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In this paper we introduce a construction for a weighted CW complex (and the associated lattice cohomology) corresponding to partially ordered sets with some additional structure. This is a generalization of the construction seen in [4] where we started from a system of subspaces of a given vector space. We then proceed to prove some basic properties of this construction that are in many ways analogous to those seen in the case of subspaces, but some aspects of the construction result in complexities not present in that scenario.
Let F be a nonempty family of graphs. A graph ๐บ is called F -free if it contains no graph from F as a subgraph. For a positive integer ๐, the planar Turรกn number of F, denoted by exp (๐, F), is the maximum number of edges in an ๐-vertex F -free planar graph.
Let ฮ๐ be the family of Theta graphs on ๐ โฅ 4 vertices, that is, graphs obtained by joining a pair of non-consecutive of a ๐-cycle with an edge. Lan, Shi and Song determined an upper bound exp (๐, ฮ6) โค 18๐/7โ36๐/7, but for large ๐, they did not verify that the bound is sharp. In this paper, we improve their bound by proving exp (๐, ฮ6) โค 18๐/โ48๐/7 and then we demonstrate the existence of infinitely many positive integer ๐ and an ๐-vertex ฮ6-free planar graph attaining the bound.
Suppose that ๐ (๐ผ, ๐ฝ) is an obtuse triangle with base length 1 and with base angles ๐ผ and ๐ฝ (where ๐ฝ > 90โฆ). In this note a tight lower bound of the sum of the areas of squares that can parallel cover ๐ (๐ผ, ๐ฝ) is given. This result complements the previous lower bound obtained for the triangles with the interior angles at the base of the measure not greater than 90โฆ.
We show that every positroid of rank ๐ โฅ 2 has a good coline. Using the definition of the chromatic number of oriented matroid introduced by J. Neลกetลil, R. Nickel, and W. Hochstรคttler, this shows that every orientation of a positroid of rank at least 2 is 3-colorable.
Let ๐ be a tree, a vertex of degree one is called a leaf. The set of all leaves of ๐ is denoted by Leaf(๐). The subtree ๐ โ Leaf(๐) of ๐ is called the stem of ๐ and denoted by Stem(๐). A tree ๐ is called a caterpillar if Stem(๐) is a path. In this paper, we give two sufficient conditions for a connected graph to have a spanning tree whose stem is a caterpillar. We also give some examples to show that these conditions are sharp.
We revisit the problem of property testing for convex position for point sets in โ๐. Our results draw from previous ideas of Czumaj, Sohler, and Ziegler (2000). First, their testing algorithm is redesigned and its analysis is revised for correctness. Second, its functionality is expanded by (i) exhibiting both negative and positive certificates along with the convexity determination, and (ii) significantly extending the input range for moderate and higher dimensions.
The behavior of the randomized tester on input set ๐ โ โ๐ is as follows: (i) if ๐ is in convex position, it accepts; (ii) if ๐ is far from convex position, with probability at least 2/3, it rejects and outputs a (๐ +2)-point witness of non-convexity as a negative certificate; (iii) if ๐ is close to convex position, with probability at least 2/3, it accepts and outputs a subset in convex position that is a suitable approximation of the largest subset in convex position. The algorithm examines a sublinear number of points and runs in subquadratic time for every fixed dimension ๐.
We prove zero density theorems for Dedekind zeta functions in the vicinity of the line Re s = 1, improving an earlier result of W. Staล.
A positive integer
Let (๐๐)๐โฅ0 and (๐๐)๐โฅ0 be the Pell and PellโLucas sequences. Let ๐ be a positive integer such that ๐ โฅ 2. In this paper, we prove that the following two Diophantine equations ๐๐ = ๐๐๐๐ + ๐๐ and ๐๐ = ๐๐๐๐ + ๐๐ with ๐, the number of digits of ๐๐ or ๐๐ in base ๐, have only finitely many solutions in nonnegative integers (๐, ๐, ๐, ๐, ๐). Also, we explicitly determine these solutions in cases 2 โค ๐ โค 10.
Grรคtzer and Lakser asked in the 1971 Transactions of the American Mathematical Society if the pseudocomplemented distributive lattices in the amalgamation class of the subvariety generated by ๐๐ โ ๐ can be characterized by the property of not having a *-homomorphism onto ๐๐ โ ๐ for 1 < ๐ < ๐.
In this article, their question from 1971 is answered.
