Some birationality criteria on 3-folds with p g > 1

We give some birationality criteria for φ _m (m = 4, 5, 6, 7) on general type 3-folds with p _g ? 2 by means of an intensive classification. In particular, we show that φ ₇ is not birational if and only if p _g (X) = 2 and X admits a genus 2 curve family of canonical degree 2/3. When the canonical volume is large, we also characterize the birationality of φ ₄, φ ₅ and φ ₆.     

Subspaces of L p , p > 2, with unconditional bases have equivalent partition and weight norms

In this note we give a simple proof that every subspace of L_p, 2 < p < ∞, with an unconditional basis has an equivalent norm determined by partitions and weights. Consequently L_p has a norm determined by partitions and weights. Received: 31 January 2005     

Analytic Extension of Non Quasi - Analytic Whitney Jets of Beurling Type

Let (M_r)_r∈_?0 be a logarithmically convex sequence of positive numbers which verifies M₀ = 1 as well as M_r ≥ 1 for every r ∈ ? and defines a non quasi - analytic class. Let moreover F be a closed proper subset of ?ⁿ. Then for every function f on ?ⁿ belonging to the non quasi - analytic (M_r)-class of Beurling type, there is an element g of the same class which is analytic on ?,ⁿ F and such that D^αf(x) = D^αg(x) for every α ∈ ?ⁿ₀ and x ∈ F.     

On the convergence of double Fourier series of functions from Lp,p > 1

It is proved that if a function from L_p, p > 1, satisfies the condition $$\frac{1}{{t \cdot \tau }}\int_0^t {\int_0^\tau {\left| {f(x + u,y + v) - f(x,y)} \right|} dudv = O\left( {\left[ {1n\frac{1}{{(t^2 + \tau ^2 )}}} \right]^{ - 2} } \right),}$$ then the double Fourier series of function f, under summation over a rectangle, converges almost everywhere.     

Geometric Realisations of Cubical Sets with Connections, and Classifying Spaces of Categories

We investigate the category of cubical sets with some additional degeneracies called connections. We prove that the realisation of a cubical set with connections is independent, up to homotopy, of whether we collapse those extra degeneracies or not and that any cubical set which is Kan admits connections. Using this type of cubical sets we define the cubical classifying space of a category and prove that this is equivalent to the simplicial one.     

On the approximation by Chebyshev splines in the metric of L p , p > 0

We prove a direct Jackson estimate for the approximation by Chebyshev splines in the classes L _p , p > 0. Translated from Ukrainskii Matematicheskii Zhurnal, Vol.50, No. 9, pp. 1193–1201, September, 1998.     

Primarite deL p (l r , 1<p,r<∞

In this article we show that if 1<p, r<∞, the spaceL _p (l _r ) is primary. If we let (h _k ) be the Haar system inL _p and (e _i ) the usual base ofl _r , we give sufficient conditions on a subsequence of (h _k ⊗e _i ) _{k ≧1,i≧1} for it to generate a space isomorphic toL _p (l _r ). We deduce the primarity ofL _p (l _r ).      

On uncomplemented subspaces ofL p , 1 <p <2

It is shown tat, for 1 <p < 2, there is an uncomplemented subspace ofL _p [0,1] that is isomorphic to Hilbert space.     

Density of systems of trigonometric polynomials in L p , 0 < p < 1

Suppose that μ is a nonnegative finite regular Borel measure on an infinite compact abelian group Γ. Let M be a finite subset of the dual group G of Γ. It is shown that if p ∈ (0, 1), then the linear span of GobM is dense in L ^p (μ). The result is extended to L ^p -spaces of operator-valued functions which are p-integrable with respect to certain operator-valued measures. An application to a theorem by Koosis is given.     

H p -spaces of martingales, 0<p≦1

For 0 <p≦1 the dual Banach space to the space of H _p -martingales is characterized as a Lipschitz class of order α=l/p?1. When p=1 and α=0 this reduces to BMO being the dual of H ₁ . In the course of the investigation a hierarchy of martingale spaces _a L _p is introduced with the property that the index p is in the range ?∞ <1/p<∞. For l≦b≦q<∞ the space L _q is a subspace of _b L _q (equality when b=q) while for q<0 the space _b L _q corresponds to a Lipschitz class of order α=?1/q.     

The Lp dual Minkowski problem for p > 1 and q > 0

General $L_p$ dual curvature measures have recently been introduced by Lutwak, Yang and Zhang [24]. These new measures unify several other geometric measures of the Brunn–Minkowski theory and the dual Brunn–Minkowski theory. $L_p$ dual curvature measures arise from qth dual intrinsic volumes by means of Alexandrov-type variational formulas. Lutwak, Yang and Zhang [24] formulated the $L_p$ dual Minkowski problem, which concerns the characterization of $L_p$ dual curvature measures. In this paper, we solve the existence part of the $L_p$ dual Minkowski problem for $p>1$ and $q>0$, and we also discuss the regularity of the solution.     

Riemann surfaces of genus g with an automorphism of order p prime and p > g

The present work completes the classification of the compact Riemann surfaces of genus g with an analytic automorphism of order p (prime number) and p > g. More precisely, we construct a parameterization space for them, we compute their groups of uniformization and we compute their full automorphism groups. Also, we give affine equations in for special cases and some implications on the components of the singular locus of the moduli space of smooth curves of genus g.     

On local Morse theory for p-area functionals,p > 2

This survey article deals with some Morse theoretic aspects for functionals defined in Sobolev Banach spaces, associated with quasilinear elliptic equations or systems, involving the p-Laplace operator, p > 2.We discuss the notion of nondegeneracy in a Banach (not Hilbert) variational framework and we present some developments concerning the critical groups estimates and the interpretation of the multiplicity of a critical point.     

Primarite DeL p(L r), 1<p,r<∞

In this article we show thatL ^p(L ^r) is primary forp andr in ]1,+∞[. If (h _k) _k≧1 denote the Haar basis, we begin with a study of the sequence (h _k ⊗h _i) and, in particular, the space generated by a subsequence of this sequence. In the first part we study the base ofL ^p(L ^r) and in the second part we show that this space is primary.     

Infinite families of 2- and 3-designs with parameters v = p + 1, k = (p − 1)/2i + 1, where p odd prime, 2e T (p − 1), e ≥ 2, 1 ≤ i ≤ e

Let p be an odd prime number such that p − 1 = 2^em for some odd m and e ≥ 2. In this article, by using the special linear fractional group PSL(2, p), for each i, 1 ≤ i ≤ e, except particular cases, we construct a 2-design with parameters v = p + 1, k = (p − 1)/2ⁱ + 1 and λ = ((p − 1)/2ⁱ+1)(p − 1)/2 = k(p − 1)/2, and in the case i = e we show that some of these 2-designs are 3-designs. Likewise, by using the linear fractional group PGL(2,p) we construct an infinite family of 3-designs with the same v k and λ = k(k − 2). These supplement a part of [4], in which we gave an infinite family of 3-designs with parameters v = q + 1, k = (q + 1)/2 = (q − 1)/2 + 1 and λ = (q + 1)(q − 3)/8 = k(k − 2)/2, where q is a prime power such that q − 1 = 2m for some odd m and q > 7. Some of the designs given in this article and in [4] fill in a few blanks in the table of Chee, Colbourn, and Kreher [2]. © 1997 John Wiley & Sons, Inc.     

Представление функций классовL p [0, 1], 0<p<1, ортогональными рядами

u . uuu u