gopakumar vafa invariants via vanishing cycles in this paper we propose an ansatz for defining gopakumar vafa invariants of calabi yau threefolds using perverse sheaves of vanishing cycles our proposal is a modification of a recent approach of kiem li which is itself based on earlier ideas of hosono saito takahashi we conjecture that these invariants are equivalent to other curve counting theories such as gromov witten theory and pandharipande thomas theory our main theorem is that for local surfaces our invariants agree with pt invariants for irreducible one cycles we also give a counter example to the kiem li conjectures where our invariants match the predicted answer finally we give examples where our invariant matches the expected answer in cases where the cycle is non reduced non planar or non primitive

proximal thresholding algorithm for minimization over orthonormal bases the notion of soft thresholding plays a central role in problems from various areas of applied mathematics in which the ideal solution is known to possess a sparse decomposition in some orthonormal basis using convex analytical tools we extend this notion to that of proximal thresholding and investigate its properties providing in particular several characterizations of such thresholders we then propose a versatile convex variational formulation for optimization over orthonormal bases that covers a wide range of problems and we establish the strong convergence of a proximal thresholding algorithm to solve it numerical applications to signal recovery are demonstrated

differentiability of the metric projection in hilbert space a study is made of differentiability of the metric projection p onto a closed convex subset k of a hilbert space h when k has nonempty interior the gateaux or fr het smoothness of its boundary can be related with some precision to gateaux or frechet differentiability properties of p for instance combining results in 3 with earlier work of r d holmes shows that k has a c2 boundary if and only if p is cl in h k and its derivative p has a certain invertibility property at each point an example in 5 shows that if the c2 condition is relaxed even slightly then p can be nondifferentiable fr het in h k

stability conditions and crepant small resolutions in this paper we describe the spaces of stability conditions on the triangulated categories associated to three dimensional crepant small resolutions the resulting spaces have chamber structures such that each chamber corresponds to a birational model together with a special fourier mukai transform we observe that these spaces are covering spaces over certain open subsets of finite dimensional vector spaces and determine their deck transformations

relaxed extragradient methods for finding minimum norm solutions of the split feasibility problem abstract in this paper we consider the split feasibility problem sfp in infinite dimensional hilbert spaces and study the relaxed extragradient methods for finding a common element of the solution set of sfp and the set fix s of fixed points of a nonexpansive mapping s combining mann s iterative method and korpelevich s extragradient method we propose two iterative algorithms for finding an element of fix s on one hand for s i the identity mapping we derive the strong convergence of one iterative algorithm to the minimum norm solution of the sfp under appropriate conditions on the other hand we also derive the weak convergence of another iterative algorithm to an element of fix s under mild assumptions

sufficient excitation and stable reduced order adaptive iir filtering a perturbed error system is used to describe the parameter and output error behaviors of reduced order application of adaptive identifier filters such as the hyperstable adaptive recursive filter harf given satisfaction of a sufficient excitation condition this error system is shown to exhibit a bounded input bounded state bibs property this implies that despite order insufficiency the output and parameter estimates of harf and similar adaptive identifier filter algorithms remain bounded

on reduced order adaptive output error identification and adaptive iir filtering the reduced order application of landau s adaptive output error identifier results in a perturbed error system where the perturbation signal is a moving average of the unmodeled portion of the unknown plant output or desired signal in adaptive filter parlance it is proven in this paper that if this perturbation signal is sufficiently small and a reduced order dimension model is sufficiently excited then the output and parameter estimates of this adaptive identifier filter remain bounded the influence of various operating conditions on this quantitatively defined bound are noted this robustness property is crucial in all real applications which due to nonlinearities and distributed effects are subject to reduced order modeling

a quadratically convergent algorithm for convex set constrained signal recovery this paper addresses the problem of recovering a signal that is constrained to lie in a convex set from linear measurements the current standard is the alternating projections paradigm pocs which has only first order convergence in general we present a quadratically convergent iterative algorithm newton algorithm for signal recovery from linear measurements and convex set constraints a new result on the existence and construction of the derivative of the projection operator onto a convex set is obtained which is used in the newton algorithm an interesting feature of the new algorithm is that each iteration requires the solution of a simpler subspace constrained reconstruction problem a computation and memory efficient version of the algorithm is also obtained by using the conjugate gradient algorithm within each newton iteration to avoid matrix inversion and storage from a computational point of view the computation per iteration of this algorithm is similar to the computation per iteration of the standard alternating projections algorithm the faster rate of convergence compared to alternating projections enables us to obtain a high resolution reconstruction with fewer computations the algorithm is thus well suited for large scale problems that typically arise in image recovery applications the algorithm is demonstrated in several applications

projective splitting methods for sums of maximal monotone operators with applications abstract splitting methods for sums of maximal monotone operators are studied in this paper by formulating the classical splitting methods including the douglas peaceman rachford splitting and the forward backward splitting into fixed point iterations general splitting methods are considered via parameterizing fixed point formulas weak convergence results for these general splitting methods are derived with the help of a demiclosedness principle moreover by employing the haugazeau like projective method the weak convergence splitting algorithms are forced to be strongly convergent finally applications to the convex feasibility and best approximation problems are made from the viewpoint of convex optimization new and improved convergence results for solving these two problems are obtained

a dual approach to linear inverse problems with convex constraints a simple constraint qualification is developed and used to derive an explicit solution to a constrained optimization problem in hilbert space a finite parameterization is obtained for the minimum norm element in the intersection of a linear variety of finite co dimension and a closed convex constraint set the result extends previous duality theorems for convex cone set constraints a fixed point iteration is presented for computing the parameters and yields a least squares solution when the variety and constraint set have empty intersection proofs rely on nearest point projections onto convex sets and the properties of monotone firmly nonexpansive and averaged mappings

proximity for sums of composite functions we propose an algorithm for computing the proximity operator of a sum of composite convex functions in hilbert spaces and investigate its asymptotic behavior applications to best approximation and image recovery are described

strong convergence of an explicit iterative algorithm for continuous pseudo contractions in banach spaces let c be a nonempty closed convex subset of a real banach space x whose norm is uniformly g teaux differentiable and t c c be a continuous pseudo contraction with a nonempty fixed point set f t for arbitrary given element u c and for t 0 1 let yt be the unique continuous path such that yt 1 t tyt tu assume that yt p f t as t 0 let n n and n be three real sequences in 0 1 satisfying the following conditions i n n n 1 ii limn n limn n 0 iii limn n1 n 0 or iii n 0 n1 n let n be a summable sequence of positive numbers for arbitrary initial datum x0 x00 c and a fixed n 0 construct elements xnm as follows xnm 1 nu nxn ntxnm m 0 1 2 suppose that there exists a least positive integer n n satisfying the following condition txnn n 1 txnn n n 1 1 n n define iteratively a sequence xn in an explicit manner as follows xn 1 xn 10 xnn n 1 nu nxn ntxnn n n 0 then xn converges strongly to a fixed point of t for all the continuous pseudo contractive mappings for which is possible to construct the sequence xn this result improves and extends a recent result of yao et al yonghong yao yeong cheng liou rudong chen strong convergence of an iterative algorithm for pseudocontractive mapping in banach spaces nonlinear anal 67 2007 3311 3317

application of benveniste s convergence results in the study of adaptive iir filtering algorithms it is shown that the weak convergence results of a benveniste et al ieee trans automat contr vol ac 25 p 1042 58 dec 1980 can be used to prove convergence of some adaptive infinite impulse response iir filtering algorithms the association of the algorithms with some ordinary differential equations for constant gains which parallels the theory of l ljung et al 1983 is suitable for constant gain adaptive filtering applications convergence proofs for a prefiltering algorithm for a simple constant gain version of the recursive maximum likelihood algorithm and for the well known simple hyperstable adaptive recursive filter sharf algorithm are given as examples

relaxing tight frame condition in parallel proximal methods for signal restoration a fruitful approach for solving signal deconvolution problems consists of resorting to a frame based convex variational formulation in this context parallel proximal algorithms and related alternating direction methods of multipliers have become popular optimization techniques to approximate iteratively the desired solution until now in most of these methods either lipschitz differentiability properties or tight frame representations were assumed in this paper it is shown that it is possible to relax these assumptions by considering a class of non necessarily tight frame representations thus offering the possibility of addressing a broader class of signal restoration problems in particular it is possible to use non necessarily maximally decimated filter banks with perfect reconstruction which are common tools in digital signal processing the proposed approach allows us to solve both frame analysis and frame synthesis problems for various noise distributions in our simulations it is applied to the deconvolution of data corrupted with poisson noise or laplacian noise by using non tight discrete dual tree wavelet representations and filter bank structures

applications of splitting algorithm to decomposition in convex programming and variational inequalities recently han and lou proposed a highly parallelizable decomposition algorithm for minimizing a strongly convex cost over the intersection of closed convex sets it is shown that their algorithm is in fact a special case of a splitting algorithm analyzed by gabay for finding a zero of the sum of two maximal monotone operators gabay s convergence analysis for the splitting algorithm is sharpened and new applications of this algorithm to variational inequalities convex programming and the solution of linear complementarily problems are proposed for convex programs with a certain separable structure a multiplier method that is closely related to the alternating direction method of multipliers of gabay mercier and of glowinski marrocco but which uses both ordinary and augmented lagrangians is obtained

three dimensional flops and noncommutative rings limburgs univ ctr dept math b 3590 diepenbeek belgium van den bergh m limburgs univ ctr dept math univ campus b 3590 diepenbeek belgium vdbergh luc ac be

s duality for surfaces with a_n type singularities we show that the generating series of euler characteristics of hilbert schemes of points on any algebraic surface with at worst a n type singularities is described by the theta series determined by integer valued positive definite quadratic forms and the dedekind eta function in particular it is a fourier development of a meromorphic modular form with possibly half integer weight the key ingredient is to apply the flop transformation formula of donaldson thomas type invariants counting two dimensional torsion sheaves on threefolds proved in the author s previous paper

perturbed projections and subgradient projections for the multiple sets split feasibility problem we study the multiple sets split feasibility problem that requires to find a point closest to a family of closed convex sets in one space such that its image under a linear transformation will be closest to another family of closed convex sets in the image space by casting the problem into an equivalent problem in a suitable product space we are able to present a simultaneous subgradients projections algorithm that generates convergent sequences of iterates in the feasible case we further derive and analyze a perturbed projection method for the multiple sets split feasibility problem and additionally furnish alternative proofs to two known results

existence of stationary points for pseudo linear regression identification algorithms the authors prove the existence of a stable transfer function satisfying the nonlinear equations characterizing an asymptotic stationary point in undermodeled cases for a class of pseudo linear regression algorithms including landau s algorithm the feintuch algorithm and s harf the proof applies to all degrees of undermodeling and assumes only that the input power spectral density function is bounded and nonzero for all frequencies and that the compensation filter is strictly minimum phase some connections to previous stability analyses for reduced order identification in this algorithm class are brought out

stable pairs and the homfly polynomial given a planar curve singularity we prove a conjecture of oblomkov shende relating the geometry of its hilbert scheme of points to the homfly polynomial of the associated algebraic link more generally we prove an extension of this conjecture due to diaconescu hua soibelman relating stable pair invariants on the conifold to the colored homfly polynomial of the algebraic link our proof uses wall crossing techniques to prove a blowup identity on the algebro geometric side we prove a matching identity for the colored homfly polynomials of a link using skein theoretic techniques

on the existence of stationary points for the steiglitz mcbride algorithm most convergence results for adaptive identification algorithms have been developed in sufficient order settings involving an unknown system with known degree reduced order settings in which the degree of the unknown system is underestimated are more common but more difficult to analyze deducing stationary points in these cases typically involves solving nonlinear equations hence the sparseness of results for reduced order cases if we allow ourselves the tractable case in which the input to an identification experiment is white noise we shall show that the steiglitz mcbride method 1965 indeed admits a stationary point in reduced order settings for which the resulting model is stable our interest in this study stems from a previous result showing an attractive a priori bound on the mismodeling error at any such stationary point

image restoration by the method of convex projections part 1ߞtheory a projection operator onto a closed convex set in hilbert space is one of the few examples of a nonlinear map that can be defined in simple abstract terms moreover it minimizes distance and is nonexpansive and therefore shares two of the more important properties of ordinary linear orthogonal projections onto closed linear manifolds in this paper we exploit the properties of these operators to develop several iterative algorithms for image restoration from partial data which permit any number of nonlinear constraints of a certain type to be subsumed automatically their common conceptual basis is as follows every known property of an original image f is envisaged as restricting it to lie in a well defined closed convex set thus m such properties place f in the intersection e0 ei of the corresponding closed convex sets e1 e2 em given only the projection operators pi onto the individual ei s i 1 m we restore f by recursive means clearly in this approach the realization of the pi s in a hilbert space setting is one of the major synthesis problems section i describes the geometrical significance of the three main theorems in considerable detail and most of the underlying ideas are illustrated with the aid of simple diagrams section ii presents rules for the numerical implementation of 11 specific projection operators which are found to occur frequently in many signal processing applications and the appendix contains proofs of all the major results

resolution limits in signal recovery resolution analysis for the problem of signal recovery from finitely many linear measurements is the subject of this paper the classical rayleigh limit serves only as a lower bound on resolution since it does not assume any recovery strategy and is based only on observed data we show that details finer than the rayleigh limit can be recovered by simple linear processing that incorporates prior information we first define a measure of resolution based on allowable levels of error that is more appropriate for current signal recovery strategies than the rayleigh definition in the practical situation in which only finitely many noisy observations are available we have to restrict the class of signals in order to make the resolution measure meaningful we consider the set of bandlimited and essentially timelimited signals since it describes most signals encountered in practice for this set we show how to precompute resolution limits from knowledge of measurement functionals signal to noise ratio passband energy concentration regions energy concentration factor and a prescribed level of error tolerance in the process we also derive an algorithm for high resolution signal recovery we illustrate the results with examples in one and two dimensions

image restoration by the method of convex projections part 2 applications and numerical results the image restoration theory discussed in a previous paper by youla and webb 1 is applied to a simulated image and the results compared with the well known method known as the gerchberg papoulis algorithm the results show that the method of image restoration by projection onto convex sets by providing a convenient technique for utilizing a priori information performs significantly better than the gerchberg papoulis method

an extragradient method for solving split feasibility and fixed point problems the purpose of this paper is to introduce and analyze an extragradient method with regularization for finding a common element of the solution set c of the split feasibility problem and the set fix s of fixed points of a nonexpansive mapping s in the setting of infinite dimensional hilbert spaces combining the regularization method and the extragradient method due to nadezhkina and takahashi n nadezhkina w takahashi weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings j optim theory appl 128 2006 191 201 we propose an iterative algorithm for finding an element of fix s email protected we prove that the sequences generated by the proposed algorithm converge weakly to an element of fix s email protected under mild conditions

stable pair invariants under blow ups we use the degeneration formula and the absolute relative correspondence tostudy the change of stable pair invariants of threefold under blow ups and obtain some closed blow up formulae which imply some interesting relations among bps state counts our results provide positive evidence for the gw dt p correspondence and also give a partial correspondence for varieties not necessarily toric or complete intersections