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Let [ · ] be the fioor function. In this paper, we show that when 1 < c < 37/36, then every sufficiently large positive integer N can be represented in the form
where p1, p2, p3 are primes close to squares.
In this article, we study a family of subgraphs of the Farey graph, denoted as ℱN for every N ∈ ℕ. We show that ℱN is connected if and only if N is either equal to one or a prime power. We introduce a class of continued fractions referred to as ℱN -continued fractions for each N > 1. We establish a relation between ℱN -continued fractions and certain paths from infinity in the graph ℱN . Using this correspondence, we discuss the existence and uniqueness of ℱN -continued fraction expansions of real numbers.
Given a finite point set P in the plane, a subset S⊆P is called an island in P if conv(S) ⋂ P = S. We say that S ⊂ P is a visible island if the points in S are pairwise visible and S is an island in P. The famous Big-line Big-clique Conjecture states that for any k ≥ 3 and l ≥ 4, there is an integer n = n(k, l), such that every finite set of at least n points in the plane contains l collinear points or k pairwise visible points. In this paper, we show that this conjecture is false for visible islands, by replacing each point in a Horton set by a triple of collinear points. Hence, there are arbitrarily large finite point sets in the plane with no 4 collinear members and no visible island of size 13.
In this article, we define the notion of a generalized open book of a n-manifold over the k−sphere Sk , k < n. We discuss Lefschetz open book embeddings of Lefschetz open books of closed oriented 4-manifolds into the trivial open book over S2 of the 7−sphere S7 . If X is the double of a bounded achiral Lefschetz fibration over D2 , then X naturally admits a Lefschetz open book given by the bounded achiral Lefschetz fibration. We show that this natural Lefschetz open book of X admits a Lefschetz open book embedding into the trivial open book over S2 of the 7−sphere S7 .
We show that if a non-degenerate PL map f : N → M lifts to a topological embedding in
This short note deals with polynomial interpolation of complex numbers verifying a Lipschitz condition, performed on consecutive points of a given sequence in the plane. We are interested in those sequences which provide a bound of the error at the first uninterpolated point, depending only on its distance to the last interpolated one.
For a lattice L of finite length n, let RCSub(L) be the collection consisting of the empty set and those sublattices of L that are closed under taking relative complements. That is, a subset X of L belongs to RCSub(L) if and only if X is join-closed, meet-closed, and whenever {a, x, b} ⊆ S, y ∈ L, x ∧ y = a, and x ∨ y = b, then y ∈ S. We prove that (1) the poset RCSub(L) with respect to set inclusion is lattice of length n + 1, (2) if RCSub(L) is a ranked lattice and L is modular, then L is 2-distributive in András P. Huhn’s sense, and (3) if L is distributive, then RCSub(L) is a ranked lattice.
In this paper, centralizing (semi-centralizing) and commuting (semi-commuting) derivations of semirings are characterized. The action of these derivations on Lie ideals is also discussed and as a consequence, some significant results are proved. In addition, Posner’s commutativity theorem is generalized for Lie ideals of semirings and this result is also extended to the case of centralizing (semi-centralizing) derivations of prime semirings. Further, we observe that if there exists a skew-commuting (skew-centralizing) derivation D of S, then D = 0. It is also proved that for any two derivations d 1 and d 2 of a prime semiring S with char S ≠ 2 and x d 1 x d 2 = 0, for all x ∈ S implies either d 1 = 0 or d 2 = 0.
We study a combinatorial notion where given a set S of lattice points one takes the set of all sums of p distinct points in S, and we ask the question: ‘if S is the set of lattice points of a convex lattice polytope, is the resulting set also the set of lattice points of a convex lattice polytope?’ We obtain a positive result in dimension 2 and a negative result in higher dimensions. We apply this to the corner cut polyhedron.
