Construction of exotic smooth structures

In this article, we construct infinitely many simply connected, nonsymplectic and pairwise nondiffeomorphic 4-manifolds starting from the elliptic surfaces E(n) and applying the sequence of knot surgery, ordinary blowups and rational blowdown. We also compute the Seiberg-Witten invariants of these manifolds.     

The Geography of Simply-Connected Symplectic Manifolds

By using the Seiberg-Witten invariant we show that the region under the Noether line in the lattice domain X is covered by minimal, simply connected, symplectic 4-manifolds.     

Über geschlossene Regelflächen im elliptischen Raum

On a ruled closed surface in the elliptic 3-space two integral invariants are considered: the aperture distance of a curve orthogonal to the generating lines of , and the aperture angle of an orthogonally circumscribed tangent surfaces. By means of these integral invariants and by considering certain ruled surfaces associated to one finds the geometric meaning of further integral invariants. If is generated by the binormals of a curve one obtains some properties of closed curves in the elliptic 3-space.     

The Index of <Emphasis Type="Italic">b</Emphasis>-Pseudodifferential Operators on Manifolds with Corners

On compact manifolds with corners of arbitrary codimension, we characterize the multi-cylindrical end (or b-type) pseudodifferential operators that are Fredholm on weighted Sobolev spaces and we compute their indices. The index formula contains the usual interior term manufactured from the local symbols of the operator and also contains boundary correction terms corresponding to eta-type invariants of the induced operators on the boundary faces.     

Cusp structures of alternating links

Summary An alternating link is canonically associated with every finite, connected, planar graph . The natural ideal polyhedral decomposition of the complement of is investigated. Natural singular geometric structures exist onS ³– , with respect to which the geometry of the cusp has a shape reflecting the combinatorics of the underlying link projection. For the class of balanced graphs, this induces a flat structure on peripheral tori modelled on the tessellation of the plane by equilateral triangles. Examples of links containing immersed, closed ₁-injective surfaces in their complements are given. These surfaces persist after most surgeries on the link, the resulting closed 3-manifolds consequently being determined by their fundamental groups.Oblatum 20-V-1991     

On the Structure of Cimpact Simply Connected Manifolds of Positive Sectional Curvature

We present a first structure theorem for compact simply connected positively curved manifolds with arbitrarily small pinching constants: For each nN and 0<1, there exists a positive number V = V(n,) such that if (M,g) is a compact simply connected n-dimensional Riemannian manifold with sectional curvature 0相似文献   

Homotopy classification of some four-dimensional manifolds

In this paper there is proved a generalization of the results of Whitehead and Pontryagin on the homotopy classification of closed, simply connected four-manifolds. Let W and M be compact four-dimensional simply connected oriented four-manifolds. By q_w is denoted the intersection index on the group H₂(W).Basic Result. THEOREM (Extension). Let the groups H₁(W)and H₁(M) be finite and suppose given a homotopy equivalence fWM. In order that f can be extended to a homotopy equivalence (W,W)(M,M), it is necessary and sufficient that there should exist an isomorphism , such that the diagram is commutative and *q_m=q_m.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 66, pp. 164–171, 1976.     

Division Algebras that Ramify only on a Plane Quartic Curve with Simply Connected Components

A central division algebra over the field of rational functions in two variables with coefficients over an algebraically closed field ramifies along a divisor on P ². If the ramification divisor of is a quartic curve which is the union of simply connected curves, we show that is a symbol algebra and satisfies the index equals exponent equation.     

Smooth curves linked to thick curves

In this paper we construct smooth irreducible spacecurves C which link geometrically by surfaces of minimal degree containingC to curves of generic embedding dimension three. This produces interesting behavior with respect to both C and . The curves link to smooth connected curves by surfaces of low degree butcannot link to smooth connected curves by surfaces of high degree.The curves C give counterexamples to a conjecture of Martin-Deschamps and Perrin.     

Construction and Properties of the t-Invariant

Special spine theory is used for constructing a new invariant of compact 3-manifolds: the t-invariant. The behavior of the invariant under (boundary) connected sum is investigated. One of the TuraevViro invariants is expressed via the t-invariant. The t-invariant is interpreted from the point of view of TQFT. The values of the t-invariant are computed for lens spaces and for all closed oriented 3-manifolds of complexity at most six. It is proved that the set of values of the t-invariant on Seifert manifolds with fixed base (which is a closed surface) and fixed number of singular fibers is finite. Bibliography: 10 titles.     

Boundary problem with discontinuous translation for two functions which are analytic in domains of different connectivity

In the space L_p(), 1 < p < we consider the problem of finding functions and which are analytic, respectively, in a doubly connected domain with boundary and a simply connected domain with boundary ( and are closed Lyapunov contours) with respect to the boundary condition , where the orientation-preserving translation and the coefficients a, b, c, d admit discontinuities at certain points of . Under specific conditions on the limit values of the derivative of the translation at the points of discontinuity we get necessary and sufficient conditions for the problem to be Noetherian and a formula for calculating its index.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 2, pp. 262–266, February, 1990.     

Existence and Uniqueness of F-Minimal Surfaces

We extend a classical result of Radó and Kneser concerning uniqueness ofminimal surfaces bounded by a given closed Jordan curve in³ to the case of extremals for certain geometric variationalintegrals. Using standard elliptic PDE theory, this gives the existenceand uniqueness of embedded F-minimal surfaces for suitable boundarycurves that project simply onto the boundary of a plane convex domain.     

Chebyshev Solution of the Nearly-Singular One-Dimensional Helmholtz Equation and Related Singular Perturbation Equations: Multiple Scale Series and the Boundary Layer Rule-of-Thumb

The one-dimensional Helmholtz equation, ² u _xx –u=f(x), arises in many applications, often as a component of three-dimensional fluids codes. Unfortunately, it is difficult to solve for 1 because the homogeneous solutions are exp(±x/), which have boundary layers of thickness O(1/). By analyzing the asymptotic Chebyshev coefficients of exponentials, we rederive the Orszag–Israeli rule [16] that Chebyshev polynomials are needed to obtain an accuracy of 1% or better for the homogeneous solutions. (Interestingly, this is identical with the boundary layer rule-of-thumb in [5], which was derived for singular functions like tanh([x–1]/).) Two strategies for small are described. The first is the method of multiple scales, which is very general, and applies to variable coefficient differential equations, too. The second, when f(x) is a polynomial, is to compute an exact particular integral of the Helmholtz equation as a polynomial of the same degree in the form of a Chebyshev series by solving triangular pentadiagonal systems. This can be combined with the analytic homogeneous solutions to synthesize the general solution. However, the multiple scales method is more efficient than the Chebyshev algorithm when is very, very tiny.     

ADIABATIC INVARIANTS OF SLOWLY VARYING THREE－DIMENSIONAL SYSTEMS AND EXISTENCE OF INVARIANT TORI OF LOTKA－VOLTERRA EQUATION

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Mixed Hodge structure in cyclic homology and algebraic K-theory, 1

Deligne defined the notion of a mixed Hodge structure (MHS) and proved that every quasiprojective variety over has a natural MHS on its cohomology. This paper establishes similar results for cyclic homology and the algebraic K-theory of simply connected quasi-projective varieties over . In the nonsimply connected case, an MHS is established on certain quotient groups of algebraic K-theory.Supported by a NSERC University Research Fellowship and operating grant.     

Analysis of Atkinson’s variable transformation for numerical integration over smooth surfaces in ℝ3

Summary Recently, a variable transformation for integrals over smooth surfaces in ³ was introduced in a paper by Atkinson. This interesting transformation, which includes a grading parameter that can be fixed by the user, makes it possible to compute these integrals numerically via the product trapezoidal rule in an efficient manner. Some analysis of the approximations thus produced was provided by Atkinson, who also stated some conjectures concerning the unusually fast convergence of his quadrature formulas observed for certain values of the grading parameter. In a recent report by Atkinson and Sommariva, this analysis is continued for the case in which the integral is over the surface of a sphere and the integrand is smooth over this surface, and optimal results are given for special values of the grading parameter. In the present work, we give a complete analysis of Atkinsons method over arbitrary smooth surfaces that are homeomorphic to the surface of the unit sphere. We obtain optimal results that explain the actual rates of convergence, and we achieve this for all values of the grading parameter.     

On solutions of elliptic differential equations with operator coefficients

It is proved that on a simply connected region the solutions of the elliptic partial differential equation have representations U(x,y) = QV(x,y), where V admits local power series expansion in the "operator variable" xI + yI and (Q,T) forms a standard pair for the monic operator polynomial A₀ + A₁ + + ^s-1A_s-1 + ^sI.This paper was written while the autor was employed at the Vrije Universiteit in Amsterdam.     

Quasi-Normal Spaces and πg-Closed Sets

We introduce the notion of g-closed sets and use it to obtain a characterization and preservation theorems of quasi-normal spaces.     

Extremal feedback perturbations for families of closed operators

Let T- S, be a family of not necessarily bounded semi-Fredholm operators, where T and S are operators acting between Banach spaces X and Y, and where S is bounded with D(S) D(T). For compact sets , as well as for certain open sets , we investigate existence and minimal rank of bounded feedback perturbations of the form F=BE such that min.ind (T-S+F)=0 for all . Here B is a given operator from a linear space Z to Y and E is some operator from X to Z.We give a simple characterization of that situation, when such regularizing feedback perturbations exist and show that for compact sets the minimal rank never exceeds max { min.ind (T-S) }+1. Moreover, an example shows that the minimal rank, in fact, may increase from max {...} to max {...}+1, if the given B enforces a certain structure of the feedbachk perturbation F.However, the minimal rank is equal to max { min.ind (T-S) }, if is an open set such that min.ind (T-S) already vanishes for all but finitely many points . We illustrate this result by applying it to the stabilization of certain infinite-dimensional dynamical systems in Hilbert space.     

h-Cobordisms between simply connected 4-manifolds

h-cobordisms between simply connected 4-manifolds are studied. It is shown that most inertial h-cobordisms have a handle decomposition with one 2-handle and one 3-handle, and h-cobordisms between nondiffeomorphic manifolds have handle decompositions with the minimal number of handles consistent with a diffeomorphism between the stabilized ends. Also the number of distinct h-cobordisms between two fixed manifolds is described in terms of isomorphisms of their quadratic forms. These results are applied to Dolgachev surfaces and the Kummer surface using recent work of Donaldson, Friedman and Morgan, and Matumoto.