<Emphasis Type="Italic">h</Emphasis>-Principles for curves and knots of constant curvature

We prove that curves of constant curvature satisfy, in the sense of Gromov, the relative -dense h-principle in the space of immersed curves in Euclidean space R ^{n ≥ 3}. In particular, in the isotopy class of any given knot f there exists a knot f͂ of constant curvature which is -close to f. More importantly, we show that if f is , then the curvature of f͂ may be set equal to any constant c which is not smaller than the maximum curvature of f. We may also require that f͂ be tangent to f along any finite set of prescribed points, and coincide with f over any compact set with an open neighborhood where f has constant curvature c. The proof involves some basic convexity theory, and a sharp estimate for the position of the average value of a parameterized curve within its convex hull. The author’s research was supported in part by NSF CAREER award DMS-0332333.     

Characterization of order 3 algebraic immersed minimal surfaces of <Emphasis Type="Italic">S</Emphasis><Superscript>3</Superscript>

In this paper we prove that if is a minimal immersion of a compact surface and , for some homogeneous polynomial f of degree 3 on R ⁴, then, M is a torus and is one of the examples given by Lawson (1970, Complete minimal surfaces in S ³. Ann. Math. 92(2), 335–374).      

Algebraic cycles on products of elliptic curves over <Emphasis Type="Italic">p</Emphasis>-adic fields

We show that for any odd prime p there is a smooth projective threefold W defined over a p-adic field such that the Chow group CH²(W)/ and the Griffiths group Griff²(W)/ are infinite for suitable primes . We further give examples of smooth projective fourfolds over these p-adic fields for which the -torsion subgroup CH³ is infinite.     

Efficient <Emphasis Type="Italic">p</Emphasis>th root computations in finite fields of characteristic <Emphasis Type="Italic">p</Emphasis>

We present a method for computing pth roots using a polynomial basis over finite fields of odd characteristic p, p ≥ 5, by taking advantage of a binomial reduction polynomial. For a finite field extension of our method requires p − 1 scalar multiplications of elements in by elements in . In addition, our method requires at most additions in the extension field. In certain cases, these additions are not required. If z is a root of the irreducible reduction polynomial, then the number of terms in the polynomial basis expansion of z ^1/p , defined as the Hamming weight of z ^1/p or , is directly related to the computational cost of the pth root computation. Using trinomials in characteristic 3, Ahmadi et al. (Discrete Appl Math 155:260–270, 2007) give is greater than 1 in nearly all cases. Using a binomial reduction polynomial over odd characteristic p, p ≥ 5, we find always.      

On rectifiable curves with <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-bounds on global curvature: self-avoidance,regularity, and minimizing knots

We discuss the analytic properties of curves γ whose global curvature function ρ _G [γ]⁻¹ is p-integrable. It turns out that the L ^p -norm is an appropriate model for a self-avoidance energy interpolating between “soft” knot energies in form of singular repulsive potentials and “hard” self-obstacles, such as a lower bound on the global radius of curvature introduced by Gonzalez and Maddocks. We show in particular that for all p > 1 finite -energy is necessary and sufficient for W ^2,p -regularity and embeddedness of the curve. Moreover, compactness and lower-semicontinuity theorems lead to the existence of -minimizing curves in given isotopy classes. There are obvious extensions to other variational problems for curves and nonlinearly elastic rods, where one can introduce a bound on to preclude self-intersections.     

Dynamically convex Finsler metrics and <Emphasis Type="Italic">J</Emphasis>-holomorphic embedding of asymptotic cylinders

We explore the relationship between contact forms on defined by Finsler metrics on and the theory developed by H. Hofer, K. Wysocki and E. Zehnder (Hofer etal. Ann. Math. 148, 197–289, 1998; Ann. Math. 157, 125–255, 2003). We show that a Finsler metric on with curvature K ≥ 1 and with all geodesic loops of length > π is dynamically convex and hence it has either two or infinitely many closed geodesics. We also explain how to explicitly construct J-holomorphic embeddings of cylinders asymptotic to Reeb orbits of contact structures arising from Finsler metrics on with K = 1, thus complementing the results obtained in Harris and Wysocki (Trans. Am. Math. Soc., to appear).      

Efficient convolution with the Newton potential in <Emphasis Type="Italic">d</Emphasis> dimensions

The paper is concerned with the evaluation of the convolution integral in d dimensions (usually d = 3), when f is given as piecewise polynomial of possibly large degree, i.e., f may be considered as an hp-finite element function. The underlying grid is locally refined using various levels of dyadically organised grids. The result of the convolution is approximated in the same kind of mesh. If f is given in tensor product form, the d-dimensional convolution can be reduced to one-dimensional convolutions. Although the details are given for the kernel the basis techniques can be generalised to homogeneous kernels, e.g., the fundamental solution of the d-dimensional Poisson equation.     

<Emphasis Type="Italic">hp</Emphasis>-Discontinuous Galerkin methods for strongly nonlinear elliptic boundary value problems

In this paper, we have analyzed a one parameter family of hp-discontinuous Galerkin methods for strongly nonlinear elliptic boundary value problems with Dirichlet boundary conditions. These methods depend on the values of the parameter , where θ = + 1 corresponds to the nonsymmetric and θ = −1 corresponds to the symmetric interior penalty methods when and f(u,∇u) = −f, that is, for the Poisson problem. The error estimate in the broken H ¹ norm, which is optimal in h (mesh size) and suboptimal in p (degree of approximation) is derived using piecewise polynomials of degree p ≥ 2, when the solution . In the case of linear elliptic problems also, this estimate is optimal in h and suboptimal in p. Further, optimal error estimate in the L ² norm when θ = −1 is derived. Numerical experiments are presented to illustrate the theoretical results. Supported by DST-DAAD (PPP-05) project.     

Orbifold index and equivariant <Emphasis Type="Italic">K</Emphasis>-homology

Let G be countable group and M be a proper cocompact even-dimensional G-manifold with orbifold quotient . Let D be a G-invariant Dirac operator on M. It induces an equivariant K-homology class and an orbifold Dirac operator on . Composing the assembly map with the homomorphism given by the representation of the maximal group C ^*-algebra induced from the trivial representation of G we define index . In the second section of the paper we show that index = index([D]) and obtain explicit formulas for this integer. In the third section we review the decomposition of in terms of the contributions of fixed point sets of finite cyclic subgroups of G obtained by W. Lück. In particular, the class [D] decomposes in this way. In the last section we derive an explicit formula for the contribution to [D] associated to a finite cyclic subgroup of G.     

On <Emphasis Type="Italic">t</Emphasis><Superscript>2</Superscript>-like solutions of certain second order differential equations

Via an integral transformation, we establish two embedding results between the Emden-Fowler type equation , t ≥ t ₀ > 0, with solutions x such that as , , and the equation , u > 0, with solutions y such that for given k > 0. The conclusions of our investigation are used to derive conditions for the existence of radial solutions to the elliptic equation , , that blow up as in the two dimensional case.      

Riesz <Emphasis Type="Italic">s</Emphasis>-Equilibrium Measures on <Emphasis Type="Italic">d</Emphasis>-Rectifiable Sets as <Emphasis Type="Italic">s</Emphasis> Approaches <Emphasis Type="Italic">d</Emphasis>

Let A be a compact set in of Hausdorff dimension d. For s ∈ (0,d) the Riesz s-equilibrium measure μ ^s is the unique Borel probability measure with support in A that minimizes

Pointwise convergence for semigroups in vector-valued <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> spaces

Suppose that {T _t  : t  ≥  0} is a symmetric diffusion semigroup on L ²(X) and denote by its tensor product extension to the Bochner space , where belongs to a certain broad class of UMD spaces. We prove a vector-valued version of the Hopf–Dunford–Schwartz ergodic theorem and show that this extends to a maximal theorem for analytic continuations of on . As an application, we show that such continuations exhibit pointwise convergence.     

<Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript>-invariants of Riemannian foliations

For a Riemannian foliation on a closed manifold M, we define L ²-spectral sequence Betti numbers and spectral sequence Novikov–Shubin invariants. The spectral sequence of the lift of to the universal covering of M is used in the definitions. These invariants are natural extensions of the L ²-Betti numbers and the Novikov–Shubin invariants of differentiable manifolds. It is shown that these numbers are invariant by foliated homotopy equivalences, and they are computed for several examples.      

On the stability of <Emphasis Type="Italic">t</Emphasis>-convex functions

Summary.  A real-valued function f defined on an open convex set is called (d, t)-convex if it satisfies

A note on <Emphasis Type="Italic">q</Emphasis>-multiplicative functions

We prove the following statement. Let , and let . Suppose that, for all and , the sequence satisfies the relation

Typical <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">q</Emphasis></Superscript>-Dimensions of Measures

For a probability measure μ on a subset of , the lower and upper L^q-dimensions of order are defined by We study the typical behaviour (in the sense of Baire’s category) of the L^q-dimensions and . We prove that a typical measure μ is as irregular as possible: for all q ≥ 1, the lower L^q-dimension attains the smallest possible value and the upper L^q-dimension attains the largest possible value.     

Extensions of linking systems with <Emphasis Type="Italic">p</Emphasis>-group kernel

We study extensions of p-local finite groups where the kernel is a p-group. In particular, we construct examples of saturated fusion systems which do not come from finite groups, but which have normal p-subgroups such that is the fusion system of a finite group. One of the tools used to do this is the concept of a “transporter system”, which is modelled on the transporter category of a finite group, and is more general than a linking system. B. Oliver is partially supported by UMR 7539 of the CNRS. J. Ventura is partially supported by FCT/POCTI/FEDER and grant PDCT/MAT/58497/2004.     

On mod <Emphasis Type="Italic">p</Emphasis> properties of Siegel modular forms

We prove two results on mod p properties of Siegel modular forms. First, we use theta series in order to construct of a Siegel modular form of weight p−1 which is congruent to 1 mod p. Second, we define a theta operator on q-expansions and show that the algebra of Siegel modular forms mod p is stable under , by exploiting the relation between and generalized Rankin-Cohen brackets.     

<Emphasis Type="Italic">K</Emphasis><Subscript>0</Subscript>-group and approximate similarity invariants of strongly irreducible operators

Let H olenote a complex separable Hilbert space and L(H) denote the collection of bounded linear operators on H. An operator T ∈ L(H) is said to be strongly irreducible if T does not commute with any nontrivial idempotent. Herrero and Jiang showed that the norm-closure of the class of all strongly irreducible operators is the class of all operators with connected spectrum. This result can be considered as an approximate inverse of the Riesz decomposition theorem. In the paper, we give a more precise charact...