Search results for "Infimum and supremum"
showing 10 items of 17 documents
On the continuous and discontinuous maximal operators
2018
Abstract In the first part of this paper we study the regularity properties of a wide class of maximal operators. These results are used to show that the spherical maximal operator is continuous W 1 , p ( R n ) ↦ W 1 , p ( R n ) , when p > n n − 1 . Other given applications include fractional maximal operators and maximal singular integrals. On the other hand, we show that the restricted Hardy–Littlewood maximal operator M λ , where the supremum is taken over the cubes with radii greater than λ > 0 , is bounded from L p ( R n ) to W 1 , p ( R n ) but discontinuous.
A solution of the minimum-time velocity planning problem based on lattice theory
2018
For a vehicle on an assigned path, we find the minimum-time speed law that satisfies kinematic and dynamic constraints, related to maximum speed and maximum tangential and transversal acceleration. We present a necessary and sufficient condition for the feasibility of the problem and a simple operator, based on the solution of two ordinary differential equations, which computes the optimal solution. Theoretically, we show that the problem feasible set, if not empty, is a lattice, whose supremum element corresponds to the optimal solution.
Free sequences and the tightness of pseudoradial spaces
2019
Let F(X) be the supremum of cardinalities of free sequences in X. We prove that the radial character of every Lindelof Hausdorff almost radial space X and the set-tightness of every Lindelof Hausdorff space are always bounded above by F(X). We then improve a result of Dow, Juhasz, Soukup, Szentmiklossy and Weiss by proving that if X is a Lindelof Hausdorff space, and $$X_\delta $$ denotes the $$G_\delta $$ topology on X then $$t(X_\delta ) \le 2^{t(X)}$$ . Finally, we exploit this to prove that if X is a Lindelof Hausdorff pseudoradial space then $$F(X_\delta ) \le 2^{F(X)}$$ .
On closures of discrete sets
2018
The depth of a topological space $X$ ($g(X)$) is defined as the supremum of the cardinalities of closures of discrete subsets of $X$. Solving a problem of Mart\'inez-Ruiz, Ram\'irez-P\'aramo and Romero-Morales, we prove that the cardinal inequality $|X| \leq g(X)^{L(X) \cdot F(X)}$ holds for every Hausdorff space $X$, where $L(X)$ is the Lindel\"of number of $X$ and $F(X)$ is the supremum of the cardinalities of the free sequences in $X$.
Extremal problems of approximation theory in fuzzy context
1999
Abstract The problem of approximation of a fuzzy subset of a normed space is considered. We study the error of approximation, which in this case is characterized by an L -fuzzy number. In order to do this we define the supremum of an L -fuzzy set of real numbers as well as the supremum and the infimum of a crisp set of L -fuzzy numbers. The introduced concepts allow us to investigate the best approximation and the optimal linear approximation. In particular, we consider approximation of a fuzzy subset in the space L p m of differentiable functions in the L q -metric. We prove the fuzzy counterparts of duality theorems, which in crisp case allows effectively to solve extremal problems of the…
Proper 1-ball contractive retractions in Banach spaces of measurable functions
2005
In this paper we consider the Wosko problem of evaluating, in an infinite-dimensional Banach space X, the infimum of all k > 1 for which there exists a k-ball contractive retraction of the unit ball onto its boundary. We prove that in some classical Banach spaces the best possible value 1 is attained. Moreover we give estimates of the lower H-measure of noncompactness of the retractions we construct. 1. Introduction Let X be an infinite-dimensional Banach space with unit closed ball B(X) and unit sphere S(X). It is well known that, in this setting, there is a retraction of B(X) onto S(X), that is, a continuous mapping R : B(X) ! S(X) with Rx = x for all x 2 S(X). In (4) Benyamini and Sternf…
Cyclic Complexity of Words
2014
We introduce and study a complexity function on words $c_x(n),$ called \emph{cyclic complexity}, which counts the number of conjugacy classes of factors of length $n$ of an infinite word $x.$ We extend the well-known Morse-Hedlund theorem to the setting of cyclic complexity by showing that a word is ultimately periodic if and only if it has bounded cyclic complexity. Unlike most complexity functions, cyclic complexity distinguishes between Sturmian words of different slopes. We prove that if $x$ is a Sturmian word and $y$ is a word having the same cyclic complexity of $x,$ then up to renaming letters, $x$ and $y$ have the same set of factors. In particular, $y$ is also Sturmian of slope equ…
Bounded Palais–Smale sequences for non-differentiable functions
2011
The existence of bounded Palais-Smale sequences (briefly BPS) for functionals depending on a parameter belonging to a real interval and which are the sum of a locally Lipschitz continuous term and of a convex, proper, lower semicontinuous function, is obtained when the parameter runs in a full measure subset of the given interval. Specifically, for this class of non-smooth functions, we obtain BPS related to mountain pass and to global infima levels. This is done by developing a unifying approach, which applies to both cases and relies on a suitable deformation lemma. © 2011 Elsevier Ltd. All rights reserved.
Some Algebraic Properties of Machine Poset of Infinite Words
2008
The complexity of infinite words is considered from the point of view of a transformation with a Mealy machine that is the simplest model of a finite automaton transducer. We are mostly interested in algebraic properties of the underlying partially ordered set. Results considered with the existence of supremum, infimum, antichains, chains and density aspects are investigated.
On two topological cardinal invariants of an order-theoretic flavour
2012
Noetherian type and Noetherian $\pi$-type are two cardinal functions which were introduced by Peregudov in 1997, capturing some properties studied earlier by the Russian School. Their behavior has been shown to be akin to that of the \emph{cellularity}, that is the supremum of the sizes of pairwise disjoint non-empty open sets in a topological space. Building on that analogy, we study the Noetherian $\pi$-type of $\kappa$-Suslin Lines, and we are able to determine it for every $\kappa$ up to the first singular cardinal. We then prove a consequence of Chang's Conjecture for $\aleph_\omega$ regarding the Noetherian type of countably supported box products which generalizes a result of Lajos S…