Let D be the ordered set of isomorphism types of finite distributive lattices, where the ordering is by embeddability. We study first-order definability in this ordered set. We prove among other things that for every finite distributive lattice D, the set {d, d} is definable, where d and d are the isomorphism types of D and its opposite (D turned upside down). We prove that the only non-identit...

Let L1 be a finite lattice with an ideal L2. Then the restriction map is a {0, 1}-homomorphism from ConL1 into ConL2. In 1986, the present authors published the converse. If D1 and D2 are finite distributive lattices, and φ : D1 → D2 is a {0, 1}-homomorphism, then there are finite lattices L1 and L2 with an embedding η of L2 as an ideal of L1, and there are isomorphisms ε1 : ConL1 → D1 and ε2 :...

By a lattice we shall always mean a distributive lattice which is bounded, i.e. has both a bottom element 0 and a top element 1. Lattice homomorphisms will always be assumed to preserve 0 and 1. By a modality on a (distributive) lattice L = (L, ∧, ∨, ≤, 0, 1) is meant a map : L → L satisfying (1) 1 = 1, (2) (x ∧ y) = x ∧ y for x, y ∈ L. The pair (L, ) will be called a modalized (distributive) l...

A resonance basis map is an important way of analyzing lattice properties in circular accelerators. A method is developed to do tracking with resonance basis maps. The speed of this method is faster than other map tracking methods, and can be dramatically increased by dropping insignificant resonance terms. This mapping method enables us to simulate and study the interplay between lattice nonli...

we propose a new cryptosystem by combing the Lissajous map, which is the asymptotic model of deterministic randomness, with the one-way coupled map lattice (OCML). The key space, the encryption efficiency, and the security are investigated. We find that the parameter sensitivity can reach the computational precision when the system size is only three, and all the lattice outputs can be treated ...

The phase diagram of the coupled sine circle map lattice exhibits a variety of interesting phenomena including spreading regions with spatiotemporal intermittency, non-spreading regions with spatial intermittency, and coherent structures termed solitons. A cellular automaton mapping of the coupled map lattice maps the spreading to non-spreading transition to a transition from a probabilistic to...

In commissioning of an e − e− collider, a great deal of effort is put to search and to remove origin of the luminosity reduction. We have studied luminosity reduction mechanisms using computer simulations to help the jobs. We consider collision among an electron bunch and a positron bunch; that is, do not consider any multi-bunch effects. One turn map of beam particles consists of the beam-beam...

We show that there is a canonical, order preserving map ψ of lattices of subgroups, which maps the lattice Sub(A) of subgroups of the ideal class group of a galois number field K into the lattice Sub(H/K) of subfields of the Hilbert class field. Furthermore, this map is a capitulation map in the sense that all the primes in the classes of A ⊂ A capitulate in ψ(A). In particular we have a new, s...

Chaotic strings are a class of non-hyperbolic coupled map lattices, exhibiting a rich structure of complex dynamical phenomena with a surprising correspondence to physical contents. In this paper we introduce different types and models for chaotic strings, where 2B-strings with finite length are considered in detail. We demonstrate possibilities to extract renormalized quantities, which are exp...