The diversity of immunoglobulin (Ig) functions may be explained by the peculiar structure of these flexible molecules. Heavy and light Ig chains are composed from compact stable immunoglobulin folds (IgF), or domains belonging to a large protein super family. Proteins composed from IgF have been found in various representatives of animal kingdom even those appeared during the very early stages ...

In this note we settle a question posed by Hobby and McKenzie in [2] on the nature of locally finite equational classes which satisfy some nontrivial congruence identity. We settle problem #14 from [2] by exhibiting a locally finite equational class V which omits types 1 and 5 and which satisfies some non-trivial congruence identity, but which contains a finite algebra having a type 4 minimal s...

We determine normal forms and ranks of tensors of border rank at most three. We present a differential-geometric analysis of limits of secant planes in a more general context. In particular there are at most four types of points on limiting trisecant planes for cominuscule varieties such as Grassmannians. We also show that the singular locus of the secant varieties σr(Seg(P × P × P)) has codime...

Dematerialization, represented by declining consumption per GDP of energy or of goods, offers some hope for rising environmental quality with development. The declining proportion of income spent on staples as affluence grows, which income elasticity <1.0 measures, makes dematerialization widespread. Further, as learning improves efficiency of resource use, the intensity of environmental impact...

An algebraic notion of representational consistency is defined. A theorem relating it to free actions is proved. A metrizability problem of the quotient (a shape space) is discussed. This leads to a new algebraic variety with a metrizability result. A concrete example is given from stereo vision.

Introduction The word brachydactyly is used to describe the hands and feet with shortened digits (brachy, meaning “short,” and daktylos, meaning “digit”) [1]. The digits themselves may be shorter than normal or they may appear small because of shortening of metacarpal and metatarsal bones in the hands or feet [1]. Most of the patients with brachydactyly do not require any specific treatment. Wh...

A collection of n distinct hyperplanes Li = {li = 0} ⊂ P , the (n − 1)-dimensional projective space over an algebraically closed field of characteristic not equal to 2, is a polar simplex of a smooth quadric Q = {q = 0}, if each Li is the polar hyperplane of the point pi = ⋂ j 6=i Lj , equivalently, if q = l 2 1 + . . .+ l 2 n for suitable choices of the linear forms li. In this paper we study ...

Let (M,ω) be a smooth symplectic algebraic variety. A (possibly singular) algebraic subvariety Y ⊂ M is said to be isotropic, resp. Lagrangian, if the tangent space, TyY , at any regular point y ∈ Y is an isotropic, resp. Lagrangian, vector subspace in the symplectic vector space TyM (we always assume Y to be reduced, but not necessarily irreducuble). The following characterisation of isotropic...

We discuss abelian variety and use it to generalize the concept of elliptic curve we discussed in the class. We state some of the basic properties of abelian variety and in particular 1-dimensional abelian variety. The main result of this report is to show that elliptic curve is that the only 1-dimensional abelian variety.

Most mathematicians have encountered modular functions. For example, when the group theorists discovered the monster group, they were surprised to find that the degrees of its irreducible representations were already encoded in the q-coefficients of the j-function. The theory of Shimura varieties grew out of the applications of modular functions and modular forms to number theory. Roughly speak...