Infinitely many elliptic curves over $$\mathbf{Q }$$ have a Galois-stable cyclic subgroup of order 4. Such subgroups come in pairs, which intersect their 2. Let $$N_j(X)$$ denote the number with at least j pairs 4, and height most X. In this article we show that $$N_1(X) = c_{1,1}X^{1/3}+c_{1,2}X^{1/6}+O(X^{0.105})$$ . We also show, as $$X\rightarrow \infty $$ , $$N_2(X)=c_{2,1}X^{1/6}+o(X^{1/1...

In the ballistic regime we study both semiclassically and quantum mechanically electron's dynamics in two-dimensional electron gas (2DEG) presence of an inhomogeneous magnetic field applied perpendicular to plane. The is constant inside four separate circular regions which are located at corners a square side length larger than diameter circles, while outside circles zero. We carry out stabilit...

Toplumların refahının temel kaynağını oluşturan çevre, günümüzde refahın önemli bir tehditcisi durumuna gelmiştir. Temel ihtiyaçlar listesinde yerini alan “kaliteli çevre” için alınacak tedbirlerin etkinliği ise çevre kirliliği ile makroekonomik değişkenler arasındaki ilişkilerin tanımlanmasına bağlıdır. Bu çalışmada, Danimarka, Macaristan, İtalya, Malta, Hollanda, Portekiz, Slovakya, İsviçre, ...

In this paper, we mainly study the critical points and zero of solutions u to a kind linear elliptic equations with nonhomogeneous Dirichlet boundary conditions in multiply connected domain Ω ℝ2. Based on delicate analysis about distributions components super-level sets {x ∈ Ω: u(x) > t} sub-level < for some t, obtain geometric structure interior point u. Precisely, let be γI external γE, where...