An extension of the peripheral group and its associated structures such as the meridian and longitude to knots of arbitrary dimension and genus is studied. The analogous structures are shown to provide a complete algebraic invariant for oriented spun tori, by using the Tube map of Satoh. The algebraic invariant also provides a constraint on equivalence classes of welded knots in the preimage of...

Reconstructing the logical network from the transition matrix is benefit for learning the logical meaning of the algebraic result from the algebraic representation of a BN. And so far there has no method to convert the matrix expression back to the logic expression for a BN with an arbitrary topology structure. Based on the canonical form and Karnaugh map, we propose a method for reconstructing...

We show that any infinite algebraic subgroup of the plane Cremona group over a perfect field is contained in maximal group. classify groups, and their subgroups rational points, up to conjugacy by birational map.

Topological quantum field theory led recently to a spectacular progress in numerical algebraic geometry. It was shown that generating functions of certain charactertistic numbers of modular spaces of stable algebraic curves with labelled points satisfy remarkable differential equations of KP -type (E. Witten, M. Kontsevich). In a later series of developments, this was generalized, partly conjec...

We will show that the singular cohomology groups of a smooth quasi-projective complex variety relative to normal crossing divisor can be described in terms delta-admissible chains. Roughly speaking, chain is simplicial semi-algebraic meeting faces properly. As an application, we Abel-Jacobi map for higher Chow cycles via example, describe Hodge realization polylog constructed by Bloch-Kriz map.

A linear algebraic group G over a field k is called a Cayley group if it admits a Cayley map, i.e., a G-equivariant birational isomorphism over k between the group variety G and its Lie algebra. A Cayley map can be thought of as a partial algebraic analogue of the exponential map. A prototypical example is the classical “Cayley transform” for the special orthogonal group SOn defined by Arthur C...

In this work, the assembly map in L-theory for the family of finite subgroups is proven to be a split injection for a class of groups. Groups in this class, including virtually polycyclic groups, have universal spaces that satisfy certain geometric conditions. The proof follows the method developed by Carlsson-Pedersen to split the assembly map in the case of torsion free groups. Here, the cont...

This is a tutorial on some aspects of toric varieties related to their potential use in geometric modeling. We discuss projective toric varieties and their ideals, as well as real toric varieties. In particular, we explain the relation between linear precision and a particular linear projection we call the algebraic moment map.

Every continuous map X → S defines, by composition, a homomorphism between the corresponding algebras of real-valued continuous functions C(S) → C(X). This paper deals with algebraic properties of the homomorphism C(S)→ C(X) in relation to topological properties of the map X → S. The main result of the paper states that a continuous map X → S between topological manifolds is a finite (branched)...

The problem of describing the homotopy groups of spheres has been fundamental to algebraic topology for around 80 years. There were periods when specific computations were important and periods when the emphasis favored theory. Many mathematical invariants have expressions in terms of homotopy groups, and at different times the subject has found itself located in geometric topology, algebra, al...