the drazin inverse of updating of a square matrix with application to perturbation formula1this project is supported by national nature science foundation of china doctoral point foundation of china and youth science foundation of universities in shanghai of china 1 the jordan canonical form of a matrix is used to derive systematically the drazin inverse of modification of a square matrix in addition to perturbation formula for the drazin inverse an open problem by campbell and meyer in 1975 has partially been solved the results extend earlier work by several authors

random constraint satisfaction easy generation of hard satisfiable instances in this paper we show that the models of random csp instances proposed by xu and li k xu w li exact phase transitions in random constraint satisfaction problems journal of artificial intelligence research 12 2000 93 103 k xu w li many hard examples in exact phase transitions with application to generating hard satisfiable instances technical report corr report cs cc 0302001 revised version in theoretical computer science 355 2006 291 302 are of theoretical and practical interest indeed these models called rb and rd present several nice features first it is quite easy to generate random instances of any arity since no particular structure has to be integrated or property enforced in such instances then the existence of an asymptotic phase transition can be guaranteed while applying a limited restriction on domain size and on constraint tightness in that case a threshold point can be precisely located and all instances have the guarantee to be hard at the threshold i e to have an exponential tree resolution complexity next a formal analysis shows that it is possible to generate forced satisfiable instances whose hardness is similar to unforced satisfiable ones this analysis is supported by some representative results taken from an intensive experimentation that we have carried out using complete and incomplete search methods

on the power of clause learning sat solvers as resolution engines in this work we improve on existing results on the relationship between proof systems obtained from conflict driven clause learning sat solvers and general resolution previous contributions such as those by beame et al 2004 hertel et al 2008 and buss et al 2008 demonstrated that variations on conflict driven clause learning sat solvers corresponded to proof systems as powerful as general resolution however the models used in these studies required either an extra degree of non determinism or a preprocessing step that is not utilized by state of the art sat solvers in practice in this paper we prove that conflict driven clause learning sat solvers yield proof systems that indeed p simulate general resolution without the need for any additional techniques moreover we show that our result can be generalized to certain other practical variations of the solvers which are based on different learning schemes and restart policies

exact thresholds for dpll on random xor sat and np complete extensions of xor sat this paper discusses a model of constraint satisfaction problems known as uniquely extendible constraint satisfaction problems this model includes and generalizes xor sat and the model includes an np complete problem that appears to share many of the threshold characteristics of random sat in this paper we find an exact threshold in the behavior of two versions of dpll on random instances of this problem one version uses the unit clause heuristic and the other uses the generalized unit clause heuristic specifically for dpll with the unit clause heuristic we prove that there is a clause density c smaller than the satisfiability threshold such that for random instances with density smaller than this threshold dpll with unit clause will find a satisfying assignment in linear time with uniformly positive probability however for random instances with density larger than this threshold dpll with unit clause will require exponential time with uniformly positive probability to find a satisfying assignment we then find the equivalent threshold density for dpll with the generalized unit clause heuristic we also prove the analog of the 2 p sat conjecture for this class of problems

a framework for space complexity in algebraic proof systems algebraic proof systems such as polynomial calculus pc and polynomial calculus with resolution pcr refute contradictions using polynomials space complexity for such systems measures the number of distinct monomials to be kept in memory while verifying a proof we introduce a new combinatorial framework for proving space lower bounds in algebraic proof systems as an immediate application we obtain the space lower bounds previously provided for pc pcr alekhnovich et al 2002 filmus et al 2012 more importantly using our approach in its full potential we prove n space lower bounds in pc pcr for random k cnfs k 4 in n variables thus solving an open problem posed in alekhnovich et al 2002 and filmus et al 2012 our method also applies to the graph pigeonhole principle which is a variant of the pigeonhole principle defined over a constant left degree expander graph

moment lyapunov exponent and stability index for linear conservative system with small random perturbation asymptotic expansion series for the moment lyapunov exponent and stability index are constructed and justified for the two dimensional linear stochastic system close to a harmonic oscillator as an example a one degree of freedom mechanical system parametrically excited in stiffness and damping is considered and several terms of the expansion are obtained

a general model and thresholds for random constraint satisfaction problems in this paper we study the relation among the parameters in their most general setting that define a large class of random csp models d k csp where d is the domain size and k is the length of the constraint scopes the model d k csp unifies several related models such as the model rb and the model k csp we prove that the model d k csp exhibits exact phase transitions if klnd increases no slower than the logarithm of the number of variables a series of experimental studies with interesting observations are carried out to illustrate the solubility phase transition and the hardness of instances around phase transitions

perturbation bound of the drazin inverse a perturbation theory for the drazin inverse a d is developed this theory is based on the jordan canonical form of a norm estimates of a e a d are derived when a d and a e are of the same rank and e is small a hard problem by campbell and meyer linear algebra appl 10 1975 77 83 is partially solved

asymptotic analysis of the lyapunov exponent and rotation number of the random oscillator and application we construct asymptotic expansions for the exponential growth rate lyapunov exponent and rotation number of the random oscillator when the noise is large small rapidly varying or slowly varying we then apply our results to problems in the stability of the random oscillator the spectrum of the one dimensional random schr6dinger operator and wave propagation in a one dimensional random medium

finding a tree structure in a resolution proof is np complete the resolution tree problem consists of deciding whether a given sequence like resolution refutation admits a tree structure this paper shows the np completeness of both the resolution tree problem and a natural generalization of the resolution tree problem that does not involve resolution

space proof complexity for random 3 cnfs abstract we investigate the space complexity of refuting 3 cnfs in resolution and algebraic systems we prove that every polynomial calculus with resolution refutation of a random 3 cnf in n variables requires with high probability n distinct monomials to be kept simultaneously in memory the same construction also proves that every resolution refutation of requires with high probability n clauses each of width n to be kept at the same time in memory this gives a n 2 lower bound for the total space needed in resolution to refute these results are best possible up to a constant factor and answer questions about space complexity of 3 cnfs the main technical innovation is a variant of hall s lemma we show that in bipartite graphs with bipartition l r and left degree at most 3 l can be covered by certain families of disjoint paths called vw matchings provided that l expands in r by a factor of 2 for 1 5

moment lyapunov exponents of the stochastic parametrical hill s equation abstract the lyapunov exponent and moment lyapunov exponents of hill s equation with frequency and damping coefficient fluctuated by white noise stochastic process are investigated a perturbation approach is used to obtain explicit expressions for these exponents in the presence of small intensity noises the results are applied to the study of the almost sure and the moment stability of the stationary solutions of the thin simply supported beam subjected to axial compressions and time varying damping which are small intensity stochastic excitations

almost sure stability of linear stochastic differential equations with jumps under the nondegenerate condition as in the diffusion case see 14 21 6 the linear stochastic jump diffusion process projected on the unit sphere is a strong feller process and has a unique invariant measure which is also ergodic using the relation between the transition probabilities of jump diffusions and the corresponding diffusions due to 22 the unique deterministic lyapunov exponent can be represented by the furstenberg khas minskii formula as an integral over the sphere or the projective space with respect to the ergodic invariant measure so that the almost sure asymptotic stability of linear stochastic systems with jumps depends on its sign the critical case of zero lyapunov exponent is discussed and a large deviations result for asymptotically stable systems is further investigated several examples are treated for illustration

perturbation of the drazin inverse for matrices with equal eigenprojections at zero abstract let a denote the eigenprojection of a matrix a corresponding to the eigenvalue 0 we characterize matrices b such that b a and derive from the results 1 error bounds for the drazin inverse of a perturbation 2 improvement of error bounds given recently by wei 10 wei and wang 11 and castro gonzalez et al 4 and 3 a new characterization of ep matrices

the perturbation theory for the drazin inverse and its applications abstract let a and e be n n matrices and b a e denote the drazin inverse of a by ad we give an upper bound for the relative error b d a d a d under certain circumstances an error bound for the solution for the singular equations ax b b r a is also considered

expander graphs and their applications a major consideration we had in writing this survey was to make it accessible to mathematicians as well as to computer scientists since expander graphs the protagonists of our story come up in numerous and often surprising contexts in both fields but perhaps we should start with a few words about graphs in general they are of course one of the prime objects of study in discrete mathematics however graphs are among the most ubiquitous models of both natural and human made structures in the natural and social sciences they model relations among species societies companies etc in computer science they represent networks of communication data organization computational devices as well as the flow of computation and more in mathematics cayley graphs are useful in group theory graphs carry a natural metric and are therefore useful in geometry and though they are just one dimensional complexes they are useful in certain parts of topology e g knot theory in statistical physics graphs can represent local connections between interacting parts of a system as well as the dynamics of a physical process on such systems the study of these models calls then for the comprehension of the significant structural properties of the relevant graphs but are there nontrivial structural properties which are universally important expansion of a graph requires that it is simultaneously sparse and highly connected expander graphs were first defined by bassalygo and pinsker and their existence first proved by pinsker in the early 70s the property of being an expander seems significant in many of these mathematical computational and physical contexts it is not surprising that expanders are useful in the design and analysis of communication networks what is less obvious is that expanders have surprising utility in other computational settings such as in the theory of error correcting codes and the theory of pseudorandomness in mathematics we will encounter e g their role in the study of metric embeddings and in particular in work around the baum connes conjecture expansion is closely related to the convergence rates of markov chains and so they play a key role in the study of monte carlo algorithms in statistical mechanics and in a host of practical computational applications the list of such interesting and fruitful connections goes on and on with so many applications we will not even

many hard examples for resolution for every choice of positive integers c and k such that k 3 and c 2 k 0 7 there is a positive number e such that with probability tending to 1 as n tends to a randomly chosen family of cn clauses of size k over n variables is unsatisfiable but every resolution proof of its unsatisfiability must generate at least 1 e n clauses

on sufficient conditions for unsatisfiability of random formulas a descriptive complexity approach to random 3 sat is initiated we show that unsatisfiability of any significant fraction of random 3 cnf formulas cannot be certified by any property that is expressible in datalog combined with the known relationship between the complexity of constraint satisfaction problems and expressibility in datalog our result implies that any constraint propagation algorithm working with small constraints will fail to certify unsatisfiability almost always our result is a consequence of designing a winning strategy for one of the players in the existential pebble game the winning strategy makes use of certain extension axioms that we introduce and hold almost surely on a random 3 cnf formula the second contribution of our work is the connection between finite model theory and propositional proof complexity to make this connection explicit we establish a tight relationship between the number of pebbles needed to win the game and the width of the resolution refutations as a consequence to our result and the known size width relationship in resolution we obtain new proofs of the exponential lower bounds for resolution refutations of random 3 cnf formulas and the pigeonhole principle

index splitting for the drazin inverse and the singular linear system abstract an index splitting of a singular matrix a for the drazin inverse and its relative iterations for the minimal p norm solution of the singular linear system ax b b r a k k ind a are presented some convergence criteria are also given and a method is developed for constructing the index splitting of matrix a

moment lyapunov exponents of a two dimensional system under bounded noise parametric excitation the moment lyapunov exponents of a two dimensional system under bounded noise parametric excitation are studied in this paper the method of regular perturbation is applied to obtain weak noise expansions of the moment lyapunov exponent lyapunov exponent and stability index in terms of the small fluctuation parameter

a note on the representation for the drazin inverse of 2 2 block matrices abstract in this note we consider representations of the drazin inverse of 2 2 block matrices under conditions weaker than those used in recent papers on the subject in particular in d s djordjevic p s stanimirovic on the generalized drazin inverse and generalized resolvent czechoslovak math j 51 126 2001 617 634 r hartwig x li y wei representations for the drazin inverse of 2 2 block matrix siam j matrix anal appl 27 2006 757 771

an improvement on the perturbation of the group inverse and oblique projection the perturbations of the group inverse a and oblique projection aa of a square matrix a have been previously studied under certain assumptions on the matrix a and a perturbation matrix e upper bounds for b bb b a a and bb aa aa where b a e are given in the literature recently upper bounds for the general case have been published by y wei appl math comp 98 1999 29 however the special cases in the literature and the continuity of the group inverse do not follow from the general upper bounds in this paper we derive new general upper bounds which not only cover all the special cases but also are sharper than wei s results such that the continuity of the group inverse directly follows a numerical example is given to illustrate the sharpness of the new general upper bounds

the np hardness of finding a directed acyclic graph for regular resolution let r be a resolution refutation given as a sequence of clauses without explicit description of the underlying dag then it is np complete to decide whether r is a regular resolution refutation

moment lyapunov exponents of a two dimensional system under combined harmonic and real noise excitations abstract the moment lyapunov exponents and the lyapunov exponents of a two dimensional system under combined excitations of harmonic and real noise which is modelled as an ornstein uhlenbeck process are studied the moment lyapunov exponents and the lyapunov exponents are important characteristics determining the moment and almost sure stability of a stochastic dynamical system the eigenvalue problem governing the moment lyapunov exponent is established a regular perturbation method is applied to solve the eigenvalue problem to obtain second order weak noise expansions of the moment lyapunov exponents the influence of the real noise excitation on the parametric resonance due to the harmonic excitation is investigated

lower bounds for the weak pigeonhole principle and random formulas beyond resolution we work with an extension of resolution called res 2 that allows clauses with conjunctions of two literals in this system there are rules to introduce and eliminate such conjunctions we prove that the weak pigeonhole principle phpncn and random unsatisfiable cnf formulas require exponential size proofs in this system this is the strongest system beyond resolution for which such lower bounds are known as a consequence to the result about the weak pigeonhole principle res log is exponentially more powerful than res 2 also we prove that resolution cannot polynomially simulate res 2 and that res 2 does not have feasible monotone interpolation solving an open problem posed by krajicek

the treewidth of proofs abstract so called ordered variants of the classical notions of pathwidth and treewidth are introduced and proposed as proof theoretically meaningful complexity measures for the directed acyclic graphs underlying proofs ordered pathwidth is roughly the same as proof space and the ordered treewidth of a proof is meant to serve as a measure of how far it is from being treelike length space lower bounds for k dnf refutations are generalized to arbitrary infinity axioms and strengthened in that the space measure is relaxed to ordered treewidth

the moment lyapunov exponent of a co dimension two bifurcation system driven by non gaussian colored noise in the present paper based on the concept of the pth moment lyapunov exponent the stochastic stability of a typical co dimension two bifurcation system that is on a three dimensional center manifold and possesses two pure imaginary eigenvalues and one zero eigenvalue and is excited by a non gaussian colored noise is investigated the non gaussian colored noise is treated as an ornstein uhlenbeck process by means of the path integral approach based on the perturbation approach and the green s functions method the second differential eigenvalue equation which governing the moment lyapunov exponent is established by solving the eigenvalue problem the weak noise asymptotic expansions for the finite pth moment lyapunov exponent are obtained and which matches the approximation of the numerical monte carlo simulations finally the conclusions are given

time space trade offs in resolution superpolynomial lower bounds for superlinear space we give the first time space trade off lower bounds for resolution proofs that apply to superlinear space in particular we show that there are formulas of size n that have resolution refutations of size and space t n n theta log n and like all formulas have another resolution refutation of space n but for which no resolution refutation can simultaneously have space s n t n o 1 and size t n o 1 in other words any substantial reduction in space results in a super polynomial increase in total size we also show somewhat stronger time space trade off lower bounds for regular resolution which are also the first to apply to superlinear space for any function t that is at most weakly exponential t n 2 o n 1 4 we give a tautology that has regular resolution proofs of size and space t n but no such proofs with space s n t n 1 omega 1 and size t n o 1 thus any polynomial reduction in space has a superpolynomial cost in size these tautologies

moment lyapunov exponents of a two dimensional system under both harmonic and white noise parametric excitations the moment lyapunov exponents and the lyapunov exponents of a 2d system under both harmonic and white noise excitations are studied the moment lyapunov exponents and the lyapunov exponents are important characteristics determining the moment and almost sure stability of a stochastic dynamical system the eigenvalue problem governing the moment lyapunov exponent is established a singular perturbation method is applied to solve the eigenvalue problem to obtain second order weak noise expansions of the moment lyapunov exponents the influence of the white noise excitation on the parametric resonance due to the harmonic excitation is investigated

moment lyapunov exponents of a two dimensional system under real noise excitation abstract the moment lyapunov exponents of a two dimensional system under real noise excitation an ornstein uhlenbeck process are studied in this paper the method of regular perturbation is applied to obtain the weak noise expansions of the moment lyapunov exponent lyapunov exponent and stability index in terms of the small fluctuation parameter

almost sure and moment lyapunov exponents for a stochastic beam equation the transverse vibrations of an euler bernoulli beam with axial tension p and axial white noise forcing are given by mytt m yt eiyxxxx pyxx 0yxxw t 0 x l with hinged endpoints y 0 t yxx 0 t y l t yxx l t 0 the solution may be written y x t n 1 sin n xl cn t where the amplitude cn t of the nth mode of vibration is given by a two dimensional linear stochastic ordinary differential equation the main results give formulas for the almost sure and moment lyapunov exponents for the spde in terms of the almost sure and moment lyapunov exponents for each of the sdes corresponding to modes which are present in the initial condition it is shown that the almost sure and moment lyapunov exponents for the spde depend sensitively on the initial distribution of energy amongst the infinitely many modes of vibration and that they cannot be well approximated using only finitely many modes of vibration

resolution for max sat max sat is the problem of finding an assignment minimizing the number of unsatisfied clauses in a cnf formula we propose a resolution like calculus for max sat and prove its soundness and completeness we also prove the completeness of some refinements of this calculus from the completeness proof we derive an exact algorithm for max sat and a time upper bound we also define a weighted max sat resolution like rule and show how to adapt the soundness and completeness proofs of the max sat rule to the weighted max sat rule finally we give several particular max sat problems that require an exponential number of steps of our max sat rule to obtain the minimal number of unsatisfied clauses of the combinatorial principle these results are based on the corresponding resolution lower bounds for those particular problems

unrestricted resolution versus n resolution abstract an n resolution proof is a resolution proof in which the resolution rule is restricted one clause to which it is applied must only consist of negative literals n resolution is complete schoning 1987 p 109 stickel 1985 p 86 we construct an infinite family of propositional formulas and show these formulas have unrestricted resolution proofs whose length is polynomially bounded in their size all n resolution proofs of these formulas are of superpolynomial length

resolution proofs of generalized pigeonhole principles we extend results of haken to give an exponential lower bound on the size of resolution proofs for propositional formulas encoding a generalized pigeonhole principle these propositional formulas express the fact that there is no one one mapping from c n objects to n objects when c 1 as a corollary resolution proof systems do not p simulate constant formula depth frege proof systems

data reductions fixed parameter tractability and random weighted d cnf satisfiability data reduction is a key technique in the study of fixed parameter algorithms in the ai literature pruning techniques based on simple and efficient to implement reduction rules also play a crucial role in the success of many industrial strength solvers understanding the effectiveness and the applicability of data reduction as a technique for designing heuristics for intractable problems has been one of the main motivations in studying the phase transition of randomly generated instances of np complete problems in this paper we take the initiative to study the power of data reductions in the context of random instances of a generic intractable parameterized problem the weighted d cnf satisfiability problem we propose a non trivial random model for the problem and study the probabilistic behavior of the random instances from the model we design an algorithm based on data reduction and other algorithmic techniques and prove that the algorithm solves the random instances with high probability and in fixed parameter polynomial time o d knm where n is the number of variables m is the number of clauses and k is the fixed parameter we establish the exact threshold of the phase transition of the solution probability and show that in some region of the problem space unsatisfiable random instances of the problem have parametric resolution proof of fixed parameter polynomial size also discussed is a more general random model and the generalization of the results to the model

on the solution of singular linear systems of algebraic equations by semiiterative methods for a square matrixt n n where i t is possibly singular we investigate the solution of the linear fixed point problemx tx c by applying semiiterative methods sim s to the basic iterationx0 n xk tck 1 c k 1 such problems arise if one splits the coefficient matrix of a linear systemax b of algebraic equations according toa m n m nonsingular which leads tox m 1nx m 1b tx c even ifx tx c is consistent there are cases where the basic iteration fails to converge namely ift possesses eigenvalues 1 with 1 or if 1 is an eigenvalue oft with nonlinear elementary divisors in these cases and also ifx tx c is incompatible we derive necessary and sufficient conditions implying that a sim tends to a vector hat x which can be described in terms of the drazin inverse of i t we further give conditions under which hat x is a solution or a least squares solution of i t x c

resolution lower bounds for the weak pigeonhole principle we prove that any resolution proof for the weak pigeonhole principle with n holes and any number of pigeons is of length 2ne for some global constant e 0 one corollary is that a certain propositional formulation of the statement np p poly does not have short resolution proofs

convergence properties of krylov subspace methods for singular linear systems with arbitrary index abstract krylov subspace methods have been recently considered to solve singular linear systems ax b in this paper we derive the necessary and sufficient conditions guaranteeing that a krylov subspace method converges to a vector adb px0 where ad is the drazin inverse of a and p is the projection p i a d a let k be the index of a we further show that a d b px 0 x 0 r a k 1 n a is a generalized least squares solution of ax b in r a k n a finally we present the convergence bounds for the quasi minimal residual algorithm qmr and transpose free quasi minimal residual algorithm tfqmr the index k of a in this paper can be arbitrary which extends to the main results of freund and hochbruck numer linear algebra appl 1 1994 403 420 that only considers the case k 1

on the automatizability of resolution and related propositional proof systems a propositional proof system is automatizable if there is an algorithm that given a tautology produces a proof in time polynomial in the size of its smallest proof this notion can be weakened if we allow the algorithm to produce a proof in a stronger system within the same time bound this new notion is called weak automatizability among other characterizations we prove that a system is weakly automatizable exactly when a weak form of the satisfiability problem is solvable in polynomial time after studying the robustness of the definition we prove the equivalence between i resolution is weakly automatizable ii res k is weakly automatizable and iii res k has feasible interpolation when k 1 in order to prove this result we show that res 2 has polynomial size proofs of the reflection principle of resolution which is a version of consistency we also show that res k for every k 1 proves its consistency in polynomial size while resolution does not in fact we show that resolution proofs of its own consistency require almost exponential size this gives a better lower bound for the monotone interpolation of res 2 and a separation from resolution as a byproduct our techniques also give us a way to obtain a large class of examples that have small resolution refutations but require relatively large width this answers a question of alekhnovich and razborov related to whether resolution is automatizable in quasipolynomial time

a new kind of tradeoffs in propositional proof complexity we exhibit an unusually strong tradeoff in propositional proof complexity that significantly deviates from the established pattern of almost all results of this kind namely restrictions on one resource width in our case imply an increase in another resource tree like size that is exponential not only with respect to the complexity of the original problem but also to the whole class of all problems of the same bit size more specifically we show that for any parameter k e k n there are unsatisfiable k cnfs that possess refutations of width o k but such that any tree like refutation of width n1 e k must necessarily have doubly exponential size exp n k this means that there exist contradictions that allow narrow refutations but in order to keep the size of such a refutation even within a single exponent it must necessarily use a high degree of parallelism our construction and proof methods combine in a non trivial way two previously known techniques the hardness escalation method based on substitution formulas and expansion this combination results in a hardness compression approach that strives to preserve hardness of a contradiction while significantly decreasing the number of its variables

moment lyapunov exponents and stochastic stability of a double beam system under compressive axial loading the lyapunov exponent and moment lyapunov exponents of two degrees of freedom linear systems subjected to white noise parametric excitation are investigated the method of regular perturbation is used to determine the explicit asymptotic expressions for these exponents in the presence of small intensity noises the lyapunov exponent and moment lyapunov exponents are important characteristics for determining the almost sure and moment stability of a stochastic dynamical system as an example we study the almost sure and moment stability of the double beam system under stochastic compressive axial loading the validity of the approximate results for moment lyapunov exponents is checked by the numerical monte carlo simulation method for this stochastic system

a switching lemma for small restrictions and lower bounds for k dnf resolution we prove a new switching lemma that works for restrictions that set only a small fraction of the variables and is applicable to formulas in disjunctive normal form dnfs with small terms we use this to prove lower bounds for the res k propositional proof system an extension of resolution which works with k dnfs instead of clauses we also obtain an exponential separation between depth d circuits of bottom fan in k and depth d circuits of bottom fan in k 1 our results for res k are as follows the 2n to n weak pigeonhole principle requires exponential size to refute in res k for k leq sqrt log n log log n for each constant k there exists a constant w k so that random w cnfs require exponential size to refute in res k for each constant k there are sets of clauses which have polynomial size res k 1 refutations but which require exponential size res k refutations

frege systems for quantified boolean logic we define and investigate frege systems for quantified boolean formulas qbf for these new proof systems we develop a lower bound technique that directly lifts circuit lower bounds for a circuit class c to the qbf frege system operating with lines from c such a direct transfer from circuit to proof complexity lower bounds has often been postulated for propositional systems but had not been formally established in such generality for any proof systems prior to this work this leads to strong lower bounds for restricted versions of qbf frege in particular an exponential lower bound for qbf frege systems operating with ac0 p circuits in contrast any non trivial lower bound for propositional ac0 p frege constitutes a major open problem improving these lower bounds to unrestricted qbf frege tightly corresponds to the major problems in circuit complexity and propositional proof complexity in particular proving a lower bound for qbf frege systems operating with arbitrary p poly circuits is equivalent to either showing a lower bound for p poly or for propositional extended frege which operates with p poly circuits we also compare our new qbf frege systems to standard sequent calculi for qbf and establish a correspondence to intuitionistic bounded arithmetic

characterising tree like frege proofs for qbf abstract we examine the tree like versions of qbf frege and extended frege systems while in the propositional setting tree like and dag like frege are equivalent we show that this is not the case for qbf frege where tree like systems are exponentially weaker this applies to the version of qbf frege where the universal reduction rule substitutes universal variables by 0 1 constants to show lower bounds for tree like qbf frege we devise a general technique that provides lower bounds for all tree like qbf systems of the form p red where p is a propositional system the lower bound is based on the semantic measure of strategy size corresponding to the size of countermodels for false qbfs we also obtain a full characterisation of hardness for tree like qbf frege lower bounds for this system either arise from a lower bound to propositional frege from a circuit lower bound or from a lower bound to strategy size

mean payoff games and propositional proofs we associate a cnf formula to every instance of the mean payoff game problem in such a way that if the value of the game is non negative the formula is satisfiable and if the value of the game is negative the formula has a polynomial size refutation in s 2 frege i e dnf resolution this reduces mean payoff games to the weak automatizability of s 2 frege and to the interpolation problem for s 2 2 frege since the interpolation problem for s 1 frege i e resolution is solvable in polynomial time our result is close to optimal up to the computational complexity of solving mean payoff games the proof of the main result requires building low depth formulas that compute the bits of the sum of a constant number of integers in binary notation and low complexity proofs of the required arithmetic properties

error bounds for the perturbation of the drazin inverse under some geometrical conditions this paper studies the drazin inverse for perturbed matrices for that given a square matrix a we consider and characterize the class of matrices b with index s such that r ba d n a d 0 n a db r a d 0 and r b s r ba d where n a and r a denote the null space and the range space of a matrix a respectively and a d denote the drazin inverse of a then we provide explicit representations for b d and bb d and upper bounds for the relative error b d a email protected a email protected and the error bb d aa email protected a numerical example illustrates that the obtained bounds are better than others given in the literature

lyapunov exponent and rotation number of two dimensional linear stochastic systems with small diffusion by means of a singular perturbation scheme and large deviation estimates we expand the lyapunov exponent and rotation number in terms of the small noise intensity varepsilon for all 2 times 2 systems dx t varepsilon ax t varepsilon dt sqrt varepsilon sum nolimits k 1 r b k x t varepsilon circ dw t k for which a is complex diagonalizable examples are given

another look at degree lower bounds for polynomial calculus abstract polynomial complexity is an algebraic proof system inspired by grobner bases which was introduced by clegg edmonds and impagliazzo alekhnovich and razborov devised a method for proving degree lower bounds for polynomial calculus we present an alternative account of their method which also encompasses a generalization due to galesi and lauria

on the perturbation of the group inverse and oblique projection for given matrices a and e which are n matrices and matrices and e we discuss upper bounds for b bb b a a and bb aa aa also a lower bound for b is derived

the error bound of the perturbation of the drazin inverse abstract let a and e be n n matrices and b a e denote the drazin inverse of a by a d in this paper we give an upper bound for the relative error b d a d a d 2 and a lower bound for b d 2 under certain circumstances the continuity properties and the derivative of the drazin inverse are also considered

an expression of the drazin inverse of a perturbed matrix given a square matrix a and its perturbation matrix e a new expression for the drazin inverse b d of b a e is derived if aa db 2 aa db 2 or b 2aa d baa d 2 based on the new expression a bound of the relative error of b d is developed some known results in the literature on the drazin inverse and the perturbation bound are included by this new formula as special cases a numerical example is given to compare the upper bounds

the intractability of resolution abstract we prove that for infinitely many disjunctive normal form propositional calculus tautologies the length of the shortest resolution proof of cannot be bounded by any polynomial of the length of the tautologies we use were introduced by cook and reckhow 1979 and encode the pigeonhole principle extended resolution can furnish polynomial length proofs of these formulas

automating resolution is np hard we show that the problem of finding a resolution refutation that is at most polynomially longer than a shortest one is np hard in the parlance of proof complexity resolution is not automatable unless p np indeed we show that it is np hard to distinguish between formulas that have resolution refutations of polynomial length and those that do not have subexponential length refutations this also implies that resolution is not automatable in subexponential time or quasi polynomial time unless np is included in subexp or qp respectively

on the width of semialgebraic proofs and algorithms in this paper we study width of semialgebraic proof systems and various cut based procedures in integer programming we focus on two important systems gomory chvatal cutting planes and lovasz schrijver lift and project procedures we develop general methods for proving width lower bounds and apply them to random k cnfs and several popular combinatorial principles like the perfect matching principle and tseitin tautologies we also show how to apply our methods to various combinatorial optimization problems we establish a supercritical trade off between width and rank that is we give an example in which small width proofs are possible but require exponentially many rounds to perform them

a note on the perturbation bound of the drazin inverse a perturbation bound for the drazin inverse a d with ind a e 1 has recently been developed however those upper bounds are not satisfied since it is not tight enough in this paper a sharper upper bounds for a e a d with weaker conditions is derived that new bound is also a generalization of a new general upper bound of the group inverse we also derive a new expression of the drazin inverse a e d with ind a e 1 and the corresponding upper bound of a e d a d in a special case numerical examples are given to illustrate the sharpness of the new bounds

hard examples for resolution exponential lower bounds are proved for the length of resolution refutations of sets of disjunctions constructed from expander graphs using the method of tseitin since these sets of clauses encode biconditionals they have short polynomial length refutations in a standard axiomatic formulation of propositional calculus

on the relative complexity of resolution refinements and cutting planes proof systems an exponential lower bound for the size of tree like cutting planes refutations of a certain family of conjunctive normal form cnf formulas with polynomial size resolution refutations is proved this implies an exponential separation between the tree like versions and the dag like versions of resolution and cutting planes in both cases only superpolynomial separations were known a urquhart bull symbolic logic 1 1995 pp 425 467 j johannsen inform process lett 67 1998 pp 37 41 p clote and a setzer in proof complexity and feasible arithmetics amer math soc providence ri 1998 pp 93 117 in order to prove these separations the lower bounds on the depth of monotone circuits of raz and mckenzie in combinatorica 19 1999 pp 403 435 are extended to monotone real circuits an exponential separation is also proved between tree like resolution and several refinements of resolution negative resolution and regular resolution actually this last separation also provides a separation between tree like resolution and ordered resolution and thus the corresponding superpolynomial separation of a urquhart bull symbolic logic 1 1995 pp 425 467 is extended finally an exponential separation between ordered resolution and unrestricted resolution also negative resolution is proved only a superpolynomial separation between ordered and unrestricted resolution was previously known a goerdt ann math artificial intelligence 6 1992 pp 169 184

on the complexity of resolution with bounded conjunctions we analyze size and space complexity of res k a family of propositional proof systems introduced by krajicek in fund math 170 1 3 2001 123 which extend resolution by allowing disjunctions of conjunctions of up to k 1 literals we show that the treelike res k proof systems form a strict hierarchy with respect to proof size and also with respect to space moreover resolution while simulating treelike res k is almost exponentially separated from treelike res k to study space complexity for general res k we introduce the concept of dynamical satisfiability which allows us to prove in a unified way all known space lower bounds for resolution and to extend them to res k

representations for the drazin inverses of 2 2 block matrices abstract let m denote a 2 2 block complex matrix a b c d where a and d are square matrices not necessarily with the same orders in this paper explicit representations for the drazin inverse of m are presented under the condition that bd i c 0 for i 0 1 n 1 where n is the order of d

instability of the harmonic oscillator with small noise we construct asymptotic expansions for the exponential growth rate lyapunov exponent and rotation number of the random oscillator when the noise is small and is defined by a temporally homogeneous markov process with a finite number of states in the case of two states the telegraph process we obtain additional terms in the expansions affording comparison between the exact values and the asymptotic formulas

regular resolution versus unrestricted resolution a resolution proof of an unsatisfiable propositional formula is called regular if and only if no variable is eliminated with the resolution rule twice on any branch of the proof tree representing the resolution proof an infinite family of unsatisfiable propositional formulas is constructed and the following shown these formulas have polynomial size unrestricted resolution proofs whereas all regular resolution proofs of these formulas are of superpolynomial length

on the stability properties of a van der pol duffing oscillator that is driven by a real noise abstract in this paper we consider a van der pol duffing oscillator that is excited parametrically by a small intensity real noise which is assumed to be an integrable function of an n dimensional ornstein uhlenbeck vector process that is an output of a linear filter system the stability properties include the moment lyapunov exponent g p x 0 and the maximal lyapunov exponent and the stability in probability are examined to study a model of enhanced generality we remove both the detailed balance condition and the strong mixing condition in the case of an arbitrary finite real number p we employ the perturbation method and a spectrum representation of the fokker planck operator of the linear filter system to construct asymptotic expansions of the pth moment lyapunov exponent and the top lyapunov exponent the same methods are also used for a nonlinear stochastic system to obtain the fpk fokker planck kolmogonov equation for the amplitude process which is identical to the one that is derived from the stochastic averaging method in the case of a broadband noise excitation on the basis of this fpk equation we also examine the almost sure stability condition of the ito stochastic differential equation for the amplitude process which matches the result that is derived from the maximal lyapunov exponent finally the method proposed by lin and cai probabilistic structural dynamics advanced theory and application mcgraw hill new york 1995 is adopted to examine the stability in probability of the amplitude process for the nonlinear ito differential equation

short proofs are narrow resolution made simple the width of a resolution proof is defined to be the maximal number of literals in any clause of the proof in this paper we relate proof width to proof length size in both general resolution and its tree like variant the following consequences of these relations reveal width as a crucial resource of resolution proofs in one direction the relations allow us to give simple unified proofs for almost all known exponential lower bounds on size of resolution proofs as well as several interesting new ones they all follow from width lower bounds and we show how these follow from natural expansion property of clauses of the input tautology in the other direction the width size relations naturally suggest a simple dynamic programming procedure for automated theorem proving one which simply searches for small width proofs this relation guarantees that the runnuing time and thus the size of the produced proof is at most quasi polynomial in the smallest tree like proof this algorithm is never much worse than any of the recursive automated provers such as dll used in practice in contrast we present a family of tautologies on which it is exponentially faster

mean square stability of a second order parametric linear system excited by a colored gaussian noise abstract the stochastic stability of a second order linear parametric oscillator whose stiffness is perturbed by a stationary zero mean colored gaussian process is investigated in this paper referring to the stability in the second response moments mean square stability the stochastic perturbation is the output of a linear filter excited by a stationary gaussian white noise stochastic process and the hypothesis of weak excitation is not formulated the first assumption allows using ito s stochastic differential calculus the moment equations of the response are written till the second order by means of ito s differential rule they form an infinite hierarchy that is the equations for the moments of order r contain moments of order larger than r in order to close the hierarchy the cumulant neglect closure method is applied herein the moment equation set is closed by neglecting the cumulants of third order nonlinear moment equations result they are linearized and the limit of stability is searched by studying the eigenvalues of the matrix of the coefficients of the linearized moment equations several numerical analyses are performed to evaluate the critical mean square value of the excitation when a system parameter is varied for the cases of both first and second order parametric excitation a first order excitation an ornstein uhlenbeck process does not cause stochastic resonance while a second order excitation does the first result was not known previously

many hard examples in exact phase transitions this paper analyzes the resolution complexity of two random constraint satisfaction problem csp models i e model rb rd for which we can establish the existence of phase transitions and identify the threshold points exactly by encoding csps into cnf formulas it is proved that almost all instances of model rb rd have no tree like resolution proofs of less than exponential size thus we not only introduce new families of csps and cnf formulas hard to solve which can be useful in the experimental evaluation of csp and sat algorithms but also propose models with both many hard instances and exact phase transitions finally conclusions are presented as well as a detailed comparison of model rb rd with the hamiltonian cycle problem and random 3 sat which respectively exhibit three different kinds of phase transition behavior in np complete problems