In this work, a new concept of upper dense sequence in interval (0,1] is introduced. There are infinitely many upper dense sequences in interval (0,1]. Using any upper dense sequence, a new decomposition theorem for fuzzy sets is established and proved. Then, using a chosen upper dense sequence as one of the necessary reference systems, infinitely many total orderings on the set of all fuzzy nu...

We develop a technique that we call Conflict Packing in the context of kernelization [8], obtaining (and improving) several polynomial kernels for editing problems on dense instances. We apply this technique on several well-studied problems: Feedback Arc Set in (Bipartite) Tournaments, Dense Rooted Triplet Inconsistency and Betweenness in Tournaments. For the former, one is given a (bipartite) ...

Our aim in this note is to present four remarkable facts about quotient sets. These observations seem to have been overlooked by the MONTHLY, despite its intense coverage of quotient sets over the years. Introduction If A is a subset of the natural numbers N = {1, 2, . . .}, then we let R(A) = {a/a : a, a ∈ A} denote the corresponding quotient set (sometimes called a ratio set). Our aim in this...

We study the problem of computing shortest paths in so-called dense distance graphs. Every planar graph G on n vertices can be partitioned into a set of O(n/r) edge-disjoint regions (called an r-division) with O(r) vertices each, such that each region has O( √ r) vertices (called boundary vertices) in common with other regions. A dense distance graph of a region is a complete graph containing a...

This paper deals with new infinite families of small dense sets in desarguesian projective planes PG(2, q). A general construction of dense sets of size about 3q2/3 is presented. Better results are obtained for specific values of q. In several cases, an improvement on the best known upper bound on the size of the smallest dense set in PG(2, q) is obtained.

and Applied Analysis 3 Definition 2.1 see 13 . A time scale T is an arbitrary nonempty closed subset of the real setR with the topology and ordering inherited from R. The forward and backward jump operators σ, ρ : T → T, and the graininess μ, ν : T → R are defined, respectively, by σ t : inf{s ∈ T : s > t}, ρ t : sup{s ∈ T : s < t}, μ t : σ t − t, ν t : t − ρ t . 2.1 The point t ∈ T is called l...

A classical theorem due to R. Phelps states that if C is a weakly compact set in a Banach space E, the strongly exposing functionals form a dense subset of the dual space E. In this paper, we look at the concrete situation where C € L1pRdq is the closed convex hull of the set of random variables Y P L1pRdq having a given law ν. Using the theory of optimal transport, we show that every random va...

It has been observed many times, both in the Monthly and elsewhere, that the set of all quotients of prime numbers is dense in the positive real numbers. In this short note we answer the related question: “Is the set of all quotients of Gaussian primes dense in the complex plane?” Quotient sets {s/t : s, t ∈ S} corresponding to subsets S of the natural numbers have been intensely studied in the...

We present several recent examples of the generic approach to nonlinear problems. In this approach, instead of considering, for instance, the convergence of an algorithm, one investigates an appropriate space of algorithms equipped with some natural complete metric and shows that convergence does indeed occur for most algorithms there. As we shall see, sometimes one can show that the complement...

This note introduces and studies an open set of PSL2(C) characters of a nonabelian free group, on which the action of the outer automorphism group is properly discontinuous, and which is strictly larger than the set of discrete, faithful convex-cocompact (i.e. Schottky) characters. This implies, in particular, that the outer automorphism group does not act ergodically on the set of characters w...