended kalman filter algorithm csran starts with no hidden neuron and builds up an appropriate number of hidden neurons resulting in a compact structure performance of the csran is evaluated using a synthetic complex valued function approximation problem two real world applications consisting of a complex quadrature amplitude modulation channel equalization and an adaptive beam forming problem since complex valued neural networks are good decision makers the decision making ability of the csran is compared with other complex valued classifiers and the best performing real valued classifier using two benchmark unbalanced classification problems from uci machine learning repository the approximation and classification results show that the csran outperforms other existing complex valued learning algorithms available in the literature

the trace identity a powerful tool for constructing the hamiltonian structure of integrable systems a new approach to hamiltonian structures of integrable systems is proposed by making use of a trace identity for a variety of isospectral problems that can be unified to one model x u it is shown that both the related hierarchy of evolution equations and the hamiltonian structure can be obtained from the same solution of the equation vx u v

decomposition of threshold functions into bounded fan in threshold functions this paper obtains explicit decomposition of threshold functions into bounded fan in threshold functions a small fan in is important to satisfy technology constraints for large scale integration by employing the concept of error in the threshold function we are able to decompose functions in email protected 1 into a network of size o n c m 2 and depth o log 2n logm where n is the number of inputs of the function and m is the fan in bound the proposed construction enables one to trade off the size and depth of the decomposition with the fan in bound combined with the work on small weight threshold functions this implies polynomial size log 2 depth bounded fan in decompositions for arbitrary threshold functions in lt d these results compare favorably with the classical decomposition which has a size o 2 n m and depth o n m we also show that the decomposition size and depth can be significantly reduced by exploiting the relationships between the input weights as examples of this strategy we demonstrate an o n 2 m size decomposition of the majority function and o n m size decompositions of an error tolerant pattern matching function and the comparison function in all these examples except for the first level all other levels use only majority functions

polynomial size ω branching programs and their computational power abstract in the following new types of branching programs so called branching programs b 2 are introduced the complexity classes related to polynomial size branching programs will be completely classified in addition to identifying a new class p bp l poly between l poly and p poly new characterizations of such fundamental space complexity classes like nl poly co nl poly and p poly are obtained using these characterizations we relate the complexity of the mentioned classes to those of some extremely restricted problems resembling the graph accessibility problem

lax representations and lax operator algebras of isospectral and nonisospectral hierarchies of evolution equations a method of constructing lax representations for isospectral and nonisospectral hierarchies of evolution equations is proposed it is shown that under some simple but essential conditions the corresponding lax operators may constitute infinite dimensional lie algebras with respect to the binary operation given in this paper furthermore the detailed analyses for the kdv couple the akns couple and a new couple of isospectral and nonisospectral hierarchies of integrable equations are presented as examples

some connections between bounded query classes and non uniform complexity let a x be the characteristic function of a consider the function cka x1 xk a x1 a xk we show that if cka can be computed in polynomial time with fewer than k queries to some set x then a p poly a generalization of this result has applications to bounded query classes circuits and enumerability in particular we obtain the following 1 assuming 3p 3p there are functions computable using f n 1 queries to sat that are not computable using f n queries to sat for f n o log n 2 if cka restricted to length n inputs can be computed by an unbounded fanin oracle circuit of size s n and depth d n with k 1 queries to some set x then a can be computed with an unbounded fanin non oracle circuit of size no k s n and depth d n o 1 3 assuming that ph 4p 4p and e 1 sat is not 2ne enumerable

algebraic structure of discrete zero curvature equations and master symmetries of discrete evolution equations an algebraic structure related to discrete zero curvature equations is established it is used to give an approach for generating master symmetries of the first degree for systems of discrete evolution equations and an answer to why there exist such master symmetries the key of the theory is to generate nonisospectral flows t l l 0 from the discrete spectral problem associated with a given system of discrete evolution equations three examples are given

retargeted least squares regression algorithm this brief presents a framework of retargeted least squares regression relsr for multicategory classification the core idea is to directly learn the regression targets from data other than using the traditional zero one matrix as regression targets the learned target matrix can guarantee a large margin constraint for the requirement of correct classification for each data point compared with the traditional least squares regression lsr and a recently proposed discriminative lsr models relsr is much more accurate in measuring the classification error of the regression model furthermore relsr is a single and compact model hence there is no need to train two class binary machines that are independent of each other the convex optimization problem of relsr is solved elegantly and efficiently with an alternating procedure including regression and retargeting as substeps the experimental evaluation over a range of databases identifies the validity of our method

log depth circuits for division and related problems we present optimal depth boolean circuits depth o log n for integer division powering and multiple products we also show that these three problems are of equivalent uniform depth and space complexity in addition we describe an algorithm for testing divisibility that is optimal for both depth and space

integrable non isospectral flows associated with the kadomtsev petviashvili equations in 2 1 dimensions the symmetries of the kadomtsev petviashvili kp equations in 2 1 dimensions yield two hierarchies of integrable non linear evolution equations nee one is the familiar family of isospectral flows the kp hierarchy the other is non isospectral and its flows have coefficients which depend linearly on x and y the spectral methods used to solve kp can be used to solve all these nee an underlying infinite dimensional lie algebra is used to determine all the lax pairs for both families and it also determines their symplectic structures constants of the motion are constructed for the non isospectral cases

on circularity circularity is an assumption that was originally introduced for the definition of the probability distribution function of complex normal vectors however this concept can be extended in various ways for nonnormal vectors the first purpose of the paper is to introduce and compare some possible definitions of circularity from these definitions it is also possible to introduce the concept of circular signals and to study whether or not the spectral representation of stationary signals introduces circular components therefore the relationships between circularity and stationarity are analyzed in detail finally the theory of linear mean square estimation for complex signals exhibits some connections with circularity and it is shown that without this assumption the estimation theory must be reformulated

symplectic structures their backlund transformations and hereditary symmetries abstract it is shown that compatible symplectic structures lead in a natural way to hereditary symmetries we recall that a hereditary symmetry is an operator valued function which immediately yields a hierarchy of evolution equations each having infinitely many commuting symmetries all generated by this hereditary symmetry furthermore this hereditary symmetry usually describes completely the soliton structure and the conservation laws of these equations this result then provide us with a method for constructing hereditary symmetries and hence exactly solvable evolution equations in addition we show how symplectic structures transform under backlund transformations this leads to a method for generating a whole class of symplectic structures from a given one several examples and applications are given illustrating the above results also the connection of our results with those of gelfand and dikii and of magri is briefly pointed out

recursion operators and bi hamiltonian structures in multidimensions ii the algebraic properties of exactly solvable evolution equations in one spatial and one temporal dimensions have been well studied in particular the factorization of certain operators called recursion operators establishes the bi hamiltonian nature of all these equations recently we have presented the recursion operator and the bi hamiltonian formulation of the kadomtsev petviashvili equation a two spatial dimensional analogue of the korteweg devries equation here we present the general theory associated with recursion operators for bi hamiltonian equations in two spatial and one temporal dimensions as an application we show that general classes of equations which include the kadomtsev petviashvili and the davey stewartson equations possess infinitely many commuting symmetries and infinitely many constants of motion in involution under two distinct poisson brackets furthermore we show that the relevant recursion operators naturally follow from the underlying isospectral eigenvalue problems

on interpolation and automatization for frege systems the interpolation method has been one of the main tools for proving lower bounds for propositional proof systems loosely speaking if one can prove that a particular proof system has the feasible interpolation property then a generic reduction can usually be applied to prove lower bounds for the proof system sometimes assuming a usually modest complexity theoretic assumption in this paper we show that this method cannot be used to obtain lower bounds for frege systems or even for tc0 frege systems more specifically we show that unless factoring of blum integers is feasible neither frege nor tc0 frege has the feasible interpolation property in order to carry out our argument we show how to carry out proofs of many elementary axioms theorems of arithmetic in polynomial sized tc0 frege as a corollary we obtain that tc0 frege as well as any proof system that polynomially simulates it is not automatizable under the assumption that factoring of blum integers is hard we also show under the same hardness assumption that the k provability problem for frege systems is hard

on the use of isospectral eigenvalue problems for obtaining hereditary symmetries for hamiltonian systems we present an algorithmic method for obtaining an hereditary symmetry the generalized squared eigenfunction operator from a given isospectral eigenvalue problem this method is applied to the n n eigenvalue problem considered by ablowitz and haberman and to the eigenvalue problem considered by alonso the relevant hamiltonian formulations are also determined finally an alternative method is presented in the case two evolution equations are related by a miura type transformation and their hamiltonian formulations are known

logspace and logtime leaf languages the computation tree of a nondeterministic machinemwith inputxgives rise to aleaf stringformed by concatenating the outcomes of all the computations in the tree in lexicographical order we may characterize problems by considering for a particular leaf language y the set of allxfor which the leaf string ofmis contained iny in this way in the context of polynomial time computation leaf languages were shown to capture many complexity classes in this paper we study the expressibility of the leaf language mechanism in the contexts of logarithmic space and of logarithmic time computation we show that logspace leaf languages yield a much finer classification scheme for complexity classes than polynomial time leaf languages capturing also many classes withinp in contrast logtime leaf languages basically behave like logtime reducibilities both cases are more subtle to handle than the polynomial time case we also raise the issue of balanced versus nonbalanced computation trees underlying the leaf language we indicate that it is a nontrivial problem to obtain information about the leaf string of a nonbalanced computation tree and present conditions under which it does not matter whether the computation tree is balanced or not

lower bounds and the hardness of counting properties rice s theorem states that all nontrivial language properties of recursively enumerable sets are undecidable borchert and stephan math logic quart 46 4 2000 489 504 started the search for complexity theoretic analogs of rice s theorem and proved that every nontrivial counting property of boolean circuits is up hard hemaspaandra and rothe theoret comput sci 244 1 2 2000 205 217 improved the up hardness lower bound to upo 1 hardness the present paper raises the lower bound for nontrivial counting properties from upo 1 hardness to fewp hardness i e from constant ambiguity nondeterminism to polynomial ambiguity nondeterminism furthermore we prove that no relativizable technique can raise this lower bound to fewp 1 ttp hardness we also prove a rice style theorem for np namely that every nontrivial language property of np sets is np hard and for a broad class of leaf language classes we prove a sufficient condition for the natural analog of rice s theorem to hold