<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.     

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.     

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.     

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).      

Remarks on the <Emphasis Type="Italic">k</Emphasis>-error linear complexity of <Emphasis Type="Italic">p</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>-periodic sequences

Recently the first author presented exact formulas for the number of 2 ⁿ -periodic binary sequences with given 1-error linear complexity, and an exact formula for the expected 1-error linear complexity and upper and lower bounds for the expected k-error linear complexity, k ≥ 2, of a random 2 ⁿ -periodic binary sequence. A crucial role for the analysis played the Chan–Games algorithm. We use a more sophisticated generalization of the Chan–Games algorithm by Ding et al. to obtain exact formulas for the counting function and the expected value for the 1-error linear complexity for p ⁿ -periodic sequences over prime. Additionally we discuss the calculation of lower and upper bounds on the k-error linear complexity of p ⁿ -periodic sequences over .      

Deforming Curves in Jacobians to Non-Jacobians I: Curves in <Emphasis Type="Italic">C</Emphasis><Superscript>(2)</Superscript>

We introduce deformation theoretic methods for determining when a curve X in a nonhyperelliptic Jacobian JC will deform with JC to a non-Jacobian. We apply these methods to a particular class of curves in the second symmetric power of C. More precisely, given a pencil of degree d on C, let X be the curve parametrizing pairs of points in divisors of (see the paper for the precise scheme-theoretical definition). We prove that if X deforms infinitesimally out of the Jacobian locus with JC then either d=4 or d=5, dim H° and C has genus 4 This material is based upon work partially supported by the National Security Agency under Grant No. MDA904-98-1-0014 and the National Science Foundation under Grant No. DMS-0071795. Any opinions, findings and conclusions or recomendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation (NSF) or the National Security Agency (NSA)     

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

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.     

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.      

A speciality theorem for curves in P<Superscript>5</Superscript>

Let be an integral projective curve. One defines the speciality index e(C) of C as the maximal integer t such that , where ω _C denotes the dualizing sheaf of . Extending a classical result of Halphen concerning the speciality of a space curve, in the present paper we prove that if is an integral degree d curve not contained in any surface of degree  < s, in any threefold of degree  < t, and in any fourfold of degree  < u, and if , then Moreover equality holds if and only if C is a complete intersection of hypersurfaces of degrees u, , and . We give also some partial results in the general case , .      

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.      

Computing pairings using <Emphasis Type="Italic">x</Emphasis>-coordinates only

To reduce bandwidth in elliptic curve cryptography one can transmit only x-coordinates of points (or x-coordinates together with an extra bit). This is called point compression. For further computation using the points one can either recover the y-coordinates by taking square roots or one can use point multiplication formulae which use x-coordinates only. We consider how to efficiently use point compression in pairing-based cryptography when the embedding degree is even. We give a method to compute compressed pairings using x-coordinates only. We also show how to compute the compressed pairings using two x-coordinates and one y-coordinate. Our methods are more efficient than taking square roots when the embedding degree is small. We implemented the algorithms in the case of embedding degree 2 curves over where (mod 4) and found that our methods can be 10–15% faster than the analogous methods using square roots.      

<Emphasis Type="Italic">r</Emphasis>-Minimal submanifolds in space forms

Let be an n-dimensional compact, possibly with boundary, submanifold in an (n + p)-dimensional space form R ^n+p (c). Assume that r is even and , in this paper we introduce rth mean curvature function S _r and (r + 1)-th mean curvature vector field . We call M to be an r-minimal submanifold if on M, we note that the concept of 0-minimal submanifold is the concept of minimal submanifold. In this paper, we define a functional of , by calculation of the first variational formula of J _r we show that x is a critical point of J _r if and only if x is r-minimal. Besides, we give many examples of r-minimal submanifolds in space forms. We calculate the second variational formula of J _r and prove that there exists no compact without boundary stable r-minimal submanifold with in the unit sphere S ^n+p . When r = 0, noting S ₀ = 1, our result reduces to Simons’ result: there exists no compact without boundary stable minimal submanifold in the unit sphere S ^n+p .      

Extremal <Emphasis Type="Italic">G</Emphasis>-invariant eigenvalues of the Laplacian of <Emphasis Type="Italic">G</Emphasis>-invariant metrics

The study of extremal properties of the spectrum often involves restricting the metrics under consideration. Motivated by the work of Abreu and Freitas in the case of the sphere S ² endowed with S ¹-invariant metrics, we consider the subsequence of the spectrum of a Riemannian manifold M which corresponds to metrics and functions invariant under the action of a compact Lie group G. If G has dimension at least 1, we show that the functional λ _k ^G admits no extremal metric under volume-preserving G-invariant deformations. If, moreover, M has dimension at least three, then the functional is unbounded when restricted to any conformal class of G-invariant metrics of fixed volume. As a special case of this, we can consider the standard O(n)-action on S ⁿ ; however, if we also require the metric to be induced by an embedding of S ⁿ in , we get an optimal upper bound on .      

Inequalities for derivatives of functions in the spaces <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>

  We obtain a new sharp inequality for the local norms of functions x ∈ L _{∞, ∞} ^r (R), namely,

Slightly larger than a graph<Emphasis Type="Italic">C</Emphasis><Superscript>*</Superscript>-algebra

To certain higher rank Cuntz algebras including the classical cases we trace a certain partial isometryU in its strong closure. AdjoiningU to we obtain a kind of uniqueness property for this largerC ^*-algebra. Its explanation is not entirely “Cuntz’s uniqueness argumentation”. The author is supported by the Austrian Research Foundation (FWF) project S8308.     

On the negative cyclic homology of <Emphasis Type="Italic">shc</Emphasis>-algebras

Let be a field of characteristic and S ¹ the unit circle. We prove that the shc-structure on a cochain algebra (A,d _A ) induces an associative product on the negative cyclic homology HC _*⁻ A. When the cochain algebra (A,d _A ) is the algebra of normalized cochains of the simply connected topological space X with coefficients in , then HC _*⁻ A is isomorphic as a graded algebra to the S ¹-equivariant cohomology algebra of LX, the free loop space of X. We use the notion of shc-formality introduced in Topology 41, 85–106 (2002) to compute the S ¹-equivariant cohomology algebras of the free loop space of the complex projective space when n + 1 = 0 [p] and of the even spheres S ²ⁿ when p = 2.      

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.     

<Emphasis Type="Italic">L</Emphasis>-functions of symmetric products of the Kloosterman sheaf over Z

The classical n-variable Kloosterman sums over the finite field F _p give rise to a lisse -sheaf Kl _n+1 on , which we call the Kloosterman sheaf. Let L _p (G _{m, F p} , Sym ^k Kl _n+1, s) be the L-function of the k-fold symmetric product of Kl _n+1. We construct an explicit virtual scheme X of finite type over Spec Z such that the p-Euler factor of the zeta function of X coincides with L _p (G _{m, F p} , Sym ^k Kl _n+1, s). We also prove similar results for and . The research of L. Fu is supported by the NSFC (10525107).     

<Emphasis Type="Italic">R</Emphasis>-bounded Representations of <Emphasis Type="Italic">L</Emphasis><Superscript>1</Superscript> (<Emphasis Type="Italic">G</Emphasis>)

We investigate R-bounded representations , where X is a Banach space and G is a lca group. Observing that Ψ induces a (strongly continuous) group homomorphism , we are then able to analyze certain classical homomorphisms U (e.g. translations in L^p (G)) from the viewpoint of R-boundedness and the theory of scalar-type spectral operators. Dedicated to the memory of H. H. Schaefer