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...

Recall that hyperbolic 3-manifolds M and N are said to be commensurable if they have a common finite sheeted covering. This is equivalent to the fundamental groups having subgroups of finite index which are conjugate in PSL(2,C). In general it is very difficult to determine if two manifolds are commensurable or not, once the most obvious invariants of commensurability (for example, the invarian...

By a tournament we mean a directed graph (T,→) such that whenever x, y are two distinct elements of T , then precisely one of the two cases, either x→ y or y → x, takes place. There is a one-to-one correspondence between tournaments and commutative groupoids satisfying ab ∈ {a, b} for all a and b: set ab = a if and only if a→ b. This makes it possible to identify tournaments with their correspo...

Consumers are often observed to purchase more than one variety of a product on a given shopping trip. The simultaneous demand for varieties is observed not only for packaged goods such as yogurt or soft drinks, but in many other product categories such as movies, music compact disks, and apparel. Multinomial (MN) choice models cannot be applied to data exhibiting the simultaneous choice of more...

Beginning with a review of some optimization problems in RKHS, and going on to a model selection problem via Likelihood Basis Pursuit (LBP).

nowadays, potato is one of the most important foods in the people¢s diet. daily consumption of potato is increasing in iran significantly; therefore the study and research on this product is essential. in order to the components of potato, three varieties of potato (diamant, aola and draga) were chosen from thirty six fields in north of khorasan province. each variety was harvested from 12 fiel...

The most important tool for classifying recognizable languages is Eilenberg’s variety theorem [1], which gives a one-to-one correspondence between (pseudo)-varieties of finite semigroups and varieties of recognizable languages. Varieties of recognizable languages are classes of recognizable languages closed under union, intersection, complement, left and right quotients and inverse morphisms. R...

The selective advantages of using light as a source of information are reflected in the diverse types of extant eyes. The physical properties of light restrict how it can be collected and processed, resulting in only eight known optical systems found in animals. Eyes develop through tissue rearrangement and differentiation. Our understanding of the source of genetic information used in developm...