Seeking invariants for blow-analytic equivalence

We introduce some blow-analytic invariants of real analytic function-germsand discuss their properties. As a consequence, we obtain, for instance, the multiplicity of function-germs is a blow-analytic invariant.     

Volume and Projective Equivalence Between Riemannian Manifolds

Let (M, g) and (M, ) be two Riemannian metrics which are pointwise projectively equivalent, i.e. they have the same geodesics as point sets. We prove that the pointwise projective equivalence is trivial, if (M, g) is a noncompact complete manifold which has at most quadratic volume growth and nonnegative total scalar curvature, and (M, ) has nonpositive Ricci curvature. Mathematics Subject Classifications (2000): 53C22, 58J05     

Projective dynamics and classical gravitation

We show that there exists a projective dynamics of a particle. It underlies intrinsically the classical particle dynamics as projective geometry underlies Euclidean geometry. In classical particle dynamics a particle moves in the Euclidean space subjected to a potential. In projective dynamics the position space has only the local structure of the real projective space. The particle is subjected to a field of projective forces. A projective force is not an element of the tangent bundle to the position space, but of some fibre bundle isomorphic to the tangent bundle. These statements are direct consequences of Appell’s remarks on the homography in mechanics, and are compatible with similar statements due to Tabachnikov concerning projective billiards. When we study Euclidean geometry we meet some particular properties that we recognize as projective properties. The same is true for the dynamics of a particle. We show that two properties in classical particle dynamics are projective properties. The fact that the Keplerian orbits close after one turn is a consequence of a more general projective statement. The fact that the fields of gravitational forces are divergence free is a projective property of these fields.      

Projective Curvature and Integral Invariants

In this paper, an extension of all Lie group actions on R ²to coordinates defined by potentials is given. This provides a new solution to the equivalence problems of curves under the projective group and two of its subgroups. The potentials correspond to integrals of higher and higher-order producing an infinite number of independent integral invariants. Applications to computer vision are discussed.     

Central invariants and Frobenius-Schur indicators for semisimple quasi-Hopf algebras

In this paper, we obtain a canonical central element ν_H for each semi-simple quasi-Hopf algebra H over any field k and prove that ν_H is invariant under gauge transformations. We show that if k is algebraically closed of characteristic zero then for any irreducible representation of H which affords the character takes only the values 0, 1 or −1, moreover if H is a Hopf algebra or a twisted quantum double of a finite group then χ(ν_H) is the corresponding Frobenius-Schur indicator. We also prove an analog of a theorem of Larson-Radford for split semi-simple quasi-Hopf algebras over any field k. Using this result, we establish the relationship between the antipode S, the values of χ(ν_H), and certain associated bilinear forms when the underlying field k is algebraically closed of characteristic zero.     

Coleman自同构群的投射极限

在这篇注记中,利用群的投射极限性质给出了有限可解群的Coleman自同构群的一个具体构造.作为应用,证明了二面体群的Coleman外自同构群或者是1或者是一个初等阿贝尔2-群.     

Coleman自同构群的投射极限

在这篇注记中,利用群的投射极限性质给出了有限可解群的Coleman自同构群的一个具体构造.作为应用,证明了二面体群的Coleman外自同构群或者是1或者是一个初等阿贝尔2-群.     

7c-Gorenstein Projective, ;Cc-Gorenstein Injectiveand Wcr-Gorenstei丨丨 Flat Modules

The authors introduce and investigate the Tc-Gorenstein projective, Lc- Gorenstein injective and Hc-Gorenstein flat modules with respect to a semidualizing module C which shares the common properties with the Gorenstein projective, injective and flat modules, respectively. The authors prove that the classes of all the Tc-Gorenstein projective or the Hc-Gorenstein flat modules are exactly those Gorenstein projective or flat modules which are in the Auslander class with respect to C, respectively, and the classes of all the Lc-Gorenstein ＇injective modules are exactly those Gorenstein injective modules which are in the Bass class, so the authors get the relations between the Gorenstein projective, injective or flat modules and the C-Gorenstein projective, injective or flat modules. Moreover, the authors consider the Tc（R）-projective and Lc（R）-injective dimensions and Tc（R）-precovers and Lc（R）-preenvelopes. Fiually, the authors study the Hc-Gorenstein flat modules and extend the Foxby equivalences.     

Versal unfolding of equivariant bifurcation problems in more general case under two equivalent groups

For the unfolding of equivariant bifurcation problems with two types of state variables in the presence of parameter symmetry,the versal unfolding theorem with respect to left-right equivalence is obtained by using the related methods and techniques in the singularity theory of smooth map-germs.The corresponding results in[4,9]can be considered as its special cases.A relationship between the versal unfolding w.r.t.left-right equivalence and the versal deformation w.r.t.contact equivalence is established.     

Joins of Projective Varieties: A Cancellation Theorem for Curves

Abstract

Let X, Y, Z be integral subvarieties of P ⁿ . Let [X; Y]??? P ⁿ denote the join. Under what conditions on X, Y and/or Z if [X; Y]?=?[X; Z], then Y?=?Z? Here, we study the case in which X, Y and Z are curves.     

CR-hypersurfaces of a conformal Kenmotsu manifold satisfying certain shape operator conditions

In this paper, conformal Kenmotsu manifolds are introduced. We consider CR-hypersurfaces of a conformal Kenmotsu manifold whose shape operator is parallel, scalar, recurrent or Lie \( \xi \)-parallel. It is proved that if the Lee vector field of a conformal Kenmotsu manifold is tangent and normal to these type of CR-hypersurfaces then the CR-hypersurfaces are totally geodesic and totally umbilic, respectively. An example of a three-dimensional conformal Kenmotsu manifold is constructed for illustration that is not Kenmotsu.     

On the Equivalence of Elementary Chains of Monoidal Transformations of the Projective Plane

We introduce the notion of curve associated with a chain of blow-ups of a complex surface. On the basis of this notion, we classify elementary chains (of length up to seven) of blow-ups of the projective plane. We prove (under an additional condition) that the ramification curve of the inverse of a polynomial mapping cannot be isolated in 4 or 5 blow-ups.     

On the complexity of recognizing a class of generalized networks

The problem of determining whether a given linear programming problem can be converted to a generalized network flow problem having no unit-weight cycles is shown to be NP-hard. The same argument also shows that the problem of determining whether a gain matroid is bicircular is NP-hard.     

On conformal invariants for elliptic systems with multiple critical exponents

In this article we study multiple solutions of the prescribed curvatures problem for a compact Riemannian manifold with smooth boundary and with indefinite signs of the energy function and nonlinearities. We generalize the conception of the critical Palais–Smale level and find new type of conformally invariant necessary and sufficient conditions for existence of multiple solutions of the problem. The second author was supported in part by grants INTAS 03-51-5007, RFBR 05-01-00370, 05-01-00515.     

Real submanifolds in complex spaces

Let(z_(11),..., z_(1N),..., z_(m1),..., z_(mN), w_(11),..., w_(mm)) be the coordinates in C~(mN) +m~2. In this note we prove the analogue of the Theorem of Moser in the case of the real-analytic submanifold M defined as follows W = ZZ~t+ O(3),where W = {w_(ij)}_(1≤i,j≤m)and Z = {z_(ij) }_(1≤i≤m, 1≤j≤N). We prove that M is biholomorphically equivalent to the model W = ZZ~t if and only if is formally equivalent to it.     

Algorithmic equivalence in quadratic programming,I: A least-distance programming problem

It is demonstrated that Wolfe's algorithm for finding the point of smallest Euclidean norm in a given convex polytope generates the same sequence of feasible points as does the van de Panne-Whinstonsymmetric algorithm applied to the associated quadratic programming problem. Furthermore, it is shown how the latter algorithm may be simplified for application to problems of this type.This work was supported by the National Science Foundation, Grant No. MCS-71-03341-AO4, and by the Office of Naval Research, Contract No. N00014-75-C-0267.     

Milnor link invariants and quantum 3-manifold invariants

Let be the 3-manifold invariant of Le, Murakami and Ohtsuki. We show that , where denotes terms of degree , if M is a homology 3-sphere obtained from by surgery on an n-component Brunnian link whose Milnor -invariants of length vanish.?We prove a realization theorem which is a partial converse to the above theorem.?Using the Milnor filtration on links, we define a new bifiltration on the vector space with basis the set of oriented diffeomorphism classes of homology 3-spheres. This includes the Milnor level 2 filtration defined by Ohtsuki. We show that the Milnor level 2 and level 3 filtrations coincide after reindexing. Received: October 23, 1998.