We study the averaging of fronts moving with positive oscillatory normal velocity, which is periodic in space and stationary ergodic in time. The problem can be formulated as the homogenization of coercive level set Hamilton-Jacobi equations with spatio-temporal oscillations. To overcome the difficulties due to the oscillations in time and the linear growth of the Hamiltonian, we first study the long time averaged behavior of the associated reachable sets using geometric arguments. The results are new for higher than one dimensions even in the space-time periodic setting.

In this paper, we introduce a new method to avoid zero reductions in Gröbner basis computation. We call this method LASyz, which stands for Lineal Algebra to compute Syzygies. LASyz uses exhaustively the information of both principal syzygies and non-trivial syzygies to avoid zero reductions. All computation is done using linear algebra techniques. LASyz is easy to understand and implement. The method does not require to compute Gröbner bases of subsequences of generators incrementally and it imposes no restrictions on the reductions allowed. We provide a complete theoretical foundation for the LASyz method and we describe an algorithm to compute Gröbner bases for zero dimensional ideals based on this foundation. A qualitative comparison with similar algorithms is provided and the performance of the algorithm is illustrated with experimental data.

Let$K$be the class of trigonometric series of power type, i.e. Taylor series $$\sum\nolimits_{n = 0}^\infty {c_n z^n } $$ for$z$=$e$^{$ix$}, whose partial sums for all$x$in$E$, where$E$is a nondenumerable subset of [0, 2π), lie on a$finite$number of circles (a priori depending on$x$) in the complex plane. The main result of this paper is that for every member of the class$K$, there exist a complex number ω, |ω|=1, and two positive integers $$\nu , \kappa , \nu < \kappa $$ , such that for the coefficients$c$_{$n$}we have: $$c_{\mu + \lambda \left( {\kappa - \nu } \right)} = c_\mu \omega ^\lambda , \mu = \nu ,\nu + 1, \ldots , \kappa - 1, \lambda = 1,2,3, \ldots $$ . Thus, every member of the class$K$has (with minor modifications) a representation of the form: $$P(x)\sum\nolimits_{n = 0}^\infty {e^{iknx} ,} $$ where P($x$) is a suitable trigonometric polynomial and$k$a positive integer. The proof is elementary but rather long. This result is closely related to a theorem of Marcinkiewicz and Zygmund on the circular structure of the set of limit points of the sequence of partial sums of ($C$, 1) summable Taylor series.

In this paper, we construct some unirational Calabi–Yau threefolds in characteristic 3. We adopt the method by Schoen, but we use quasi-elliptic surfaces instead of elliptic surfaces. We find new examples which do not admit a lifting to characteristic zero.

No abstract uploaded!