Sign-changing solutions of nonlinear elliptic equations

In this survey article, we recall some known results on existence and multiplicity of sign-changing solutions of elliptic equations. Methods for obtaining sign-changing solutions developed in the last two decades will also be briefly revisited.      

Subdivision schemes for the fair discretization of the spherical motion group

We present two subdivision schemes for the fair discretization of the spherical motion group. The first one is based on the subdivision of the 600-cell according to the tetrahedral/octahedral subdivision scheme in [S. Schaefer, J. Hakenberg, J. Warren, Smooth subdivision of tetrahedral meshes, in: R. Scopigno, D. Zorin (Eds.), Eurographics Symposium on Geometry Processing, 2004, pp. 151–158]. The second presented subdivision scheme is based on the spherical kinematic mapping. In the first step we discretize an elliptic linear congruence by the icosahedral discretization of the unit sphere. Then the resulting lines of the elliptic three-space are discretized such that the difference in the maximal and minimal elliptic distance between neighboring grid points becomes minimal.     

Zeta functions of a class of elliptic curves over a rational function field of characteristic two

We show how to calculate the zeta functions and the orders of Tate-Shafarevich groups of the elliptic curves with equation over the rational function field , where is a power of 2. In the range , , odd of degree , the largest values obtained for are (one case), (one case) and (three cases). We observe and discuss a remarkable pattern for the distributions of signs in the functional equation and of fudge factors at places of bad reduction. These imply strong restrictions on the precise form of the Langlands correspondence for GL over local or global fields of characteristic two.

     

Calculation of values of -functions associated to elliptic curves

We calculated numerically the values of -functions of four typical elliptic curves in the critical strip in the range . We found that all the non-trivial zeros in this range lie on the critical line and are simple except the one at . The method we employed in this paper is the approximate functional equation with incomplete gamma functions in the coefficients. For incomplete gamma functions, we continued them holomorphically to the right half plane , which enables us to calculate for large . Furthermore we remark that a relation exists between Sato-Tate conjecture and the generalized Riemann Hypothesis.

     

Design Problem of Laminated Plates with Specified Characteristics

The problem of design of laminated plates with specified stiffness and strength characteristics is considered. The starting design problem is reduced to the convexcombination problems, which is solved by the convolution method. The following design problems are solved: design with allowance for strength, design of a laminated plate of unconstrained thickness, and design for approximately specified characteristics.     

Strongly indefinite systems with critical Sobolev exponents

We consider an elliptic system of Hamiltonian type on a bounded domain. In the superlinear case with critical growth rates we obtain existence and positivity results for solutions under suitable conditions on the linear terms. Our proof is based on an adaptation of the dual variational method as applied before to the scalar case.

     

Adsorption of anti-human IgG to plasma-polymerized allylamine film formed on a silver plate

Increase in the sensitivity of an immunosensor or biosensor requires the immobilization of a large amount of proteins. Good materials must be developed for this protein adsorption. Allylamine thin film was formed on a flat silver plate by plasma polymerization; this plate is referred to as Ag(ALAM). Ag(ALAM) films were characterized by SEM, FT–IR and ESCA. Investigations on adsorption and desorption of F(ab')₂ anti-human IgG (hIgG) on to Ag and Ag(ALAM) revealed the following: (1) The adsorption isotherm of F(ab')₂ anti-hIgG on to AG(ALAM) or Ag was of a Langmuir type. The binding constant (K_b) and saturation binding (A_b) for this protein on to Ag(ALAM) were 8.93 l/mol and 181.8 nmol/m², respectively, and for Ag, were 7.71 l/mole and 87.9 nmol/m². (2) The extent of desorption of labeled F(ab')₂ anti-hIgG absorbed on the two plates was the same. (3) Ag(ALAM) and Ag were used as solid phases in a two-site immunoradiometric assay of human serum IgG. The dose-response on AG(ALAM) occurred at lower concentration, and was of greater magnitude than that on Ag.     

On ring class eigenspaces of Mordell-Weil groups of elliptic curves over global function fields

If E is a non-isotrivial elliptic curve over a global function field F of odd characteristic we show that certain Mordell-Weil groups of E have 1-dimensional χ-eigenspace (with χ a complex ring class character) provided that the projection onto this eigenspace of a suitable Drinfeld-Heegner point is non-zero. This represents the analogue in the function field setting of a theorem for elliptic curves over Q due to Bertolini and Darmon, and at the same time is a generalization of the main result proved by Brown in his monograph on Heegner modules. As in the number field case, our proof employs Kolyvagin-type arguments, and the cohomological machinery is started up by the control on the Galois structure of the torsion of E provided by classical results of Igusa in positive characteristic.