We consider an abstract equation F (λ, u) = 0 with one parameter λ, where F ∈ C(R × X,Y ), p ≥ 2 is a nonlinear differentiable mapping, and X,Y are Banach spaces. We apply Lyapunov-Schmidt procedure and Morse Lemma to obtain a “double” saddle-node bifurcation theorem near a degenerate point with a two-dimensional kernel. It is shown that the solution set of the equation is the union of two para...

Consider genus g curves that admit degree d covers of an elliptic curve. Varying a branch point, we get a one-parameter family W of simply branched covers. Varying the target elliptic curve, we get another one-parameter family Y of covers that have a unique branch point. We investigate the geometry of W and Y by using admissible covers to study their slopes, genera and components. The results c...

Let F (x, y) = ax + bxy + cy be a positive definite binary quadratic form with discriminant Δ whose base points lie on the line x = −1/m for an integer m ≥ 2, let p be a prime number and let Fp be a finite field. Let EF : y = ax + bx + cx be an elliptic curve over Fp and let CF : ax + bx + cx ≡ 0(mod p) be the cubic congruence corresponding to F . In this work we consider some properties of pos...

We have just finished our study of plane and space curves, or, if you like, of vectorvalued functions of a scalar variable. The next unit of the course focuses on the differential calculus of surfaces in three-dimensional space: these include the case of scalar-valued functions of a vector variable, i.e., z = f(x, y). It turns out that that surfaces in R are most analogous to curves in the plan...

A simple and parsimonious model which originated from the Weibull frequency distribution was proposed to describe nonlinear survival curves of spores. This model was suitable for downward concavity curves (Bacillus cereus and Bacillus pumilus), as well as for upward concavity curves (Clostridium botulinum). It was shown that traditional F values calculated from this new model were no longer add...

By combining experiments with simple model calculations, we obtain new insight in spin transport through hybrid, CoFeB/Al2O3(1.5 nm)/tris(8-hydroxyquinoline)aluminium (Alq3)/Co spin valves. We have measured the characteristic changes in the I-V behavior as well as the intrinsic loss of magnetoresistance at the onset of multiple-step tunneling. In the regime of multiple-step tunneling, under the...

Sarabia et al. (1999) present a basic model to create ordered families of Lorenz curves, along with basic theorems that describe the conditions for the models to satisfy the definition of the Lorenz curve. Using these basic models, they suggest a family which includes several well-known Lorenz models as special cases. This paper first shows that their basic theorems can be generalized. The pape...

OOset curves have diverse engineering applications , which have consequently motivated extensive research concerning various ooset techniques. OOset research in the early 1980s focused on approximation techniques to solve immediate application problems in practice. This trend continued until 1988, when Hoschek 1, 2] applied non-linear optimization techniques to the ooset approximation problem. ...

The contact structure of two meromorphic curves gives a factorization of their jacobian. Section 1: Introduction Let J(F,G) = J(X,Y )(F,G) be the jacobian of F = F (X,Y ) and G = G(X,Y ) with respect to X and Y , i.e., let J(F,G) = FXGY − FY GX where subscripts denote partial derivatives. Here, to begin with, F and G are plane curves, i.e., polynomials in X and Y over an algebraically closed gr...

Let Y be a closed oriented 3-manifold with a contact form such that all Reeb orbits are nondegenerate. The embedded contact homology (ECH) index associates an integer to each relative 2-dimensional homology class of surfaces whose boundary is the difference between two unions of Reeb orbits. This integer determines the relative grading on ECH; the ECH differential counts holomorphic curves in t...