Tadeusz Krasiński

THE LOJASIEWICZ EXPONENT OF AN ISOLATED WEIGHTED HOMOGENEOUS SURFACE SINGULARITY

Mei-Chi Shaw) We give an explicit formula for the Lojasiewicz exponent of an isolated weighted homogeneous surface singularity in terms of its weights. From the formula we get that the Lojasiewicz exponent is a topological invariant of these singularities.

The jump of the Milnor number in the X 9 singularity class

The jump of the Milnor number of an isolated singularity f0 is the minimal non-zero difference between the Milnor numbers of f0 and one of its deformations (fs). We prove that for the singularities in the X9 singularity class their jumps are equal to 2.

Arithmetical Analysis of Biomolecular Finite Automaton

In the paper we present a theoretical analysis of extension of the finite automaton built on DNA introduced by the Shapiro team to an arbitrary number of states and symbols. In the implementation we use a new idea of several restriction enzymes instead of one. We give arithmetical conditions for the existence of such extensions in terms of ingredients used in the implementation.

Towards an autonomous multistate biomolecular devices built on DNA

A major challenge in DNA computing area is to design autonomous and programmable biomolecular devices built on DNA. The significant achievement in the field of DNA nanodevices was a laboratory implementation of the 2-state biomolecular finite automaton based on one restriction enzyme FokI [3]. Although this practical implementation represents a proof of concept for autonomous computing with DNA molecules, it has a limited computational power. The restriction enzyme FokI enables construction an automata with at most 3-states. We propose to use several restriction enzymes (instead of one) which act autonomously in a test tube to construct more powerful finite state machines. It enables to build any finite nondeterministic automata or even push-down automata. The autonomous operation of the automaton is based on alternating cleavages of DNA molecules by several restriction enzymes. We illustrate this new idea by presenting a laboratory implementation of a particular case of finite automata. In this experiment two restriction endonucleases act autonomously on DNA in one test tube. This approach may be used (in the future) to build nanomachines, even push-down automata (made of DNA molecules) which may be applied in medicine, pharmacy or biotechnology.

Jumps of Milnor Numbers of Brieskorn–Pham Singularities in Non-degenerate Families

The jump of the Milnor number of an isolated singularity

On the Łojasiewicz exponent for analytic curves

f_{0}

On the Łojasiewicz exponent at infinity for polynomial mappings of ℂ² into ℂ² and components of polynomial automorphisms of ℂ²

f0 is the minimal non-zero difference between the Milnor numbers of