Properties of Solutions of Generalized Sturm–Liouville Discrete Equations

We consider discrete Sturm–Liouville-type equations of the form $$\begin{aligned} \varDelta (r_n\varDelta x_n)=a_nf(x_{\sigma (n)})+b_n. \end{aligned}$$ Δ ( r n Δ x n ) = a n f ( x σ ( n ) ) + b n . We present a theory of asymptotic properties of solutions which allows us to control the degree of approximation. Namely, we establish conditions under which for a given sequence y which solves the equation $$\varDelta (r_n\varDelta y_n)=b_n$$ Δ ( r n Δ y n ) = b n , the above equation possesses a solution x with the property $$x_n=y_n+\mathrm {o}(u_n)$$ x n = y n + o ( u n ) , where u is a given positive, nonincreasing sequence. The obtained results are applied to the study of asymptotically periodic solutions. Moreover, these results also allow us to obtain some nonoscillation criteria for the classical Sturm–Liouville equation.

Simple Additive Weighting Method Equipped with Fuzzy Ranking of Evaluated Alternatives

From the perspective of each evaluation criterion, any decision alternative is evaluated by means of trapezoidal ordered fuzzy numbers (TrOFN). This approach is justified in the way that some criteria are linguistically evaluated. In this paper, decision alternatives are evaluated using oriented fuzzy Simple Additive Weighting (OF-SAW) scoring function. The ranking of alternatives may be defined by means of a nonincreasing sequence of defuzzified values of a scoring function. Any defuzzification procedure distorts ordered fuzzy numbers in a way that information on imprecision and orientation is lost. This undermines the credibility of the determined alternatives’ ranking. The main purpose of this paper is to avoid the defuzzification stage in the OF-SAW method. Thus, the OF-SAW method is equipped with fuzzy scoring order. This OF-SAW method is described as a negotiation scoring system. We study an empirical example of the OF-SAW application and rank some negotiation offers. Here, we focus on the effects of replacing the defuzzified scoring function by a fuzzy one. The obtained conclusions are generalized for the case of any decision alternatives.

Stick number of tangles

An [Formula: see text]-string tangle is a pair [Formula: see text] such that [Formula: see text] is a disjoint union of properly embedded [Formula: see text] arcs in a topological [Formula: see text]-ball [Formula: see text]. And an [Formula: see text]-string tangle is said to be trivial (or rational)[Formula: see text], if it is homeomorphic to [Formula: see text] as a pair, where [Formula: see text] is a 2-disk, [Formula: see text] is the unit interval and each [Formula: see text] is a point in the interior of [Formula: see text]. A stick tangle is a tangle each of whose arcs consists of finitely many line segments, called sticks. For an [Formula: see text]-string stick tangle its stick-order is defined to be a nonincreasing sequence [Formula: see text] of natural numbers such that, under an ordering of the arcs of the tangle, each [Formula: see text] denotes the number of sticks constituting the [Formula: see text]th arc of the tangle. And a stick-order [Formula: see text] is said to be trivial, if every stick tangle of the order [Formula: see text] is trivial. In this paper, restricting the [Formula: see text]-ball [Formula: see text] to be the standard 3-ball, we give the complete list of trivial stick-orders.

NONHOMOGENEOUS PATTERNS ON NUMERICAL SEMIGROUPS

Patterns on numerical semigroups are multivariate linear polynomials, and they are said to admit a numerical semigroup if evaluating the pattern at any nonincreasing sequence of elements of the semigroup gives integers belonging to the semigroup. In a first approach, only homogeneous patterns were analyzed. In this contribution we study conditions for a nonhomogeneous pattern to admit a nontrivial numerical semigroup, and particularize this study to the case the independent term of the pattern is a multiple of the multiplicity of the semigroup. Moreover, for the so-called strongly admissible patterns, the set of numerical semigroups admitting these patterns with fixed multiplicity m forms an m-variety, which allows us to represent this set in a tree and to describe minimal sets of generators of the semigroups in the variety with respect to the pattern. Furthermore, we characterize strongly admissible patterns having a finite associated tree.

A SHORT NOTE ON THE THOM–BOARDMAN SYMBOLS OF DIFFERENTIABLE MAPS

It is well known that Thom–Boardman symbols are realized by nonincreasing sequences of nonnegative integers. A natural question is whether the converse is also true. In this paper we answer this question affirmatively, that is, for any nonincreasing sequence of nonnegative integers, there is at least one map-germ with the prescribed sequence as its Thom–Boardman symbol.