A Hochschild Homology Euler Characteristic for Circle Actions

We define an S¹-Euler characteristic, _S ¹(X), of a circle action on a compact manifold or finite complex X. It lies in the first Hochschild homology group HH ₁(G) where G is the fundamental group of X. This _S ¹(X) is analogous in many ways to the ordinary Euler characteristic. One application is an intuitively satisfying formula for the Euler class (integer coefficients) of the normal bundle to a smooth circle action without fixed points on a manifold. In the special case of a three-dimensional Seifert fibered space, this formula is particularly effective.     

Euler Characteristic and Fixed-Point Theorems

We propose a geometric definition of the Euler characteristic (M) for the class of compact epi-Lipschitzian sets MRⁿ and we provide existence theorems of (generalized) equilibria for set-valued mappings F when the domain M of F is neither assumed to be convex, nor smooth but has a nonzero Euler characteristic.     

Linear Scheme for Finite Element Solution of Nonlinear Parabolic-Elliptic Problems with Nonhomogeneous Dirichlet Boundary Condition

The computation of nonlinear quasistationary two-dimensional magnetic fields leads to a nonlinear second order parabolic-elliptic initial-boundary value problem. Such a problem with a nonhomogeneous Dirichlet boundary condition on a part ₁ of the boundary is studied in this paper. The problem is discretized in space by the finite element method with linear functions on triangular elements and in time by the implicit-explicit method (the left-hand side by the implicit Euler method and the right-hand side by the explicit Euler method). The scheme we get is linear. The strong convergence of the method is proved under the assumptions that the boundary is piecewise of class C³ and the initial condition belongs to L₂ only. Strong monotonicity and Lipschitz continuity of the form a(v,w) is not an assumption, but a property of this form following from its physical background.     

The Euler obstruction of a function on a determinantal variety and on a curve

Given an analytic function germ f: (X, 0) → C on an isolated determinantal singularity or on a reduced curve, we present formulas relating the local Euler obstruction of f to the vanishing Euler characteristic of the fiber X ∩ f^-1(0) and to the Milnor number of f. Restricting ourselves to the case where X is a complete intersection, we obtain an easy way to calculate the local Euler obstruction of f as the difference between the dimension of two algebras.     

Characteristic Foliations of Spheres Embedded in the Standard Overtwisted Structure (R3, ζ1)

The characteristic foliation of a sphere embedded in the standard tight contact structure (R³, ₀) is unique up to isotopy. We show that any Morse-Smale foliation on the sphere with null Euler class, is, up to isotopy, the characteristic foliation of a sphere embedded in the standard overtwisted contact structure (R³, ₁). We thus have a new way of looking at the two standard structures as opposites in the world of contact structures.     

The Lagrangian Gauss image of a compact surface in Minkowski 3-Space

A generic compact surfaceQ in Minkowski 3-Space is naturally stratified by the loci where the orthogonal line bundle is tangent to the next lower stratum,SP D ⁰ Q M³. To each component inD ⁰ we associate a light-like hypersurface and in turn a Lagrangian loop in the cotangent bundle of the circle. We then establish an inequality relating the Euler characteristic of the indefinite component ofQ with the total Gauß-Maslov index of the associated Lagrangian loops.     

Local existence and blow-up criterion for the Euler equations in Besov spaces of weak type

We introduce Besov type function spaces, based on the weak L ^p -spaces instead of the standard L ^p -spaces, and prove a local-in-time unique existence and a blow-up criterion of solutions in those spaces for the Euler equations of perfect incompressible fluid in . For the proof, we establish the Beale-Kato-Majda type logarithmic inequality and commutator type estimates in our weak spaces. Abbreviate title: Euler equations in Besov spaces of weak type     

A Stereological Version of the Gauss–Bonnet Formula

We give a stereological version of the Gauss–Bonnet formula in order to compute the Euler characteristic of a domain with boundary in a smooth orientable surface in ³, by looking at contacts with a 'sweeping' plane.     

Unstable Manifolds of Euler Equations

We consider a steady state v₀ of the Euler equation in a fixed bounded domain in ?ⁿ. Suppose the linearized Euler equation has an exponential dichotomy of unstable and center‐stable subspaces. By rewriting the Euler equation as an ODE on an infinite‐dimensional manifold of volume‐preserving maps in W^{k, q} the unstable (and stable) manifolds of v₀ are constructed under a certain spectral gap condition that is satisfied for both two‐dimensional and three‐dimensional examples. In particular, when the unstable subspace is finite dimensional, this implies the nonlinear instability of v₀ in the sense that arbitrarily small W^{k, q} perturbations can lead to L² growth of the nonlinear solutions. © 2013 Wiley Periodicals, Inc.     

A study of semiiterative methods for nonsymmetric systems of linear equations

Summary Given a nonsingular linear systemA x=b, a splittingA=M–N leads to the one-step iteration (1)x _m =T X _m–1 +c withT:=M ^–1N andc:=M ^–1 b. We investigate semiiterative methods (SIM's) with respect to (1), under the assumption that the eigenvalues ofT are contained in some compact set of , with 1. There exist SIM's which are optimal with respect to , but, except for some special sets , such optimal methods are not explicitly known in general. Using results about maximal convergence of polynomials and uniformly distributed nodes from approximation and function theory, we describe here SIM's which are asymptotically optimal with respect to . It is shown that Euler methods, extensively studied by Niethammer-Varga [NV], are special SIM's. Various algorithms for SIM's are also derived here. A 1-1 correspondence between Euler methods and SIM's, generated by generalized Faber polynomials, is further established here. This correspondence gives that asymptotically optimal Euler methods are quite near the optimal SIM's.Dedicated to Professor Karl Zeller (Universität Tübingen) on the occasion of his sixtieth birthday (December 28, 1984)     

Geometrical properties of some Euler and circular cubics. Part 1

This sequel to our earlier paper (1995) continues the investigation of the Euler cubic curves therein defined, with particular reference to perspectivities and associated conics. Study of the circular cubic in this pencil, the Neuberg cubic, brings with it some discussion of the properties of circular cubics in general.

Glossary

Named points Bennett See Isoptic - Brocard Isogonal points P,Q in a triangle ABC for which all six angles PBC, PCA, PAB, QCB, QAC, QBA are equal - De Longchamps Image of orthocentre H in circumcentre O - Euler Point on circumcircle whose Simson line is parallel to OH - Fermat Points F for which BFC = CFA = AFB (two such) - Hessian Points of intersection of the Apollonius circles; their pedal triangles are equilateral - lsogonic See Fermat - Isodynamic See Hessian - Isoptic Point at which the 4 circumcircles of the component triangles of a quadrangle subtend equal angles - Lemoine Isogonal conjugate (K) of centroid G - Neuberg Point on circumcircle whose Simson line is perpendicular to OH - Steiner Point on circumcircle whose Simson line is parallel to OK - Symmedian See Lemoine; common point of symmedian lines Point pairs Isogonal conjugates Points P, ¯P such that their joins to each vertex of a triangle form an angle having the same bisectors as the angle of the triangle there. They are polar conjugate points with respect to all conies through the tritangent centres - Isotomic conjugates Points P, ¯P such that their joins to each vertex meet the opposite side at the ends of a segment with the same midpoint as the side. They are polar conjugates with respect to all conies through the centroid and the vertices of the anticomplementary triangle (see below) Named lines Brocard axis OK, mediator of join of Brocard points, containing Hessian points - Cevians of P PA, PB, and PC - Euler OH, containing also centroid G, 9-point centre and De Longchamps point - Simson line Join of feet of perpendiculars from a point of the circumcircle to sides BC, CA, AB - Symmedians Reflexions of medians in the angle bisectors Named curves Conies Apollonius circles Three circles, one through each vertex for which the other two vertices are inverse - Jerabek's hyperbola Rectangular, through ABCHO; isogonal conjugate of OH - Kiepert's hyperbola Rectangular, through ABCHG; isogonal conjugate of OK - Nine-point circle Through 3 diagonal points and 6 midpoints of the sides of quadrangle ABCH - Steiner ellipse Touching sides at midpoints - Tritangent circles Touching the three sides of a triangle, either externally or internally Cubics Darboux Auto-isogonal with pivot De Longchamps point - Euler Member of pencil of auto-isogonal cubics with pivot on Euler line - Feuerbach Euler cubic with pivot at Nine-point centre - Lucas Locus of isotomic conjugate points whose join contains the isotomic conjugate of H - McCay Euler cubic with pivot at circumcentre O - Neuberg Locus of isogonal conjugates whose join is parallel to OH; Euler cubic with pivot at infinity on OH - Ortho Euler cubic with pivot at orthocentre H - Thomson Euler cubic with pivot at centroid G Triangles associated with a point P, given base triangle ABC Cevian LMN, where AP meets BC at L, &c - Anticevian Triangle LMN for which ABC is the Cevian triangle of P; [AL, PL] = –l, &c - Complementary ABC, where A is the midpoint of BC, &c. BC is parallel to CB, &c - Anticomplementary ABC, where BAC is a line parallel to CB, &c - Pedal DEF, where PD is perpendicular to B C, &c - Antipedal Triangle DEF with respect to which ABC is the pedal triangle of P. EAF is perpendicular to AP, &c - Cyclopedal XYZ, where AP meets circumcircle again at X,&c     

On a class of compact homogeneous spaces I

Le K be a compact connected Lie group, L be a connected closed subgroup of K. It is well known that L is a subgroup of maximal rank of K if and only if the Euler characteristic of the manifold M = K/L is positive. Such homogeneous spaces M have been classified in [7, 10]. However, their topological classification was unknown. This classification is obtained in the present article. We show tha two compact homogeneous spaces M = K/L and M = K/L of positive Euler characteristic are diffeomorphic if and only if the graded rings H ^*(M,Z) and H ^*(M,Z) are isomorphic. We also obtain the rational homotopy classification of such homogeneous spaces which is not equivalent to the differential one. These results were announced in [15].     

Orbifold Euler characteristics and the number of commutingm-tuples in the symmetric groups

Generating functions for the number of commutingm-tuples in the symmetric groups are obtained. We define a natural sequence of orbifold Euler characteristics for a finite groupG acting on a manifoldX. Our definition generalizes the ordinary Euler characteristic ofX/G and the string-theoretic orbifold Euler characteristic. Our formulae for commutingm-tuples underlie formulae that generalize the results of Macdonald and Hirzebruch-Höfer concerning the ordinary and string-theoretic Euler characteristics of symmetric products.Supported partly by a grant from the Ford Foundation.     

A Geometric approximation to the euler equations: the Vlasov–Monge–Ampère system

This paper studies the Vlasov–Monge–Ampère system (VMA), a fully non-linear version of the Vlasov–Poisson system (VP) where the (real) Monge–Ampère equation det ²/x_i x_j = substitutes for the usual Poisson equation. This system can be derived as a geometric approximation of the Euler equations of incompressible fluid mechanics in the spirit of Arnold and Ebin. Global existence of weak solutions and local existence of smooth solutions are obtained. Links between the VMA system, the VP system and the Euler equations are established through rigorous asymptotic analysis.     

Existence of weak solutions of 2-D Euler equations with initial vorticityω 0∈L(log+ L) α (α>0)

It is proved that there exist global weak solutions of 2-D Euler equations inR ² under the assumption that the initial vorticity belongs to a kind of wider spaces,L ¹L(log⁺ L) (>0), which are Orlicz spaces containing spacesL ^p L ¹,L(log⁺ L) L (>1/2) and so on. This result improves on that of [2], [4], [11]. Moreover, these solutions are obtained by vanishing the viscosity term of Navier-Stokes equations.     

Statistical solutions of Euler's equations

Solutions of nonstationary Euler equations on a torus in R³ are investigated. For a broad class of random initial velocity fields a measure in L₂(× [0, T]) is constructed, the moments of which for any finite T are solutions of the corresponding chain of moment equations.Translated from Problemy Matematicheskogo Analiza, No. 11, pp. 27–37, 1990.     

On the Limit as the Density Ratio Tends to Zero for Two Perfect Incompressible Fluids Separated by a Surface of Discontinuity

We study the asymptotic limit as the density ratio ρ^?/ρ⁺ → 0, where ρ⁺ and ρ^? are the densities of two perfect incompressible 2-D/3-D fluids, separated by a surface of discontinuity along which the pressure jump is proportional to the mean curvature of the moving surface. Mathematically, the fluid motion is governed by the two-phase incompressible Euler equations with vortex sheet data. By rescaling, we assume the density ρ⁺ of the inner fluid is fixed, while the density ρ^? of the outer fluid is set to ε. We prove that solutions of the free-boundary Euler equations in vacuum are obtained in the limit as ε → 0.     

The hypertree poset and the ℓ<Superscript>2</Superscript>-Betti numbers of the motion group of the trivial link

We give explicit formulae for the Euler characteristic and ²-cohomology of the group of motions of the trivial link, or isomorphically the group of free group automorphisms that send each standard generator to a conjugate of itself. The method is primarily combinatorial and ultimately relies on a computation of the Möbius function for the poset of labelled hypertrees.Partially supported by NSF grant no. DMS-0101506Partially supported by an AMS Centennial Research Fellowship     

Asymptotic behavior of the solutions to the damped compressible Euler equations with vacuum

The asymptotic behavior of solutions of the damped compressible Euler equations is conjectured to obey to the famous porous media equations (PMES). The previous works on this topic concern the case away from vacuum where the system is strictly hyperbolic. In present paper, we prove that the L^∞ entropy weak solution with vacuum, obtained by the compensated compactness theory, converges strongly in space to the unique similarity solution of the related PME, as time goes to infinity.