computing with words via turing machines a formal approach computing with words cw as a methodology means computing and reasoning by the use of words in place of numbers or symbols which may conform more to humans perception when describing real world problems in this paper as a continuation of a previous paper we aim to develop and deepen a formal aspect of cw according to the previous paper the basic point of departure is that cw treats certain formal modes of computation with strings of fuzzy subsets instead of symbols as their inputs specifically 1 we elaborate on cw via turing machine tm models showing the time complexity is at least exponential if the inputs are strings of words 2 a negative result of 6 not holding is verified which indicates that the extension principle for cw via tms needs to be re examined 3 we discuss cw via context free grammars and regular grammars and the extension principles for cw via these formal grammars are set up 4 some equivalences between fuzzy pushdown automata respectively fuzzy finite state automata fuzzy context free grammars respectively fuzzy regular grammars are demonstrated in the sense that the inputs are instead strings of words 5 some instances are described in detail summarily formal aspect of cw is more systematically established more deeply dealt with while some new problems also emerge

linear time variable systems balancing and model reduction a uniformly balanced realization for linear time variable systems is defined this representation is characterized by the fact that its controllability and observability gramians are equal and diagonal existence and uniqueness of the uniformly balanced realization is studied such a framework has many remarkable properties and leads to a novel method for approximating time variable systems where the subsystems of the balanced realization can be taken as a reduced model the reduced model is examined from the point of view of stability controllability and observability

numerical range and functional calculus in hilbert space we prove an inequality related to polynomial functions of a square matrix involving the numerical range of the matrix we also show extensions valid for bounded and also unbounded operators in hilbert spaces which allow the development of a functional calculus

nonlinear schrodinger equation with inhomogeneous dirichlet boundary data in this article we study the following nonlinear schrodinger equation iut u g u p 1u in a domain rn with initial condition u x 0 x and the dirichlet boundary condition u x t q x t on where q are given smooth functions the nonlinear term contributes a negative term to the energy i e g 0 we present the existence theorem for a global solution of finite energy when p 1 2 n

well posedness for nonlinear schrodinger equations with boundary forces in low dimensions by strichartz estimates abstract in this paper we study the well posedness of solutions for nonlinear schrodinger equations on one and two dimensional domains with boundary where the boundary is disturbed by an external inhomogeneous type of dirichlet or neumann force we first prove the local existence of solutions at the energy level for quadratic and superquadratic sources using the strichartz estimates on domains secondly we obtain conditional uniqueness and local stability then we prove the boundedness of solutions in the energy space to pass from the local theory to the global theory regarding subquadratic sources we appeal to classical methods and trudinger s inequality to prove the uniqueness which combined with the existence of weak energy solutions mass and energy inequalities eventually implies the continuity of solutions in time

maps preserving the joint numerical radius distance of operators abstract denote the joint numerical radius of an m tuple of bounded operators a a 1 a m by w a we give a complete description of maps f satisfying w a b w f a f b for any two m tuples of operators a a 1 a m and b b 1 b m we also characterize linear isometries for the joint numerical radius and maps preserving the joint numerical range of a

local distinguishability of quantum states in infinite dimensional systems we investigate local distinguishability of quantum states by use of convex analysis of joint numerical range of operators on a hilbert space we show that any two orthogonal pure states are distinguishable by local operations and classical communications even for infinite dimensional systems an estimate of the local discrimination probability is also given for some families of more than two pure states

computation in finitary stochastic and quantum processes abstract we introduce stochastic and quantum finite state transducers as computation theoretic models of classical stochastic and quantum finitary processes formal process languages representing the distribution over a process behaviors are recognized and generated by suitable specializations we characterize and compare deterministic and nondeterministic versions summarizing their relative computational power in a hierarchy of finitary process languages quantum finite state transducers and generators are a first step toward a computation theoretic analysis of individual repeatedly measured quantum dynamical systems they are explored via several physical systems including an iterated beam splitter an atom in a magnetic field and atoms in an ion trap a special case of which implements the deutsch quantum algorithm we show that these systems behaviors and so their information processing capacity depends sensitively on the measurement protocol

computational geometry of positive definiteness abstract in matrix computations such as in factoring matrices hermitian and preferably positive definite elements are occasionally required related problems can often be cast as those of existence of respective elements in a matrix subspace for two dimensional matrix subspaces first results in this regard are due to finsler to assess positive definiteness in larger dimensional cases the task becomes computational geometric for the joint numerical range in a natural way the hermitian element of the frobenius norm one with the maximal least eigenvalue is found to this end extreme eigenvalue computations are combined with ellipsoid and perceptron algorithms

characterizations of quantum automata we define q quantum finite automata qqfas and q quantum regular grammars qqrgs and verify that they are exactly equivalent to those measure once quantum finite automata mo qfas in the literature in particular we define q quantum pushdown automata qqpdas and qpdas that are at least as powerful as those defined by moore and crutchfield and especially we focus on demonstrating the equivalence between qqpdas and qpdas also we discuss some of the properties of languages accepted by qqpdas for example every cut point language accepted by qqpda is independent of the cut point

monotonicity method applied to the complex ginzburg landau and related equations abstract global existence of unique strong solutions is established for the complex ginzburg landau equation tu i u i u p 1u u 0 where 0 0 r p 1 and 1 2 p p 1 the key is a new inequality in monotonicity methods it is based on the sectorial estimates of in lp 1 and the nonlinear operator u u p 1u appearing in the equation the key inequality also yields the global existence of unique strong solutions of the nonlinear schrodinger type equation with monotone nonlinearity tu i u u p 1u 0 for all p 1

quantum automata and quantum grammars abstract to study quantum computation it might be helpful to generalize structures from language and automata theory to the quantum case to that end we propose quantum versions of finite state and push down automata and regular and context free grammars we find analogs of several classical theorems including pumping lemmas closure properties rational and algebraic generating functions and greibach normal form we also show that there are quantum context free languages that are not context free so qcfl cfl

quantum finite automata we first outline main steps and achievements along bertoni s research path in quantum finite automata theory from the very basic definitions of the models of quantum finite automata throughout the investigation of their computational and descriptional power next we choose to focus on bertoni s studies on quantum finite automata descriptional complexity in particular we expand on a statistical framework for the synthesis of succinct quantum finite automata discussing its adaptation to the case of multiperiodic events and languages we then improve such a framework to obtain even more succinct quantum finite automata for some multiperiodic languages finally we introduce some promise problems for multiperiodic inputs showing that even on this class of problems the descriptional power of quantum finite automata greatly outperforms that of equivalent classical finite automata we build small size monte carlo quantum finite automata for multiperiodic languages we analyze the modular architecture of such monte carlo quantum finite automata we solve promise problems on multiperiodic inputs by small quantum finite automata

flat portions on the boundary of the davis wielandt shell of 3 by 3 matrices abstract let t be an n n matrix the davis wielandt shell of t is defined as the set w t t t x tx x t tx x c r x x 1 in this paper we discuss the existence of flat portions on the boundary of the davis wielandt shell of 3 by 3 matrices

the generalized numerical range of a set of matrices abstract for a given set of n n matrices f we study the union of the c numerical ranges of the matrices in the set f denoted by w c f we obtain basic algebraic and topological properties of w c f and show that there are connections between the geometric properties of w c f and the algebraic properties of c and the matrices in f furthermore we consider the starshapedness and convexity of the set w c f in particular we show that if f is the convex hull of two matrices such that w c a and w c b are convex then the set w c f is star shaped we also investigate the extensions of the results to the joint c numerical range of an m tuple of matrices

regular languages accepted by quantum automata in this paper we analyze some features of the behaviour of quantum automata in particular we prove that the class of languages recognized by quantum automata with isolated cut point is the class of reversible regular languages as a more general result we give a bound on the inverse error that implies the regularity of the language accepted by a quantum automaton

global existence and open loop exponential stabilization of weak solutions for nonlinear schrodinger equations with localized external neumann manipulation abstract in this paper we consider the nonlinear schrodinger equations nls with focusing defocusing interior source and possibly nonlinear damping on a bounded regular domain in the euclidean space moreover it is assumed that the solutions are subject to external neumann boundary manipulation on one portion of the boundary our aim is to obtain global existence of the weak solutions under various assumptions on the sign of the source and powers of the nonlinearities in the case of a linear damping we also prove that solutions decay exponentially under the assumption that the neumann type control at the boundary decays in a similar manner

characterizations of one way general quantum finite automata generally unitary transformations limit the computational power of quantum finite automata qfa in this paper we study a generalized model named one way general quantum finite automata 1gqfa in which each symbol in the input alphabet induces a trace preserving quantum operation instead of a unitary transformation two different kinds of 1gqfa will be studied measure once one way general quantum finite automata mo 1gqfa where a measurement deciding to accept or reject is performed at the end of a computation and measure many one way general quantum finite automata mm 1gqfa where a similar measurement is performed after each trace preserving quantum operation on reading each input symbol we characterize the measure once model from three aspects the closure property the language recognition power and the equivalence problem we prove that mo 1gqfa recognize with bounded error precisely the set of all regular languages our results imply that some models of quantum finite automata proposed in the literature which were expected to be more powerful still cannot recognize non regular languages we prove that mm 1gqfa also recognize only regular languages with bounded error thus mm 1gqfa and mo 1gqfa have the same language recognition power in sharp contrast with traditional mo 1qfa and mm 1qfa the former being strictly less powerful than the latter finally we present a necessary and sufficient condition for two mm 1gqfa to be equivalent

classification of joint numerical ranges of three hermitian matrices of size three abstract the joint numerical range w f of three hermitian 3 by 3 matrices f f 1 f 2 f 3 is a convex and compact subset in r 3 generically we find that w f is a three dimensional oval assuming dim w f 3 every one or two dimensional face of w f is a segment or a filled ellipse we prove that only ten configurations of these segments and ellipses are possible we identify a triple f for each class and illustrate w f using random matrices and dual varieties

well posedness and energy decay estimates in the cauchy problem for the damped defocusing schrodinger equation abstract in this paper we study the existence at the h 1 level as well as the stability for the damped defocusing schrodinger equation in r d the considered damping coefficient is time dependent and may vanish at infinity to prove the existence we employ the method devised by ozsari kalantarov and lasiecka 27 which is based on monotone operators theory in particular when d 1 or d 2 we obtain the uniqueness decay estimates for the l 2 level and h 1 l p 2 level energies are established with the help of direct multipliers method coupled with refined energy estimates and a lower semi continuity argument

hierarchy and equivalence of multi letter quantum finite automata multi letter quantum finite automata qfas are a new one way qfa model proposed recently by belovs rosmanis and smotrovs a belovs a rosmanis j smotrovs multi letter reversible and quantum finite automata in proceedings of the 13th international conference on developments in language theory dlt 2007 harrachov czech republic in lecture notes in computer science vol 4588 springer berlin 2007 pp 60 71 and they showed that multi letter qfas can accept with no error some regular languages a b b that are unacceptable by the one way qfas in this paper we continue to study multi letter qfas we mainly focus on two issues 1 we show that k 1 letter qfas are computationally more powerful than k letter qfas that is k 1 letter qfas can accept some regular languages that are unacceptable by any k letter qfa a comparison with the one way qfas is made by some examples 2 we prove that a k 1 letter qfa a 1 and another k 2 letter qfa a 2 are equivalent if and only if they are n 1 n 2 4 k 1 equivalent and the time complexity of determining the equivalence of two multi letter qfas using this method is o n 1 2 k 2n 4 kn 8 where n 1 and n 2 are the numbers of states of a 1 and a 2 respectively and k max k 1 k 2 some other issues are addressed for further consideration

determining the equivalence for one way quantum finite automata two quantum finite automata are equivalent if for any input string x the two automata accept x with equal probability in this paper we first focus on determining the equivalence for one way quantum finite automata with control language cl 1qfas defined by bertoni et al and then as an application we address the equivalence problem for measure many one way quantum finite automata mm 1qfas introduced by kondacs and watrous more specifically we obtain that i two cl 1qfas a 1 and a 2 with control languages regular languages l 1 and l 2 respectively are equivalent if and only if they are c 1n 1 2 c 2n 2 2 1 equivalent where n 1 and n 2 are the numbers of states in a 1 and a 2 respectively and c 1 and c 2 are the numbers of states in the minimal dfas that recognize l 1 and l 2 respectively furthermore if l 1 and l 2 are given in the form of dfas with m 1 and m 2 states respectively then there exists a polynomial time algorithm running in time o m 1n 1 2 m 2n 2 2 4 that takes as input a 1 and a 2 and determines whether they are equivalent ii as an application of item i two mm 1qfas a 1 and a 2 with n 1 and n 2 states respectively are equivalent if and only if they are 3n 1 2 3n 2 2 1 equivalent furthermore there is a polynomial time algorithm running in time o 3n 1 2 3n 2 2 4 that takes as input a 1 and a 2 and determines whether a 1 and a 2 are equivalent

analytic models and ambiguity of context free languages we establish that several classical context free languages are inherently ambiguous by proving that their counting generating functions when considered as analytic functions exhibit some characteristic form of transcendental behaviour to that purpose we survey some general results on elementary analytic properties and enumerative uses of algebraic functions in relation to formal languages in particular the paper contains a general density theorem for unambiguous context free languages

crouzeix s conjecture and perturbed jordan blocks abstract crouzeix conjectured in m crouzeix bounds for analytical functions of matrices integr equ oper theory 48 2004 461 477 that for any square matrix a and any polynomial p p a 2 max p z z w a where w a is the field of values of a and denotes the spectral norm in this paper we show that the conjecture holds for matrices of the form 1 1 where and are complex numbers the technique for 1 and for 1 is to show that if g is a bijective conformal mapping from the field of values of such a matrix a to the unit disk mapping to 0 then g a is similar to a contraction via a similarity transformation with condition number at most 2

quantum computational networks the theory of quantum computational networks is the quantum generalization of the theory of logic circuits used in classical computing machines quantum gates are the generalization of classical logic gates a single type of gate the universal quantum gate together with quantum unit wires is adequate for constructing networks with any possible quantum computational property

the normalized numerical range and the davis wielandt shell abstract for a given n by n matrix a its normalized numerical range f n a is defined as the range of the function f n a x x a x a x x on the complement of ker a we provide an explicit description of this set for the case when a is normal or n 2 this extension of earlier results for particular cases of 2 by 2 matrices by gevorgyan and essentially hermitian matrices of arbitrary size by a stoica and one of the authors was achieved due to the fresh point of view at f n a as the image of the davis wielandt shell dw a under a certain non linear mapping h r 3 c

a proof of crouzeix s conjecture for a class of matrices abstract crouzeix s conjecture is that for any square matrix a and any polynomial p we have p a 2 max p z z w a where w a is the field of values of a and denotes the spectral norm in this paper we show that the conjecture holds for the matrices of the form 1 n 1 n where c and 1 n c n

a remark on the blowing up of solutions to the cauchy problem for nonlinear schrodinger equations on considere des solutions de iu t u u p 1 u u o u 0 ou x appartient a un domaine lisse cr n et on demontre que sous de bonnes conditions sur p n et u 0 h 2 h 0 1 u t l 2 eclate en un temps fini

m form numerical range and the computation of the structured singular value an equivalent expressions for the structures singular value is proposed leading to an alternative algorithm for its computation the approach is based to the geometric properties of the m form numerical range of certain tuples of matrices similar to previously considered schemes the algorithm proposed is proved to give the correct value for block structures of size up to three for larger sizes insight is gained into the question of the possible gap between the structured singular value and its known upper bound

the star shapedness of a generalized numerical range let hn be the set of all n n hermitian matrices and hnm be the set of all m tuples of n n hermitian matrices for a a1 am hnm and for any linear map l hnm rl we define the l numerical range of a by wl a l u a1u u amu u cn n u u in in this paper we prove that if l 3 n l and a1 am are simultaneously unitarily diagonalizable then wl a is star shaped with star center at l tra1nin tramnin

strengths and weaknesses of quantum computing recently a great deal of attention has been focused on quantum computation following a sequence of results bernstein and vazirani in proc 25th annual acm symposium theory comput 1993 pp 11 20 siam j comput 26 1997 pp 1277 1339 simon in proc 35th annual ieee symposium foundations comput sci 1994 pp 116 123 siam j comput 26 1997 pp 1340 1349 shor in proc 35th annual ieee symposium foundations comput sci 1994 pp 124 134 suggesting that quantum computers are more powerful than classical probabilistic computers following shor s result that factoring and the extraction of discrete logarithms are both solvable in quantum polynomial time it is natural to ask whether all of np can be efficiently solved in quantum polynomial time in this paper we address this question by proving that relative to an oracle chosen uniformly at random with probability 1 the class np cannot be solved on a quantum turing machine qtm in time o 2 n 2 we also show that relative to a permutation oracle chosen uniformly at random with probability 1 the class np cap conp cannot be solved on a qtm in time o 2 n 3 the former bound is tight since recent work of grover in it proc 28 th annual acm symposium theory comput 1996 shows how to accept the class np relative to any oracle on a quantum computer in time o 2 n 2

perturbation of the q numerical radius of a weighted shift operator abstract we formulate the taylor series expansion for the q numerical radius of a weighted shift operator with periodic weights near q 0 coefficients up to the fourth order in the expansion are found via the perturbation theory of hermitian matrices

weakly damped focusing nonlinear schrodinger equations with dirichlet control abstract in this article we consider the weakly damped focusing nonlinear schrodinger equations on bounded domains at the natural h 1 energy level with dirichlet control acting on a portion of the boundary we introduce the dynamic extension method for homogenizing the inhomogeneous boundary input then we construct approximate solutions using monotone operator theory a hidden trace regularity is proved to control the norm of the solutions in a global sense this allows the use of compactness techniques by which we prove the existence of weak solutions finally using multiplier techniques we prove the exponential decay of solutions under the assumption that the boundary control also decays in a similar fashion

qualitative properties of solutions for nonlinear schrodinger equations with nonlinear boundary conditions on the half line in this paper we study the interaction between a nonlinear focusing robin type boundary source a nonlinear defocusing interior source and a weak damping term for nonlinear schrodinger equations posed on the infinite half line we construct solutions with negative initial energy satisfying a certain set of conditions which blow up in finite time in the h1 sense we obtain a sufficient condition relating the powers of nonlinearities present in the model which allows construction of blow up solutions in addition to the blow up property we also discuss the stabilization property and the critical exponent for this model

differential topology of numerical range abstract the numerical range of an n n matrix also known as its field of values is reformulated as the image of a smooth quadratic mapping from the n 1 dimensional complex projective space to the complex plane this paper investigates the numerical range from the perspective of differential topology morse theory more specifically the boundary of the range is interpreted as a rank 1 critical value curve and its sharp points are interpreted as rank 0 critical values more importantly the map is shown to have additional critical value curves in the interior of the numerical range these additional curves are shown to have such singularity phenomena as cusps and swallow tails to be the caustic envelopes of families of lines and to exhibit the so called normal bifurcation when an eigenvalue becomes unitarily decoupled

on the crossing rule abstract the paper gives conditions on a family of matrices which guarantee that some matrix in the family will have a multiple eigenvalue in particular the main theorem states exactly which dimensions admit k dimensional subspaces of matrices for which all nonzero elements have distinct eigenvalues this question arises naturally in the theory of first order hyperbolic systems of partial differential equations the main theorem in this context tells exactly for which integers n an n x n system in k space variables may be strictly hyperbolic author

davis wielandt shell and q numerical range abstract we describe the boundary of the q numerical range of a square matrix using its davis wielandt shell the result is used to generate an algorithm for plotting the q numerical range of the square matrix computations of the q numerical ranges of a special class of matrices are explicitly given

the calculi of emergence computation dynamics and induction abstract defining structure and detecting the emergence of complexity in nature are inherently subjective though essential scientific activities despite the difficulties these problems can be analyzed in terms of how model building observers infer from measurements the computational capabilities embedded in nonlinear processes an observer s notion of what is ordered what is random and what is complex in its environment depends directly on its computational resources the amount of raw measurement data of memory and of time available for estimation and inference the discovery of structure in an environment depends more critically and subtlely though on how those resources are organized the descriptive power of the observer s chosen or implicit computational model class for example can be an overwhelming determinant in finding regularity in data this paper presents an overview of an inductive framework hierarchical machine reconstruction in which the emergence of complexity is associated with the innovation of new computational model classes complexity metrics for detecting structure and quantifying emergence along with an analysis of the constraints on the dynamics of innovation are outlined illustrative examples are drawn from the onset of unpredictability in nonlinear systems finitary nondeterministic processes and cellular automata pattern recognition they demonstrate how finite inference resources drive the innovation of new structures and so lead to the emergence of complexity

boundary controllability for the semilinear schrodinger equations on riemannian manifolds we study the boundary exact controllability for the semilinear schrodinger equation defined on an open bounded connected set of a complete n dimensional riemannian manifold m with metric g we prove the locally exact controllability around the equilibria under some checkable geometrical conditions our results show that exact controllability is geometrical characters of a riemannian metric given by the coefficients and equilibria of the semilinear schrodinger equation we then establish the globally exact controllability in such a way that the state of the semilinear schrodinger equation moves from an equilibrium in one location to an equilibrium in another location

joint numerical range and its generating hypersurface in this paper we study the joint numerical range of m tuples of hermitian matrices via their generating hypersurfaces an example is presented which shows the invalidity of an analogous kippenhahn theorem for the joint numerical range of three hermitian matrices

the constrained bilinear form and the c numerical range abstract let v be an n dimentional unitary space with inner product and s the set x v x x 1 for any a hom v v and q c with q 1 we define w a q ax y x y s x y q if q 1 then w a q is just the classical numerical range ax x x s the convexity of which is well known another generalization of the numerical range is the c numerical range which is defined to be the set w c a tr cu au u unitary where c hom v v in this note we prove that w a q is always convex and that wc a is convex for all a if rank c 1 or n 2

quantum mechanical hamiltonian models of turing machines quantum mechanical hamiltonian models which represent an aribtrary but finite number of steps of any turing machine computation are constructed here on a finite lattice of spin 1 2 systems different regions of the lattice correspond to different components of the turing machine plus recording system successive states of any machine computation are represented in the model by spin configuration states both time independent and time dependent hamiltonian models are constructed here the time independent models do not dissipate energy or degrade the system state as they evolve they operate close to the quantum limit in that the total system energy uncertainty computation speed is close to the limit given by the time energy uncertainty relation however the model evolution is time global and the hamiltonian is more complex the time dependent models do not degrade the system state also they are time local and the hamiltonian is less complex

strict convexity of the joint c numerical range abstract let h h 1 h 2 h m be an m tuple of n n hermitian matrices and c c 1 c 2 c n r n the joint c numerical range w c h of h is defined for x x 1 x 2 x m r m the form associated with h is defined as f h t x det ti n x 1 h 1 x 2 h 2 x m h m we prove that the convex hull of w c h is strictly convex if the roots of f h t x 0 have constant multiplicities for all x on the unit sphere in r m the result is illustrated with several examples

local exact controllability and stabilizability of the nonlinear schrodinger equation on a bounded interval this paper studies the exact controllability and the stabilization of the cubic schrodinger equation posed on a bounded interval both internal and boundary controls are considered and the results are given first in a periodic setting and next with dirichlet resp neumann boundary conditions it is shown that the systems with either an internal control or a boundary control are locally exactly controllable in the classical sobolev space h s for any s geq0 it is also shown that the systems with an internal stabilization are locally exponentially stabilizable in h s for any s geq0

joint numerical ranges quantum maps and joint numerical shadows abstract we associate with a k tuple of hermitian n n matrices a probability measure on r k supported on their joint numerical range the joint numerical shadow of these matrices when k 2 we recover the numerical range and the numerical shadow of the complex matrix corresponding to a pair of hermitian matrices we apply this material to the theory of quantum information thus we show that quantum maps on the set of quantum states defined by kraus operators satisfying the identity resolution assumption shrink joint numerical ranges

computing power of turing machines in the framework of unsharp quantum logic we present recursion theoretical characterizations of the computational power of turing machines in the framework of unsharp quantum logic for unsharp quantum logic let a lattice ordered quantum multiple valued qmv algebra be its truth value lattice when lattice ordered qmv algebras satisfy a locally finite condition that is every non zero element has a finite order and its meet operation is computable we prove that 1 0 1 0 l d t e or l w t e 2 0 where l d t e respectively l w t e denotes the class of languages accepted by these turing machines in the depth first method respectively the width first method for sharp quantum logic similar results are obtained for a general orthomodular truth value lattice

qrt fifo automata breadth first grammars and their relations abstract some classes of queue automata deterministic and nondeterministic i e machines equipped with one or more fifo tapes and corresponding languages are considered for quasi real time qrt deterministic machines recognition power closure properties and counting capabilities are studied for such machines by showing that the language l j j j 1 l j l j ba n 1 b a n j ca n j b can be recognized with j queues but not with fewer an infinite hierarchy theorem which contradicts the known results for nondeterministic machines is proved restricted palindromes can be recognized with two queues we introduce a generative system for some queue languages the breadth first context free grammars bcf which are the same as context free grammars but for the ordering of terminals which is breadth first i e the least recently produced nonterminal symbol must be rewritten first bcf languages are recognized essentially by stateless queue automata they are semilinear closed with respect to permutation and homomorphism but not with respect to intersection with regular sets a periodicity property pumping lemma is proved for bcf languages whence comparisons with other families are obtained finally a homomorphic characterization of any queue language is presented l h r b where h is a homomorphism nonerasing if l is qrt r a regular set and b a bcf language the result can be extended to multiqueue automata

dense quantum coding and quantum finite automata we consider the possibility of encoding m classical bits into many fewer n quantum bits qubits so that an arbitrary bit from the original m bits can be recovered with good probability we show that nontrivial quantum codes exist that have no classical counterparts on the other hand we show that quantum encoding cannot save more than a logarithmic additive factor over the best classical encoding the proof is based on an entropy coalescence principle that is obtained by viewing holevo s theorem from a new perspective in the existing implementations of quantum computing qubits are a very expensive resource moreover it is difficult to reinitialize existing bits during the computation in particular reinitialization is impossible in nmr quantum computing which is perhaps the most advanced implementation of quantum computing at the moment this motivates the study of quantum computation with restricted memory and no reinitialization that is of quantum finite automata it was known that there are languages that are recognized by quantum finite automata with sizes exponentially smaller than those of corresponding classical automata here we apply our technique to show the surprising result that there are languages for which quantum finite automata take exponentially more states than those of corresponding classical automata

an inhomogeneous boundary value problem for nonlinear schrodinger equations abstract we consider a nonlinear schrodinger equation in a domain r n with the inhomegeneous dirichlet boundary condition u q where q is a given smooth function the nonlinear term contributes a positive term to the energy we prove the existence of global solutions of finite energy

reduction of joint c numerical ranges abstract let h 1 h 2 h m be an m tuple of n by n hermitian matrices and c r n we investigate the reduction problem of the joint c numerical range w c h 1 h 2 h m in particular for m 3 and n 3 we prove that w c h 1 h 2 h 3 is attainable by the joint numerical range w k 1 k 2 k 3 for some triple hermitian matrices k 1 k 2 k 3

a survey of the s lemma in this survey we review the many faces of the s lemma a result about the correctness of the s procedure the basic idea of this widely used method came from control theory but it has important consequences in quadratic and semidefinite optimization convex geometry and linear algebra as well these were all active research areas but as there was little interaction between researchers in these different areas their results remained mainly isolated here we give a unified analysis of the theory by providing three different proofs for the s lemma and revealing hidden connections with various areas of mathematics we prove some new duality results and present applications from control theory error estimation and computational geometry

the complex structured singular value a tutorial introduction to the complex structured singular value is presented with an emphasis on the mathematical aspects of the based methods discussed here have been useful for analysing the performance and robustness properties of linear feedback systems several tests for robust stability and performance with computable bounds for transfer functions and their state space realizations are compared and a simple synthesis problem is studied uncertain systems are represented using linear fractional transformations lfts which naturally unify the frequency domain and state space methods

between the von neumann inequality and the crouzeix conjecture abstract a new concept of a deformed numerical range w t is introduced here t is a bounded linear operator or a matrix and 1 is a parameter each w t is a closed convex set that contains the spectrum of t furthermore w t is decreasing with respect to and w 2 t coincides with the numerical range it is also shown that w t is contained in the closed unit disc if and only if t has a unitary dilation in the sense of nagy foias the spectral constant of w t is investigated it is shown that it is monotone and continuous with respect to the parameter

pauli group classification and joint higher rank numerical range abstract in mousavi and salemi 2015 17 authors showed that joint higher rank numerical range of triple of pauli group elements is often convex and characterized it exactly in the case that it is convex in the present paper those results are derived in a different manner also n qubit pauli group using simultaneously unitary similarity are classified which may generally be used to characterize joint higher rank numerical range of any hermitian m tuple in the n qubit pauli group besides some more results about joint numerical range are here derived

exact boundary controllability of the nonlinear schrodinger equation abstract this paper studies the exact boundary controllability of the semi linear schrodinger equation posed on a bounded domain r n with either the dirichlet boundary conditions or the neumann boundary conditions it is shown that if s n 2 or 0 s n 2 with 1 n 2 2 s or s 0 1 with n 2 then the systems are locally exactly controllable in the classical sobolev space h s around any smooth solution of the cubic schrodinger equation

analogies and differences between quantum and stochastic automata abstract we analyze some features of the behaviour of quantum automata providing analogies and differences with the corresponding stochastic models in particular we prove there is a quantum automaton where the change of state depends on unitary transformations defined by matrices with nonnull amplitudes that accepts a non regular language with cut point zero and inverse error polynomially bounded stochastic automata with matrices having nonnull elements and with polynomial bounds on the inverse error recognize only regular languages the class of stochastic languages contains the class of quantum languages quantum languages are empty or contain an infinite number of words the class of quantum languages is not closed under complementation

determination of equivalence between quantum sequential machines quantum sequential machines qsms that may be viewed as a quantum variant of stochastic sequential machines ssms are one of the important quantum computing models a very crucial result on ssms is that two ssms with n and n states respectively and the same input and output alphabets are equivalent if and only if they are n n 1 equivalent therefore gudder asked whether or not it holds for qsms as an open problem in this direction since the further study is closely related to this result qiu demonstrated that in qsms this result does not hold and therefore answered in part the problem however as qiu indicated the sufficient and necessary conditions for the equivalence between two qsms are still not discovered in this paper we show that if the condition of n n 1 equivalence is appropriately relaxed then we can give a sufficient and necessary condition for justifying the equivalence between two qsms more precisely we show that any two qsms m and m are equivalent if and only if they are n n 2 equivalent where n and n are respectively the numbers of states in m and m we therefore solve the open problem suggested by gudder and provide a basic result for further developing qsms as well we discuss strongly factorizable qsms and present the conditions for the equivalence between two strongly factorizable qsms

a note on quantum sequential machines quantum sequential machines qsms are a quantum version of stochastic sequential machines ssms recently we showed that two qsms m 1 and m 2 with n 1 and n 2 states respectively are equivalent iff they are n 1 n 2 2 equivalent l z li d w qiu determination of equivalence between quantum sequential machines theoretical computer science 358 2006 65 74 however using this result to check the equivalence is likely to need exponential expected time in this note we consider the time complexity of deciding the equivalence between qsms and related problems the main results are as follows 1 we present a polynomial time algorithm for deciding the equivalence between qsms and if two qsms are not equivalent this algorithm will produce an input output pair with length not more than n 1 n 2 2 2 we improve the bound for the equivalence between qsms from n 1 n 2 2 to n 1 2 n 2 2 1 by employing moore and crutchfield s method c moore j p crutchfield quantum automata and quantum grammars theoretical computer science 237 2000 275 306 also quant ph 9707031 1997 in addition by viewing mo 1qfas as a special case of qsms we briefly discuss the equivalence between mo 1qfas where the method used and the result obtained are slightly different from those given by koshiba t koshiba polynomial time algorithms for the equivalence for one way quantum finite automata in proceedings of the 12th international symposium on algorithms and computation isaac 2001 christchurch new zealand in lecture notes in computer science vol 2223 springer berlin 2001 pp 268 278

rapid solution of problems by quantum computation a class of problems is described which can be solved more efficiently by quantum computation than by any classical or stochastic method the quantum computation solves the problem with certainty in exponentially less time than any classical deterministic computation

automata theory based on quantum logic reversibilities and pushdown automata automata theory based on quantum logic called l valued finite automata l vfas may be viewed as a logical approach to quantum computing this work is mainly divided into two parts one part deals with reversibility of l vfas and the other establishes a basic framework of l valued pushdown automata l vpdas first we provide some preliminaries concerning quantum logic and l vfas and we prove a useful property of l valued successor and source operators then we clarify the relationships between various reversibilities closely related to quantum finite automata in the literature in particular we define a reversibility of l vfas which is termed as retrievability and we clarify the relationships between a number of different fashions regarding retrievability of l vfas we prove that some of them are equivalent but for the others to be equivalent the truth value set is required to satisfy a certain condition this is an essential difference from the classical situation afterwards we introduce l vpdas and show that the class of the languages accepted by l vpdas by empty stack coincides with that accepted by l vpdas by final state finally we provide some examples of l vfas and conclude with some remarks

two way finite automata with quantum and classical states we introduce 2 way finite automata with quantum and classical states 2qcfa s this is a variant on the 2 way quantum finite automata 2qfa model which may be simpler to implement than unrestricted 2qfa s the internal state of a 2qcfa may include a quantum part that may be in a mixed quantum state but the tape head position is required to be classical we show two languages for which 2qcfa s are better than classical 2 way automata first 2qcfa s can recognize palindromes a language that cannot be recognized by 2 way deterministic or probabilistic finite automata second in polynomial time 2qcfa s can recognize anbn n n a language that can be recognized classically by a 2 way probabilistic automaton but only in exponential time

joint ranges of hermitian matrices and simultaneous diagonalization abstract let a a 1 a k be a k tuple of heritian operators on an n diensional inner product space x with unit sphere s u x u u 1 with 1 k as the corresponding quadratic forms i e j x x a j x for all x x j 1 k we study ranges of over various subsets t of x the interplay between geometric properties of t and algebraic properties of a is our main concern in particular we characterize those a for which s is identical with the convex hull of the joint spectrum of a defined by a c k a j u j u for some u s the closely related problem of diagonalizing several hermitian operators simultaneously is also studied

convexity of the joint numerical range topological and differential geometric viewpoints abstract we investigate the convexity of the joint numerical range of m tuples of n n hermitian matrices the methods come from differential geometry and the differential and algebraic topology our main result is a sufficient condition for convexity of the joint numerical range for arbitrary m and n modulo a mild technical assumption this condition is formulated in terms of the largest eigenvalue of an associated family of hermitian matrices parameterized by the m 1 dimensional unit sphere the condition is that the eigenvalue has constant multiplicity we show that the constant multiplicity condition is in fact a criterion for the stable convexity of numerical ranges as a byproduct of our main result we obtain a new proof of the celebrated toeplitz hausdorff theorem and of its extension the numerical range of any triple of hermitian n n matrices is convex if n 2

uniform decay rates for the energy of weakly damped defocusing semilinear schrodinger equations with inhomogeneous dirichlet boundary control abstract in this paper we study the open loop stabilization as well as the existence and regularity of solutions of the weakly damped defocusing semilinear schrodinger equation with an inhomogeneous dirichlet boundary control first of all we prove the global existence of weak solutions at the h 1 energy level together with the stabilization in the same sense it is then deduced that the decay rate of the boundary data controls the decay rate of the solutions up to an exponential rate secondly we prove some regularity and stabilization results for the strong solutions in h 2 sense the proof uses the direct multiplier method combined with monotonicity and compactness techniques the result for weak solutions is strong in the sense that it is independent of the dimension of the domain the power of the nonlinearity and the smallness of the initial data however the regularity and stabilization of strong solutions are obtained only in low dimensions with small initial and boundary data

quantum theory the church turing principle and the universal quantum computer it is argued that underlying the church turing hypothesis there is an implicit physical assertion here this assertion is presented explicitly as a physical principle every finitely realizable physical system can be perfectly simulated by a universal model computing machine operating by finite means classical physics and the universal turing machine because the former is continuous and the latter discrete do not obey the principle at least in the strong form above a class of model computing machines that is the quantum generalization of the class of turing machines is described and it is shown that quantum theory and the universal quantum computer are compatible with the principle computing machines resembling the universal quantum computer could in principle be built and would have many remarkable properties not reproducible by any turing machine these do not include the computation of non recursive functions but they do include quantum parallelism a method by which certain probabilistic tasks can be performed faster by a universal quantum computer than by any classical restriction of it the intuitive explanation of these properties places an intolerable strain on all interpretations of quantum theory other than everett s some of the numerous connections between the quantum theory of computation and the rest of physics are explored quantum complexity theory allows a physically more reasonable definition of the complexity or knowledge in a physical system than does classical complexity theory

differential properties of the numerical range map of pairs of matrices abstract let a a 1 a 2 be a pair of hermitian operators in c n and a a1 ia2 we investigate certain differential properties of the numerical range map n a x a 1 x x a 2 x x with the aim of better understanding the nature of the numerical range w a of a for example the joint eigenvalues of a correspond to the stationary points of n a i e points where the derivative n a vanishes moreover points x where rank n a x 2 get mapped by n a into the interior w a of w a for n 2 it turns out that if a1 and a2 have no common invariant subspace then the image under n a of the set 1 a consisting of those points x with rank n a x 1 is precisely the boundary w a of w a and the image under n a of the rank 2 points for n a is precisely w a there are no rank 0 points for n a as a consequence for n 2 we have that a1a2 a2a1 iff 1 a n a 1 w a

singular points of the algebraic curves associated to unitary bordering matrices abstract let a be an n n complex matrix a ternary form associated to a is defined as the homogeneous polynomial f a t x y det t i n x a y a we prove for a unitary boarding matrix a the ternary form f a t x y is strongly hyperbolic and the algebraic curve f a t x y 0 has no real singular points as a consequence we obtain that the higher rank numerical range of a unitary boarding matrix is strictly convex

state succinctness of two way finite automata with quantum and classical states two way finite automata with quantum and classical states 2qcfa were introduced by ambainis and watrous in 2002 in this paper we study state succinctness of 2qcfa for any email protected z and any e 1 2 we show that 1 there is a promise problem a e q m which can be solved by a 2qcfa with one sided error e in a polynomial expected running time with a constant number that depends neither on m nor on e of quantum states and o email protected classical states whereas the sizes of the corresponding deterministic finite automata dfa two way nondeterministic finite automata 2nfa and polynomial expected running time two way probabilistic finite automata 2pfa are at least 2m 2 logm and logm b3 respectively 2 there exists a language l t w i n m wcw email protected a b w m over the alphabet s a b c which can be recognized by a 2qcfa with one sided error e in an exponential expected running time with a constant number of quantum states and o email protected classical states whereas the sizes of the corresponding dfa 2nfa and polynomial expected running time 2pfa are at least 2 m m and m b3 respectively where b is a constant

on global solutions to the initial boundary value problem for the damped nonlinear schrodinger equations where a 0 and q is a bounded domain in rn with smooth boundary x when a 0 undamped case the global existence of smooth solutions to the initial boundary value problem l 3 has been proved only when n 2 and f lu12 w lulp with p 2 brezis and gallouet cl with 2 i p i3 m tsutsumi 9 higher dimensional cases are very difficult pribl s and remain unproved as far as the author knows if we add a linear damping term iau a 0 then the mass ilull iztr1 and the energy ivu x t l dx j f lu x t l dx decay exponentially as t cc as is seen below which may assert the global existence of small smooth solutions even if the nonlinear term f lul u does not have a proper sign it is our purpose to establish the global existence and decay estimates of smooth solutions of the problem l 3 with small data there are a number of works on the nonlinear evolution equations see 3 5 71 and their references in the following spirit energy and decay estimates give global existence of solutions if the damping term is not added there is no hope that the decay estimates of solutions in some f q p 2 2 can be obtained since

sets of matrices with given joint numerical range abstract the following question is addressed to what extent the n tuple of m m hermitian matrices is determined by its joint numerical range the cases m 2 n arbitrary and m n 3 are considered in detail