how to project onto an isotone projection cone abstract the solution of the complementarity problem defined by a mapping f r n r n and a cone k r n consists of finding the fixed points of the operator p k i f where p k is the projection onto the cone k and i stands for the identity mapping for the class of isotone projection cones cones admitting projections isotone with respect to the order relation they generate and f satisfying certain monotonicity properties the solution can be obtained by iterative processes see g isac a b nemeth projection methods isotone projection cones and the complementarity problem j math anal appl 153 1 1990 258 275 and s z nemeth iterative methods for nonlinear complementarity problems on isotone projection cones j math anal appl 350 1 2009 340 347 these algorithms require computing at each step the projection onto the cone k in general computing the projection mapping onto a cone k is a difficult and computationally expensive problem in this note it is shown that the projection of an arbitrary point onto an isotone projection cone in r n can be obtained by projecting recursively at most n 1 times into subspaces of decreasing dimension this emphasizes the efficiency of the algorithms mentioned above and furnishes a handy tool for some problems involving special isotone projection cones as for example the non negative monotone cones occurring in reconstruction problems see e g section 5 13 in j dattorro convex optimization and euclidean distance geometry meboo 2005 v2009 04 11

solvability of variational inequalities on hilbert lattices this paper provides a systematic solvability analysis for generalized variational inequalities on separable hilbert lattices by contrast to a large part of the existing literature our approach is lattice theoretic and is not based on topological fixed point theory this allows us to establish the solvability of certain types of generalized variational inequalities without requiring the involved set valued maps be hemicontinuous or monotonic some of our results generalize those obtained in the context of nonlinear complementarity problems in earlier work and appear to have scope for applications this is illustrated by means of several applications to fixed point theory optimization and game theory

twisting the fake monster superalgebra abstract we calculate the twisted denominator identities of the fake monster superalgebra corresponding to elements of 2 aut e 8 of odd order they give many new infinite product identities and new examples of generalized kac moody superalgebras we also obtain generalizations of jacobi s identity on theta functions

projection methods isotone projection cones and the complementarity problem one studies some iterative methods for solving complementarity problems explicit or implicit in hilbert spaces the iterative methods presented are based on the fixed point theory on the coincidence equations on convex cones and on some properties of projection operators onto a special cone named the isotone projection cone the ordering defined by the cone is essential for the complementarity theory the idea is to use systematically the ordering in the iterative methods for solving complementarity problems in this sense one shows that the isotone projection cone is a good instrument

some p properties for linear transformations on euclidean jordan algebras a real square matrix is said to be a p matrix if all its principal minors are positive it is well known that this property is equivalent to the nonsign reversal property based on the componentwise product of vectors the order p property based on the minimum and maximum of vectors uniqueness property in the standard linear complementarity problem lipschitzian homeomorphism property of the normal map corresponding to the nonnegative orthant in this article we extend these notions to a linear transformation defined on a euclidean jordan algebra we study some interconnections between these extended concepts and specialize them to the space s n of all n n real symmetric matrices with the semidefinite cone s n and to the space r n with the lorentz cone

lattice like operations and isotone projection sets abstract by using some lattice like operations which constitute extensions of ones introduced by m s gowda r sznajder and j tao for self dual cones a new perspective is gained on the subject of isotonicity of the metric projection onto the closed convex sets the results of this paper are wide range generalizations of some results of the authors obtained for self dual cones the aim of the subsequent investigations is to put into evidence some closed convex sets for which the metric projection is isotonic with respect to the order relation which give rise to the above mentioned lattice like operations the topic is related to variational inequalities where the isotonicity of the metric projection is an important technical tool for euclidean sublattices this approach was considered by g isac and respectively by h nishimura and e a ok

carved visual hulls for image based modeling this article presents a novel method for acquiring high quality solid models of complex 3d shapes from multiple calibrated photographs after the purely geometric constraints associated with the silhouettes found in each image have been used to construct a coarse surface approximation in the form of a visual hull photoconsistency constraints are enforced in three consecutive steps 1 the rims where the surface grazes the visual hull are first identified through dynamic programming 2 with the rims now fixed the visual hull is carved using graph cuts to globally optimize the photoconsistency of the surface and recover its main features 3 an iterative local refinement step is finally used to recover fine surface details the proposed approach has been implemented and experiments with seven real data sets are presented along with qualitative and quantitative comparisons with several state of the art image based modeling algorithms

multiview stereo via volumetric graph cuts and occlusion robust photo consistency this paper presents a volumetric formulation for the multiview stereo problem which is amenable to a computationally tractable global optimization using graph cuts our approach is to seek the optimal partitioning of 3d space into two regions labeled as object and empty under a cost functional consisting of the following two terms 1 a term that forces the boundary between the two regions to pass through photo consistent locations and 2 a ballooning term that inflates the object region to take account of the effect of occlusion on the first term we use an occlusion robust photo consistency metric based on normalized cross correlation which does not assume any geometric knowledge about the reconstructed object the globally optimal 3d partitioning can be obtained as the minimum cut solution of a weighted graph

a duality between the metric projection onto a convex cone and the metric projection onto its dual abstract if k and l are mutually dual pointed convex cones in r n with the metric projections onto them denoted by p k and p l respectively then the following two assertions are equivalent i p k is isotone with respect to the order induced by k i e v u k implies p k v p k u k ii p l is subadditive with respect to the order induced by l i e p l u p l v p l u v l for any u v r n

finding solutions of implicit complementarity problems by isotonicity of the metric projection abstract isac and nemeth g isac and a b nemeth projection method isotone projection cones and the complementarity problem j math anal app 153 258 275 1990 proved that solving a coincidence point equation fixed point problem in turn solves the corresponding implicit complementarity problem nonlinear complementarity problem and they exploited the isotonicity of the metric projection onto isotone projection cones to solve implicit complementarity problems nonlinear complementarity problems defined by these cones in this paper the notion of isotone projection cones is employed and an iterative algorithm is presented in connection with an implicit complementarity problem on isotone projection cones it is proved that if the sequence generated through the defined algorithm is convergent then its limit is a solution of the coincidence point equation and thus solves the implicit complementarity problem sufficient conditions are given for this sequence to be convergent for implicit complementarity problems defined by isotone projection cones the question of finding nonzero solutions of these problems is also studied

solving cone constrained convex programs by differential inclusions a differential inclusion is designed for solving cone constrained convex programs the method is of subgradient projection type it involves projection penalties and lagrangian relaxation nonsmooth data can be accommodated a novelty is that multipliers converge monotonically upwards to equilibrium levels an application to stochastic programming is considered

shading based surface detail recovery under general unknown illumination reconstructing the shape of a 3d object from multi view images under unknown general illumination is a fundamental problem in computer vision high quality reconstruction is usually challenging especially when fine detail is needed and the albedo of the object is non uniform this paper introduces vertex overall illumination vectors to model the illumination effect and presents a total variation tv based approach for recovering surface details using shading and multi view stereo mvs behind the approach are the two important observations 1 the illumination over the surface of an object often appears to be piecewise smooth and 2 the recovery of surface orientation is not sufficient for reconstructing the surface which was often overlooked previously thus we propose to use tv to regularize the overall illumination vectors and use visual hull to constrain partial vertices the reconstruction is formulated as a constrained tv minimization problem that simultaneously treats the shape and illumination vectors as unknowns an augmented lagrangian method is proposed to quickly solve the tv minimization problem as a result our approach is robust stable and is able to efficiently recover high quality surface details even when starting with a coarse model obtained using mvs these advantages are demonstrated by extensive experiments on the state of the art mvs database which includes challenging objects with varying albedo

on the kodaira dimension of orthogonal modular varieties we prove that up to scaling there are only finitely many integral lattices l of signature 2 n with n ge 21 or n 17 such that the modular variety defined by the orthogonal group of l is not of general type in particular when n ge 108 every modular variety defined by an arithmetic group for a rational quadratic form of signature 2 n is of general type we also obtain similar finiteness in n ge 9 for the stable orthogonal groups as a byproduct we derive finiteness of lattices admitting reflective modular form of bounded vanishing order which proves a conjecture of gritsenko and nikulin

projection onto simplicial cones by picard s method abstract by using moreau s decomposition theorem for projecting onto cones the problem of projecting onto a simplicial cone is reduced to finding the unique solution of a nonsmooth system of equations it is shown that picard s method applied to the system of equations associated with the problem of projecting onto a simplicial cone generates a sequence that converges linearly to the solution of the system numerical experiments are presented making the comparison between picard s and semi smooth newton s methods to solve the nonsmooth system associated with the problem of projecting a point onto a simplicial cone

solving nonlinear complementarity problems by isotonicity of the metric projection abstract the main motivation for introducing the notion of isotone projection cones was to solve nonlinear complementarity problems the notion of isotone projection cones is introduced by this paper in a similar fashion iterative methods for finding solutions of complementarity problems on isotone projection cones are presented the problem of finding nonzero solutions of these problems is also considered

isotone retraction cones in hilbert spaces abstract a mapping is called isotone if it is monotone increasing with respect to the order induced by a pointed closed convex cone finding the pointed closed convex generating cones for which the projection mapping onto the cone is isotone is a difficult problem which was analyzed in isac and nemeth 1986 1990 1992 1 2 3 4 5 such cones are called isotone projection cones in particular it was shown that any isotone projection cone is latticial isac 1990 2 this problem is extended by replacing the projection mapping with continuous retractions onto the cone by introducing the notion of sharp mappings it is shown that a pointed closed convex generating cone is latticial if and only if there is a continuous retraction onto the cone whose complement is sharp several particular cases are considered and examples are given

domination of quadratic forms we study domination of quadratic forms in the abstract setting of ordered hilbert spaces our main result gives a characterization in terms of the associated forms this generalizes and unifies various earlier works along the way we present several examples

a duality between the metric projection onto a convex cone and the metric projection onto its dual in hilbert spaces abstract if k and l are mutually dual closed convex cones in a hilbert space h with the metric projections onto them denoted by p k and p l respectively then the following two assertions are equivalent i p k is isotone with respect to the order induced by k i e v u k implies p k v p k u k ii p l is subadditive with respect to the order induced by l i e p l u p l v p l u v l for any u v h this extends the similar result of a b nemeth and the author for euclidean spaces the extension is essential because the proof of the result for euclidean spaces is essentially finite dimensional and seemingly cannot be extended for hilbert spaces the proof of the result for hilbert spaces is based on a completely different idea which uses extended lattice operations

iterative methods for variational and complementarity problems in this paper we study both the local and global convergence of various iterative methods for solving the variational inequality and the nonlinear complementarity problems included among such methods are the newton and several successive overrelaxation algorithms for the most part the study is concerned with the family of linear approximation methods these are iterative methods in which a sequence of vectors is generated by solving certain linearized subproblems convergence to a solution of the given variational or complementarity problem is established by using three different yet related approaches the paper also studies a special class of variational inequality problems arising from such applications as computing traffic and economic spatial equilibria finally several convergence results are obtained for some nonlinear approximation methods

a new projection method for variational inequality problems we propose a new projection algorithm for solving the variational inequality problem where the underlying function is continuous and satisfies a certain generalized monotonicity assumption e g it can be pseudomonotone the method is simple and admits a nice geometric interpretation it consists of two steps first we construct an appropriate hyperplane which strictly separates the current iterate from the solutions of the problem this procedure requires a single projection onto the feasible set and employs an armijo type linesearch along a feasible direction then the next iterate is obtained as the projection of the current iterate onto the intersection of the feasible set with the halfspace containing the solution set thus in contrast with most other projection type methods only two projection operations per iteration are needed the method is shown to be globally convergent to a solution of the variational inequality problem under minimal assumptions preliminary computational experience is also reported

fixed point approach for complementarity problems abstract in this paper we use the fixed point technique to suggest a new unified and general algorithm for computing the approximate solution of a nonlinear complementarity problem of finding u such that u 0 tu a u 0 u tu a u 0 where t is a continuous mappping from r n into itself and a is a non linear transformation from r n into itself this algorithm includes many existing algorithms for complementarity problems as special cases convergence properties are also discussed and analyzed

on the nonlinear complementarity problem abstract we consider and study an algorithm for a new class of complementarity problems of finding u such that u 0 tu a u 0 u tu a u 0 where t is a continuous mapping and a is a diagonally nonlinear mapping from r n into itself furthermore it is proved that the approximate solution obtained by the iterative scheme converges to the exact solution special cases are also discussed

a construction of antisymmetric modular forms for weil representations we give coefficient formulas for antisymmetric vector valued cusp forms with rational fourier coefficients for the weil representation associated to a finite quadratic module the forms we construct always span all cusp forms in weight at least three these formulas are useful for computing explicitly with theta lifts

a semi smooth newton method for a special piecewise linear system with application to positively constrained convex quadratic programming in this paper a special piecewise linear system is studied it is shown that under a mild assumption the semi smooth newton method applied to this system is well defined and the method generates a sequence that converges linearly to a solution besides we also show that the generated sequence is bounded for any starting point and a formula for any accumulation point of this sequence is presented as an application we study the convex quadratic programming problem under positive constraints the numerical results suggest that the semi smooth newton method achieves accurate solutions to large scale problems in few iterations

multi view stereo reconstruction of dense shape and complex appearance we address the problem of estimating the three dimensional shape and complex appearance of a scene from a calibrated set of views under fixed illumination our approach relies on a rank condition that must be satisfied when the scene exhibits specular diffuse reflectance characteristics this constraint is used to define a cost functional for the discrepancy between the measured images and those generated by the estimate of the scene rather than attempting to match image to image directly minimizing such a functional yields the optimal estimate of the shape of the scene represented by a dense surface as well as its radiance represented by four functions defined on such a surface these can be used to generate novel views that capture the non lambertian appearance of the scene

iterative methods for nonlinear complementarity problems on isotone projection cones in this paper we present a recursion related to a nonlinear complementarity problem defined by a closed convex cone in a hilbert space and a continuous mapping defined on the cone if the recursion is convergent then its limit is a solution of the nonlinear complementarity problem in the case of isotone projection cones sufficient conditions are given for the mapping so that the recursion to be convergent

automorphic products of singular weight for simple lattices we classify the simple even lattices of square free level and signature 2 n n ge 4 a lattice is called simple if the space of cusp forms of weight 1 n 2 for the dual weil representation of the lattice is trivial for a simple lattice every formal principal part obeying obvious conditions is the principal part of a vector valued modular form using this we determine all holomorphic borcherds products of singular weight arising from vector valued modular forms with non negative principal part for the simple lattices we construct the corresponding vector valued modular forms by eta products and compute expansions of the automorphic products at different cusps