Arbitrary Cardinality Research Articles

On graphs of central episturmian words

Episturmian sequences are a natural extension of Sturmian sequences to the case of finite alphabets of arbitrary cardinality. In this paper, we are interested in central episturmian words, or simply, epicentral words, i.e., the palindromic prefixes of standard episturmian sequences. An epicentral word admits a variety of faithful representations including as a directive word, as a certain type of period vector, as a Parikh vector, as a certain type of Fine and Wilf extremal word, as a suitable modular matrix, and as a labeled graph. Various interconnections between the different representations of an epicentral word are analyzed. In particular, we investigate the structure of the graphs of epicentral words proving some curious and surprising properties.

Covariant Genetic Dynamics

We present a covariant form for the dynamics of a canonical GA of arbitrary cardinality, showing how each genetic operator can be uniquely represented by a mathematical object - a tensor - that transforms simply under a general linear coordinate transformation. For mutation and recombination these tensors can be written as tensor products of the analogous tensors for one-bit strings thus giving a greatly simplified formulation of the dynamics. We analyze the three most well known coordinate systems -- string, Walsh and Building Block - discussing their relative advantages and disadvantages with respect to the different operators, showing how one may transform from one to the other, and that the associated coordinate transformation matrices can be written as a tensor product of the corresponding one-bit matrices. We also show that in the Building Block basis the dynamical equations for all Building Blocks can be generated from the equation for the most fine-grained block (string) by a certain projection ("zapping").

Metric Spaces with Linear Extensions Preserving Lipschitz Condition

We study a new bi-Lipschitz invariant λ( M ) of a metric space M ; its finiteness means that Lipschitz functions on an arbitrary subset of M can be linearly extended to functions on M whose Lipschitz constants are expanded by a factor controlled by λ( M ). We prove that λ( M ) is finite for several important classes of metric spaces. These include metric trees of arbitrary cardinality, groups of polynomial growth, Gromov-hyperbolic groups, certain classes of Riemannian manifolds of bounded geometry and the finite direct sums of arbitrary combinations of these objects. On the other hand we construct an example of a two-dimensional Riemannian manifold M of bounded geometry for which λ( M ) = ∞.

Classification from a Computable Viewpoint

Classification is an important goal in many branches of mathematics. The idea is to describe the members of some class of mathematical objects, up to isomorphism or other important equivalence, in terms of relatively simple invariants. Where this is impossible, it is useful to have concrete results saying so. In model theory and descriptive set theory, there is a large body of work showing that certain classes of mathematical structures admit classification while others do not. In the present paper, we describe some recent work on classification in computable structure theory.Section 1 gives some background from model theory and descriptive set theory. From model theory, we give sample structure and non-structure theorems for classes that include structures of arbitrary cardinality. We also describe the notion of Scott rank, which is useful in the more restricted setting of countable structures. From descriptive set theory, we describe the basic Polish space of structures for a fixed countable language with fixed countable universe. We give sample structure and non-structure theorems based on the complexity of the isomorphism relation, and on Borel embeddings.Section 2 gives some background on computable structures. We describe three approaches to classification for these structures. The approaches are all equivalent. However, one approach, which involves calculating the complexity of the isomorphism relation, has turned out to be more productive than the others. Section 3 describes results on the isomorphism relation for a number of mathematically interesting classes—various kinds of groups and fields. In Section 4, we consider a setting similar to that in descriptive set theory. We describe an effective analogue of Borel embedding which allows us to make distinctions even among classes of finite structures. Section 5 gives results on computable structures of high Scott rank. Some of these results make use of computable embeddings. Finally, in Section 6, we mention some open problems and possible directions for future work.

Going-Down Implies Generalized Going-Down

It is proved that if a unital homomorphism f of commutative rings satisfies the going-down property GD, then f also satisfies the infinitistic variant, the generalized going-down property GGD (that is, f exhibits going-down behavior for prime ideal chains of arbitrary cardinality).

Decompositions of infinite graphs: Part II circuit decompositions

We characterize the graphs that admit a decomposition into circuits, i.e. finite or infinite connected 2-regular graphs. Moreover, we show that, as is the case for the removal of a closed eulerian subgraph from a finite graph, removal of a non-dominated eulerian subgraph from a (finite or infinite) graph does not change its circuit-decomposability or circuit-indecomposability. For cycle-decomposable graphs, we show that in any end which contains at least n + 1 pairwise edge-disjoint rays, there are n edge-disjoint rays that can be removed from the graph without altering its cycle-decomposability. We also generalize the notion of the parity of the degree of a vertex to vertices of infinite degree, and in this way extend the well-known result that eulerian finite graphs are circuit-decomposable to graphs of arbitrary cardinality.

Amalgamating infinite latin squares

A finite latin square is an n × n matrix whose entries are elements of the set { 1 , … , n } and no element is repeated in any row or column. Given equivalence relations on the set of rows, the set of columns, and the set of symbols, respectively, we can use these relations to identify equivalent rows, columns and symbols, and obtain an amalgamated latin square. There is a set of natural equations that have to be satisfied by an amalgamated latin square. Using these equations we can define the notion of an outline latin square and it follows easily that an amalgamated latin square is an outline latin square. Hilton (Math. Programming Stud. 13 (1980) 68) proved that the opposite implication holds as well, that is, every outline latin square is an amalgamated latin square. In this paper, we present a generalization of that result to infinite latin squares with the sets of rows, columns and symbols of arbitrary cardinality.

Containment for c, s, t operators on a binary relation

The operators c, s and t are complement, symmetric and transitive closure of a binary relation. If u and v denote finite sequences of these operators then we define u ≤ v iff for every binary relation \( uR \subseteq vR \). We find the distinct representative and containment between these sequences. The asymmetric operator is not one of these. There are 54 representatives for binary relations, 20 for transitive relations, and 10 for symmetric relations. There are 26 component types of a binary relation, 10 for transitive relations, and 6 for symmetric relations. There are 16 connected types of a binary relation, 8 for transitive relations, and 4 for symmetric relations. We study well founded relations. Total relations may not be contractible but well founded ones are. The complement of (a Hasse diagram of) a non-empty partial order of arbitrary cardinality is contractible. Ordered sets are naturally homotopy equivalent to partially ordered sets. There are 10 relations which can have arbitrary polyhedral homotopy type and 42 are either contractible or the homotopy type of a wedge of n-spheres. The homotopy type of two relations is not determined.

Bases in max-algebra

For n-tuples over the algebraic system ( R,⊕,⊗)=( R, max,+) , concepts such as linear dependence, space and basis may be defined by analogy with classical linear algebra. Whenever a space is finitely generated, it possesses a basis and all its bases are trivially related and therefore have the same cardinality. However, for any given n>2, spaces with bases of arbitrary cardinality may be constructed, as well as spaces with no basis.

Theorems on sets not belonging to algebras

Let A 1 , … , A n , A n + 1 \mathcal {A}_1,\dots , \mathcal {A}_n, \mathcal {A}_{n+1} be a finite sequence of algebras of sets given on a set X X , ⋃ k = 1 n A k ≠ P ( X ) \bigcup _{k=1}^n \mathcal {A}_k \ne \mathfrak {P}(X) , with more than 4 3 n \frac {4}{3}n pairwise disjoint sets not belonging to A n + 1 \mathcal {A}_{n+1} . It has been shown in the author’s previous articles that in this case ⋃ k = 1 n + 1 A k ≠ P ( X ) \bigcup _{k=1}^{n+1} \mathcal {A}_k \ne \mathfrak {P}(X) . Let us consider, instead of A n + 1 \mathcal {A}_{n+1} , a finite sequence of algebras A n + 1 , … , A n + l \mathcal {A}_{n+1}, \dots , \mathcal {A}_{n+l} . It turns out that if for each natural i ≤ l i \le l there exist no less than 4 3 ( n + l ) − l 24 \frac {4}{3}(n+l)- \frac {l}{24} pairwise disjoint sets not belonging to A n + i \mathcal {A}_{n+i} , then ⋃ k = 1 n + l A k ≠ P ( X ) \bigcup _{k=1}^{n+l} \mathcal {A}_k \ne \mathfrak {P}(X) . Besides this result, the article contains: an essentially important theorem on a countable sequence of almost σ \sigma -algebras (the concept of almost σ \sigma -algebra was introduced by the author in 1999), a theorem on a family of algebras of arbitrary cardinality (the proof of this theorem is based on the beautiful idea of Halmos and Vaughan from their proof of the theorem on systems of distinct representatives), a new upper estimate of the function v ( n ) \mathfrak {v}(n) that was introduced by the author in 2002, and other new results.

On robust matrix completion with prescribed eigenvalues

Matrix completion with prescribed eigenvalues is a special kind of inverse eigenvalue problems. Thus far, only a handful of specific cases concerning its existence and construction have been studied in the literature. The general problem where the prescribed entries are at arbitrary locations with arbitrary cardinalities proves to be challenging both theoretically and computationally. This paper investigates some continuation techniques by recasting the completion problem as an optimization of the distance between the isospectral matrices with the prescribed eigenvalues and the affine matrices with the prescribed entries. The approach not only offers an avenue to solving the completion problem in its most general setting but also makes it possible to seek a robust solution that is least sensitive to perturbation.

On linear spaces and matroids of arbitrary cardinality

In this paper, we study linear spaces of arbitrary finite dimension on some (possibly infinite) set. We interpret linear spaces as simple matroids and study the problem of erecting some linear space of dimension n to some linear space of dimension n + 1 if possible. Several examples of some such erections are studied; in particular, one of these erections is computed within some infinite iteration process.

GENERATING THE FULL TRANSFORMATION SEMIGROUP USING ORDER PRESERVING MAPPINGS

Fo ra linearly ordered set X we consider the relative rank of the semigroup of all order preserving mappings OX on X modulo the full transformation semigroup TX .I no ther words, we ask what is the smallest cardinality of a set A of mappings such thatOX ∪ A �= TX .W he nX is countably infinite or well-ordered (of arbitrary cardinality) we show that this number is one, while when X = (the set of real numbers) it is uncountable.

Gradient flow methods for matrix completion with prescribed eigenvalues

Matrix completion with prescribed eigenvalues is a special type of inverse eigenvalue problem. The goal is to construct a matrix subject to both the structural constraint of prescribed entries and the spectral constraint of prescribed spectrum. The challenge of such a completion problem lies in the intertwining of the cardinality and the location of the prescribed entries so that the inverse problem is solvable. An intriguing question is whether matrices can have arbitrary entries at arbitrary locations with arbitrary eigenvalues and how to complete such a matrix. Constructive proofs exist to a certain point (and those proofs, such as the classical Schur–Horn theorem, are amazingly elegant enough in their own right) beyond which very few theories or numerical algorithms are available. In this paper the completion problem is recast as one of minimizing the distance between the isospectral matrices with the prescribed eigenvalues and the affined matrices with the prescribed entries. The gradient flow is proposed as a numerical means to tackle the construction. This approach is general enough that it can be used to explore the existence question when the prescribed entries are at arbitrary locations with arbitrary cardinalities.

Monodic Packed Fragment with Equality is Decidable

We prove decidability of satisfiability of sentences of the monodic packed fragment of first-order temporal logic with equality and connectives Until and Since, in models with various flows of time and domains of arbitrary cardinality. We also prove decidability over models with finite domains, over flows of time including the real order.

Dedekind completion as a method for constructing new Scott domains

Many operations exist for constructing Scott-domains. This paper presents Dedekind completion as a new operation for constructing such domains and outlines an application of the operation. Dedekind complete Scott domains are of particular interest when modeling versions of λ calculus that allow quantification over sets of arbitrary cardinality. Hence, it is of interest when constructing models of powerful specification languages and when using λ calculus as a foundation for mathematics.

On the complexity measures of genetic sequences.

It is well known that the regulatory regions of genomes are highly repetitive. They are rich in direct, symmetric and complemented repeats, and there is no doubt about the functional significance of these repeats. Among known measures of complexity, the Ziv-Lempel complexity measure reflects most adequately repeats occurring in the text. But this measure does not take into account isomorphic repeats. By isomorphic repeats we mean fragments that are identical (or symmetric) modulo some permutation of the alphabet letters. In this paper, two complexity measures of symbolic sequences are proposed that generalize the Ziv-Lempel complexity measure by taking into account any isomorphic repeats in the text (rather than just direct repeats as in Ziv-Lempel). The first of them, the complexity vector, is designed for small alphabets such as the alphabet of nucleotides. The second is based on a search for the longest isomorphic fragment in the history of sequence synthesis and can be used for alphabets of arbitrary cardinality. These measures have been used for recognition of structural regularities in DNA sequences. Some interesting structures related to the regulatory region of the human growth hormone are reported.

A note on the infinitary action of triangular norms and conorms

In this note the action of triangular norms and conorms on subsets of the unit interval with arbitrary cardinality is defined, and the properties of these generalized operations are also considered.

On prequojections and their duals

The paper is devoted to the class of Frechet spaces which are called prequojections. This class appeared in a natural way in the structure theory of Frechet spaces. The structure of prequojections was studied by G. Metafune and V.B. Moscatelli, who also gave a survey of the subject. Answering a question of these authors we show that their result on duals of prequojections cannot be generalized from the separable case to the case of spaces of arbitrary cardinality. We also introduce a special class of prequojections, we call them canonical, and show that in the main result of G. Metafune and V.B. Moscatelli on the existence of a prequojection with a given dual we may require this prequojection to be a canonical one.

Some Semilattices of Semigroups Each Having One Idempotent

We give a structure theorem for semigroups of arbitrary cardinality which are semilattices of t-semigroups, semilattices of groups and semilattices of PJ-semigroups each having only one idempotent.