on the rank minimization problem over a positive semidefinite linear matrix inequality we consider the problem of minimizing the rank of a positive semidefinite matrix subject to the constraint that an affine transformation of it is also positive semidefinite our method for solving this problem employs ideas from the ordered linear complementarity theory and the notion of the least element in a vector lattice this problem is of importance in many contexts for example in feedback synthesis problems and such an example is also provided

weighted inequalities for commutators of fractional integrals on spaces of homogeneous type abstract let 0 1 b be a bmo function and i b m the commutator of order m for the fractional integral we prove two type of weighted l p inequalities for i b m in the context of the spaces of homogeneous type the first one establishes that for a weights the operator i b m is bounded in the weighted l p norm by the maximal operator m m m where m is the fractional maximal operator and m m is the hardy littlewood maximal operator iterated m times the second inequality is a consequence of the first one and shows that the operator i b m is bounded from l p m p m m 1 p w x d x to l p w x d x where m 1 p is the integer part of m 1 p and no condition on the weight w is required from the first inequality we also obtain weighted l p l q estimates for i b m generalizing the classical results of muckenhoupt and wheeden for the fractional integral operator

surfaces in three dimensional digital images this is one of a series of reports on the digital geometry of three dimensional images such as those produced by computed tomography in this report we define simple surface points and simple closed surfaces and show that any connected collection of simple surface points form a simple closed surface thus proving a three dimensional analog of the two dimensional jordan curve theorem we also show that the converse is not a theorem in contrast to the two dimensional case and discuss more complex surface types finally we show that the two dimensional analog of our definition of simple closed surface characterizes simple closed curves but that several other characterizations of 2 d curves when extended to 3 d are not adequate to characterize surfaces

weighted sobolev morrey estimates for hypoelliptic operators with drift on homogeneous groups abstract let x 0 x 1 x q be left invariant real vector fields on the homogeneous group g and assume that x 1 x q are homogeneous of degree one and x 0 is homogeneous of degree two in this paper we consider the following hypoelliptic operator with drift l i j 1 q a i j x i x j a 0 x 0 where a i j is a constant matrix satisfying the elliptic condition in r q and a 0 0 and obtain weighted sobolev morrey estimates and morrey estimates with two weights of l by establishing boundedness of several singular integrals and interpolation lemmas on g

surface area estimation of digitized 3d objects using quasi monte carlo methods a novel and efficient quasi monte carlo method for estimating the surface area of digitized 3d objects in the volumetric representation is presented it operates directly on the original digitized objects without any surface reconstruction procedure based on the cauchy crofton formula from integral geometry the method estimates the surface area of a volumetric object by counting the number of intersection points between the object s boundary surface and a set of uniformly distributed lines generated with low discrepancy sequences using a clustering technique we also propose an effective algorithm for computing the intersection of a line with the boundary surface of volumetric objects a number of digitized objects are used to evaluate the performance of the new method for surface area measurement

representation of enclosing surfaces from simple voxelized objects by means of a chain code a chain code for representing three dimensional 3d simple objects is defined once digitalized any solid composed of voxels and homeomorphic to the sphere can be described by means of a codified sequence of faces in the enclosing surface this sequence is obtained from a hamiltonian cycle in the face adjacency graph of such a surface for the proposed code each chain element takes one of nine possible values and the length of a chain is determined by the number of faces in the surface since this code only considers relative changes of direction the descriptor is invariant under rotation and translation we also show some simple operations over the chain to make this descriptor invariant under mirroring and complement transformations finally we present some results of this code applied to large objects and demonstrate its convenience over other codes part of the relevance of this work is the lossless compact representation of 3d objects by using a single chain regardless of its position and orientation

uncertainty principle estimates for vector fields abstract we derive weighted norm estimates for integral operators of potential type and for their related maximal operators these operators are generalizations of the classical fractional integrals and fractional maximal functions the norm estimates are derived in the context of a space of homogeneous type the conditions required of the weight functions involve generalizations of the fefferman phong r bump condition the results improve some earlier ones of the same kind and they also extend to homogeneous spaces some estimates that were previously known to hold only in the classical euclidean setting

on cone invariant linear matrix inequalities an exact solution for a special class of cone preserving linear matrix inequalities lmis is developed by using a generalized version of the classical perron frobenius theorem the optimal value is shown to be equal to the spectral radius of an associated linear operator this allows for a much more efficient computation of the optimal solution using for instance power iteration type algorithms this particular lmi class appears in the computation of upper bounds for some generalizations of the structured singular value spl mu spherical spl mu and in a class of rank minimization problems previously studied examples and comparisons with existing techniques are provided

lowest rank solutions of continuous and discrete lyapunov equations over symmetric cone abstract the low rank solutions of continuous and discrete lyapunov equations are of great importance but generally difficult to compute in control system analysis and design fortunately mesbahi and papavassilopoulos 1997 30 showed that with the semidefinite cone constraint the lowest rank solutions of the discrete lyapunov inequality can be efficiently solved by a linear semidefinite programming in this paper we further show that the lowest rank solutions of both the continuous and discrete lyapunov equations over symmetric cone are unique and can be exactly solved by their convex relaxations the symmetric cone linear programming problems therefore they are polynomial time solvable since the underlying symmetric cone is a more general algebraic setting which contains the semidefinite cone as a special case our results also answer an open question proposed by recht fazel and parrilo 2010 36

endpoint estimates for commutators of singular integral operators abstract we prove endpoint estimates for commutatotrs of singular integrals with bmo functions we first show that they satisfy l log l type inequalities and then that there is a hardy space type estimate where the usual atomic hardy space is replaced by an appropriate subspace

multiple relaxation time lattice boltzmann model for the convection and anisotropic diffusion equation a lattice boltzmann model with a multiple relaxation time mrt collision operator for the convection diffusion equation is presented the model uses seven discrete velocities in three dimensions d3q7 model the off diagonal components of the relaxation time matrix which originate from the rotation of the principal axes enable us to take into account full anisotropy of diffusion an asymptotic analysis of the model equation with boundary rules for the dirichlet and neumann type specified flux conditions is carried out to show that the model is first and second order accurate in time and space respectively the results of the analysis are verified by several numerical examples it is also shown numerically that the error of the mrt model is less sensitive to the variation of the relaxation time coefficients than that of the classical bgk model in addition an alternative treatment for the neumann type boundary condition that improves the accuracy on a curved boundary is presented along with a numerical example of a spherical boundary

some open problems in matrix theory arising in linear systems and control control theory has long provided a rich source of motivation for developments in matrix theory accordingly we discuss some open problems in matrix theory arising from theoretical and practical issues in linear systems theory and feedback control the problems discussed include robust stability matrix exponent induced norms stabilizability and pole assignability and nonstandard matrix equations a substantial number of references are included to acquaint matrix theorists with problems and trends in this application area feedback control theory has long provided a rich source of motivation for developments in matrix theory the purpose of this paper is to discuss several open problems in matrix theory that arise from theoretical and practical issues in feedback control theory and the associated area of linear systems theory many of these problems are remarkably simple to state are of intense interest in control theory and applications and yet remain unsolved besides leading to the resolution of these problems it is hoped that this paper will help to stimulate increased interaction between matrix and control theorists accord

rank minimization with applications to image noise removal abstract rank minimization problem has a wide range of applications in different areas however since this problem is np hard and non convex the frequently used method is to replace the matrix rank minimization with nuclear norm minimization nuclear norm is the convex envelope of the matrix rank and it is more computationally tractable matrix completion is a special case of rank minimization problem in this paper we consider directly using matrix rank as the regularization term instead of nuclear norm in the cost function for matrix completion problem the solution is analyzed and obtained by a hard thresholding operation on the singular values of the observed matrix then by exploiting patch based nonlocal self similarity scheme we apply the proposed rank minimization algorithm to remove white gaussian additive noise in images gamma multiplicative noise is also removed in logarithm domain the experimental results illustrate that the proposed algorithm can remove noises in images more efficiently than nuclear norm can do and the results are also competitive with those obtained by using the existing state of the art noise removal methods in the literature

the complex structured singular value a tutorial introduction to the complex structured singular value is presented with an emphasis on the mathematical aspects of the based methods discussed here have been useful for analysing the performance and robustness properties of linear feedback systems several tests for robust stability and performance with computable bounds for transfer functions and their state space realizations are compared and a simple synthesis problem is studied uncertain systems are represented using linear fractional transformations lfts which naturally unify the frequency domain and state space methods

approximation of the joint spectral radius using sum of squares we provide an asymptotically tight computationally efficient approximation of the joint spectral radius of a set of matrices using sum of squares sos programming the approach is based on a search for an sos polynomial that proves simultaneous contractibility of a finite set of matrices we provide a bound on the quality of the approximation that unifies several earlier results and is independent of the number of matrices additionally we present a comparison between our approximation scheme and earlier techniques including the use of common quadratic lyapunov functions and a method based on matrix liftings theoretical results and numerical investigations show that our approach yields tighter approximations

an input output approach to structured stochastic uncertainty we consider linear time invariant systems with exogenous stochastic disturbances and in feedback with structured stochastic uncertainties this setting encompasses linear systems with both additive and multiplicative noise our concern is to characterize second order properties such as mean square stability mss and performance a purely input output treatment of these systems is given without recourse to state space models and thus the results are applicable to certain classes of distributed systems we derive necessary and sufficient conditions for mss in terms of the spectral radius of a linear matrix operator whose dimension is that of the number of uncertainties rather than the dimension of any underlying state space models our condition is applicable to the case of correlated uncertainties and reproduces earlier results for uncorrelated uncertainties for cases where state space realizations are given linear matrix inequality equivalents of the input output conditions are given

on the rank minimization problem and its control applications we present several new results on the rank minimization problem over sets which are defined by linear matrix inequalities we proceed to apply some of these results to propose a computational procedure for determining lower and upper bounds for the minimal order dynamic output feedback which stabilizes a given linear time invariant plant

weighted inequalities and pointwise estimates for the multilinear fractional integral and maximal operators abstract in this article we prove weighted norm inequalities and pointwise estimates between the multilinear fractional integral operator and the multilinear fractional maximal as a consequence of these estimations we obtain weighted weak and strong inequalities for the multilinear fractional maximal operator or function in particular we extend some results given in carro et al 2005 7 to the multilinear context on the other hand we prove weighted pointwise estimates between the multilinear fractional maximal operator m b associated to a young function b and the multilinear maximal operators m m 0 t b t 1 n m n m n m as an application of these estimate we obtain a direct proof of the l p l q boundedness results of m b for the case b t t and b k t t 1 log t k when 1 q 1 p n we also give sufficient conditions on the weights involved in the boundedness results of m b that generalizes those given in moen 2009 22 for b t t finally we prove some boundedness results in banach function spaces for a generalized version of the multilinear fractional maximal operator

digital representation schemes for 3d curves digital representation of three dimensional 3 d curves is considered two dimensional representation schemes are first reviewed then methods applicable to curves in three and higher dimensional spaces are described desirable properties of curve representation schemes are identified and used as a basis for qualitative and quantitative comparison of the various methods that have been suggested conclusions concerning the selection of a digital 3 d curve representation scheme for shape analysis and coding are drawn it is shown that grid intersect quantization is a poor choice for curve representation in 3 d space and that cube quantization that leads to 6 connected chain codes meets all the identified requirements and should be preferred

adaptive estimation of normals and surface area for discrete 3 d objects application to snow binary data from x ray tomography estimating the normal vector field on the boundary of discrete three dimensional objects is essential for rendering and image measurement problems most of the existing algorithms do not provide an accurate determination of the normal vector field for shapes that present edges here we propose a new and simple computational method in order to obtain accurate results on all types of shapes whatever their local convexity degree the presented method is based on the gradient vector field analysis of the object distance map this vector field is adaptively filtered around each surface voxel using angle and symmetry criteria so that as many relevant contributions as possible are accounted for this optimizes the smoothing of digitization effects while preserving relevant details of the processed numerical object thanks to the precise normal field obtained a projection method can be proposed to immediately derive the surface area from a raw discrete object an empirical justification of the validity of such an algorithm in the continuous limit is also provided some results on simulated data and snow images from x ray tomography are presented compared to the marching cubes and convex hull results and discussed