Isotone Maps as Maps of Congruences. II. Concrete Maps

Let L be a lattice and let L ₁, L ₂ be sublattices of L. Let be a congruence relation of L ₁. We extend to L by taking the smallest congruence......     

A Note on Weakly Isotone Maps and Common Solutions for Differential Systems

In this short note we examine the connection between weakly isotone maps and common solutions for first order Cauchy problems in R^n and, as a rule, in Banach lattices.     

Random Logistic Maps. I

Let {C _i} ₀ be a sequence of independent and identically distributed random variables with vales in [0, 4]. Let {X _n} ₀ be a sequence of random variables with values in [0, 1] defined recursively by X _n+1=C _n+1 X _n(1–X _n). It is shown here that: (i) E ln C ₁<0X _n0 w.p.1. (ii) E ln C ₁=0X _n0 in probability (iii) E ln C ₁>0, E |ln(4–C ₁)| such that (0, 1)=1 and is invariant for {X _n}. (iv) If there exits an invariant probability measure such that {0}=0, then E ln C ₁>0 and – ln(1–x) (dx)=E ln C ₁. (v) E ln C ₁>0, E |ln(4–C ₁)|< and {X _n} is Harris irreducible implies that the probability distribution of X _n converges in the Cesaro sense to a unique probability distribution on (0, 1) for all X ₀0.     

Idempotent Block Splitting on Partial Partitions,I: Isotone Operators

Image segmentation algorithms can be modelled as image-guided operators (maps) on the complete lattice of partitions of space, or on the one of partial partitions (i.e., partitions of subsets of the space). In particular region-splitting segmentation algorithms correspond to block splitting operators on the lattice of partial partitions, in other words anti-extensive operators that act by splitting each block independently. This first paper studies in detail block splitting operators and their lattice-theoretical and monoid properties; in particular we consider their idempotence (a requirement in image segmentation). We characterize block splitting openings (kernel operators) as operators splitting each block into its connected components according to a partial connection; furthermore, block splitting openings constitute a complete sublattice of the complete lattice of all openings on partial partitions. Our results underlie the connective approach to image segmentation introduced by Serra. The second paper will study two classes of non-isotone idempotent block splitting operators, that are also relevant to image segmentation.     

Isotone mapping of free lattices

An arbitrary partially ordered set P and an isotone mapping of this set into a complete lattice L are considered. It is proved that the mapping can be extended to an isotone mapping of the lattice FL(P) into the lattice L.Translated from Matematicheskie Zametki, Vol. 4, No. 4, pp. 405–416, October, 1968.The author is deeply indebted to L. A. Skornyakov, under whose guidance the work has been done.     

Abstract Logics, Logic Maps, and Logic Homomorphisms

What is a logic? Which properties are preserved by maps between logics? What is the right notion for equivalence of logics? In order to give satisfactory answers we generalize and further develop the topological approach of [4] and present the foundations of a general theory of abstract logics which is based on the abstract concept of a theory. Each abstract logic determines a topology on the set of theories. We develop a theory of logic maps and show in what way they induce (continuous, open) functions on the corresponding topological spaces. We also establish connections to well-known notions such as translations of logics and the satisfaction axiom of institutions [5]. Logic homomorphisms are maps that behave in some sense like continuous functions and preserve more topological structure than logic maps in general. We introduce the notion of a logic isomorphism as a (not necessarily bijective) function on the sets of formulas that induces a homeomorphism between the respective topological spaces and gives rise to an equivalence relation on abstract logics. Therefore, we propose logic isomorphisms as an adequate and precise notion for equivalence of logics. Finally, we compare this concept with another recent proposal presented in [2]. This research was supported by the grant CNPq/FAPESB 350092/2006-0.     

Weakly Regular Semigroups of Isotone Transformations

Let X be a partially ordered set and O(X) be the semigroup of all mappings X → X that preserve the order, i.e., x ≤ y ? xα ≤ yα for all x, y ∈ X. It is proved that the semigroup O(X) is weakly regular in the wide sense if and only if at least one of the following conditions holds: (1) X is a quasi-complete chain; (2) the elements of X are not comparable pairwise; (3) X = Y ∪ Z, where y < z for y ∈ Y, z ∈ Z; (4) X = Y ∪ Z, where y ₀ ∈ Y, z ₀ ∈ Z, and y ₀ < z for z ∈ Z, y < z0 for y ∈ Y; (5) X = {a, c} ∪ B, where a < b < c for b ∈ B; (6) X = {1, 2, 3, 4, 5, 6}, where 1 < 4, 1 < 5, 2 < 5, 2 < 6, 3 < 4, 3 < 6. Moreover, if X is a quasi-ordered set but not partially ordered, then the semigroup O(X) is weakly regular in the wide sense if and only if x ≤ y for all x, y ∈ X.     

Classification of Plane Congruences in 4 C4 (I)

Nondegenerate plane congruences in the four-dimensional complex projective space with degenerate general focal conic are classified by using the focal method due to Corrado Segre.     

Isotone solutions of parametric linear complementarity problems

The parametric linear complementarity problem is given by the conditions:q + p + Mz 0, 0,z 0,z ^T (q + p + Mz) = 0. Under the assumption thatM is a P-matrix, Cottle proved that the solution mapz() of the above problem is montonically nondecreasing in the parameter for every nonnegativeq and everyp if and only ifM is a Minkowski matrix. This paper examines whether a similar result holds in various other settings including a nonlinear case.     

Abstract multiparameter theory,I

In this paper we continue our investigation of multiparameter spectral theory. Let H₁,…, H_k be separable Hilbert spaces and H = ?_{r = 1}^kH_r, be their tensor product. In each space H_r we have densely defined self-adjoint operators T_r and continuous Hermitian operators V_rs. The multiparameter eigenvalue problem concerns eigenvalues λ = (λ₁,…, λ_n) ?R^k and eigenvectors $\text{? = ?}{}_1\text{? ··· ? ?}{}_{\mathrm{k}}\text{? H}$ such that $\text{T}{}_{\mathrm{r}}\text{?}{}_{\mathrm{r}}\text{+ ∑}{}_{\mathrm{s\; =\; 1}}{}^{\mathrm{k}}\text{λ}{}_{\mathrm{s}}\text{V}{}_{\mathrm{rs}}\text{?}{}_{\mathrm{r}}\text{= 0}$. We develop a spectral theory for such systems leading to a Parseval equality and generalized eigenvector expansion. The results are applied to a k × k system of linked secondorder differential equations.     

Congruences on an inverse Bruck-Reilly monoid Part I: Non-group congruences

Communicated by N. R. Reilly     

Lattice tensor products. III - Congruences

G. Grätzer and F. Wehrung introduced the lattice tensor product, A?B, of the lattices Aand B. In Part I of this paper, we showed that for any finite lattice A, we can "coordinatize" A?B, that is, represent A?,B as a subset A of B ^A, and provide an effective criteria to recognize the A-tuples of elements of B that occur in this representation. To show the utility of this coordinatization, we prove, for a finite lattice A and a bounded lattice B, the isomorphism Con A ≌ (Con A)B>, which is a special case of a recent result of G. Grätzer and F. Wehrung and a generalization of a 1981 result of G. Grätzer, H. Lakser, and R.W. Quackenbush.     

A Fixed-Point Theorem and Equilibria of Abstract Economies with Weakly Upper Semicontinuous Set-Valued Maps

The main purpose of this paper is to introduce the notion of weakly upper semicontinuous set-valued maps and to establish a new fixed-point theorem. The set-valued maps with an approximating upper semicontinuous selection property are also defined. Next, we use our fixed-point result to obtain equilibrium existence in abstract economies with two constraints, which provide a natural scenario for potential applications of our approach to general equilibrium theory. In this regard, we set models of economies with asymmetric informed agents, who are able to improve their initial information through market signals. These economies offer examples in which the informational feasibility requirement represents an additional constraint.     

On Thermodynamics of Rational Maps. II: Non-Recurrent Maps

The pressure function p(t) of a non-recurrent map is real analyticon some interval (0,t_*) with t_* strictly greater than the dimensionof the Julia set. The proof is an adaptation of the well knowntower techniques to the complex dynamics situation. In general,p(t) need not be analytic on the whole positive axis.     

Quantum Hamiltonians with Weak Random Abstract Perturbation. I. Initial Length Scale Estimate

We study random Hamiltonians on finite-size cubes and waveguide segments of increasing diameter. The number of random parameters determining the operator is proportional to the volume of the cube. In the asymptotic regime where the cube size, and consequently the number of parameters as well, tends to infinity, we derive deterministic and probabilistic variational bounds on the lowest eigenvalue, i.e., the spectral minimum, as well as exponential off-diagonal decay of the Green function at energies above, but close to the overall spectral bottom.