space complexity in propositional calculus we study space complexity in the framework of propositional proofs we consider a natural model analogous to turing machines with a read only input tape and such popular propositional proof systems as resolution polynomial calculus and frege systems we propose two different space measures corresponding to the maximal number of bits and clauses monomials that need to be kept in the memory simultaneously we prove a number of lower and upper bounds in these models as well as some structural results concerning the clause space for resolution and frege systems

analysis of a semigroup approach in the inverse problem of identifying an unknown parameters abstract this article presents a semigroup approach to the mathematical analysis of the inverse parameter problems of identifying the unknown parameters p t and q in the linear parabolic equation u t x t u xx qu x x t p t u x t with dirichlet boundary conditions u 0 t 0 u 1 t 1 the main purpose of this paper is to investigate the distinguishability of the input output mapping p h 1 2 0 t via semigroup theory in this paper it is shown that if the nullspace of the semigroup t t consists of only zero function then the input output mapping has the distinguishability property it is also shown that the types of the boundary conditions and the region on which the problem is defined play an important role in the distinguishability property of the mapping moreover under the light of the measured output data u x 0 t f t the unknown parameter p t at x t 0 0 and the unknown coefficient q are determined via the input data furthermore it is shown that measured output data f t can be determined analytically by an integral representation hence the input output mapping p h 1 2 0 t is given explicitly interms of the semigroup

pp is as hard as the polynomial time hierarchy in this paper two interesting complexity classes pp and oplus text p are compared with ph the polynomial time hierarchy it is shown that every set in ph is polynomial time turing reducible to a set in pp and ph is included in text bp cdot oplus text p as a consequence of the results it follows that text pp subseteq text ph or oplus text p subseteq text ph implies a collapse of ph a stronger result is also shown every set in pp ph is polynomial time turing reducible to a set in pp

the complexity of partial derivatives abstract let l denote the nonscalar complexity in k x 1 x n we prove l x 1 x n 3l using this we determine the complexity of single power sums single elementary symmetric functions the resultant and the discriminant as root functions up to order of magnitude also we linearly reduce matrix inversion to computing the determinant

random self reducibility of complete sets this paper generalizes the previous formal definitions of random self reducibility it is shown that even under a very general definition sets that are complete for any level of the polynomial hierarchy are not nonadaptively random self reducible unless the hierarchy collapses in particular np complete sets are not nonadaptively random self reducible unless the hierarchy collapses at the third level by contrast we show that sets complete for the classes pp and text mod m text p are random self reducible

identification of the unknown diffusion coefficient in a linear parabolic equation by the semigroup approach in this article we study the semigroup approach for the mathematical analysis of the inverse coefficient problems of identifying the unknown coefficient k x in the linear parabolic equation ut x t k x ux x t x with dirichlet boundary conditions u 0 t 0 u 1 t 1 main goal of this study is to investigate the distinguishability of the input output mappings k c1 0 t k c1 0 t via semigroup theory in this paper we show that if the null space of the semigroup t t consists of only zero function then the input output mappings and have the distinguishability property moreover the values k 0 and k 1 of the unknown diffusion coefficient k x at x 0 and x 1 respectively can be determined explicitly by making use of measured output data boundary observations f t k 0 ux 0 t or and h t k 1 ux 1 t in addition to these the values k 0 and k 1 of the unknown coefficient k x at x 0 and x 1 respectively are also determined via the input data furthermore it is shown that measured output data f t and h t can be determined analytically by an integral representation hence the input output mappings k c1 0 t k c1 0 t are given explicitly in terms of the semigroup finally by using all these results we construct the local representations of the unknown coefficient k x at the end points x 0 and x 1

on the effective nullstellensatz let mathbb k be an algebraically closed field and let x subset mathbb k m be an n dimensional affine variety assume that f1 fk are polynomials which have no common zeros on x we estimate the degrees of polynomials a i in mathbb k x such that 1 ki 1aifi on x our estimate is sharp for k n and nearly sharp for k n now assume that f1 fk are polynomials on x let i f 1 dots f k subset mathbb k x be the ideal generated by fi it is well known that there is a number e i the noether exponent such that ie i i we give a sharp estimate of e i in terms of n deg x and deg fi we also give similar estimates in the projective case finally we obtain a result from the elimination theory if f 1 dots f n in mathbb k x 1 dots x n is a system of polynomials with a finite number of common zeros then we have the following optimal elimination phi i x i sum n j 1 f jg ij quad i 1 dots n where deg f jg ij le prod n i 1 deg f i

a geometric effective nullstellensatz the purpose of this paper is to present a geometric theorem which clarifies and extends in several directions work of brownawell kollar and others on the effective nullstellensatz specifically we work on an arbitrary smooth complex projective variety x with the previous classical results corresponding to the case when x is projective space in this setting we prove a local effective nullstellensatz for ideal sheaves and a corresponding global division theorem for adjoint type bundles we also make explicit the connection with the intersection theory of fulton and macpherson finally constructions involving products of prime ideals that appear in earlier work are replaced by geometrically more natural conditions involving orders of vanishing along subvarieties much of the previous activity in this area has been algebraic in nature and seems perhaps not well known in detail among geometers therefore we have felt it worthwhile to include here a rather extended introduction we start with an overview of the questions and earlier work on them then we present the set up and statement of our main theorem we conclude with a series of examples which can be read before the general result of what it yields in special cases

on the restoration of translational invariance in the critical quantum ising model on a fibonacci chain the spectra of quasiperiodically modulated quantum ising chains are investigated for moderate values of the modulation strength a unified approach based on the lifting of the degeneracy between the left and right moving eigenmodes is proposed the local scaling behaviour reflecting partial restoration of translational invariance is established both the conformally invariant scaling for the lower modes and the multi fractal scaling for the higher ones are described in terms of the same quantity a simple but accurate calculational scheme is constructed where the rearrangement in the level spectra is due to the interaction between left and right moving modes of the periodic system as induced by the quasiperiodic modulation

circuit complexity proof complexity and polynomial identity testing the ideal proof system we introduce a new and natural algebraic proof system whose complexity measure is essentially the algebraic circuit size of nullstellensatz certificates this enables us to exhibit close connections between effective nullstellensatze proof complexity and algebraic circuit complexity in particular we show that any super polynomial lower bound on any boolean tautology in our proof system implies that the permanent does not have polynomial size algebraic circuits vnp vp we also show that super polynomial lower bounds on the number of lines in polynomial calculus proofs imply the permanent versus determinant conjecture note that there was no proof system prior to ours for which lower bounds on an arbitrary tautology implied any complexity class lower bound our proof system helps clarify the relationships between previous algebraic proof systems in doing so we highlight the importance of polynomial identity testing pit in proof complexity in particular we use pit to illuminate ac0 p frege lower bounds which have been open for nearly 30 years with no satisfactory explanation as to their apparent difficulty finally we explain the obstacles that must be overcome in any attempt to extend techniques from algebraic circuit complexity to prove lower bounds in proof complexity using the algebraic structure of our proof system we propose a novel route to such lower bounds although such lower bounds remain elusive this proposal should be contrasted with the difficulty of extending ac0 p circuit lower bounds to ac0 p frege lower bounds

sharp effective nullstellensatz the usual proofs of this result however give no information about the g1 s for instance they give no bound on their degrees this question was first considered by g hermann h she used elimination theory to get a bound on the degree of the gi s which was doubly exponential in the number of variables her results were later improved in mw and in th all these produce bounds that are doubly exponential in the number of variables a major breakthrough was achieved by brownawell b1 who proved the following result

local critical behaviour at aperiodic surface extended perturbation in the ising quantum chain the surface critical behaviour of the semi infinite one dimensional quantum ising model in a transverse field is studied in the presence of an aperiodic surface extended modulation the perturbed couplings are distributed according to a generalized fredholm sequence leading to a marginal perturbation and varying surface exponents the surface magnetic exponents are calculated exactly whereas the expression of the surface energy density exponent is conjectured from a finite size scaling study the system displays surface order at the bulk critical point above a critical value of the modulation amplitude it may be considered as a discrete realization of the hilhorst van leeuwen model

critical behaviour in aperiodic systems the author considers layered systems with aperiodically modulated couplings and study the effect of inhomogeneity on the critical behaviour using scaling arguments a relevance irrelevance type criterion is formulated and for relevant inhomogeneities the scaling form of singular quantities is determined the author has performed exact calculations for the surface magnetization of aperiodic quantum ising chains and for a 1 1 dimensional directed polymer in a longitudinal aperiodic environment the results for both systems are in accord with scaling considerations for a marginal aperiodicity the critical exponents are found to be non universal

improved kinetic renormalisation group approach to diffusion limited aggregation an improved kinetic renormalisation group approach to diffusion limited aggregation is presented this approach is based on the growth process itself and accounts for the dispersity in the growth probabilities it yields the multifractal spectrum d q with better values at smaller q on the 2d square lattice the authors find df 1 694 for the fractal dimensions of the cluster and that of its interface and d 1 1 01 for the information dimension the former agrees with the simulation results df approximately 1 70 and the latter compares very well with the exact value d 1 1

an inverse radiative coefficient problem arising in a two dimensional heat conduction equation with a homogeneous dirichlet boundary condition in a circular section abstract in this study we investigate the well posedness of the solution of an optimal control problem related to a nonlinear inverse coefficient problem problems of this type have important applications in several fields of applied science unlike other terminal control problems the observation data are only given for a fixed direction rather than for the whole domain which may make the conjugate theory for parabolic equations ineffective moreover the coefficients in our model are singular so we propose some weighted sobolev spaces based on the optimal control framework the problem is transformed into an optimization problem and the existence of the minimizer is established after deducing the necessary conditions that must be satisfied by the minimizer we prove the uniqueness and stability of the minimizer following a minor modification of the cost functional and imposing some a priori regularity conditions on the forward operator we obtain the convergence of the minimizer for the noisy input data considered in this study the results obtained in this study are interesting and useful and they can be extended to more general parabolic equations with singular coefficients

lower bounds in a parallel model without bit operations we define a natural and realistic model of parallel computation called the pram model without bit operations it is like the usual pram model the main difference being that no bit operations are provided it encompasses virtually all known parallel algorithms for weighted combinatorial optimization and algebraic problems in this model we prove that for some large enough constant b the mincost flow problem for graphs with n vertices cannot be solved deterministically or with randomization in sqrt n b expected time using 2 sqrt n b processors this is so even if we restrict every cost and capacity to be an integer nonnegative if it is a capacity of bitlength at most an for some large enough constant a a similar lower bound is also proved for the max flow problem it follows that these problems cannot be solved in our model deterministically or with randomization in omega n c expected time with 2 omega n c processors where c is an appropriate positive constant and n is the total bitlength of the input since these problems were known to be p complete this provides concrete support for the belief that p completeness implies high parallel complexity and for the p not nc conjecture our lower bounds also extend to the pram model with limited bit operations which provides instructions for parity and left or right shift by one bit our proof is based on basic algebraic geometry so we investigate if the algebrogeometric approach could also work for the p versus nc problem our results support this possibility and a close analysis of the limitation of our technique in this context suggests that such a proof of p not nc should somehow use geometric invariant theory in a deep way

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

multi parameters identification problem for a degenerate parabolic equation abstract this paper investigates an inverse problem of simultaneously reconstructing the initial value and source coefficient in a degenerate parabolic equation problems of this type have important applications in several fields of applied science being different from other inverse coefficient problems in classical parabolic equations non degenerate the principal coefficient in the mathematical model discussed in the paper may vanish on both extremities of the domain on the basis of carleman estimate the uniqueness and conditional stability of solution for the original problem are established then an iteration algorithm of landweber type is designed to obtain the numerical solution and some typical numerical experiments are also performed numerical results show that the proposed method is stable and the unknown coefficients are recovered quite well

ising model on an icosahedral quasilattice the ising model on the three dimensional icosahedral quasilattice is studied by use of a monte carlo simulation the authors treat both cases where the spins are located on the vertices of the lattice and the centres of the lattice they investigate the critical phenomena on the basis of finite size scaling it is shown that the critical exponents are universal among regular lattices and quasilattices the critical temperatures of both models are found to be higher than that of the simple cubic lattice though the average coordination numbers of the three lattices are all six

cook s versus valiant s hypothesis we first investigate the dependency of our complexity concepts on the field k we prove that for algebraically closed fields k the truth of valiant s hypothesis vp vnp depends at most on the characteristic of k

a scaling method for asymptotic analysis of power series a new method is proposed for asymptotic analysis of power series in the method the author uses the relation that an n term power series of a singular quantity with critical exponent beta behaves asymptotically like n beta at the phase transition point the method is tested on known series with satisfactory results and used to analyse the power series of the two dimensional three state potts model of enting in good agreement with den nijs conjecture

phase transition in aperiodic quantum ising models phase transition in the aperiodic quantum ising models were studied in which the transverse magnetic fields are uniform and the couplings are arranged according to the aperiodic sequences that are generated by two types of two tile inflation rules with initial condition a p b q a a a m11 b m12 b b a m21 b m22 m12 m21 not equal 0 and a a a m11 b m12 b b b m21 a m22 m12 m22 not equal 0 when the inflation rules have the pisot vijayaraghavan pv property in number theory the generated aperiodic systems undergo phase transition at the critical points when the inflation rules do not have the pv property it is found that only by changing the initial conditions the inflation rules can produce aperiodic systems exhibiting phase transition as well as aperiodic systems which do not exhibit any phase transition

geometric complexity theory i an approach to the p vs np and related problems we suggest an approach based on geometric invariant theory to the fundamental lower bound problems in complexity theory concerning formula and circuit size specifically we introduce the notion of a partially stable point in a reductive group representation which generalizes the notion of stability in geometric invariant theory due to mumford geometric invariant theory springer verlag berlin 1965 then we reduce fundamental lower bound problems in complexity theory to problems concerning infinitesimal neighborhoods of the orbits of partially stable points we also suggest an approach to tackle the latter class of problems via construction of explicit obstructions

walks on the penrose lattice the authors performed extensive monte carlo simulations for different types of walks random walks ideal chains and self avoiding walks on the penrose quasilattice the critical exponent nu for each process is found to be the same as for periodic two dimensional lattices thus universality seems to hold also for the penrose tiling

sharp estimates for the arithmetic nullstellensatz we present sharp estimates for the degree and the height of the polynomials in the nullstellensatz over the integer ring zz the result improves previous work of philippon berenstein yger and krick pardo we also present degree and height estimates of intrinsic type which depend mainly on the degree and the height of the input polynomial system as an application we derive an effective arithmetic nullstellensatz for sparse polynomial systems the proof of these results relies heavily on the notion of local height of an affine variety defined over a number field we introduce this notion and study its basic properties

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

geometric complexity theory ii towards explicit obstructions for embeddings among class varieties in k d mulmuley and m sohoni siam j comput 31 2001 pp 496 526 henceforth referred to as part i we suggested an approach to the p vs np and related lower bound problems in complexity theory through geometric invariant theory in particular it reduces the arithmetic characteristic zero version of the np not subseteq p conjecture to the problem of showing that a variety associated with the complexity class np cannot be embedded in a variety associated with the complexity class p we shall call these class varieties associated with the complexity classes p and np this paper develops this approach further reducing these lower bound problems which are all nonexistence problems to some existence problems specifically to proving the existence of obstructions to such embeddings among class varieties it gives two results towards explicit construction of such obstructions the first result is a generalization of the borel weil theorem to a class of orbit closures which include class varieties the second result is a weaker form of a conjectured analogue of the second fundamental theorem of invariant theory for the class variety associated with the complexity class nc these results indicate that the fundamental lower bound problems in complexity theory are in turn intimately linked with explicit construction problems in algebraic geometry and representation theory the results here were announced in k d mulmuley and m sohoni in advances in algebra and geometry hyderabad 2001 hindustan book agency new delhi india 2003 pp 239 261

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

arithmetic circuits the chasm at depth four gets wider in their paper on the chasm at depth four agrawal and vinay have shown that polynomials in m variables of degree o m which admit arithmetic circuits of size 2 o m also admit arithmetic circuits of depth four and size 2 o m this theorem shows that for problems such as arithmetic circuit lower bounds or black box derandomization of identity testing the case of depth four circuits is in a certain sense the general case in this paper we show that smaller depth four circuits can be obtained if we start from polynomial size arithmetic circuits for instance we show that if the permanent of nxn matrices has circuits of size polynomial in n then it also has depth 4 circuits of size n o n l o g n if the original circuit uses only integer constants of polynomial size then the same is true for the resulting depth four circuit these results have potential applications to lower bounds and deterministic identity testing in particular for sums of products of sparse univariate polynomials we also use our techniques to reprove two results on the existence of nontrivial boolean circuits of constant depth for languages in logcfl reduction to polylogarithmic depth for arithmetic circuits of polynomial size and polynomially bounded degree

nonperiodic ising quantum chains and conformal invariance in a recent paper luck investigated the critical behavior of one dimensional ising quantum chains with coupling constants modulated according to general nonperiodic sequences in this note we take a closer look at the case where the sequences are obtained from two letter substituion rules and at the consequences of luck s results at criticality they imply that only for a certain class of substitution rules is the long distance behavior still described by thec 1 2 conformal field theory of a free majorana fermion as for the periodic ising quantum chain whereas the general case does not lead to a conformally invariant scaling limit

fast parallel computation of polynomials using few processors it is shown that any multivariate polynomial of degree d that can be computed sequentially in c steps can be computed in parallel in o log d log c log d steps using only cd o 1 processors

percolation in two dimensional quasicrystals by monte carlo simulations the authors examine whether or not two dimensional quasi periodic lattices belong to the same universality class as periodic lattices and they study bond and site percolation in penrose tiling and its dual lattice by making use of monte carlo simulations for the sake of comparison they also investigate percolation in periodic systems such as a square kagome and dice lattice for all these lattices they evaluate several critical exponents such as alpha related to the total number of clusters beta related to the percolation strength gamma related to the mean cluster size nu related to the correlation length tau related to cluster sizes and the fractal dimension d their results indicate that universality holds in two dimensional lattices with or without periodicity where the coordination number could be either single valued or multi valued

the complexity of the word problems for commutative semigroups and polynomial ideals abstract any decision procedure for the word problems for commutative semigroups and polynomial deals inherently requires computational storage space growing exponentially with the size of the problem instance to which the procedure is applied this bound is achieved by a simple procedure for the semigroup problem