varieties of nilpotent lie algebras of dimension less than six the set of all nilpotent ide algebra multiplications on a fixed vector space of dimension n over a field k forms an affine variety n isomorphism classes of these lie algebras are orbits in i under the change of basis action of gl n k by examining the orbit structure of n for small n we hope to gain a new perspective on the classification of nilpotent lie algebras in this paper we determine completely the orbit structure of the variety n when n 6 5 and k c we also show that the homomorphic image of a nilpotent lie algebra is in the closure of the orbit of that lie algebra taking the direct product with an abelian lie algebra to make the dimension right some of the orbit structure of n for n 6 and other geometric properties of n were described by vergne in vl v2 v3 recently carles has used vergne s results to study the variety of all lie algebras cl our motivation for studying the classification of nilpotent lie algebras is to gain information about the classification of nilpotent groups the correspondence between nilpotent lie algebras and nilpotent groups is described in go

generalizing the hough transform to detect arbitrary shapes abstract the hough transform is a method for detecting curves by exploiting the duality between points on a curve and parameters of that curve the initial work showed how to detect both analytic curves 1 2 and non analytic curves 3 but these methods were restricted to binary edge images this work was generalized to the detection of some analytic curves in grey level images specifically lines 4 circles 5 and parabolas 6 the line detection case is the best known of these and has been ingeniously exploited in several applications 7 8 9 we show how the boundaries of an arbitrary non analytic shape can be used to construct a mapping between image space and hough transform space such a mapping can be exploited to detect instances of that particular shape in an image furthermore variations in the shape such as rotations scale changes or figure ground reversals correspond to straightforward transformations of this mapping however the most remarkable property is that such mappings can be composed to build mappings for complex shapes from the mappings of simpler component shapes this makes the generalized hough transform a kind of universal transform which can be used to find arbitrarily complex shapes

heuristics for the fixed cost median problem we describe in this paper polynomial heuristics for three important hard problems the discrete fixed cost median problem the plant location problem the continuous fixed cost median problem in a euclidean space and the network fixed cost median problem with convex costs the heuristics for all the three problems guarantee error ratios no worse than the logarithm of the number of customer points the derivation of the heuristics is based on the presentation of all types of median problems discussed as a set covering problem

contractions of low dimensional lie algebras theoretical background of continuous contractions of finite dimensional lie algebras is rigorously formulated and developed in particular known necessary criteria of contractions are collected and new criteria are proposed a number of requisite invariant and semi invariant quantities are calculated for wide classes of lie algebras including all low dimensional lie algebras an algorithm that allows one to handle one parametric contractions is presented and applied to low dimensional lie algebras as a result all one parametric continuous contractions for both the complex and real lie algebras of dimensions not greater than 4 are constructed with intensive usage of necessary criteria of contractions and with studying correspondence between real and complex cases levels and colevels of low dimensional lie algebras are discussed in detail properties of multiparametric and repeated contractions are also investigated

on detection of multiple object instances using hough transforms hough transform based methods for detecting multiple objects use nonmaxima suppression or mode seeking to locate and distinguish peaks in hough images such postprocessing requires the tuning of many parameters and is often fragile especially when objects are located spatially close to each other in this paper we develop a new probabilistic framework for object detection which is related to the hough transform it shares the simplicity and wide applicability of the hough transform but at the same time bypasses the problem of multiple peak identification in hough images and permits detection of multiple objects without invoking nonmaximum suppression heuristics our experiments demonstrate that this method results in a significant improvement in detection accuracy both for the classical task of straight line detection and for a more modern category level pedestrian detection problem

3 lie algebras with an ideal n we define the hypo nilpotent ideal in n lie algebras and obtain all solvable 3 lie algebras with an m dimensional simplest filiform 3 lie algebra as a maximal hypo nilpotent ideal we prove that the dimension of such solvable 3 lie algebras is at most m 2 and there is no solvable 3 lie algebra with the simplest filiform 3 lie algebra as the nilradical

robust object detection with interleaved categorization and segmentation this paper presents a novel method for detecting and localizing objects of a visual category in cluttered real world scenes our approach considers object categorization and figure ground segmentation as two interleaved processes that closely collaborate towards a common goal as shown in our work the tight coupling between those two processes allows them to benefit from each other and improve the combined performance the core part of our approach is a highly flexible learned representation for object shape that can combine the information observed on different training examples in a probabilistic extension of the generalized hough transform the resulting approach can detect categorical objects in novel images and automatically infer a probabilistic segmentation from the recognition result this segmentation is then in turn used to again improve recognition by allowing the system to focus its efforts on object pixels and to discard misleading influences from the background moreover the information from where in the image a hypothesis draws its support is employed in an mdl based hypothesis verification stage to resolve ambiguities between overlapping hypotheses and factor out the effects of partial occlusion an extensive evaluation on several large data sets shows that the proposed system is applicable to a range of different object categories including both rigid and articulated objects in addition its flexible representation allows it to achieve competitive object detection performance already from training sets that are between one and two orders of magnitude smaller than those used in comparable systems

closed form line segment extraction using the hough transform this paper proposes a novel closed form solution to complete line segment extraction given a voting angle in image space the voting distribution is analyzed and two functional relationships are deduced regarding the corresponding column in hough space voting along the distance axis is considered as being a random variable and voting values in cells are considered as forming a probability distribution statistical characteristics of this distribution are used to fit a quadratic polynomial curve and a linear curve direction length and width of a line segment are simultaneously computed in a closed form based on coefficients of fitted quadratic polynomial curves the midpoint of a line segment is determined based on the fitted linear curve the method is tested on simulated and real world images results show that the proposed closed form solution is feasible in the presence of quantization errors or image noise highlightswe proposed a method for obtaining the complete set of line segment parameters the direction length and width of a line segment are extracted simul taneously in a closed form we provided a complete theoretical derivation of voting variance with respect to voting angle the midpoint of a line segment is determined by the coefficients of the fitted linear curve parallel crossing and aligned line segments are discussed by analysing events in image space and hough space

minimizing energies with hierarchical costs computer vision is full of problems elegantly expressed in terms of energy minimization we characterize a class of energies with hierarchical costs and propose a novel hierarchical fusion algorithm hierarchical costs are natural for modeling an array of difficult problems for example in semantic segmentation one could rule out unlikely object combinations via hierarchical context in geometric model estimation one could penalize the number of unique model families in a solution not just the number of models a kind of hierarchical mdl criterion hierarchical fusion uses the well known expansion algorithm as a subroutine and offers a much better approximation bound in important cases

a coarse to fine strategy for multiclass shape detection multiclass shape detection in the sense of recognizing and localizing instances from multiple shape classes is formulated as a two step process in which local indexing primes global interpretation during indexing a list of instantiations shape identities and poses is compiled constrained only by no missed detections at the expense of false positives global information such as expected relationships among poses is incorporated afterward to remove ambiguities this division is motivated by computational efficiency in addition indexing itself is organized as a coarse to fine search simultaneously in class and pose this search can be interpreted as successive approximations to likelihood ratio tests arising from a simple naive bayes statistical model for the edge maps extracted from the original images the key to constructing efficient hypothesis tests for multiple classes and poses is local oring in particular spread edges provide imprecise but common and locally invariant features natural tradeoffs then emerge between discrimination and the pattern of spreading these are analyzed mathematically within the model based framework and the whole procedure is illustrated by experiments in reading license plates

on 3 lie algebras with abelian ideals and subalgebras abstract in this paper we study the maximal dimension l of abelian subalgebras and the maximal dimension l of abelian ideals of m dimensional 3 lie algebras l over an algebraically closed field we show that these dimensions do not coincide if the field is of characteristic zero even for nilpotent 3 lie algebras we then prove that 3 lie algebras with l m 2 are 2 step solvable see definition in section 2 furthermore we give a precise description of these 3 lie algebras with one or two dimensional derived algebras in addition we provide a classification of 3 lie algebras with l dim l 2 we also obtain the classification of 3 lie algebras with l dim l 1 and with their derived algebras of one dimension

numerical computations with the trace formula and the selberg eigenvalue conjecture we verify the selberg eigenvalue conjecture for congruence groups of small squarefree conductor improving on a result of huxley m n huxley introduction to kloostermania in elementary and analytic theory of numbers banach center publ 17 warsaw 1985 217 306 the main tool is the selberg trace formula which unlike previous geometric methods allows for treatment of cases where the eigenvalue 1 4 is present we present a few other sample applications including the classification of even 2 dimensional galois representations of small squarefree conductor

powers of the euler product and commutative subalgebras of a complex simple lie algebra if mathfrak g is a complex simple lie algebra and k does not exceed the dual coxeter number of mathfrak g then the absolute value of the kth coefficient of the dim mathfrak g power of the euler product may be given by the dimension of a subspace of wedge k mathfrak g defined by all abelian subalgebras of mathfrak g of dimension k this has implications for all the coefficients of all the powers of the euler product involved in the main results are dale peterson s 2rank theorem on the number of abelian ideals in a borel subalgebra of mathfrak g an element of type and my heat kernel formulation of macdonald s function theorem a set dalcove of special highest weights parameterized by all the alcoves in a weyl chamber generalizing young diagrams of null m core when mathfrak g text lie mathit sl m mathbb c and the homology and cohomology of the nil radical of the standard maximal parabolic subalgebra of the affine kac moody lie algebra

on the asymptotic behavior of counting functions associated to degenerating hyperbolic riemann surfaces abstract in this article we will study what we call weighted counting functions on hyperbolic riemann surfaces of finite volume ifmis compact then we define the weighted counting function forw 0 to be n m w t n t t n w where n is the set of eigenvalues of the laplacian which acts on the space of smooth functions onm ifmis non compact then we define the weighted counting functionnm w t via the inverse laplace transform from which one can express the weighted counting function in terms of spectral data associated tom see proposition 5 1 and remark 5 2 using the convergence results from jl2 concerning the regularized heat trace on finite volume hyperbolic riemann surfaces we shall prove the following results t heorem let mldenote a degenerating family of compact or non compact hyperbolic riemann surfaces of finite volume which converges to the non compact hyperbolic surface m0 let s ldenote the set of lengths of pinching geodesics on the family of surfaces ml for w 0and t 1 4 let g l w t 2 w 1 16 1 2 n 1 l k s 1 l k sinfh nl k 2 t 1 4 nl k 2 2 w 1 2 2 j w 1 2 nl k t 1 4 where js x is the j bessel function and x is the standard gamma function for0 t 3 2andt 0 we have n m l w t g l w t n m 0 w t o 1 as l 0 b for w 0andt 0 we have n m l w t g l w t o l k s l log 1 l k as l 0 c for w 0andt 1 4 we have g l w t 2 w 1 t 1 4 w 1 2 4 1 2 w 3 2 l k s l log 1 l k o 1 as l 0 observe that by takingt

on foundation of the generalized nambu mechanics we outline basic principles of a canonical formalism for the nambu mechanics a generalization of hamiltonian mechanics proposed by yoichiro nambu in 1973 it is based on the notion of a nambu bracket which generalizes the poisson bracket a binary operation on classical observables on the phase space to the multiple operation of higher ordern 3 nambu dynamics is described by the phase flow given by nambu hamilton equations of motion a system of ode s which involvesn 1 hamiltonians we introduce the fundamental identity for the nambu bracket a generalization of the jacobi identity as a consistency condition for the dynamics we show that nambu bracket structure defines a hierarchy of infinite families of subordinated structures of lower order including poisson bracket structure which satisfy certain matching conditions the notion of nambu bracket enables us to define nambu poisson manifolds phase spaces for the nambu mechanics which turn out to be more rigid than poisson manifolds phase spaces for the hamiltonian mechanics we introduce the analog of the action form and the action principle for the nambu mechanics in its formulation dynamics of loops n 2 dimensional chains for the generaln ary case naturally appears we discuss several approaches to the quantization of nambu mechanics based on the deformation theory path integral formulation and on nambu heisenberg commutation relations in the latter formalism we present an explicit representation of the nambu heisenberg relation in then 3 case we emphasize the role ternary and higher order algebraic operations and mathematical structures related to them play in passing from hamilton s to nambu s dynamical picture

region competition unifying snakes region growing and bayes mdl for multiband image segmentation we present a novel statistical and variational approach to image segmentation based on a new algorithm named region competition this algorithm is derived by minimizing a generalized bayes minimum description length mdl criterion using the variational principle the algorithm is guaranteed to converge to a local minimum and combines aspects of snakes balloons and region growing the classic snakes balloons and region growing algorithms can be directly derived from our approach we provide theoretical analysis of region competition including accuracy of boundary location criteria for initial conditions and the relationship to edge detection using filters it is straightforward to generalize the algorithm to multiband segmentation and we demonstrate it on gray level images color images and texture images the novel color model allows us to eliminate intensity gradients and shadows thereby obtaining segmentation based on the albedos of objects it also helps detect highlight regions

constructing simple stable descriptions for image partitioning a new formulation of the image partitioning problem is presented construct a complete and stable description of an image in terms of a specified descriptive language that is simplest in the sense of being shortest we show that a descriptive language limited to a low order polynomial description of the intensity variation within each region and a chain code like description of the region boundaries yields intuitively satisfying partitions for a wide class of images the advantage of this formulation is that it can be extended to deal with subsequent steps of the image understanding problem or to deal with other attributes such as texture in a natural way by augmenting the descriptive language experiments performed on a variety of both real and synthetic images demonstrate the superior performance of this approach over partitioning techniques based on clustering vectors of local image attributes and standard edge detection techniques

on skew symmetric maps on lie algebras for certain lie subalgebras l of associative algebras q we characterize the multi additive skew symmetric maps f l n q which are covariant under the action of l

classification of orbit closures of 4 dimensional complex lie algebras abstract let l n c be the variety of complexn dimensional lie algebras the groupgln c acts on it via change of basis an orbito under this action consists of all structures isomorphic to the aim of this paper is to give a complete classification of orbit closures of 4 dimensional lie algebras i e determining all o where l 4 c starting with a classification of complex lie algebras of dimensionn 4 we study the behavior of several lie algebra invariants under degeneration i e under transition to the orbit closure as a corollary we will show that all degenerations in l 3 c can be realized via a one parameter subgroup but this is not the case in l 4 c

demisting the hough transform for 3d shape recognition and registration in applying the hough transform to the problem of 3d shape recognition and registration we develop two new and powerful improvements to this popular inference method the first intrinsic hough solves the problem of exponential memory requirements of the standard hough transform by exploiting the sparsity of the hough space the second minimum entropy hough explains away incorrect votes substantially reducing the number of modes in the posterior distribution of class and pose and improving precision our experiments demonstrate that these contributions make the hough transform not only tractable but also highly accurate for our example application both contributions can be applied to other tasks that already use the standard hough transform

on selberg s small eigenvalue conjecture and residual eigenvalues we show that selberg s eigenvalue conjecture concern ing small eigenvalues of the automorphic laplacian for congruence groups is equivalent to a conjecture about the non existence of residual eigenvalues for a perturbed system we prove this using a combination of methods from asymptotic perturbation theory and number theory