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Abstract Strong extrema of functions on some quasi-metric and compact spaces are studied. It is shown that the maximum cardinalities of the set of strong extrema of discontinuous functions can be larger than for continuous ones. We construct quasi-metric spaces with uncountable weight, on which any function has no more than a countable set of strong extrema. [ABSTRACT FROM AUTHOR]

Let G be a graph with adjacency matrix A ( G ) and let D ( G ) be the diagonal matrix of the degrees of G . For any real α ∈ [ 0 , 1 ] , Nikiforov (2017) [7] defined the matrix A α ( G ) as A α ( G ) = α D ( G ) + ( 1 − α ) A ( G ) . Let u and v be two vertices of a connected graph G . Suppose that u and v are connected by a path w 0 ( = v ) w 1 ⋯ w s − 1 w s ( = u ) where d ( w i ) = 2 for 1 ≤ i ≤ s − 1 . Let G p , s , q ( u , v ) be the graph obtained by attaching the paths P p to u and P q to v . Let s = 0 , 1 . Nikiforov and Rojo (2018) [9] conjectured that ρ α ( G p , s , q ( u , v ) ) < ρ α ( G p − 1 , s , q + 1 ( u , v ) ) if p ≥ q + 2 . In this paper, we confirm the conjecture. As applications, firstly, the extremal graph with maximal A α -spectral radius with fixed order and cut vertices is characterized. Secondly, we characterize the extremal tree which attains the maximal A α -spectral radius with fixed order and matching number. These results generalize some known results. [ABSTRACT FROM AUTHOR]

Let C v ( k ; T ) be the number of closed walks of length k starting at vertex v in a tree T . We prove that for any tree T with a given degree sequence π , the vector C ( k ; T ) ≡ ( C v ( k ; T ), v  ∈  V ( T )) is weakly majorized by the vector C ( k ; T π * ) ≡ ( C v ( k ; T π * ) , v ∈ V ( T π * ) ) , where T π * is the greedy tree with the degree sequence π . In addition, for two trees degree sequences π and π ′, if π is majorized by π ′, then C ( k ; T π * ) is weakly majorized by C ( k ; T π ′ * ) . [ABSTRACT FROM AUTHOR]

Abstract: The family of subsets of is 2-union free if it does not contain three different sets A, B, C such that , and is union free if it does not contain different sets , , …, such that . Let , be the maximum size of a union free, union free family, respectively. Simple construction shows that , which slightly improves the lower bound obtained by Kleitman if n is odd. Another more complicated construction further improves this lower bound. Union free family of m subsets of naturally corresponds to an matrix with independent rows. The result is related to maximum number m of independent rows in some matrix with n columns. [Copyright &y& Elsevier]

Abstract: We prove two cases of a strong version of Dejean''s conjecture involving extremal letter frequencies. The results are that there exist an infinite -free word over a 5 letter alphabet with letter frequency and an infinite -free word over a 6 letter alphabet with letter frequency . [Copyright &y& Elsevier]

Abstract: In 1958, Claude Berge studied the domination number of a graph and showed that every graph satisfies , where is the degree of vertex . We show here that every graph G satisfies the inequality , where stands for any of the three parameters (irredundance number), (domination number) and (size of a smallest maximal independent set), and is the size of a largest matching in the subgraph of G induced by the non-neighbours of . This improves on Berge''s and other inequalities. We then give a characterization of the trees T that achieve equality in the above inequality for the parameters and , and of the regular graphs G that achieve equality for . [Copyright &y& Elsevier]

Abstract: The performance of a predictive controller is typically poor when the true plant evolution deviates significantly from that predicted by the model. A robust control approach that considers model uncertainty explicitly is then needed. However, it is often difficult to find a single input profile that works well for the range of uncertainty considered. Thus, multiple input profiles, i.e. one for each realization of the uncertainty, need to be determined. Unfortunately, this is computationally extremely expensive. This paper proposes an alternative approach that is based on neighboring extremals, where the multiple input profiles are computed using a simple feedback law, thereby reducing considerably the computational burden. The idea is illustrated via the simulation of a continuous stirred tank reactor and an inverted pendulum on a cart. [Copyright &y& Elsevier]