singular kahler einstein metrics we study degenerate complex monge ampere equations of the form omega dd c f n e t f mu where omega is a big semi kahler form on a compact kahler manifold x of dimension n t in r and mu f omega n is a positive measure with density f in l p x omega n p 1 we prove the existence and unicity of continuous o plurisubharmonic solutions in case x is projective and omega psi omega where psi x to v is a proper birational morphism to a normal projective variety omega in ns r v is an ample class and mu has only algebraic singularities we prove that the solution is smooth in the regular locus of the equation we use these results to construct singular kahler einstein metrics of non positive curvature on projective klt pairs in particular on canonical models of algebraic varieties of general type

comparisons of several algorithms for toeplitz matrix recovery in this paper we study algorithms for toeplitz matrix recovery inspired by the singular value thresholding svt algorithm for matrix completion and the alternating directions iterative method we first propose a new mean value algorithm for toeplitz matrix recovery then we apply our idea to the augmented lagrange multiplier alm algorithm for matrix recovery and put forward four modified alm algorithms for toeplitz matrix recovery convergence analysis of the new algorithms is discussed all the iterative matrices generated by the five algorithms keep a toeplitz structure that ensures the fast singular value decomposition svd of toeplitz matrices compared with the original algorithms our algorithms are far superior in the time of svd as well as the cpu time

stein compacts in levi flat hypersurfaces we explore connections between geometric properties of the levi foliation of a levi flat hypersurface m and holomorphic convexity of compact sets in m or bounded in part by m applications include extendability of cauchy riemann functions solvability of the b equation approximation of cauchy riemann and holomorphic functions and global regularity of the neumann operator

strong q convexity in uniform neighborhoods of subvarieties in coverings of complex spaces the main result is that for any projective compact analytic subset y of dimension q 0 in a reduced complex space x there is a neighborhood of y such that for any covering space upsilon colon widehat x to x in which widehat y equiv upsilon 1 y has no noncompact connected analytic subsets of pure dimension q with only compact irreducible components there exists a c exhaustion function varphi on widehat x which is strongly q convex on widehat omega upsilon 1 omega outside a uniform neighborhood of the q dimensional compact irreducible components of widehat y

minimal surfaces in minimally convex domains in this paper we prove that every conformal minimal immersion of a compact bordered riemann surface m into a minimally convex domain d r can be approximated uniformly on compacts in m m bm by proper complete conformal minimal immersions m d see theorems 1 1 1 7 and 1 9 we also obtain a rigidity theorem for complete immersed minimal surfaces of finite total curvature contained in a minimally convex domain in r see theorem 1 16 and we characterize the minimal surface hull of a compact set k in r for any n 3 by sequences of conformal minimal disks whose boundaries converge to k in the measure theoretic sense see corollary 5 6

complexity of equivalence problems for concurrent systems of finite agents abstract a concurrent system of synchronous communicating agents is assembled from simpler sequential agents by parallel composition and hiding for example hide a1 alin p1 p2 pn describes the system of communicating agentsp1 pnin which the communication eventsa1 alare hidden consider descriptions of two systemspandqof synchronously communicating finite state agents assume that one wants to check whetherp qfor one of the commonly used equivalence we show that this question is pspace hard for all equivalences that lie between strong bisimulation and trace equivalences for some equivalences exponential lower and upper bounds are proven we also show that this problem is np hard and co np hard even for a class of very simple finite agents

multiple representations to compute orthogonal eigenvectors of symmetric tridiagonal matrices in this paper we present an o nk procedure algorithm mr3 for computing k eigenvectors of an n n symmetric tridiagonal matrix t a salient feature of the algorithm is that a number of different ldlt products l unit lower triangular d diagonal are computed in exact arithmetic each ldlt is a factorization of a translate of t we call the various ldlt products representations of t and roughly speaking there is a representation for each cluster of close eigenvalues the unfolding of the algorithm for each matrix is well described by a representation tree we present the tree and use it to show that if each representation satisfies three prescribed conditions then the computed eigenvectors are orthogonal to working accuracy and have small residual norms with respect to the original matrix t

strongly pseudoconvex domains as subvarieties of complex manifolds in this paper we obtain existence and approximation results for closed complex subvarieties that are normalized by strongly pseudoconvex stein domains our sufficient condition for the existence of such subvarieties in a complex manifold x is expressed in terms of the morse indices and the number of positive levi eigenvalues of an exhaustion function on x examples show that our conditions cannot be weakened in general we obtain optimal results for subvarieties of this type in complements of compact complex submanifolds with griffiths positive normal bundle in the projective case these generalize classical theorems of remmert bishop and narasimhan concerning proper holomorphic maps and embeddings to bbb c n bbb p n backslash bbb p n 1

some q convexity properties of coverings of complex manifolds let c be a nowhere dense compact analytic subset of a 2 dimensional complex manifold x a result of napier is the construction of a neighborhood v of c such that for any covering space open image in new window in which open image in new window is holomorphically convex there is a c exhaustion function on open image in new window which is strictly plurisubharmonic on 1 v away from the compact irreducible components of open image in new window col oiu and v j itu obtained a q convex version of this result for c a q dimensional fiber of a proper surjective submersion x y the goal of this paper is a similar version for cprojective

attractor stability in nonuniform boolean networks the attractor graph of a boolean network f has the attractors of f as vertices there is a directed edge between attractors a i and a j if there is a state x a i and an index k such that x when perturbed in the kth coordinate yields a state x in the basin of attractor a j kauffman 2007 proposed to use the strongly connected components of the attractor graph called ergodic sets to model possible targets for cell differentiation however computational follow up work by him and others has revealed that most boolean networks have a single strongly connected component in an overwhelming number of cases and two strongly connected components in rare cases as a proof of concept we explicitly construct classes of hierarchical nonuniform boolean networks where the attractor graph may have an arbitrary number of ergodic sets the construction uses threshold functions and networks that are combinations of connected cliques earlier work does not address the structure of ergodic sets building on the previous construction we show how to generate boolean networks where ergodic sets possess binary hyper cubes q 2 m as sub graphs although the focus of this paper is to demonstrate that there are boolean networks with a large number of ergodic sets and that ergodic sets can be arbitrarily large we remark that networks we construct have community structure and the results may therefore be relevant to application areas ranging from biology to social science the results also provide a starting point for additional studies into the attractor structure of boolean networks their basins of attractions and conditions for stability under state noise

general purpose computation with neural networks a survey of complexity theoretic results we survey and summarize the literature on the computational aspects of neural network models by presenting a detailed taxonomy of the various models according to their complexity theoretic characteristics the criteria of classification include the architecture of the network feedforward versus recurrent time model discrete versus continuous state type binary versus analog weight constraints symmetric versus asymmetric network size finite nets versus infinite families and computation type deterministic versus probabilistic among others the underlying results concerning the computational power and complexity issues of perceptron radial basis function winner take all and spiking neural networks are briefly surveyed with pointers to the relevant literature in our survey we focus mainly on the digital computation whose inputs and outputs are binary in nature although their values are quite often encoded as analog neuron states we omit the important learning issues

one way cellular automata on cayley graphs the notion of one dimensional one way cellular automata has been introduced to model cellular automata with only a one way communication between two neighbor cells in this paper we generalize this notion to cellular automata working on different communication graphs we present some sufficient conditions for a cellular automaton to be stimulated by a one way cellular automaton having the same underlying graph and we give some bounds on the simulation time of this mimic

predecessors and garden of eden configurations in parallel dynamical systems on maxterm and minterm boolean functions abstract in this work we solve the classical predecessor problems for parallel dynamical systems on maxterm and minterm boolean functions actually we solve analytically the predecessor existence problem by giving a characterization to have a predecessor for any given configuration as a consequence we also get a characterization of the garden of eden configurations of these systems moreover the structure of the predecessors found out allows us to give a solution to the unique predecessor problem the coexistence of predecessors problem and the number of predecessors problem

q completeness of subsets in complex projective space the importance of exhaustion functions in complex analysis arises mainly from the theorems of andreott i and grauert 2 and the applications of morse theory the first give vanishing theorems for cohomology groups of coherent sheaves the second imply topological statements if x is a compact complex manifold and y an analytic subset of x the vanishing of the relative homotopy groups rh x y for i d i m x q follows from the q completeness of x y which is by definition the same as the existence of a q convex exhaustion function these theorems are called lefschetz theorems the first of this kind was the lefschetz theorem on hyperplane sections which can be proved by using the fact that the complement of a hypersurface in projective space is stein because there is a 1 convex exhaustion function it is natural to ask whether there is an exhaustion function of p a with certain properties if one has a lefschetz theorem if a is a q codimensional complex submanifold of p we have h p a 7z 0 4 and even i p a 0 9 for i n 2 q 1 so we are led to the question if p a is 2 q 1 complete and the proof of this assertion is the aim of this paper

equidistribution of zeros of holomorphic sections in the non compact setting we consider tensor powers l n of a positive hermitian line bundle l h l over a non compact complex manifold x in the compact case b shiffman and s zelditch proved that the zeros of random sections become asymptotically uniformly distributed as n with respect to the natural measure coming from the curvature of l under certain boundedness assumptions on the curvature of the canonical line bundle of x and on the chern form of l we prove a non compact version of this result we give various applications including the limiting distribution of zeros of cusp forms with respect to the principal congruence subgroups of sl 2 and to the hyperbolic measure the higher dimensional case of arithmetic quotients and the case of orthogonal polynomials with weights at infinity we also give estimates for the speed of convergence of the currents of integration on the zero divisors

on the limit sets of cellular automata the limit sets of cellular automata defined by wolfram play an important role in applications of cellular automata to complex systems a number of results on limit sets are proved considering both finite and infinite configurations of cellular automata the main concern of this paper is with testing membership and essential emptiness of limit sets for linear and two dimensional cellular automata

cohomology of q convex spaces in top degrees it is shown that every strongly q complete subvariety of a complex analytic space has a fundamental system of strongly q complete neighborhoods as a consequence we find a simple proof of ohsawa s result that every non compact irreducible n dimensional analytic space is strongly n complete finally it is shown that l cohomology theory readily implies ohsawa s hodge decomposition and lefschetz isomorphism theorems for absolutely q convex manifolds

attraction radii in binary hopfield nets are hard to compute we prove that it is an np hard problem to determine the attraction radius of a stable vector in a binary hopfield memory network and even that the attraction radius is hard to approximate under synchronous updating the problems are already np hard for two step attraction radii direct one step attraction radii can be computed in polynomial time

analytic cohomology groups in top degrees of zariski open sets in mathbb p n we show that if a is a closed analytic subset of mathbb p n of pure codimension q then h i mathbb p n setminus a mathcal f are finite dimensional for every coherent algebraic sheaf mathcal f and every i geq n left frac n 1 q right if n 1 geq 2q rm we show that h n 2 mathbb p n setminus a mathcal f 0

two modified augmented lagrange multiplier algorithms for toeplitz matrix compressive recovery abstract in this paper we propose two modified augmented lagrange multiplier algorithms by mean value for toeplitz matrix compressive recovery in the algorithms the mean value modification makes the iteration matrices keep the toeplitz structure which contributes to reduce the svd time and cpu time numerical experiments show that the proposed algorithms achieve better precision than the augmented lagrange multiplier method especially when the matrix e is less sparse convergence analysis of the proposed algorithms is also given in detail

inferring local transition functions of discrete dynamical systems from observations of system behavior abstract we consider the problem of inferring the local transition functions of discrete dynamical systems from observed behavior our focus is on synchronous systems whose local transition functions are threshold functions we assume that the topology of the system is known and that the goal is to infer a threshold value for each node so that the system produces the observed behavior we show that some of these inference problems are efficiently solvable while others are np complete even when the underlying graph of the dynamical system is a simple path we identify a fixed parameter tractable problem in this context we also consider constrained versions of threshold inference problems where the input includes a set of equality or inequality constraints which specify pairs of nodes which must have the same threshold value or different threshold values we present algorithmic and complexity results for several constrained threshold inference problems

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

the hartogs extension theorem on n 1 complete complex spaces employing morse theory for the global control of monodromy and the method of analytic discs for local extension we establish a version of the global hartogs extension theorem in a singular setting for every domain d of an n 1 complete normal complex space x of pure dimension n 2 and for every compact set k in d such that d k is connected holomorphic or meromorphic functions in d k extend holomorphically or meromorphically to d normality is an unvavoidable assumption for holomorphic extension but we show that meromorphic extension holds on a reduced globally irreducible not necessarily normal x of pure dimension n 2 provided that the regular part of d k is connected

modeling and analyzing social network dynamics using stochastic discrete graphical dynamical systems motivated by applications such as the spread of epidemics and the propagation of influence in social networks we propose a formal model for analyzing the dynamics of such networks our model is a stochastic version of discrete graphical dynamical systems using this model we formulate and study the computational complexity of two fundamental problems called reachability and predecessor existence problems which arise in the context of social networks we also address other problems that deal with the time evolution of such stochastic dynamical systems further we point out the implications of our results to problems for other computational models such as hopfield networks communicating finite state machines and systolic arrays in particular our polynomial time algorithms for the predecessor existence problem for stochastic dynamical systems imply similar results for one dimensional finite cellular automata

singular value decompositions of complex symmetric matrices abstract if the complex square matrix a is symmetric i e a at then it has a symmetric singular value decomposition a q qt an algorithm is presented for the computation of this decomposition

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

on defining functions and cores for unbounded domains i we show that every strictly pseudoconvex domain omega with smooth boundary in a complex manifold mathcal m admits a global defining function i e a smooth plurisubharmonic function varphi u rightarrow mathbb r defined on an open neighbourhood u subset mathcal m of overline omega such that omega varphi 0 d varphi ne 0 on b omega and varphi is strictly plurisubharmonic near b omega we then introduce the notion of the core mathfrak c omega of an arbitrary domain omega subset mathcal m as the set of all points where every smooth and bounded from above plurisubharmonic function on omega fails to be strictly plurisubharmonic if omega is not relatively compact in mathcal m then in general mathfrak c omega is nonempty even in the case when mathcal m is stein it is shown that every strictly pseudoconvex domain omega subset mathcal m with smooth boundary admits a global defining function that is strictly plurisubharmonic precisely in the complement of mathfrak c omega we then investigate properties of the core in particular we prove that the core is always 1 pseudoconcave

a fast symmetric svd algorithm for square hankel matrices abstract this paper presents an o n 2 log n algorithm for computing the symmetric singular value decomposition of square hankel matrices of order n in contrast with existing o n 3 svd algorithms the algorithm consists of two stages first a complex square hankel matrix is reduced to a complex symmetric tridiagonal matrix using the block lanczos method in o n 2 log n flops second the singular values and singular vectors of the symmetric tridiagonal matrix resulted from the first stage are computed in o n 2 flops the singular vector matrix is given in the form of a product of three or two unitary matrices the performance of our algorithm is demonstrated by comparing it with the svd subroutines in matlab and lapack

a relative oka grauert principle for holomorphic submersions over 1 convex spaces this paper presents the proof of the relative oka grauert principle for holomorphic submersions over 1 convex spaces using conic neighbourhoods of holomorphic sections over 1 convex spaces a proof of a version of cartan s theorem a for 1 convex spaces is also given

an o n2 algorithm for the bidiagonal svd the relatively robust representations rrr algorithm computes the eigendecomposition of a symmetric tridiagonal matrix t with an o n2 complexity this article discusses how this method can be extended to the bidiagonal svd b u vt it turns out that using the rrr algorithm as a black box to compute btb v 2vt and bbt u 2ut separately may give poor results for bv u the use of the standard jordan wielandt representation can fail as well if clusters of tiny singular values are present a solution is to work on btb and to keep factorizations of bbt implicitly we introduce a set of coupling transformations which allow us to replace the representation u 1 bv by a more stable representation u xv where x is a diagonal matrix numerical results of our implementation are compared with the lapack routines dstegr dbdsqr and dbdsdc

predecessor existence problems for finite discrete dynamical systems we study the predecessor existence problem for finite discrete dynamical systems given a finite discrete dynamical system s and a configuration c the predecessor existence or pre problem is to determine whether there is a configuration c such that s has a transition from c to c in addition to the decision version we also study the following variants the predecessor existence or pre problem counting the number of predecessors the unique predecessor existence or upre problem deciding whether there is a unique predecessor and the ambiguous predecessor existence or apre problem given a configuration c and a predecessor c of c deciding whether there is a different predecessor c of c general techniques are presented for simultaneously characterizing the computational complexity of the pre problem and its three variants our hardness results are based on the concept of simultaneous reductions single transformations that can be used to simultaneously prove the hardness of the different variants of the pre problem for their respective complexity classes our easiness results are based on dynamic programming and they extend the previous results on pre problem for one dimensional cellular automata the hardness results together with the easiness results provide a tight separation between easy and hard instances further the results imply similar bounds for other classes of finite discrete dynamical systems including discrete hopfield and recurrent neural networks concurrent state machines systolic networks and one and two dimensional cellular automata our results extend the earlier results of green sutner and orponen on the complexity of the predecessor existence problem and its variants

on the n concavity of covering spaces with parameters x t where x and t are connected complex manifolds of dimensions n m and m respectively is a proper and surjective holomorphic submersion and is a covering map thus x can be regarded as a family of n dimensional complex manifolds over t we will use the same definition of q convexity as in 1 for the precise formulation see the next section r green and h wu 7 proved that a connected non compact complex manifold is n complete in 5 m col oiu and v v j itu proved the following

plurisubharmonic functions in calibrated geometry and q convexity let m be a kahler manifold an integrable function varphi on m is called q plurisubharmonic if the current dd c varphi wedge omega q 1 is positive we prove that varphi is q plurisubharmonic if and only if varphi is subharmonic on all q dimensional complex subvarieties we prove that a q plurisubharmonic function is q convex and admits a local approximation by smooth q plurisubharmonic functions for any closed subvariety z subset m dim mathbb c z leq q 1 there exists a strictly q plurisubharmonic function in a neighbourhood of z this result is known for q convex functions this theorem is used to give a new proof of sibony s lemma on integrability of positive closed p p forms which are integrable outside of a complex subvariety of codimension p 1

strongly plurisubharmonic exhaustion functions on 1 convex spaces the conclusion obtained from a stein space by blowing up finitely many points means precisely that there exist a compact analytic set s c x with d imxs 0 for any x s a stein space 14 a finite set a c y and a proper holomorphic map p x y inducing a biholomorphism x s y a and which satisfies p c x cr r following a customary terminology x is called an 1 convex space s its exceptional set and y the remmert reduction of x theorem 1 1 was conjectured by fornaess narashimhan in 5 in this paper we prove the converse

inversion of 2d cellular automata some complexity results abstract in this paper we prove the co np completeness of the following decision problem given a two dimensional cellular automaton a even with von neumann neighborhood is a injective when restricted to finite configurations not greater than its length in order to prove this result we introduce two decision problems concerning respectively turing machines and tilings that we prove np complete then we present a transformation of problems concerning tilings into problems concerning cellular automata

a linear representation of dynamics of boolean networks a new matrix product called semi tensor product of matrices is reviewed using it a matrix expression of logic is proposed where a logical variable is expressed as a vector a logical function is expressed as a multiple linear mapping under this framework a boolean network equation is converted into an equivalent algebraic form as a conventional discrete time linear system analyzing the transition matrix of the linear system formulas are obtained to show a the number of fixed points b the numbers of cycles of different lengths c transient period for all points to enter the set of attractors and d basin of each attractor the corresponding algorithms are developed and used to some examples

convexity properties of coverings of 1 convex manifolds we show the following result let x be a 1 convex manifold a its exceptional set k dima and p y x any covering then y can be exhausted by an increasing sequence y nu nu in mathbb n of smoothly bounded k convex domains

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

holomorphic flexibility properties of complex manifolds we obtain results on approximation of holomorphic maps by algebraic maps the jet transversality theorem for holomorphic and algebraic maps between certain classes of manifolds and the homotopy principle for holomorphic submersions of stein manifolds to certain algebraic manifolds

elements of a theory of simulation iii equivalence of sds in the development of mathematical foundations for a theory of simulation a certain class of discrete sequential dynamical systems sds is of particular importance these systems which we refer to as sds consist of a a graph y with vertex set 1 2 n where each vertex has associated a binary state b a vertex labeled set of functions f i y f 2 n f 2 n and c a permutation p s n the function f i y update the state of vertex i as a function of the states of vertex i and its y neighbors and leaves all other states invariant by composing these functions in the order given by p we obtain the sds f y p i 1 nf p i y f 2 n f 2 n in this paper we give a combinatorial upper bound for the number of non equivalent sds for a given graph and we compute this bound explicitly for certain classes of graphs we give a full characterization of invertible sds and analyze the set of fixed points of sequential and parallel cellular automata further we introduce the concept of maj type sds and show that systems of this class only have fixed points as attractors finally we analyze sds that are fixed point free for arbitrary base graphs

a transcendental approach to kollar s injectivity theorem ii we treat kollar s injectivity theorem from the analytic or differential geometric viewpoint more precisely we give a curvature condition which implies kollar type cohomology injectivity theorems our main theorem is formulated for a compact kahler manifold but the proof uses the space of harmonic forms on a zariski open set with a suitable complete kahler metric we need neither covering tricks desingularizations nor leray s spectral sequence

a lanczos bidiagonalization algorithm for hankel matrices abstract this paper presents a fast algorithm for bidiagonalizing a hankel matrix an m n hankel matrix is reduced to a real bidiagonal matrix in o m n n log m n floating point operations flops using the lanczos method with modified partial orthogonalization and reset schemes to improve its stability performance improvement is achieved by exploiting the hankel structure as fast hankel matrix vector multiplication is used the accuracy and efficiency of the algorithm are demonstrated by our numerical experiments