The equation on weakly pseudo-convex CR manifolds of dimension 3

Let M be a smooth, compact, orientable, weakly pseudoconvex manifold of dimension 3, embedded in (N2), of codimension one or more, and endowed with the induced CR structure. Assuming that the tangential Cauchy-Riemann operator has closed range in L²(M) in order to rule out the Rossi example, we push regularity up to show has closed range in H^s(M) for all s>0. We then use the Szegö projection to show there is a smooth solution for the problem given smooth data. The results are obtained via microlocalization by piecing together estimates for functions and (0,1) forms that hold on different microlocal regions.     

The implicitization problem for

We develop in this paper methods for studying the implicitization problem for a rational map defining a hypersurface in , based on computing the determinant of a graded strand of a Koszul complex. We show that the classical study of Macaulay resultants and Koszul complexes coincides, in this case, with the approach of approximation complexes and we study and give a geometric interpretation for the acyclicity conditions.Under suitable hypotheses, these techniques enable us to obtain the implicit equation, up to a power, and up to some extra factor. We give algebraic and geometric conditions for determining when the computed equation defines the scheme theoretic image of , and, what are the extra varieties that appear. We also give some applications to the problem of computing sparse discriminants.     

On solutions of matrix equations and

With the help of the Kronecker map, a complete, general and explicit solution to the Yakubovich matrix equation V−AVF=BW, with F in an arbitrary form, is proposed. The solution is neatly expressed by the controllability matrix of the matrix pair (A,B), a symmetric operator matrix and an observability matrix. Some equivalent forms of this solution are also presented. Based on these results, explicit solutions to the so-called Kalman–Yakubovich equation and Stein equation are also established. In addition, based on the proposed solution of the Yakubovich matrix equation, a complete, general and explicit solution to the so-called Yakubovich-conjugate matrix is also established by means of real representation. Several equivalent forms are also provided. One of these solutions is neatly expressed by two controllability matrices, two observability matrices and a symmetric operator matrix.     

On matrix equations and

With the help of the concept of Kronecker map, an explicit solution for the matrix equation X−AXF=C is established. This solution is neatly expressed by a symmetric operator matrix, a controllability matrix and an observability matrix. In addition, the matrix equation is also studied. An explicit solution for this matrix equation is also proposed by means of the real representation of a complex matrix. This solution is neatly expressed by a symmetric operator matrix, two controllability matrices and two observability matrices.     

On the mean value of

Let χ be the Dirichlet character modulo q3 and L(s,χ) denote the corresponding Dirichlet L-function. The mean value of is studied and a few asymptotic formulae are given. Hybrid mean value of , general Kloosterman sums and general quadratic Gauss sums are considered.     

A characterization of the natural embedding of the split Cayley hexagon in by intersection numbers

In this paper, we prove that a set of q⁵+q⁴+q³+q²+q+1 lines of with the properties that (1) every point of is incident with either 0 or q+1 elements of , (2) every plane of is incident with either 0, 1 or q+1 elements of , (3) every solid of is incident with either 0, 1, q+1 or 2q+1 elements of , and (4) every hyperplane of is incident with at most q³+3q²+3q members of , is necessarily the set of lines of a regularly embedded split Cayley generalized hexagon in .     

The global attractor of the damped forced Ostrovsky equation

The existence of the global attractor of the damped forced Ostrovsky equation in is proved for the forces in . Moreover, the global attractor of the equation in is actually a compact set in .     

The operator equation

The solution of the linear operator equation:A^n-1X+A^n-2XB++AXB^n-2+XB^n-1=Y is given by if the spectra of A and B are in the sector {z:z≠0,-π/n<argz<π/n}.     

On the nonlinear matrix equation

Based on fixed point theorems for monotone and mixed monotone operators in a normal cone, we prove that the nonlinear matrix equation always has a unique positive definite solution. A conjecture which is proposed in [X.G. Liu, H. Gao, On the positive definite solutions of the matrix equation X^s±A^TX^-tA=I_n, Linear Algebra Appl. 368 (2003) 83–97] is solved. Multi-step stationary iterative method is proposed to compute the unique positive definite solution. Numerical examples show that this iterative method is feasible and effective.     

Global well-posedness and scattering for the defocusing -subcritical Hartree equation in

We prove the global well-posedness and scattering for the defocusing -subcritical (that is, 2<γ<3) Hartree equation with low regularity data in , d3. Precisely, we show that a unique and global solution exists for initial data in the Sobolev space with s>4(γ−2)/(3γ−4), which also scatters in both time directions. This improves the result in [M. Chae, S. Hong, J. Kim, C.W. Yang, Scattering theory below energy for a class of Hartree type equations, Comm. Partial Differential Equations 33 (2008) 321–348], where the global well-posedness was established for any s>max(1/2,4(γ−2)/(3γ−4)). The new ingredients in our proof are that we make use of an interaction Morawetz estimate for the smoothed out solution Iu, instead of an interaction Morawetz estimate for the solution u, and that we make careful analysis of the monotonicity property of the multiplier m(ξ)ξ^p. As a byproduct of our proof, we obtain that the H^s norm of the solution obeys the uniform-in-time bounds.     

Generalized rough approximations in

In this paper we will treat a generalization of inner and outer approximations of fuzzy sets, which we will call -inner and -outer approximations respectively ( being any finite set of rational numbers in [0,1]). In particular we will discuss the case of those fuzzy sets which are definable in the logic by means of step functions from the hypercube [0,1]^k and taking value in an arbitrary (finite) subset of . Then, we will show that if a fuzzy set is definable as truth table of a formula of , then both its -inner and -outer approximation are definable as truth table of formulas of . Finally, we will introduce a generalization of abstract approximation spaces and compare our approach with the notion of fuzzy rough set.     

On the recursive sequence

Our aim in this paper is to investigate the global asymptotic stability of all positive solutions of the higher order nonlinear difference equation

where B, C and α, β, γ are positive, k {1, 2, 3, … }, and the initial conditions x_−2k+1, … , x₋₁, x₀ are positive real numbers. We show that the unique positive equilibrium of the equation is globally asymptotically stable and has some basins that depend on certain conditions posed on the coefficients. Our concentration is on invariant intervals, the character of semicycles, and the boundedness of the above mentioned equation. Our final comments are about informative examples.     

Global well-posedness of Korteweg–de Vries equation in

We prove that the Korteweg–de Vries initial-value problem is globally well-posed in and the modified Korteweg–de Vries initial-value problem is globally well-posed in . The new ingredient is that we use directly the contraction principle to prove local well-posedness for KdV equation in H^−3/4 by constructing some special resolution spaces in order to avoid some ‘logarithmic divergence’ from the high–high interactions. Our local solution has almost the same properties as those for H^s (s>−3/4) solution which enable us to apply the I-method to extend it to a global solution.     

Compactness in

This paper is concerned with compactness for some topologies on the collection of bounded linear operators on Banach spaces. New versions of the Eberlein–Šmulian theorem and Day's lemma in the collection are established. Also we obtain a partial solution of the dual problem for the quasi approximation property, that is, it is shown that for a Banach space X if X^** is separable and X^* has the quasi approximation property, then X has the quasi approximation property.     

Coboundaries in

Let T be an ergodic automorphism of a probability space, f a bounded measurable function, . It is shown that the property that the probabilities μ(|S_n(f)|>n) are of order n^−p roughly corresponds to the existence of an approximation in L^∞ of f by functions (coboundaries) g−g○T, gL^p. Similarly, the probabilities μ(|S_n(f)|>n) are exponentially small iff f can be approximated by coboundaries g−g○T where g have finite exponential moments.

Résumé

Soit T un automorphisme ergodique d'un espace probabilisé, f une fonction bornée mesurable et . Une correspondance est établie entre l'existence de l'estimation des probabilités μ(|S_n(f)|>n) d'ordre n^−p et l'existence de l'approximation dans L^∞ de la fonction f par des cobords g−g○T où g est “presque” dans L^p. De manière similaire, les probabilités μ(|S_n(f)|>n) sont d'ordre e^−cn, pour un certain c>0, n=1,2… , si et seulement si f admet une approximation dans L^∞ par des cobords g−g○T avec g ayant des moments exponentiels.     

Semifields of order with left nucleus and center

In [G. Marino, O. Polverino, R. Trombetti, On -linear sets of PG(3,q³) and semifields, J. Combin. Theory Ser. A 114 (5) (2007) 769–788] it has been proven that there exist six non-isotopic families (i=0,…,5) of semifields of order q⁶ with left nucleus and center , according to the different geometric configurations of the associated -linear sets. In this paper we first prove that any semifield of order q⁶ with left nucleus , right and middle nuclei and center is isotopic to a cyclic semifield. Then, we focus on the family by proving that it can be partitioned into three further non-isotopic families: , , and we show that any semifield of order q⁶ with left nucleus , right and middle nuclei and center belongs to the family .     

Trace formulae for irreducible polynomials over with minimal order roots in

The extension field where q is a prime divisor of (P−1), has a unique structure. This paper describes this unique structure and uses it to derive formulas relating the trace values for elements in . These formulas can be refined for certain elements to produce a formula for the trace.     

On Rado numbers for

For all integers m3 and all natural numbers a₁,a₂,…,a_m−1, let n=R(a₁,a₂,…,a_m−1) represent the least integer such that for every 2-coloring of the set {1,2,…,n} there exists a monochromatic solution to

a₁x₁+a₂x₂++a_m−1x_m−1=x_m.