act on the upper half-plane by linear fractional transformations. We may form the quotient Y0(N) = Γ0(N)\H, a Riemann surface with finitely many cusps. Compactifying gives a curve X0(N) which is in fact defined over Q; the map z → (j(z), j(Nz)) ∈ AC realizes Y0(N) as a (highly singular) plane curve with Q-coefficients. Over a general field k of characteristic zero, the k-points of the curve X0(...

We consider magnetic flux moving in superconductors with periodic pinning arrays. We show that sample heating by moving vortices produces negative differential resistivity (NDR) of both N and S type (i.e., N- and S-shaped) in the voltage-current characteristic (VI curve). The uniform flux flow state is unstable in the NDR region of the VI curve. Domain structures appear during the NDR part of t...

We prove that the isomorphism class of the tame fundamental group of a smooth, connected curve over an algebraically closed eld k of characteristic p > 0 determines the genus g and the number n of punctures of the curve, unless (g, n) = (0, 0), (0, 1). Moreover, assuming g = 0, n > 1, and that k is the algebraic closure of the prime eld Fp, we prove that the isomorphism class of the tame fundam...

Let $N$ be a compact, connected, nonorientable surface of genus $g$ with $n$ boundary components. $\mathcal {C}(N)$ the curve complex $N$. We prove that if $(g, n) \neq (1,2)$ and $g + n 4$, then there is an exhaustion

Four different vector fields on (x,y)-plane A vector field on a sphere ♣ Curves. A smooth curve is a continuous curve (no gaps or discontinuities) with no corners (no abrupt changes in slope at a point such as you would get from the intersection of two lines). A continuous and differentiable produces a smooth function. A path C is called closed if its initial and final points are the same point...

Diagnosing and quantifying plant-parasitic nematodes is critical for efficient nematode management. Several studies have been performed intending to demonstrate nematode quantification via real-time quantitative PCR. However, most of the studies used dilution of DNA templates to make standard curves, while few studies used samples with different nematode numbers to make the standard curve, resu...

This paper establishes an isomorphism between the Bar-Natan skein module of the solid torus with a particular boundary curve system and the homology of the (n, n) Springer variety. The results build on Khovanov’s work with crossingless matchings and the cohomology of the (n, n) Springer variety. We also give a formula for comultiplication in the Bar-Natan skein module for this specific three-ma...