With a given holomorphic section of a Hermitian vector bundle, one can associate a residue current by means of Cauchy–Fantappiè–Leray type formulas. In this paper we define products of such residue currents. We prove that, in the case of a complete intersection, the product of the residue currents of a tuple of sections coincides with the residue current of the direct sum of the sections.

A moment problem is presented for a class of signed measures which are termed pseudo-positive. Our main result says that for every pseudo-positive definite functional (subject to some reasonable restrictions) there exists a representing pseudo-positive measure.

In this paper, we study existence of global entropy weak solutions to a critical-case unstable thin lm equation in one-dimensional case ht + @x(h n @xxxh) + @x(h n+2@xh) = 0; where n  1. There exists a critical mass Mc = 2 p 6 3 found by Witelski etal. (2004 Euro. J. of Appl. Math. 15, 223{256) for n = 1. We obtain global existence of a non-negative entropy weak solution if initial mass is less than Mc. For n  4, entropy weak solutions are positive and unique. For n = 1,a nite time blow-up occurs for solutions with initial mass larger than Mc.For the Cauchy problem with n = 1 and initial mass less than Mc, we show that at least one of the following long-time behavior holds: the second moment goes to innity as the time goes to innity or h(
; tk) * 0 in L1 (R) for some subsequence tk ! 1.

In this paper, we consider a class of optimization problems of minimizing a (at least) twice continuously dierentiable function (probably nonconvex) f(x) : Rn ! R over a product of multiple balls/spheres constraints. Upon rescaling the balls/spheres, we cast without loss of generality such class of minimization problems in the following form: (BCOP) 8>< >: min x2Rn f(x) s:t: ci(x) := kx[i]k2 􀀀 1 = 0; i 2 E; ci(x) := kx[i]k2 􀀀 1  0; i 2 I; where E = f1; 2; : : : ;m1g, I = fm1 + 1;m1 + 2; : : : ;mg, x[i] 2 Rpi , x = (xT [1]; xT [2]; : : : ; xT [m])T , n = Pm i=1 pi, and k
k stands for the `2 vector norm. Here, we introduce the notation x[i] 2 Rpi to represent the ith subvector of x 2 Rn and formulate the product of multiple ball/sphere constraints as a set of equality and inequality constraints. To simplify subsequent presentation, we call the above programming the ball/sphere constrained optimization problem (BCOP). We emphasize that this problem does not allow overlap among the variables x[i] and therefore the constraints are separable. However, these variables x[i] may be linked together through the objective function f(x).

In this Letter we propose that for Lax integrable nonlinear partial differential equations the natural concept of weak solutions is implied by the compatibility condition for the respective distributional Lax pairs. We illustrate our proposal by comparing two concepts of weak solutions of the modified Camassa-Holm equation pointing out that in the <i>peakon</i> sector (a family of non-smooth solitons) only one of them, namely the one obtained from the distributional compatibility condition, supports the time invariance of the Sobolev <i>H</i>¹ norm.