Some Invariants of Admissible Homotopies of Space Curves

A regular homotopy of a generic curve in a three-dimensional projective space is called admissible if it defines a generic one-parameter family of curves in which every curve has neither self-intersections nor inflection points, is not tangent to a smooth part of its evolvent, and has no tangent planes osculating with the curve at two different points. We indicate some invariants of admissible homotopies of space curves and prove, in particular, that the curve cannot be deformed in the class of admissible homotopies into a curve without flattening points.     

Tangential limit values of a biharmonic poisson integral in a disk

Let C ₀ be a curve in a disk D={|z|<1} that is tangent to the circle at the point z=1, and let C _θ be the result of rotation of this curve about the origin z=0 by an angle θ. We construct a bounded function biharmonic in D that has a zero normal derivative on the boundary and for which the limit along C _θ does not exist for all θ, 0≤θ≤2π.     

Zero density of L-functions related to Maass forms

Let f(z) be a Hecke-Maass cusp form for SL ₂(?), and let L(s, f) be the corresponding automorphic L-function associated to f. For sufficiently large T, let N(σ, T) be the number of zeros ρ = β +iγ of L(s, f) with |γ| ? T, β ? σ, the zeros being counted according to multiplicity. In this paper, we get that for 3/4 ? σ ? 1 ? ?, there exists a constant C = C(?) such that N(σ,T) ? T ^2(1?σ)/σ(logT) ^C , which improves the previous results.     

A priori estimates for one-dimensional singular integral operators with continuous coefficients

Let the contour γ consist of a finite number of simple closed pairwise nonintersecting curves, satisfying a Lyapunov condition, let S be the operator of singular integration in spacel _p , (γ) (1 <p < ∞), and leta (t), b (t) εC (γ) 1 <p ₁. <p < ∞. The necessary and sufficient condition for A = aI+ bS to be a Φ-operator in space L_p(γ) is that, for all?ε L_p(γ), ∥?∥_p ? const (∥ A?_p + ∥ ? ∥_p1), where ∥?∥_p = ∥?∥L_p (γ).     

An example of the Weierstrass semigroup of a pointed curve on K3 surfaces

Let X be a K3 surface with Picard number one which is given by a double cover π:X→?². Let C be a smooth curve on X with π ^?1 π(C)=C which is not the ramification divisor of π, and let P be a ramification point of π| _C :C→π(C). In this paper, in the case where the intersection multiplicity at π(P) of the curve π(C) and the tangent line at π(P) on π(C) is equal to deg(π(C)) or deg(π(C))?1, we investigate the Weierstrass semigroup of the pointed curve (C,P).     

Cantor-Lebesgue theorem for double trignometric series

Let ∥·∥ be a norm in R² and let γ be the unit sphere induced by this norm. We call a segment joining points x,y ε R² rational if (x₁ ? y₁)/(x₂ ? y₂) or (x₂ ? y₂)/(x₁ ? y₁) is a rational number. Let γ be a convex curve containing no rational segments. Satisfaction of the condition $$T_\nu (x) = \sum\nolimits_{\parallel n\parallel = \nu } {c_n e^{2\pi i(n_1 x_1 + n_2 x_2 )} } \to 0(\nu \to \infty )$$ in measure on the set e? [- 1/2,1/2)×[- 1/2, 1/2) =T² of positive planar measure implies ∥T _v ∥L₄ (T²) → 0(v → ∞). if, however, γ contains a rational segment, then there exist a sequence of polynomials {T _v } and a set E ? T², ¦E¦ > 0, such that T _v (x) → 0(v → ∞) on E; however, ¦c_n¦ ? 0 for ∥n∥ → ∞.     

On the zeros of the Davenport–Heilbronn function

Let N ₀(T) be the number of zeros of the Davenport–Heilbronn function in the interval [1/2, 1/2+ i T]. It is proved that N ₀(T) ? T (ln T)^1/2+1/16?ε, where ε is an arbitrarily small positive number.     

A boundary uniqueness theorem for holomorphic functions of several complex variables

If D ? Cⁿ is a region with a smooth boundary and M ? ?D is a smooth manifold such that for some point p ∈ M the complex linear hull of the tangent plane T_p(M) coincides with Cⁿ, then for each functionf ε A(D) the conditionf¦M=0 implies thatf=0 in D.     

Improved gradient method for monotone and lipschitz continuous mappings in banach spaces

Let C be a nonempty closed convex subset of a 2-uniformly convex and uniformly smooth Banach space E and {A_n}_(n∈N) be a family of monotone and Lipschitz continuos mappings of C into E~*. In this article, we consider the improved gradient method by the hybrid method in mathematical programming [10] for solving the variational inequality problem for{A_n} and prove strong convergence theorems. And we get several results which improve the well-known results in a real 2-uniformly convex and uniformly smooth Banach space and a real Hilbert space.     

Fixed point iteration processes for asymptotic pointwise nonexpansive mappings in Banach spaces

Let X be a uniformly convex Banach space with the Opial property. Let T:C→C be an asymptotic pointwise nonexpansive mapping, where C is bounded, closed and convex subset of X. In this paper, we prove that the generalized Mann and Ishikawa processes converge weakly to a fixed point of T. In addition, we prove that for compact asymptotic pointwise nonexpansive mappings acting in uniformly convex Banach spaces, both processes converge strongly to a fixed point.     

A note on rational normal curves totally tangent to a Hermitian variety

Let q be a power of a prime integer p, and let X be a Hermitian variety of degree q + 1 in the n-dimensional projective space. We count the number of rational normal curves that are tangent to X at distinct q + 1 points with intersection multiplicity n. This generalizes a result of Segre on the permutable pairs of a Hermitian curve and a smooth conic.     

Uniqueness Theorems for CR Functions

Let M be a CR manifold embedded in ?^s of arbitrary codimension. M is called generic if the complex hull of the tangent space in all points of M is the whole ?^s. M is minimal (in sense of Tumanov) in p ? M if there does not exist any CR submanifold of M passing through p with the same CR dimension as M but of smaller dimension. Let M be generic and minimal in some point p ? M and N be a generic submanifold of M passing through p. We prove that a continuous CR function on M vanishes identically in some neigbourhood of p if its restriction to N either vanishes in p faster then some function with non-integrable logarithm or it vanishes on a subset of N of positive measure.     

Lagrangian 4-planes in holomorphic symplectic varieties of K3[4]-type

We classify the cohomology classes of Lagrangian 4-planes ?⁴ in a smooth manifold X deformation equivalent to a Hilbert scheme of four points on a K3 surface, up to the monodromy action. Classically, the Mori cone of effective curves on a K3 surface S is generated by nonnegative classes C, for which (C, C) ≥ 0, and nodal classes C, for which (C, C) = ?2; Hassett and Tschinkel conjecture that the Mori cone of a holomorphic symplectic variety X is similarly controlled by “nodal” classes C such that (C, C) = ?γ, for (·,·) now the Beauville-Bogomolov form, where γ classifies the geometry of the extremal contraction associated to C. In particular, they conjecture that for X deformation equivalent to a Hilbert scheme of n points on a K3 surface, the class C = ? of a line in a smooth Lagrangian n-plane ? ⁿ must satisfy (?,?) = ?(n + 3)/2. We prove the conjecture for n = 4 by computing the ring of monodromy invariants on X, and showing there is a unique monodromy orbit of Lagrangian 4-planes.     

The Kernel of the Eisenstein Ideal

Let N be a prime number, and let J₀(N) be the Jacobian of the modular curve X₀(N). Let T denote the endomorphism ring of J₀(N). In a seminal 1977 article, B. Mazur introduced and studied an important ideal I⊆T, the Eisenstein ideal. In this paper we give an explicit construction of the kernel J₀(N)[I] of this ideal (the set of points in J₀(N) that are annihilated by all elements of I). We use this construction to determine the action of the group Gal(Q/Q) on J₀(N)[I]. Our results were previously known in the special case where N−1 is not divisible by 16.     

Resolution of veronese embedding of plane curves

Let C be a smooth (irreducible) curve of degree d in ?². Let ?² ? ?⁵ be the Veronese embedding and let I _C denote the homogeneous ideal of C on ?⁵. In this note we explicitly write down the minimal free resolution of I _C for d ≥ 2.     

Estimation effective de la perte de positivité dans la régularisation des courants

Let (X,ω) be a compact complex Hermitian manifold, and let T?γ be a d-closed (1,1) almost positive current on X. A variant of Demailly's regularization-of-currents theorem states that T is the weak limit of a sequence of (1,1)-currents T_m with analytic singularities of coefficient 1/m, lying in the same cohomology class as T, whose Lelong numbers converge to those of T, and with a loss of positivity decaying to zero. We prove that if the (1,1)-form γ is assumed to be closed and C^∞, the regularizing currents T_m can be chosen such that $\text{T}{}_{\mathrm{m}}\text{?γ?}\text{C}\text{m}$ for a constant C>0 independent of m. To cite this article: D. Popovici, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Deformation of CR-manifolds,minimal points and CR-manifolds with the microlocal analytic extension property

LetM be a smooth CR-manifold embedded into ? ⁿ . Letp be a point inM and letC be a small truncated cone inM (in suitable Euclidean coordinates onM) with vertexp which “symmetry axis” is a real vector in the complex tangent space. Then one can deformM into a smooth CR-manifoldM _d letting fixed all points outsideC in such a way thatp is a minimal point ofM _d . This result is used to give a new proof of the fact that wedge extendability of continuous CR-functions propagates along the CR-orbits of a CR-manifold. It allows also to prove the following natural result which was conjectured by Trepreau. LetM be a smooth generic CR-manifold in ? ⁿ . SupposeM consists of one single CR-orbit. Then each continuous CR-function onM is wedge extendable at any point ofM. Uniqueness theorems for continuous CR-functions are derived.