多项式微分系统的不变直线的不同斜率的最大个数

本文证明了公式β（ｎ）＝σ（ｎ－１）＋１其中α（ｎ－１）是ｎ－１次多项式微分系统的不为直线的最多条数，βｎ）是ｎ次多项式微分系统的不变直线的不同斜率的最大个数。这里假设所讨论的多项式系统只有限条不变直线。     

Charge-ordered ferromagnetic phase in La(0.5)Ca(0.5)MnO3

Mixed-valent manganites are noted for their unusual magnetic, electronic and structural phase transitions. For example, the La(1-x)Ca(x)MnO(3) phase diagram shows that below transition temperatures in the range 100-260 K, compounds with 0.2 < x < 0.5 are ferromagnetic and metallic, whereas those with 0.5 < x < 0.9 are antiferromagnetic and charge ordered. In a narrow region around x = 0.5, these totally dissimilar ground states are thought to coexist. It has been shown that charge order and charge disorder can coexist in the related compound, La(0.25)Pr(0.375)Ca(0.375)MnO(3). Here we present electron microscopy data for La(0.5)Ca(0.5)MnO(3) that shed light on the distribution of these coexisting phases, and uncover an additional, unexpected phase. Using electron holography and Fresnel imaging, we find micrometre-sized ferromagnetic regions spanning several grains coexisting with similar-sized regions with no local magnetization. Holography shows that the ferromagnetic regions have a local magnetization of 3.4 +/- 0.2 Bohr magnetons per Mn atom (the spin-aligned value is 3.5 micro (B) per Mn). We use electron diffraction and dark-field imaging to show that charge order exists in regions with no net magnetization and, surprisingly, can also occur in ferromagnetic regions.