interpreting perspective images a fundamental problem in computer vision is how to determine the 3 d spatial orientation of curves and surfaces appearing in an image the problem is generally underconstrained and is complicated by the fact that metric properties such as orientation and length are not invariant under projection under perspective projection the correct model for most real images the transform is nonlinear and therefore hard to invert two constructive methods are presented the first finds the orientation of parallel lines and planes by locating vanishing points and vanishing lines the second determines the orientation of planes by backprojection of two intrinsic properties of contours angle magnitude and curvature

maps between non commutative spaces let j be a graded ideal in a not necessarily commutative graded k algebra a a 0 a 1 in which dim k a i for all i we show that the map a a j induces a closed immersion i proj nc a j proj nc a between the non commutative projective spaces with homogeneous coordinate rings a and a j we also examine two other kinds of maps between non commutative spaces first a homomorphism a b between not necessarily commutative n graded rings induces an affine map proj nc b u proj nc a from a non empty open subspace u c proj nc b second if a is a right noetherian connected graded algebra not necessarily generated in degree one and a n is a veronese subalgebra of a there is a map proj nc a proj nc a n we identify open subspaces on which this map is an isomorphism applying these general results when a is a quotient of a weighted polynomial ring produces a non commutative resolution of a closed subscheme of a weighted projective space

structure and motion from line segments in multiple images this paper presents a new method for recovering the three dimensional structure of a scene composed of straight line segments using the image data obtained from a moving camera the recovery algorithm is formulated in terms of an objective function which measures the total squared distance in the image plane between the observed edge segments and the projections perspective of the reconstructed lines this objective function is minimized with respect to the line parameters and the camera positions to obtain an estimate for the structure of the scene the effectiveness of this approach is demonstrated quantitatively through extensive simulations and qualitatively with actual image sequences the implementation is being made publicly available

noncommutative instantons and twistor transform recently n nekrasov and a schwarz proposed a modification of the adhm construction of instantons which produces instantons on a noncommutative deformation of 4 in this paper we study the relation between their construction and algebraic bundles on noncommutative projective spaces we exhibit one to one correspondences between three classes of objects framed bundles on a noncommutative 2 certain complexes of sheaves on a noncommutative 3 and the modified adhm data the modified adhm construction itself is interpreted in terms of a noncommutative version of the twistor transform we also prove that the moduli space of framed bundles on the noncommutative 2 has a natural hyperkahler metric and is isomorphic as a hyperkahler manifold to the moduli space of framed torsion free sheaves on the commutative 2 the natural complex structures on the two moduli spaces do not coincide but are related by an so 3 rotation finally we propose a construction of instantons on a more general noncommutative 4 than the one considered by nekrasov and schwarz a q deformed 4

the classification of 4 dimensional leibniz algebras abstract this paper is a contribution to the development of the non associative algebras theory more precisely this work deals with the classification of the complex 4 dimensional leibniz algebras note that the classification of 4 dimensional nilpotent complex leibniz algebras was obtained in albeverio et al 2006 1 therefore we will only consider non nilpotent case in this work

estimation of 3d structure and motion from image corners abstract this paper deals with motion estimation from image corner correspondences in two cases the orthogonal corner and the general corner with known space angles the contribution of the paper is in three folds first the three dimensional structure of a corner is recovered easily from its image by introducing a new coordinate system second it is shown that the one corner and two points correspondences over two views are sufficient to uniquely determine the motion i e the rotation and translation third experiments using both simulated data and real images are conducted which present good results

a note on canonical forms for matrix congruence abstract canonical forms for matrix congruence for general matrices are exhibited as an easy consequence of results presented in a recent paper by r c thompson on pencils of symmetric and skew symmetric matrices we also show how to exhibit standard equivalence class representatives in the orbits of the semiorthogonal groups acting naturally on skew symmetric forms

the geometry of arithmetic noncommutative projective lines abstract let k be a perfect field and let k k be a finite extension of fields an arithmetic noncommutative projective line is a noncommutative space of the form proj s k v where v be a k central two sided vector space over k of rank two and s k v is the noncommutative symmetric algebra generated by v over k defined by m van den bergh 26 we study the geometry of these spaces more precisely we prove they are integral we classify vector bundles over them we classify them up to isomorphism and we classify isomorphisms between them using the classification of isomorphisms we compute the automorphism group of an arithmetic noncommutative projective line

canonical matrices of isometric operators on indefinite inner product spaces abstract we give canonical matrices of a pair a b consisting of a nondegenerate form b and a linear operator a satisfying b ax ay b x y on a vector space over f in the following cases f is an algebraically closed field of characteristic different from 2 or a real closed field and b is symmetric or skew symmetric f is an algebraically closed field of characteristic 0 or the skew field of quaternions over a real closed field and b is hermitian or skew hermitian with respect to any nonidentity involution on f these classification problems are wild if b may be degenerate we use a method that admits to reduce the problem of classifying an arbitrary system of forms and linear mappings to the problem of classifying representations of some quiver this method was described in v v sergeichuk classification problems for systems of forms and linear mappings math ussr izv 31 3 1988 481 501

betti numbers of mcm modules over the cone of an elliptic normal curve abstract we apply orlov s equivalence to derive formulas for the betti numbers of maximal cohen macaulay modules over the cone an elliptic curve e x embedded into p n 1 by the full linear system o n x for n 3 the answers are given in terms of recursive sequences these results are applied to give a criterion of co koszulity in the last two sections of the paper we apply our methods to study the cases n 1 2 geometrically these correspond to the embedding of an elliptic curve into a weighted projective space the singularities of the corresponding cones are called minimal elliptic they were studied by k saito 13 where he introduced the notation e 8 for n 1 e 7 for n 2 and e 6 for the cone over a smooth cubic that is for the case n 3 for the singularities e 7 and e 8 we obtain formulas for the betti numbers and the numerical invariants of mcm modules analogous to the case of a plane cubic

canonical forms for mixed symmetric skewsymmetric quaternion matrix pencils canonical forms are described for pairs of quaternionic matrices or equivalently matrix pencils where one matrix is symmetric and the other matrix is skewsymmetric under strict equivalence and symmetry respecting congruence the symmetry is understood in the sense of a fixed involutory antiautomorphism of the skew field of the real quaternions the involutory antiautomorphism is assumed to be nonstandard i e other than the quaternionic conjugation some applications are developed such as canonical forms for quaternionic matrices under symmetry respecting congruence and canonical forms for matrices that are skewsymmetric with respect to a nondegenerate symmetric or skewsymmetric quaternion valued inner product

congruences of a square matrix and its transpose abstract it is known that any square matrix a over any field is congruent to its transpose at stas for some nonsingular s moreover s can be chosen such that s2 i that is s can be chosen to be involutory we show that a and at are congruent over any field f of characteristic not two with involution a a the involution can be the identity a t s t as for some nonsingular s moreover s can be chosen such that s s i that is s can be chosen to be coninvolutory the short and simple proof is based on sergeichuk s canonical form for congruence math ussr izvestiya 31 3 1988 481 it follows that any matrix a over f can be represented as a eb in which e is coninvolutory and b is symmetric

characterizing nilpotent leibniz algebras by a new bound on their second homologies abstract let l be a finite dimensional nilpotent leibniz algebra such that dim l n and dim l 2 m 0 in this paper we prove dim h l 2 l n m 2 n m m 2 where h l 2 l is the second leibniz homology of l as a consequence for a non abelian nilpotent leibniz algebra l we find that s l n 1 2 1 dim h l 2 l 0 furthermore we determine all finite dimensional nilpotent leibniz algebras with s l less than or equal to three

self adjoint operators and pairs of hermitian forms over the quaternions abstract we classify self adjoint operators and pairs of hermitian forms over the real quaternions by providing canonical matrix representations in the preliminaries we discuss the jordan canonical form theorem for quaternionic linear endomorphisms

a structure theorem for leibniz homology abstract presented is a structure theorem for the leibniz homology hl of an abelian extension of a simple real lie algebra g as applications results are stated for affine extensions of the classical lie algebras sl n r so n r and sp n r furthermore hl h is calculated when h is the lie algebra of the poincare group as well as the lie algebra of the affine lorentz group the structure theorem identifies all of these in terms of g invariants

a regularization algorithm for matrices of bilinear and sesquilinear forms abstract over a field or skew field f with an involution a a possibly the identity involution each singular square matrix a is congruent to a direct sum s as b j n 1 j n p 1 n 1 n p inwhich s is nonsingular and s s t b is nonsingular and is determined by a up to congruence and the ni by ni singular jordan blocks j n i and their multiplicities are uniquely determined by a we give a regularization algorithm that needs only elementary row operations to construct such a decomposition if f c respectively f r we exhibit a regularization algorithm that uses only unitary respectively real orthogonal transformations and a reduced form that can be achieved via a unitary congruence or congruence respectively a real orthogonal congruence the selfadjoint matrix pencil a a is decomposed by our regularization algorithm into the direct sum s a a s b b j n 1 j n 1 j n p j n p with selfadjoint summands

non commutative hopf algebra of formal diffeomorphisms abstract this paper deals with two hopf algebras which are the non commutative analogues of two different groups of formal power series the first group is the set of invertible series with the group law being multiplication of series while the second is the set of formal diffeomorphisms with the group law being a composition of series the motivation to introduce these hopf algebras comes from the study of formal series with non commutative coefficients invertible series with non commutative coefficients still form a group and we interpret the corresponding new non commutative hopf algebra as an alternative to the natural hopf algebra given by the co ordinate ring of the group which has the advantage of being functorial in the algebra of coefficients for the formal diffeomorphisms with non commutative coefficients this interpretation fails because in this case the composition is not associative anymore however we show that for the dual non commutative algebra there exists a natural co associative co product defining a non commutative hopf algebra moreover we give an explicit formula for the antipode which represents a non commutative version of the lagrange inversion formula and we show that its coefficients are related to planar binary trees then we extend these results to the semi direct co product of the previous hopf algebras and to series in several variables finally we show how the non commutative hopf algebras of formal series are related to some renormalization hopf algebras which are combinatorial hopf algebras motivated by the renormalization procedure in quantum field theory and to the renormalization functor given by the double tensor algebra on a bi algebra

quantum planes of weight 1 1 n 1 in this article we study the categories tails a arising from artin schelter regular algebras of global dimension 3 of weight 1 1 n where n is an integer greater than 1 we determine generators and relations for all such algebras and this information is used to classify the points and lines in tails a

noncommutative projective schemes an analogue of the concept of projective scheme is defined for noncommutative n graded algebras using the quotient category c of graded right a modules modulo its full subcategory of torsion modules we define proj a c a s where a is the object corresponding to the module aa and s is the autoequivalence defined by the shift of degrees the triples equivalent to proj a for a right noetherian graded algebra a are characterized in terms of a condition on extensions

some cohomologically rigid solvable leibniz algebras abstract in this paper we describe solvable leibniz algebras whose quotient algebra by one dimensional ideal is a lie algebra with rank equal to the length of the characteristic sequence of its nilpotent radical we prove that such leibniz algebra is unique and centerless also it is proved that the first and the second cohomology groups of the algebra with coefficients in the adjoint representation is trivial

pencils of complex and real symmetric and skew matrices abstract this expository paper establishes the canonical forms under congruence for pairs of complex or real symmetric or skew matrices the treatment is in the spirit of the well known book of gantmacher on matrix theory and may be regarded as a supplement to gantmacher s chapters on pencils of matrices

unbiased estimation and statistical analysis of 3 d rigid motion from two views the problem of estimating 3d rigid motion from point correspondences over two views is formulated as nonlinear least squares ls optimization and the statistical behaviors of the errors in the solution are analyzed by introducing a realistic model of noise described in terms of the covariance matrices of n vectors it is shown that the ls solution based on the epipolar constraint is statistically biased the geometry of this bias is described in both quantitative and qualitative terms finally an unbiased estimation scheme is presented and random number simulations are conducted to observe its effectiveness

determination of camera location from 2 d to 3 d line and point correspondences a method for the determination of camera location from two dimensional 2 d to three dimensional 3 d straight line or point correspondences is presented with this method the computations of the rotation matrix and the translation vector of the camera are separable first the rotation matrix is found by a linear algorithm using eight or more line correspondences or by a nonlinear algorithm using three or more line correspondences where the line correspondences are either given or derived from point correspondences then the translation vector is obtained by solving a set of linear equations based on three or more line correspondences or two or more point correspondences eight 2 d to 3 d line correspondences or six 2 d to 3 d point correspondences are needed for the linear approach three 2 d to 3 d line or point correspondences for the nonlinear approach good results can be obtained in the presence of noise if more than the minimum required number of correspondences are used

leibniz algebras with small derived ideal abstract a nilpotent leibniz algebra l is called extra special if dim z l dim l 2 1 in this paper the structure the second homology and the capability of extra special leibniz algebras are completely determined

subspaces of non commutative spaces this paper concerns the closed points closed subspaces open subspaces weakly closed and weakly open subspaces and effective divisors on a non commutative space

canonical matrices of bilinear and sesquilinear forms abstract canonical matrices are given for i bilinear forms over an algebraically closed or real closed field ii sesquilinear forms over an algebraically closed field and over real quaternions with any nonidentity involution and iii sesquilinear forms over a field f of characteristic different from 2 with involution possibly the identity up to classification of hermitian forms over finite extensions of f the canonical matrices are based on any given set of canonical matrices for similarity over f a method for reducing the problem of classifying systems of forms and linear mappings to the problem of classifying systems of linear mappings is used to construct the canonical matrices this method has its origins in representation theory and was devised in v v sergeichuk classification problems for systems of forms and linear mappings math ussr izv 31 1988 481 501

hopf dense galois extensions with applications abstract let h be a finite dimensional hopf algebra and let a be a left h module algebra motivated by the study of the isolated singularities of a h and the endomorphism ring end a h a we introduce the concept of hopf dense galois extensions in this paper hopf dense galois extensions yield certain equivalences between the quotient categories over a and a h a special class of hopf dense galois extensions consists of the so called densely group graded algebras which are weaker versions of strongly graded algebras a weaker version of dade s theorem holds for densely group graded algebras as applications we recover the classical equivalence of the noncommutative projective scheme over a noetherian n graded algebra a and its d th veronese subalgebra a d respectively hopf dense galois extensions are also applied to the study of noncommutative graded isolated singularities

canonical forms for complex matrix congruence and congruence abstract canonical forms for congruence and congruence of square complex matrices were given by horn and sergeichuk in linear algebra appl 389 2004 347 353 based on sergeichuk s paper math ussr izvestiya 31 3 1988 481 501 which employed the theory of representations of quivers with involution we use standard methods of matrix analysis to prove directly that these forms are canonical our proof provides explicit algorithms to compute all the blocks and parameters in the canonical forms we use these forms to derive canonical pairs for simultaneous congruence of pairs of complex symmetric and skew symmetric matrices as well as canonical forms for simultaneous congruence of pairs of complex hermitian matrices

integral non commutative spaces abstract a non commutative space x is a grothendieck category mod x we say x is integral if there is an indecomposable injective x module e x such that its endomorphism ring is a division ring and every x module is a subquotient of a direct sum of copies of e x a noetherian scheme is integral in this sense if and only if it is integral in the usual sense we show that several classes of non commutative spaces are integral we also define the function field and generic point of an integral space and show that these notions behave as one might expect

recovery of the three dimensional shape of an object from a single view given a single picture which is a projection of a three dimensional scene onto the two dimensional picture plane we usually have definite ideas about the 3 d shapes of objects to do this we need to use assumptions about the world and the image formation process since there exist a large number of shapes which can produce the same picture the purpose of this paper is to identify some of these assumptions mostly geometrical ones by demonstrating how the theory and techniques which exploit such assumptions can provide a systematic shape recovery method the method consists of two parts the first is the application of the origami theory which models the world as a collection of plane surfaces and recovers the possible shapes qualitatively the second is the technique of mapping image regularities into shape constraints for recovering the probable shapes quantitatively actual shape recovery from a single view is demonstrated for the scenes of an object such as a box and a chair given a single image the method recovers the 3 d shapes of an object in it and generates images of the same object as we would see it from other directions