faster exact solutions for some np hard problems this paper considers a number of np complete problems and provides faster algorithms for solving them the solutions are based on a recursive partitioning of the problem domain and careful elimination of some of the branches along the search without actually checking them the time complexity of the proposed algorithms is of the form o 2en for constant 0 e 1 where n is the output size of the problem in particular such algorithms are presented for the exact sat and exact hitting set problems with e 0 3212 and for the exact 3sat problem with e 0 2072 both algorithms improve on previous ones proposed in the literature

redundancy in logic i cnf propositional formulae a knowledge base is redundant if it contains parts that can be inferred from the rest of it we study some problems related to the redundancy of a cnf formula in particular any cnf formula can be made irredundant by deleting some of its clauses what results is an irredundant equivalent subset we study the complexity of problems related to irredundant equivalent subsets verification checking existence of an irredundant equivalent subset with a given size checking necessary and possible presence of clauses in irredundant equivalent subsets and uniqueness we also consider the problem of redundancy with different definitions of equivalence

heat content asymptotics of a riemannian manifold with boundary abstract let m be a compact riemannian manifold with a smooth c boundary m suppose m has initial temperature 1 at time t 0 and suppose that m is kept at temperature 0 for all t 0 we study the asymptotic behavior of the amount of heat in m as t 0 we generalize these results to arbitrary operators of laplace type with dirichlet boundary conditions and to arbitrary smooth initial conditions

hamiltonian control of quantum dynamical semigroups stabilization and convergence speed we consider finite dimensional markovian open quantum systems and characterize the extent to which time independent hamiltonian control may allow to stabilize a target quantum state or subspace and optimize the resulting convergence speed for a generic lindblad master equation we introduce a dissipation induced decomposition of the associated hilbert space and show how it serves both as a tool to analyze global stability properties for given control resources and as the starting point to synthesize controls that ensure rapid convergence the resulting design principles are illustrated in realistic markovian control settings motivated by quantum information processing including quantum optical systems and nitrogen vacancy centers in diamond

asymptotics of the heat exchange abstract let k be a compact subset in euclidean space r m and let e k t denote the total amount of heat in r m k at time t if k is kept at fixed temperature 1 for all t 0 and if r m k has initial temperature 0 for two disjoint compact subsets k 1 and k 2 we define the heat exchange h k 1 k 2 t e k 1 t e k 2 t e k 1 k 2 t we obtain the leading asymptotic behaviour of h k 1 k 2 t as t 0 under mild regularity conditions on k 1 and k 2

nonclassical eigenvalue asymptotics abstract the leading asymptotics for the growth of the number of eigenvalues of the two dimensional dirichlet laplacian in the regions x y x y 1 and for x y all of which are non weyl because of infinite phase space volumes are computed along the way a general inequality on quantum partition functions computed in a kind of born oppenheimer approximation is proved

a kind of logical compilation for knowledge bases the forward chaining algorithm is perhaps the best known algorithm in expert systems however it is not complete because it cannot compute the two valued consequence literals of a propositional knowledge base i e set of rules with negations if the user wants to compute them he must use a particular algorithm which often takes much time we propose a compilation system for knowledge bases which we call logical compilation which allows us to compute the two valued consequence literals of a knowledge base i e set of rules using a forward chaining on the compiled base with any extensional knowledge base i e set of basic facts added we also use this compilation in a wide propositional calculus and solve the or problem in rule conclusion we present several methods with their benefits to make this compilation and we give properties on knowledge bases to avoid this compilation finally we give a theorem which defines incremental compilation

complexity classes without machines on complete languages for up this paper develops techniques for studying complexity classes that are not covered by known recursive enumerations of machines often counting classes probabilistic classes and intersection classes lack such enumerations concentrating on the counting class up we show that there are relativizations for which upa has no complete languages and other relativizations for which pb upb npb and upb has complete languages among other results we show that p up if and only if there exists a set s in p of boolean formulas with at most one satisfying assignment such that s sat is not in p p up coup if and only if there exists a set s in p of uniquely satisfiable boolean formulas such that no polynomial time machine can compute the solutions for the formulas in s if up has complete languages then there exists a set r in p of boolean formulas with at most one satisfying assignment so that sat r is complete for up finally we indicate the wide applicability of our techniques to counting and probabilistic classes by using them to examine the probabilistic class bpp there is a relativized world where bppa has no complete languages if bpp has complete languages then it has a complete language of the form b majority where b p and majority f f is true for at least half of all assignments is the canonical pp complete set

on the spectrum of the dirichlet laplacian for horn shaped regions in rn with infinite volume abstract an inequality for trace et d is proven where d is the dirichlet laplacian for horn shaped regions d in rn the results of rozenbljum and simon for the leading asymptotics for the growth of the number of eigenvalues of the two dimensional dirichlet laplacian in the regions x y x y 1 0 are easily recovered an example of a horn shaped region in r2 where that asymptotics is exponential is given

ground state stabilization of quantum finite level systems by dissipation control by dissipation or environment engineering constitutes an important methodology within quantum coherent control which was proposed to improve the robustness and scalability of quantum control systems the system environment coupling often considered to be detrimental to quantum coherence also provides the means to steer the system to desired states this paper aims to develop the theory for engineering of the dissipation based on a ground state lyapunov stability analysis of open quantum systems via a heisenberg picture approach in particular lyapunov stability conditions expressed as operator inequalities allow a purely algebraic treatment of the environment engineering problem which facilitates the integration of quantum components into a large scale quantum system and draws an explicit connection to the classical theory of vector lyapunov functions and decomposition aggregation methods for control of complex systems this leads to tractable algebraic conditions concerning the ground state stability and scalability of quantum systems the implications of the results in relation to dissipative quantum computing and state engineering are also discussed in this paper

trace properties of the dirichlet laplacian on obtient des bornes superieures et inferieures pour tr exp ht en termes de la fonction r x

computing functions with parallel queries to np the class 2p of languages polynomial time truth table reducible to sets in np has a wide range of different characterizations we consider several functional versions of 2p based on these characterizations we show that in this way the three function classes fllognp fplognp and fp np are obtained in contrast to the language case the function classes seem to all be different we give evidence in support of this fact by showing that fllognp coincides with any of the other classes then l p and that the equality of the classes fplognp and fp np would imply that the number of nondeterministic bits needed for the computation of any problem in np can be reduced by a polylogarithmic factor and that the problem can be computed deterministically with a subexponential time bound of order 2no 1 log log n

heat content asymptotics with singular initial temperature distributions abstract we study the heat content asymptotics with either dirichlet or robin boundary conditions where the initial temperature exhibits radial blowup near the boundary we show that there is a complete small time asymptotic expansion and give explicit geometrical formulas for the first few terms in the expansion

stabilizing feedback controls for quantum systems no quantum measurement can give full information on the state of a quantum system hence any quantum feedback control problem is necessarily one with partial observations and can generally be converted into a completely observed control problem for an appropriate quantum filter as in classical stochastic control theory here we study the properties of controlled quantum filtering equations as classical stochastic differential equations we then develop methods using a combination of geometric control and classical probabilistic techniques for global feedback stabilization of a class of quantum filters around a particular eigenstate of the measurement operator

optimization problems and replica symmetry breaking in finite connectivity spin glasses a formalism capable of handling the first step of hierarchical replica symmetry breaking rsb in finite connectivity models is introduced the emerging order parameter is claimed to be a probability distribution over the space of field distributions or equivalently magnetization distributions inside the cluster of states the approach is shown to coincide with previous works in the replica symmetric case and in the two limiting cases m 0 and 1 where m is parisi s break point as an application to the study of optimization problems the gs properties of the random 3 satisfiability problem are investigated and we present a first rsb solution improving replica symmetric results

ultracontractivity and the heat kernel for schrodinger operators and dirichlet laplacians abstract abstract connections between integral kernels of positivity preserving semigroups and suitable lp contractivity properties are established then these questions are studied for the semigroups generated by v and h the dirichlet laplacian for an open connected region as an application under a suitable hypothesis sobolev estimates are proved valid up to of the form x c 0 x h k 2 where 0 is the unique positive l2 eigenfunction of h

heat flow heat content and the isoparametric property let m be a riemannian manifold and omega a compact domain of m with smooth boundary we study the solution of the heat equation on omega having constant unit initial conditions and dirichlet boundary conditions the purpose of this paper is to study the geometry of domains for which at any fixed value of time the normal derivative of the solution heat flow is a constant function on the boundary we express this fact by saying that such domains have the constant flow property in constant curvature spaces known examples of such domains are given by geodesic balls and more generally by domains whose boundary is connected and isoparametric the question is are they all like that this problem is the analogous for the heat equation of the classical serrin s problem for harmonic domains in this paper we give an affirmative answer to the above question in fact we prove more generally that if a domain in an analytic riemannian manifold has the constant flow property then every component of its boundary is an isoparametric hypersurface for space forms we also relate the order of vanishing of the heat content with fixed boundary data with the constancy of the r mean curvatures of the boundary and with the isoparametric property finally we discuss the constant flow property in relation to other well known overdetermined problems involving the laplace operator like the serrin problem or the schiffer problem

moderate deviations for the volume of the wiener sausage for a 0 let wa t be the a neighbourhood of standard brownian motion in rd starting at 0 and observed until time t it is well known that elwa t l ikat t oc for d 3 with kea the newtonian capacity of the ball with radius a we prove that lm t d 2 d log p wa t l oc this is markedly different from the optimal strategy for large deviations iwa t i 5 is nonanalytic at some critical point in 0 kia above which it follows a pure power law this crossover is associated with a collapse transition in the optimal strategy we also derive the analogous moderate deviation result for d 2 in this case eiwa t 2wrt logt t oc and we prove that

bose einstein condensation in satisfiability problems abstract this paper is concerned with the complex behavior arising in satisfiability problems we present a new statistical physics based characterization of the satisfiability problem specifically we design an algorithm that is able to produce graphs starting from a k sat instance in order to analyze them and show whether a bose einstein condensation occurs we observe that analogously to complex networks the networks of k sat instances follow bose statistics and can undergo bose einstein condensation in particular k sat instances move from a fit get rich network to a winner takes all network as the ratio of clauses to variables decreases and the phase transition of k sat approximates the critical temperature for the bose einstein condensation finally we employ the fitness based classification to enhance sat solvers e g chainsat and obtain the consistently highest performing sat solver for cnf formulas and therefore a new class of efficient hardware and software verification tools

analysis and synthesis of attractive quantum markovian dynamics we propose a general framework for investigating a large class of stabilization problems in markovian quantum systems building on the notions of invariant and attractive quantum subsystem we characterize attractive subspaces by exploring the structure of the invariant sets for the dynamics our general analysis results are exploited to assess the ability of open loop hamiltonian and output feedback control strategies to synthesize markovian generators which stabilize a target subsystem subspace or pure state in particular we provide an algebraic characterization of the manifold of stabilizable pure states in arbitrary finite dimensional markovian systems that leads to a constructive strategy for designing the relevant controllers implications for stabilization of entangled pure states are addressed by example

dirichlet spectrum and heat content abstract let m be a complete riemannian manifold and d m a smoothly bounded domain with compact closure we use brownian motion to study the relationship between the dirichlet spectrum of d and the heat content asymptotics of d central to our investigation is a sequence of invariants associated to d defined using exit time moments we prove that our invariants determine that part of the spectrum corresponding to eigenspaces which are not orthogonal to constant functions that our invariants determine the heat content asymptotics associated to the manifold and that when the manifold is a generic domain in euclidean space the invariants determine the dirichlet spectrum

on the index of geometrical operators for riemannian manifolds with boundary abstract we use heat equation methods and invariance theory to compute the index of the classical elliptic complexes with pseudo differential boundary conditions of atiyah patodi singer type where the structure are not product near the boundary

on the unique satisfiability problem unique sat is the problem of deciding whether a given boolean formula has exactly one satisfying truth assignment this problem is a typical moreover complete representative of a natural class of problems about unique solutions all these problems belong to the class dife l1 l2 l1 lz np studied by papadimitriou and yannakakis we consider the relationship between these two classes particularly whether unique sat is dife complete it is if np co np we construct an oracle relative to which unique sat is not complete for dif and another oracle relative to which unique sat is complete for dif e whereas np v co np

on the comptational complexity of determining polyatomic structures by x rays the problem of recovering the structure of crystalline materials from their discrete x rays is of fundamental interest in many practical applications an important special case concerns determining the position of atoms of several different types in the integer lattice given the number of each type lying on each line parallel to some lattice directions we show that the corresponding consistency problem is np complete for any two or more different fixed directions when six or more types of atoms are involved

area versus capacity and independence in the crushed ice model we study the heat flow from an infinite union b of disjoint closed balls in euclidean space rm kept at constant temperature 1 while the surrounding medium is at temperature 0 at time 0 we obtain conditions on the geometry of b such that the heat flow for small time is governed by the area of the boundary of b

unique optimal solutions complexity results for identifying and locating dominating codes abstract we investigate the complexity of four decision problems dealing with the uniqueness of a solution in a graph uniqueness of an r locating dominating code with bounded size u ldcr uniqueness of an optimal r locating dominating code u oldcr uniqueness of an r identifying code with bounded size u idcr uniqueness of an optimal r identifying code u oidcr for any fixed integer r 1 in particular we describe a polynomial reduction from unique satisfiability of a boolean formula u sat to u oldcr and from u sat to u oidcr for u ldcr and u idcr we can do even better and prove that their complexity is the same as that of u sat up to polynomials consequently all these problems are np hard and u ldcr and u idcr belong to the class dp

heat content and hardy inequality for complete riemannian manifolds abstract upper bounds are obtained for the heat content of an open set d in a complete riemannian manifold provided the dirichlet laplace beltrami operator satisfies a strong hardy inequality and the distance function on d satisfies an integrability condition

a hierarchy of propositional horn formulas a hierarchy of propositional horn formulas is introduced the levels hk and hk of the hierarchy are defined by way of the number of alternations between players in a certain game related to the satisfiability of horn formulas the satisfiability problems for formulas from a given level of the hierarchy are shown to be complete in nspace log n a certain relationship between the hierarchy and the bounded depth circuits is exhibited using it we show that for some hk and hk formulas the equivalent formulas in the lower levels of the hierarchy must be exponentially longer

modeling and control of quantum systems an introduction the scope of this work is to provide a self contained introduction to a selection of basic theoretical aspects in the modeling and control of quantum mechanical systems as well as a brief survey on the main approaches to control synthesis while part of the existing theory especially in the open loop setting stems directly from classical control theory most notably geometric control and optimal control a number of tools specifically tailored for quantum systems have been developed since the 1980s in order to take into account their distinctive features the probabilistic nature of atomic scale physical systems the effect of dissipation and the irreversible character of the measurements have all proved to be critical in feedback design problems the relevant dynamical models for both closed and open quantum systems are presented along with the main results on their controllability and stability a brief review of several currently available control design methods is meant to provide the interested reader with a roadmap for further studies

propositional circumscription and extended closed world reasoning are p p 2 complete abstract circumscription and the closed world assumption with its variants are well known nonmonotonic techniques for reasoning with incomplete knowledge their complexity in the propositional case has been studied in detail for fragments of propositional logic one open problem is whether the deduction problem for arbitrary propositional theories under the extended closed world assumption or under circumscription is p 2 complete i e complete for a class of the second level of the polynomial hierarchy we answer this question by proving these problems p 2 complete and we show how this result applies to other variants of closed world reasoning

on the hardness of approximate reasoning abstract many ai problems when formalized reduce to evaluating the probability that a propositional expression is true in this paper we show that this problem is computationally intractable even in surprisingly restricted cases and even if we settle for an approximation to this probability we consider various methods used in approximate reasoning such as computing degree of belief and bayesian belief networks as well as reasoning techniques such as constraint satisfaction and knowledge compilation that use approximation to avoid computational difficulties and reduce them to model counting problems over a propositional domain we prove that counting satisfying assignments of propositional languages is intractable even for horn and monotone formulae and even when the size of clauses and number of occurrences of the variables are extremely limited this should be contrasted with the case of deductive reasoning where horn theories and theories with binary clauses are distinguished by the existence of linear time satisfiability algorithms what is even more surprising is that as we show even approximating the number of satisfying assignments i e approximating approximate reasoning is intractable for most of these restricted theories we also identify some restricted classes of propositional formulae for which efficient algorithms for counting satisfying assignments can be given

complexity of generalized satisfiability counting problems abstract the class of generalized satisfiability problems introduced in 1978 by schaefer presents a uniform way of studying the complexity of satisfiability problems with special conditions the complexity of each decision and counting problem in this class depends on the involved logical relations in 1979 valiant defined the class p the class of functions definable as the number of accepting computations of a polynomial time nondeterministic turing machine clearly all satisfiability counting problems belong to this class p we prove a dichotomy theorem for generalized satisfiability counting problems that is if all logical relations involved in a generalized satisfiability counting problem are affine then the number of satisfying assignments of this problem can be computed in polynomial time otherwise this function is p complete this gives us a comparison between decision and counting generalized satisfiability problems we can determine exactly the polynomial satisfiability decision problems whose number of solutions can be computed in polynomial time and also the polynomial satisfiability decision problems whose counting counterparts are already p complete moreover taking advantage of a similar dichotomy result proved in 1978 by schaefer for generalized satisfiability decision problems we get as a corollary the implication that the counting counterpart of each np complete generalized satisfiability decision problem is p complete

np is as easy as detecting unique solutions abstract for every known np complete problem the number of solutions of its instances varies over a large range from zero to exponentially many it is therefore natural to ask if the inherent intractability of np complete problem is caused by this wide variation we give a negative answer to this question using the notion of randomized polynomial time reducibility we show that the problems of distinguishing between instances of sat having zero or one solution or of finding solutions to instances of sat having a unique solution are as hard as sat under randomized reductions several corollaries about the difficulty of specific problems follow for example computing the parity of the number of solutions of a sat formula is shown to be np hard and deciding if a sat formula has a unique solution is shown to be d p hard under randomized reduction central to the study of cryptography is the question as to whether there exist np problems whose instances have solutions that are unique but are hard to find our result can be interpreted as strengthening the belief that such problems exist

an o 2 0 304n algorithm for solving maximum independent set problem a faster algorithm for finding a maximum independent set in a graph is presented the algorithm is an improved version of the one by tarjan and trojanowski 7 a technique to further accelerate this algorithm is also described

new methods for 3 sat decision and worst case analysis abstract we prove the worst case upper bound 1 5045 n for the time complexity of 3 sat decision where n is the number of variables in the input formula introducing new methods for the analysis as well as new algorithmic techniques we add new 2 and 3 clauses called blocked clauses generalizing the extension rule of extended resolution our methods for estimating the size of trees lead to a refined measure of formula complexity of 3 clause sets and can be applied also to arbitrary trees

diffusions on graphs poisson problems and spectral geometry we study diffusions variational principles and associated boundary value problems on directed graphs with natural weightings we associate to certain subgraphs domains a pair of sequences each of which is invariant under the action of the automorphism group of the underlying graph we prove that these invariants differ by an explicit combinatorial factor given by stirling numbers of the first and second kind we prove that for any domain with a natural weighting these invariants determine the eigenvalues of the laplace operator corresponding to eigenvectors with nonzero mean as a specific example we investigate the relationship between our invariants and heat content asymptotics expressing both as special values of an analog of a spectral zeta function

gaussian bounds for the dirichlet heat kernel we obtain a gaussian lower bound for the heat kernel associated to d t where d is the dirichlet laplacian for an open set d in rm

algorithms for four variants of the exact satisfiability problem we present four polynomial space and exponential time algorithms for variants of the exact satisfiability problem first an o 1 1120n where n is the number of variables time algorithm for the np complete decision problem of exact 3 satisfiability and then an o 1 1907n time algorithm for the general decision problem of exact satisfiability the best previous algorithms run in o 1 1193n and o 1 2299n time respectively for the p complete problem of counting the number of models for exact 3 satisfiability we present an o 1 1487n time algorithm we also present an o 1 2190n time algorithm for the general problem of counting the number of models for exact satisfiability presenting a simple reduction we show how this algorithm can be used for computing the permanent of a 0 1 matrix

heat content asymptotics with singular data we study the asymptotic behaviour of the heat content on a compact riemannian manifold with boundary and with singular specific heat and singular initial temperature distributions assuming the existence of a complete asymptotic series we determine the first three terms in that series in addition to the general setting the interval is studied in detail this article is part of a special issue of journal of physics a mathematical and theoretical in honour of stuart dowker s 75th birthday devoted to applications of zeta functions and other spectral functions in mathematics and physics

polynomial terse sets abstract let a be a set and k n be such that we wish to know the answers to x1 a x2 a xk a for various k tuples x1 x2 xk if this problem requires k queries to a in order to be solved in polynomial time then a is called polynomial terse or pterse we show the existence of both arbitrarily complex pterse and non pterse sets and that p np iff every np complete set is pterse we also show connections with p immunity p selective p generic sets and the boolean hierarchy in our framework unique satisfiability and a variation of it called ksat is in some sense close to satisfiability

on the asymptotics of the heat equation and bounds on traces associated with the dirichlet laplacian we use the feynman kac formula and a decomposition of the brownian bridge to obtain pointwise estimates on the diagonal elements of the heat kernel we also compute bounds on trace et d where d is the dirichlet laplacian for i a bounded region d in rn with smooth boundary or ii an unbounded region d in r2 with finite area

asymptotically short term behavior of solutions to one dimensional diffusion processes consider the problem of solving the diffusion equation u s p x u 0 x p t 0 t x x u 0 t s u p t s 0 t 0 1 u x 0 s u x 0 w x 1 where p x is a smooth positive function on 0 p and u g l 0 p r 0 if we define the operator a d a a l 0 p r via d a w s p x w x dx for all w g d a s w 2 1 0 p r l w 1 1 0 p r 0

language equations with complementation decision problems systems of language equations of the form xi i x1 xn 1 i n are studied here every i may contain the operations of concatenation and complementation the properties of having solutions and of having a unique solution are given mathematical characterizations as decision problems the former is np complete while the latter is pspace hard and is in co re and its decidability remains in general open uniqueness becomes decidable in the case of a unary alphabet where it is us complete and in the case of linear concatenation where it is l complete

on the computational complexity of the forcing chromatic number we consider vertex colorings of graphs in which adjacent vertices have distinct colors a graph is s chromatic if it is colorable in s colors and any coloring of it uses at least s colors the forcing chromatic number f chi g of an s chromatic graph g is the smallest number of vertices which must be colored so that with the restriction that s colors are used every remaining vertex has its color determined uniquely we estimate the computational complexity of force g relating it to the complexity class us introduced by blass and gurevich inform control 55 1982 pp 80 88 we prove that recognizing whether f chi g le2 is us hard with respect to polynomial time many one reductions moreover this problem is conp hard even under the promises that f chi g le3 and g is 3 chromatic on the other hand recognizing whether f chi g le k for each constant k is reducible to a problem in us via a disjunctive truth table reduction similar results are obtained also for forcing variants of the clique and the domination numbers of a graph

generating hard satisfiability problems abstract we report results from large scale experiments in satisfiability testing as has been observed by others testing the satisfiability of random formulas often appears surprisingly easy here we show that by using the right distribution of instances and appropriate parameter values it is possible to generate random formulas that are hard that is for which satisfiability testing is quite difficult our results provide a benchmark for the evaluation of satisfiability testing procedures

counting models for 2 sat and 3 sat formulae we here present algorithms for counting models and max weight models for 2sat and 3sat formulae they use polynomial space and run in o 1 2561n and o 1 6737n time respectively where n is the number of variables this is faster than the previously best algorithms for counting nonweighted models for 2sat and 3sat which run in o 1 3247n and o 1 6894n time respectively in order to prove these time bounds we develop new measures of formula complexity allowing us to conveniently analyze the effects of certain factors with a large impact on the total running time we also provide an algorithm for the restricted case of separable 2sat formulae with fast running times for well studied input classes for all three algorithms we present interesting applications such as computing the permanent of sparse 0 1 matrices

complexity and expressive power of deterministic semantics for datalog abstract deterministic models are partial stable models which are not contradicted by any other stable model i e m is a deterministic model if there is no stable model n such that m n is not an interpretation for instanced the well founded model which coincides with the intersection of all partial stable models is a deterministic model as the well founded model is deterministic and unique for each program well founded model semantics has been proposed as the canonical deterministic semantics for partial stable models but the well founded model is not the unique deterministic model indeed the family of deterministic partial stable models is not in general a singleton and admits a minimum the well founded model and a maximum the max deterministic model the latter model is another candidate for a deterministic semantics the aim of this paper is to study the complexity and the expressive power of deterministic semantics in coherence with the deterministic nature of the model the expressive power of max deterministic semantics is shown to be able to express problems with unique solutions whereas the well founded model only captures a proper subset of the queries computable in polynomial time the so called fixpoint queries

heat flow and brownian motion for a region in r2 with a polygonal boundary consider the following heat conduction problem letd be an open bounded and connected set in euclidean spacer 2 with a polygonal boundary suppose thatd has temperature 1 at timet 0 while the boundary is kept at temperature 0 for all timet 0 we obtain the asymptotic behaviour for the amount of heat ind at timet up too e q t ast 0

the complexity of counting in sparse regular and planar graphs we show that a number of graph theoretic counting problems remain cal np hard indeed cal p complete in very restricted classes of graphs in particular we prove that the problems of counting matchings vertex covers independent sets and extremal variants of these all remain hard when restricted to planar bipartite graphs of bounded degree or regular graphs of constant degree we obtain corollaries about counting cliques in restricted classes of graphs and counting satisfying assignments to restricted classes of monotone 2 cnf formulae to achieve these results a new interpolation based reduction technique which preserves properties such as constant degree is introduced

number of models and satisfiability of sets of clauses abstract we present a way of calculating the number of models of propositional formulas represented by sets of clauses the complexity of such a procedure is o n k where k is the length of clauses and n is the number of variables in the clauses the value of 2 is approximately 1 619 value of 3 is approximately 1 840 and the value of k approaches 2 when k is large further we apply the theory on satisfiability problems especially on the 3 sat problems the complexity of the 3 sat problems is o n where n is the number of variables in the clauses the value of is approximately 1 571 which is better than the results in schiermeyer 1993 and monien and schiermeyer 1985

heat content and brownian motion for some regions with a fractal boundary letd be an open bounded set in euclidean space m m 2 3 with boundary d supposed has temperature 0 at timet 0 while d is kept at temperature 1 for allt 0 we use brownian motion to obtain estimates for the solution of corresponding heat equation and to obtain results for the asymptotic behaviour ofe d t the amount of heat ind at timet ast 0 for the triadic von koch snowflakek our results imply that c 1 t 1 mathbf mathbf log mathbf 2 log mathbf 3 leqq mathbf e k t leqq ct 1 mathbf mathbf log mathbf 2 log mathbf 3 mathbf 0 leqq t leqq c 1 for some constantc 1

growth of heat trace and heat content asymptotic coefficients abstract we show in the smooth category that the heat trace asymptotics and the heat content asymptotics can be made to grow arbitrarily rapidly in the real analytic context however this is not true and we establish universal bounds on their growth

the hybrid spectral problem and robin boundary conditions the hybrid spectral problem where the field satisfies dirichlet conditions d on part of the boundary of the relevant domain and neumann n on the remainder is discussed in simple terms a conjecture for the c1 coefficient is presented and the conformal determinant on a 2 disc where the d and n regions are semi circles is derived comments on higher coefficients are made a separable second order hemisphere hybrid problem is introduced that involves robin boundary conditions and leads to logarithmic terms in the heat kernel expansion which are evaluated explicitly

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

the complexity of very simple boolean formulas with applications the concepts of textbf sat hardness and textbf sat completeness modulo npolylogn time and linear size reducibility denoted by textbf sat hard npolylogn n and textbf sat complete npolylogn n respectively are introduced regardless of whether textbf p textbf np or textbf p neq textbf np it is shown that intuitivelyeach textbf sat hard npolylogn n problem requires essentially at least as much deterministic time as andeach textbf sat complete npolylogn n problem requires essentially the same deterministic time as the satisfiability problem for 3cnf formulas it is proved that the leqq satisfiability tautology unique satisfiability equivalence and minimization problems are already textbf sat complete npolylogn n for very simple boolean formulas and for very simple systems of boolean equations these completeness results are used to characterize the deterministic time complexities of a number of problems for lattices propositional calculi

lyapunov control methods of closed quantum systems according to special geometric or physical meanings the paper summarizes three lyapunov functions in controlling closed quantum systems and their controller designing processes specially for the average value based method the paper gives the generalized condition of the largest invariant set in the original reference and develops the construction method of the imaginary mechanical quantity for the error based method this paper gives its strict mathematical proof train of thought on the asymptotic stability and the corresponding physical meaning also we study the relations among the three lyapunov functions and give a unified form of these lyapunov functions finally we compare the control effects of three lyapunov methods by doing some simulation experiments

unambiguous computation boolean hierarchies and sparse turing complete sets this paper studies for up two topics that have been intensely studied for np boolean hierarchies and the consequences of the existence of sparse turing complete sets unfortunately as is often the case the results for np draw on special properties of np that do not seem to carry over straightforwardly to up for example it is known for np and more generally for any class containing sigma and emptyset and closed under union and intersection that the symmetric difference hierarchy the boolean hierarchy and the boolean closure all are equal we prove that closure under union is not needed for this claim for any class cal k that contains sigma and emptyset and is closed under intersection e g up us and fewp the symmetric difference hierarchy over cal k the boolean hierarchy over cal k and the boolean closure of cal k all are equal on the other hand we show that two hierarchies the hausdorff hierarchy and the nested difference hierarchy which in the np case are equal to the boolean closure fail to be equal for the up case in some relativized worlds regarding sparse turing complete sets for up we prove that if up has sparse turing complete sets then the levels of the unambiguous polynomial hierarchy are simpler than one would otherwise expect they collapse one level in terms of their location in the promise unambiguous polynomial hierarchy we obtain related results under the weaker assumption that up has sparse turing hard sets

the complexity of planar counting problems we prove the p hardness of the counting problems associated with various satisfiability graph and combinatorial problems when restricted to planar instances these problems include 3sat 1 3sat 1 ex3sat minimum vertex cover minimum dominating set minimum feedback vertex set x3c partition into triangles and clique cover we also prove the np completeness of the ambiguous satisfiability problems j b saxe two papers on graph embedding problems tech report cmu cs 80 102 dept of computer science carnegie mellon univ pittsburgh pa 1980 and the dp completeness with respect to random polynomial reducibility of the unique satisfiability problems l g valiant and v v vazirani np is as easy as detecting unique solutions in proc 17th acm symp on theory of computing 1985 pp 458 463 associated with several of the above problems when restricted to planar instances previously very few p hardness results no sf np hardness results and no dp completeness results were known for counting problems ambiguous satisfiability problems and unique satisfiability problems respectively when restricted to planar instances assuming sf p neq np one corollary of the above results is that there are no epsilon approximation algorithms for the problems of maximizing or minimizing a linear objective function subject to a planar system of linear inequality constraints over the integers

a specifically fully nonlinear perturbation of the heat equations in exterior domains there is extensive literature on solving nonlinear evolution equations with small initial data by means of energy estimates and l l estimates this approach has been applied to 1 1 in case g r cf l 3 in earlier work to solve such a fully nonlinear problem one established an a priori c2 estimate by using energy estimates which must be considered in a version of hilbert spaces hence there had to be attached the unpleasant requirement that f e ck and a is small in w g fl w g for k and i large enough we also refer to 4 where bandle and levine found that the problem

h infty control of linear quantum stochastic systems the purpose of this paper is to formulate and solve a h infin controller synthesis problem for a class of noncommutative linear stochastic systems which includes many examples of interest in quantum technology the paper includes results on the class of such systems for which the quantum commutation relations are preserved such a requirement must be satisfied in a physical quantum system a quantum version of standard classical dissipativity results are presented and from this a quantum version of the strict bounded real lemma is derived this enables a quantum version of the two riccati solution to the h infin control problem to be presented this result leads to controllers which may be realized using purely quantum purely classical or a mixture of quantum and classical elements this issue of physical realizability of the controller is examined in detail and necessary and sufficient conditions are given our results are constructive in the sense that we provide explicit formulas for the hamiltonian function and coupling operator corresponding to the controller

complexity of unique list colorability given a list l v for each vertex v we say that the graph g is l colorable if there is a proper vertex coloring of g where each vertex v takes its color from l v the graph is uniquelyk list colorable if there is a list assignment l such that l v k for every vertex v and the graph has exactly one l coloring with these lists mahdian and mahmoodian m mahdian e s mahmoodian a characterization of uniquely 2 list colorable graphs ars combin 51 1999 295 305 gave a polynomial time characterization of uniquely 2 list colorable graphs answering an open question from m ghebleh e s mahmoodian on uniquely list colorable graphs ars combin 59 2001 307 318 m mahdian e s mahmoodian a characterization of uniquely 2 list colorable graphs ars combin 51 1999 295 305 we show that uniquely 3 list colorable graphs are unlikely to have such a nice characterization since recognizing these graphs is s 2 p complete

heat flow and hardy inequality in complete riemannian manifolds with singular initial conditions upper bounds are obtained for the heat content of an open set d with singular initial condition f on a complete riemannian manifold provided i the dirichlet laplace beltrami operator satisfies a strong hardy inequality and ii f satisfies an integrability condition precise asymptotic results for the heat content are obtained for an open bounded and connected set d in euclidean space with c2 boundary and with initial condition f x x 0 2 where x is the distance from x to the boundary of d

bounded real properties for a class of annihilation operator linear quantum systems this paper considers the bounded real properties for a class of linear quantum systems which can be defined by complex quantum stochastic differential equations in terms of annihilation operators only the paper considers complex quantum versions of the bounded real lemma the strict bounded real lemma and the lossless bounded real lemma for the class of quantum systems under consideration it is shown that the question of physical realizability is related to the bounded real and lossless bounded real properties

separating complexity classes with tally oracles abstract long and selman 1986 proved that for most familiar pairs of complexity classes separating the classes with a tally oracle is no easier than truly separating the classes refuting a claim in the literature we prove for the first time that for many familiar pairs of complexity classes separating the classes with a tally oracle is easier than truly separating the classes

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

on the power of two way random generators and the impossibility of deterministic poly space simulations borodin cook and pippenger inform and control 58 1983 96 114 proved that both probabilistic acceptors and transducers working in space s n log n can be simulated by deterministic machines in o s n 2 space the definition of probabilistic computations uses one way read only random tape borodin et al asked is it possible to extend our simulation results to the case of a two way read only oracle head in the same vein furst lipton and stockmeyer inform and control 64 1985 43 51 suggested that it could be a difference between two way and one way random tape for space bounded probabilistic computations where the space bound is much less than the length of y it could matter y denoting the random tape inscription in this paper we give a full characterization of two way random space classes that answers both questions karpinski and verbeek there is no polynomial deterministic space simulation of probabilistic space with two way random tape generator inform and control 67 1985 158 162 proved that there is no polynomial deterministic space simulation of two way random space without giving any recursive upper bound the results of this paper solved the open questions of karpinski and verbeek 1985 karpinski and verbeek 1985 and gave full characterization in sense of upper and lower deterministic space classes they are proved precisely exponentially more powerful than the corresponding one way classes

the heat content asymptotics for variable geometries let be a time dependent family of riemannian metrics on a manifold m with a smooth boundary let be the initial temperature of m and let be the specific heat of m impose dirichlet or neumann boundary conditions and let be the resulting total heat energy content of m as one can expand in an asymptotic series in half integer powers of the parameter t we determine for in terms of geometric quantities this extends previous results from the autonomous setting where the metric was independent of the parameter t to a dynamic setting where the metric is permitted to be time dependent

feedback stabilization of discrete time quantum systems subject to non demolition measurements with imperfections and delays we consider a controlled quantum system whose finite dimensional state is governed by a discrete time nonlinear markov process in open loop the measurements are assumed to be quantum non demolition qnd the eigenstates of the measured observable are thus the open loop stationary states they are used to construct a closed loop supermartingale playing the role of a strict control lyapunov function the parameters of this supermartingale are calculated by inverting a metzler matrix that characterizes the impact of the control input on the kraus operators defining the markov process the resulting state feedback scheme taking into account a known constant delay provides the almost sure convergence to the target state this convergence is ensured even in the case where the filter equation results from imperfect measurements corrupted by random errors with conditional probabilities given as a left stochastic matrix closed loop simulations corroborated by experimental data illustrate the interest of such nonlinear feedback scheme for the photon box a cavity quantum electrodynamics system

heat kernel bounds on manifolds with cusps abstract we describe a method of obtaining pointwise upper bounds for the heat kernel of a riemannian manifold with cusps we apply our results to a class of approximately hyperbolic manifolds by which we mean manifolds which have bounded geometry with respect to a hyperbolic structure with cusps these manifolds include all asymptotically hyperbolic manifolds for such manifolds we obtain upper bounds on the heat kernels which we believe to be nearly optimal

heat equation on the arithmetic von koch snowflake we investigate the asymptotic behaviour of the heat content as the time t 0 for an s adic von koch snowflake generated by a square we show that the heat content satisfies a functional equation which after appropriate transformations takes the form of an inhomogeneous renewal equation we obtain the structure of the solution of this equation in the arithmetic case up to an exponentially small remainder in t