conformal geometry and its applications on 3d shape matching recognition and stitching three dimensional shape matching is a fundamental issue in computer vision with many applications such as shape registration 3d object recognition and classification however shape matching with noise occlusion and clutter is a challenging problem in this paper we analyze a family of quasi conformal maps including harmonic maps conformal maps and least squares conformal maps with regards to 3d shape matching as a result we propose a novel and computationally efficient shape matching framework by using least squares conformal maps according to conformal geometry theory each 3d surface with disk topology can be mapped to a 2d domain through a global optimization and the resulting map is a diffeomorphism i e one to one and onto this allows us to simplify the 3d shape matching problem to a 2d image matching problem by comparing the resulting 2d parametric maps which are stable insensitive to resolution changes and robust to occlusion and noise therefore highly accurate and efficient 3d shape matching algorithms can be achieved by using the above three parametric maps finally the robustness of least squares conformal maps is evaluated and analyzed comprehensively in 3d shape matching with occlusion noise and resolution variation in order to further demonstrate the performance of our proposed method we also conduct a series of experiments on two computer vision applications i e 3d face recognition and 3d nonrigid surface alignment and stitching

graded morita theory for infinite groups abstract having shown that not all graded rings are graded equivalent via classical morita theory to a skew group ring we extend classical morita theory which is based on rings with identity to a generalized graded morita theory for rings with local units this enables us to give necessary and sufficient conditions for graded morita equivalence between two rings graded by a group we show that the strongly graded property is a graded morita invariant and we show that a graded ring is graded morita equivalent to a skew group ring namely r g g if and only if it is strongly graded these results extend the classical cohen montgomery duality theory to rings graded by infinite groups the fundamental tools include rings with local units and the results of boisen

the multivariate complex normal distribution a generalization the multivariate complex normal distribution usually employed in the literature is a special case since certain restrictions have been imposed on the covariances of the real and imaginary parts of its variables a more general distribution is proposed of which the usual distribution is shown to be a special case

on dubrovin valuation rings in crossed product algebras let f be a field and let v be a valuation ring in f if a is a central simple f algebra then v can be extended to a dubrovin valuation ring in a in this paper we consider the structure of dubrovin valuation rings with center v in crossed product algebras k f g f where k f is a finite galois extension with galois group g unramified over v and f is a normalized two cocycle in the case where v is indecomposed in k we introduce a family of orders naturally associated to f examine their basic properties and determine which of these orders is dubrovin in the case where v is decomposed we determine the structure in the case of certain special discrete finite rank valuations

reduction techniques for strongly graded rings and finite representation type abstract we present reduction techniques for studying the category of lattices over strongly graded orders in particular we apply these techniques in order to reduce the problem of classifying those strongly graded orders with finite representation type to the case where the coefficient ring is a maximal order in a division ring

group graded rings and duality we give an alternative construction of the duality between finite group actions and group gradings on rings which was shown by cohen and montgomery in 1 this duality is then used to extend known results on skew group rings to corresponding results for large classes of group graded rings finally we modify the construction slightly to handle infinite groups introduction in the first section we give an alternate construction of the duality between finite group actions and group gradings on rings which was shown by cohen and montgomery in 1 this duality is then used to extend results on skew group rings and their modules to corresponding results for large classes of groupgraded rings in the second section we modify the construction slightly to handle infinite groups and apply it to a theorem on prime and semiprime infinite crossed products i would like to thank my thesis advisor d s passman for his guidance and encouragement throughout the writing of this paper 1 finite group graded rings and the smash product let g be a multiplicative group an associative ring with identity is said to be g graded if 1 r e r x xec is a direct sum of additive subgroups r x with r x r y c r xy it follows that necessarily 1r e r 1 so that each r x is a unitary r l bimodule if r e r we write r x for the component of r in r x so that r exgcr x r is said to be strongly g graded if r x r y r xy for all x y e g throughout this section r is assumed to be g graded where g is a finite group with igi n let mc r denote the set of n x n matrices over r with the rows and columns indexed by the elements of g if a c mg r we write ax y for the entry in the x y position of a then if a 1 c mc r the matrix product a 8 is given by 2 x x y e xx zaz y z e c if u c g is any subset of g let r u ex eur x in particular r r g now suppose h c g is a subgroup of g we define r h c mc r by r h a e ms r ax y e r xhy l that is r h ex ygcr xhy l e x y where e x y is the matrix unit with 1r in the x y position and zeroes elsewhere note that r g mc r received by the editors september 27 1984 1980 mathematics subject classification primary 16a03 c 1985 american mathematical society 0002 9947 85 1 00 25 per page

automorphisms of tiled orders abstract let be a tiled r order we give a description of autr as the semidirect product of inn and a certain subgroup of aut q where q is the link graph of additionally we give criteria for determining when an element of aut q belongs to this subgroup in terms of the exponent matrix for

crossed product algebras defined by separable extensions abstract we generalize the classical construction of crossed product algebras defined by finite galois field extensions to finite separable field extensions by studying properties of rings graded by groupoids we are able to calculate the jacobson radical of these algebras we use this to determine when the analogous construction of crossed product orders yield azumaya maximal or hereditary orders in a local situation thereby we generalize results by haile larson and sweedler

weakly azumaya algebras abstract we define a class of algebras weakly azumaya algebras which includes both azumaya algebras and weak crossed products cf haile 1982 j algebra 74 270 279 1983 j algebra 91 521 539 and haile et al 1983 amer j math 105 no 3 689 814 just as with azumaya algebras these have a rank function whose values at localizations of the center are always a square after a general description of these algebras we specialize to the case where the center is a field f the jacobson radical need not be 0 and we prove a wedderburn principal theorem for these algebras our class is closed under extension of scalars and under tensor products and yields an interesting monoid which generalizes the brauer group our monoid is a union of groups called stalks in each of which the unit element is represented by an algebra called an idempotent algebra the ideal structure of members of the same stalk is the same a given stalk is not torsion but the kernel of the restriction map to the algebraic closure is torsion modulo the idempotent algebras at the end we consider low dimensional examples in detail

combinatorial ricci flows on surfaces we show that the analogue of hamilton s ricci flow in the combinatorial setting produces solutions which converge exponentially fast to thurston s circle packing on surfaces as a consequence a new proof of thurston s existence of circle packing theorem is obtained as another consequence ricci flow suggests a new algorithm to find circle packings

spectral generalized multi dimensional scaling multidimensional scaling mds is a family of methods that embed a given set of points into a simple usually flat domain the points are assumed to be sampled from some metric space and the mapping attempts to preserve the distances between each pair of points in the set distances in the target space can be computed analytically in this setting generalized mds is an extension that allows mapping one metric space into another that is mds into target spaces in which distances are evaluated numerically rather than analytically here we propose an efficient approach for computing such mappings between surfaces based on their natural spectral decomposition where the surfaces are treated as sampled metric spaces the resulting spectral gmds procedure enables efficient embedding by incorporating smoothness of the metric structure into the problem thereby substantially reducing the complexity involved in its solution while practically overcoming its non convex nature the method is compared to existing techniques that compute dense correspondence between shapes numerical experiments of the proposed method demonstrate its efficiency and accuracy compared to state of the art approaches especially when isometry invariance is a dominant property

the group of automorphisms of certain subalgebras of matrix algebras the purpose of this note is to find and characterize the full group of automorphisms of certain prime algebras which are subalgebras of matrix algebras over commutative rings here we extend the methods which we used in 6 to characterize automorphisms of certain upper triangular matrix rings for a central simple k algebra a the group of k automorphisms of a is extensively treated in the literature 5 theorem 6 21 other cases where the automorphism group of a ring is known are the automorphism group of a polynomial ring in two indeterminates over a field and the free algebra in two variables over a field 4 chap 61 finally if a is a prime p z algebra with center c then every c automorphism of a is x inner 7 example 3 91

computing surface uniformization using discrete beltrami flow in this paper we propose a novel algorithm for computing surface uniformization for surfaces with arbitrary topology according to the celebrated uniformization theorem all riemann surfaces can be classified as elliptic parabolic or hyperbolic our algorithm is able to work on all these cases by first constructing an initial map onto an appropriate domain such as a sphere or a polygon in the plane mathbb r 2 or the hyperbolic disk d and then morphing the diffeomorphism based on the discrete beltrami flow algorithm for high genus surfaces both the final mapping and the target domain are unknown which presents a challenge in general each such surface can be conformally mapped onto d modulo a discrete subgroup of all fractional linear transforms on d a conformal copy of the surface which is also its uniformization domain can be visualized as a fundamental polygon in d corresponding to the discrete subgroup where the generators of the discrete subgroup map each side of the fundament

hereditary crossed product orders over discrete valuation rings in this paper we consider weak crossed product orders af sx with coefficients in the integral closure s of a discrete valuation ring r in a tamely ramified galois extension of the field of fractions of r in the first section we compute the jacobson radical of af when s is local and we give a characterization of the hereditarity of the order in terms of the cocycle values in the second section we prove again in the local case that every in the inertia group for s r must belong to g f 1 is a unit of s in the final section we compute the jacobson radical in the general case s is semilocal and show how the hereditarity of af can be determined locally under an additional hypothesis

shape analysis of planar multiply connected objects using conformal welding shape analysis is a central problem in the field of computer vision in 2d shape analysis classification and recognition of objects from their observed silhouettes are extremely crucial but difficult it usually involves an efficient representation of 2d shape space with a metric so that its mathematical structure can be used for further analysis although the study of 2d simply connected shapes has been subject to a corpus of literatures the analysis of multiply connected shapes is comparatively less studied in this work we propose a representation for general 2d multiply connected domains with arbitrary topologies using conformal welding a metric can be defined on the proposed representation space which gives a metric to measure dissimilarities between objects the main idea is to map the exterior and interior of the domain conformally to unit disks and circle domains unit disk with several inner disks removed using holomorphic 1 forms a set of diffeomorphisms of the unit circle mathbb s 1 can be obtained which together with the conformal modules are used to define the shape signature a shape distance between shape signatures can be defined to measure dissimilarities between shapes we prove theoretically that the proposed shape signature uniquely determines the multiply connected objects under suitable normalization we also introduce a reconstruction algorithm to obtain shapes from their signatures this completes our framework and allows us to move back and forth between shapes and signatures with that a morphing algorithm between shapes can be developed through the interpolation of the beltrami coefficients associated with the signatures experiments have been carried out on shapes extracted from real images results demonstrate the efficacy of our proposed algorithm as a stable shape representation scheme

frobenius and separable functors for the category of entwined modules over cowreaths ii applications abstract let h be a quasi hopf algebra we apply results obtained in 8 to give necessary and sufficient conditions for the forgetful functor from doi hopf modules two sided hopf modules or yetter drinfeld modules over h to representations of the underlying algebra to be frobenius resp separable we show that in some situations these conditions reduce to the unimodularity and or co semisimplicity of the quasi hopf algebra h

epsilon strongly graded rings separability and semisimplicity abstract we introduce the class of epsilon strongly graded rings and show that it properly contains both the class of strongly graded rings and the class of unital partial crossed products we determine precisely when an epsilon strongly graded ring is separable over its principal component thereby we simultaneously generalize a result for strongly group graded rings by n st sescu van den bergh and van oystaeyen and a result for unital partial crossed products by bagio lazzarin and paques we also show that the class of unital partial crossed products appears in the class of epsilon strongly graded rings in a fashion similar to how the classical crossed products present themselves in the class of strongly graded rings thereby we obtain in the special case of unital partial crossed products a short proof of a general result by dokuchaev exel and simon concerning when graded rings can be presented as partial crossed products we also provide some interesting classes of examples of separable epsilon strongly graded rings with finite as well as infinite grading groups in particular we obtain an answer to a question raised by le bruyn van den bergh and van oystaeyen in 1988

the ricci flow on the 2 sphere the classical uniformization theorem interpreted differential geomet rically states that any riemannian metric on a 2 dimensional surface ispointwise conformal to a constant curvature metric thus one can con sider the question of whether there is a natural evolution equation whichconformally deforms any metric on a surface to a constant curvature met ric the primary interest in this question is not so much to give a newproof of the uniformization theorem but rather to understand nonlinearparabolic equations better especially those arising in differential geome try a sufficiently deep understanding of parabolic equations should yieldimportant new results in riemannian geometry the question in the preceding paragraph has been studied by richardhamilton 3 and brad osgood ralph phillips and peter sarnak 6 in 3 hamilton studied the following equation which we refer to as

variational principles for circle patterns and koebe s theorem we prove existence and uniqueness results for patterns of circles with prescribed intersection angles on constant curvature surfaces our method is based on two new functionals one for the euclidean and one for the hyperbolic case we show how colin de verdiere s bragger s and rivin s functionals can be derived from ours

price s theorem for complex variates price s 1958 theorem is derived for complex valued variates the derivation differs from the existing derivation in two respects first the normal variates are not assumed to be circularly complex thus the result is more general second the characteristic function of the complex variates is not used

separability and triangulated categories abstract we prove that the category of modules over a separable ring object in a tensor triangulated category admits a unique structure of triangulated category which is compatible with the original one this applies in particular to etale algebras more generally we do this for exact separable monads

computing quasiconformal maps using an auxiliary metric and discrete curvature flow surface mapping plays an important role in geometric processing which induces both area and angular distortions if the angular distortion is bounded the mapping is called a quasiconformal mapping qc mapping many surface mappings in our physical world are quasiconformal the angular distortion of a qc mapping can be represented by the beltrami differentials according to qc teichmuller theory there is a one to one correspondence between the set of beltrami differentials and the set of qc surface mappings under normalization conditions therefore every qc surface mapping can be fully determined by the beltrami differential and reconstructed by solving the so called beltrami equation in this work we propose an effective method to solve the beltrami equation on general riemann surfaces the solution is a qc mapping associated with the prescribed beltrami differential the main strategy is to define an auxiliary metric am on the domain surface such that the original qc mapping becomes conformal under the auxiliary metric the desired qc mapping can then be obtained by using the conventional conformal mapping method in this paper we first formulate a discrete analogue of qc mappings on triangular meshes then we propose an algorithm to compute discrete qc mappings using the discrete yamabe flow method to the best of our knowledge it is the first work to compute the discrete qc mappings for general riemann surfaces especially with different topologies numerically the discrete qc mapping converges to the continuous solution as the mesh grid size approaches to 0 we tested our algorithm on surfaces scanned from real life with different topologies experimental results demonstrate the generality and accuracy of our auxiliary metric method

estimation of the autocorrelation function of complex gaussian stationary processes by amplitude clipped signals the normalized autocorrelation function of a gaussian process may be recovered from second order moments of their polarity through the arcsin law by analogy it is possible to calculate the normalized autocorrelation function of a circularly complex gaussian process from the knowledge of moments of its instantaneous phase in the present paper two estimators of the normalized autocorrelation function based on the phase only are presented their theoretical accuracy is evaluated and compared to the accuracy of the direct estimate

on probability density functions for complex variables complex random variables arise naturally in many settings and their properties are of general interest past work on complex variables has mainly focused on their second order structure as well as that of their conjugates whereas the main purpose of this correspondence is to clarify the concept of a density function for a complex random variable and to discuss its properties two different functions play the role that the density of a real univariate random variable holds only one of these two functions can be correctly interpreted as a density but both functions clarify the nature of a complex variable the role played by the complex conjugate of the variable in this formulation is clarified and the complex scalar nature of z is discussed as the properties of complex random variables are most naturally specified in terms of the complex quantities directly and given in terms of the distribution of the complex variables rather than formulated in terms of the real and imaginary parts ensuring that an interpretable complex formulation exists is important

on the differential graded eilenberg moore construction abstract the eilenberg moore construction for modules over a differential graded monad is used to study a question of balmer regarding existence of an exact adjoint pair representing an exact monad a bousfield like localization for differential graded categories is realized as a special case of this construction using drinfeld quotients as applications we study some example coming from g equivariant triangulated categories and twisted derived categories

a new invariant for c over r almost invertible cohomology theory and the classification of idempotent cohomology classes and algebras by partially ordered sets with a galois group action

weak crossed product orders over valuation rings abstract let f be a field let v be a valuation ring of f of arbitrary krull dimension rank let k be a finite galois extension of f with group g and let s be the integral closure of v in k let f g g k 0 be a normalized two cocycle such that f g g s 0 but we do not require that f should take values in the group of multiplicative units of s one can construct a crossed product v order a f g s x with multiplication given by x s x s f x for s s g we characterize semihereditary and dubrovin crossed product orders under mild valuation theoretic assumptions placed on the nature of the extension k f

picard groups and infinite matrix rings we describe a connection between the picard group of a ring with local units t and the picard group of the unital overring end tt using this connection we show that the three groups pic r pic fm r and pic rfm r are isomorphic for any unital ring r furthermore eaclh ele ment of pic rfm r arises from an automorphism of rfm r which yields an isomorphsm between pic rfm r and out rfm r as one applica tion we extend a classical result of rosenberg and zelinsky by showing that the group outr rfm r is abelian for any commutative unital ring r

ricci flow for 3d shape analysis ricci flow is a powerful curvature flow method which is invariant to rigid motion scaling isometric and conformal deformations we present the first application of surface ricci flow in computer vision previous methods based on conformal geometry which only handle 3d shapes with simple topology are subsumed by the ricci flow based method which handles surfaces with arbitrary topology we present a general framework for the computation of ricci flow which can design any riemannian metric by user defined curvature the solution to ricci flow is unique and robust to noise we provide implementation details for ricci flow on discrete surfaces of either euclidean or hyperbolic background geometry our ricci flow based method can convert all 3d problems into 2d domains and offers a general framework for 3d shape analysis we demonstrate the applicability of this intrinsic shape representation through standard shape analysis problems such as 3d shape matching and registration and shape indexing surfaces with large nonrigid anisotropic deformations can be registered using ricci flow with constraints of feature points and curves we show how conformal equivalence can be used to index shapes in a 3d surface shape space with the use of teichmuller space coordinates experimental results are shown on 3d face data sets with large expression deformations and on dynamic heart data

graded equivalence theory with applications we develop the foundations for graded equivalence theory and apply them to investigate properties such as primeness finite representation type and vertex theory of graded rings the key fact that we prove is that for any two g graded rings r and s such that there is a category equivalence from gr r to gr s that commutes with suspensions then for any subgroup h of g the categories gr h g r and gr h g s of modules graded by the g set of right cosets are also equivalent

covers and envelopes over gr gorenstein rings abstract in this paper we study the existence of gorenstein injective envelopes and gorenstein projective and flat covers in the category of graded modules and we relate them with the corresponding envelopes and covers in the category of modules

genus zero surface conformal mapping and its application to brain surface mapping we developed a general method for global conformal parameterizations based on the structure of the cohomology group of holomorphic one forms for surfaces with or without boundaries gu and yau 2002 gu and yau 2003 for genus zero surfaces our algorithm can find a unique mapping between any two genus zero manifolds by minimizing the harmonic energy of the map in this paper we apply the algorithm to the cortical surface matching problem we use a mesh structure to represent the brain surface further constraints are added to ensure that the conformal map is unique empirical tests on magnetic resonance imaging mri data show that the mappings preserve angular relationships are stable in mris acquired at different times and are robust to differences in data triangulation and resolution compared with other brain surface conformal mapping algorithms our algorithm is more stable and has good extensibility