Saturated Sets Research Articles

Outer Strong Blocking Sets

Strong blocking sets, introduced first in 2011 in connection with saturating sets, have recently gained a lot of attention due to their correspondence with minimal codes. In this paper, we dig into the geometry of the concatenation method, introducing the concept of outer strong blocking sets and their coding theoretical counterpart. We investigate their structure and provide bounds on their size. As a byproduct, we improve the best-known upper bound on the minimum size of a strong blocking set. Finally, we present a geometric construction of small strong blocking sets, whose computational cost is significantly smaller than the previously known ones.

Similarities and differences between specification and non-uniform specification

Abstract Pavlov [Adv. Math.295 (2016), 250–270; Nonlinearity32 (2019), 2441–2466] studied the measures of maximal entropy for dynamical systems with weak versions of specification property and found the existence of intrinsic ergodicity would be influenced by the assumptions of the gap functions. Inspired by these, in this article, we study the dynamical systems with non-uniform specification property. We give some basic properties these systems have and give an assumption for the gap functions to ensure the systems have the following five properties: CO-measures are dense in invariant measures; for every non-empty compact connected subset of invariant measures, its saturated set is dense in the total space; ergodic measures are residual in invariant measures; ergodic measures are connected; and entropy-dense. In addition, we will give examples to show the assumption is optimal.

Emergence Behavior Versus Physical-Like Behavior

In this paper, we study the dynamical complexity of points with emergence behavior but without weak face behavior, especially for points without physical-like behavior in certain dynamical systems such as transitive Anosov systems. We use the tools of saturated sets to prove that these points show strong dynamical complexity in the sense of entropy, density and distributional chaos. We obtain some observations of those results related to irregular sets and level sets. These results strengthen the previous results of [Catsigeras et al., 2019; Hou et al., 2023].

Incoherent J/ψ production at large |t| identifies the onset of saturation at the LHC

We predict that the onset of gluon saturation can be uniquely identified using incoherent J/ψ production in Pb–Pb collisions at currently accessible energies of the LHC. The diffractive incoherent photo-production of a J/ψ vector meson off a hadron provides information on the partonic structure of the hadron. Within the Good-Walker approach it specifically measures the variance over possible target configurations of the hadronic colour field. For this process then, gluon saturation sets in when the cross section reaches a maximum, as a function of the centre-of-mass energy of the photon-hadron system (W), and then decreases. We benchmark the energy-dependent hot-spot model against data from HERA and the LHC and demonstrate a good description of the available data. We show that the study of the energy dependence of the incoherent production of J/ψ allows us to pinpoint the onset of saturation effects by selecting the region of Mandelstam-t around 1 GeV2 where the contribution of hot spots is dominant. We predict the onset of saturation in a Pb target to occur for W around a few hundred GeV. This can be measured with current data in ultra-peripheral Pb–Pb collisions at the LHC.

Saturating linear sets of minimal rank

Saturating sets are combinatorial objects in projective spaces over finite fields that have been intensively investigated in the last three decades. They are related to the so-called covering problem of codes in the Hamming metric. In this paper, we consider the recently introduced linear version of such sets, which is, in turn, related to the covering problem in the rank metric. The main questions in this context are how small the rank of a saturating linear set can be and how to construct saturating linear sets of small rank. Recently, Bonini, Borello, and Byrne provided a lower bound on the rank of saturating linear sets in a given projective space, which is shown to be tight in some cases. In this paper, we provide construction of saturating linear sets meeting the lower bound and we develop a link between the saturating property and the scatteredness of linear sets. The last part of the paper is devoted to show some parameters for which the bound is not tight.

New Type of Characterizations of Pre-Open (Closed and Continuous) Mappings

In this paper, new characterizations of pre-open (pre-closed, pre-continuous) maps will be obtained using the map η# induced via a map η between any two topological spaces. Further, these maps are characterized in terms of saturated sets under some sufficient conditions.

Numerical fitting of Extrapolated semisolid Magnetization transfer Reference (NEMR) signals: Improved detection of ischemic stroke.

To propose a novel Numerical fitting method of the Extrapolated semisolid Magnetization transfer Reference (NEMR) signal for quantifying the CEST effect. Modified two-pool Bloch-McConnell equations were used to numerically fit the magnetization transfer (MT) and direct water saturation (DS) signals at far off-resonance frequencies, which was subsequently extrapolated into the frequency range of amide proton transfer (APT) and nuclear Overhauser enhancement (NOE) pools. Then the subtraction of the fitted two-pool z-spectrum and the experimentally acquired z-spectrum yielded APT# and NOE# signals mostly free of MT and DS contamination. Several strategies were used to accelerate the NEMR fitting. Furthermore, the proposed NEMR method was compared with the conventional extrapolated semisolid magnetization transfer reference (EMR) and magnetization transfer ratio asymmetry (MTRasym ) methods in simulations and stroke patients. The combination of RF downsampling, MT lineshape look-up table, and conversion of MATLAB code to C code accelerated the NEMR fitting by over 2700-fold. Monte-Carlo simulations showed that NEMR had higher accuracy than EMR and eliminated the requirement of the steady-state condition. In ischemic stroke patients, the NEMR maps at 1 μT removed hypointense artifacts seen on EMR and MTRasym images, and better depicted stroke lesions than EMR. For NEMR, NOE# yielded significantly (p < 0.05) stronger signal contrast between stroke and normal tissues than APT# at 1 μT. The proposed NEMR method is suitable for arbitrary saturation settings and can remove MT and DS contamination from the CEST signal for improved detection of ischemic stroke.

High pointwise emergence via saturated sets

In this article, we prove that the set of points whose pointwise emergence have same degree in with repect to polynomial has strong dynamical complexity in sense of entropy, residuality and density, and distributional chaos for a β-shift, a C 1 diffeomorphism having a nontrival homoclinic class, or a diffeomorphism preserving a hyperbolic ergodic measure. In this process, we construct nonempty compact connected set with fixed box-counting dimension in the space of invariant measures. Then combining with the relation between pointwise emergence of saturated set and box-counting dimension, we get the goal from two frameworks about dynamical complexity of saturated set. One framework was introduced by C. Pfister and W. Sullivan and can be applicable to transitive Anosov diffeomorphisms, mixing expanding maps, β-shifts, transitive subshifts of finite type, mixing sofic subshifts, and mixing S-gap shifts. The other framework is applicable to general homoclinic classes and diffeomorphisms preserving hyperbolic ergodic measures. We also get some combined observations with irregular set.

Small Strong Blocking Sets by Concatenation

Strong blocking sets and their counterparts, minimal codes, have attracted much attention in the past few years. Combining the concatenating construction of codes with a geometric insight into the minimality condition, we explicitly provide infinite families of small strong blocking sets, whose size is linear in the dimension of the ambient projective spaces. As a byproduct, small saturating sets are obtained.

Topological structures on saturated sets, optimal orbits and equilibrium states

In this paper, we investigate topological complexity of saturated set from the viewpoints of upper capacity entropy and packing entropy. We obtain that if a topological dynamical system $ (X, T) $ satisfies almost product property, then for each nonempty compact convex subset $ K $ of invariant measures, the following two formulas hold for the saturated set $ G_K $:$ \begin{equation*} h_{top}^{UC}(T, G_{K}) = h_{top}(T, X)\ \mathrm{and}\ h_{top}^{P}(T, G_{K}) = \sup\{h(T, \mu):\mu\in K\}, \end{equation*} $where $ h_{top}^{UC}(T, G_{K}) $ is the upper capacity entropy of $ T $ on $ G_{K}, $ $ h_{top}^{P}(T, G_{K}) $ is the packing entropy of $ T $ on $ G_{K}, $ and $ h(T, \mu) $ is the Kolmogorov-Sinai entropy of $ \mu $. As applications, for a transitive Anosov diffeomorphism on a compact manifold, we use the above results to quantify topological complexity of optimal orbits and equilibrium states for both real and matrix valued potentials.

Upper capacity entropy and packing entropy of saturated sets for amenable group actions

Let $ (X, G) $ be a $ G $-action topological system, where $ G $ is a countable infinite discrete amenable group and $ X $ a compact metric space. In this paper we study the upper capacity entropy and packing entropy for systems with weaker version of specification. We prove that the upper capacity always carries full entropy while there is a variational principle for packing entropy of saturated sets.

Packing Entropy of Saturated Sets for Nonuniformly Hyperbolic Systems

In this paper, we prove that for a nonuniformly hyperbolic system [Formula: see text], and for every convex and compact subset [Formula: see text] with the same hyperbolic rate, the packing entropy of saturated set [Formula: see text] equals the supremum of the metric entropy of all measures in K. Here [Formula: see text] denotes the set of all [Formula: see text]-invariant Borel probability measures supported on [Formula: see text]. As an application, we describe the size of the set of mean Li–Yorke pairs.

On the equivalence of all models for (∞,2)$(\infty,2)$‐categories

The goal of this paper is to provide the last equivalence needed in order to identify all known models for ( ∞ , 2 ) $(\infty,2)$ -categories. We do this by showing that Verity's model of saturated 2-trivial complicial sets is equivalent to Lurie's model of ∞ $\infty$ -bicategories, which, in turn, has been shown to be equivalent to all other known models for ( ∞ , 2 ) $(\infty,2)$ -categories. A key technical input is given by identifying the notion of ∞ $\infty$ -bicategories with that of weak ∞ $\infty$ -bicategories, a step which allows us to understand Lurie's model structure in terms of Cisinski–Olschok's theory. Several of our arguments use tools coming from a new theory of outer (co)-Cartesian fibrations, further developed in a companion paper. In the last part of the paper, we construct a homotopically fully faithful scaled simplicial nerve functor for 2-categories, give two equivalent descriptions of it, and show that the homotopy 2-category of an ∞ $\infty$ -bicategory retains enough information to detect thin 2-simplices.

Dominance of parent ion formation in the ultrafast, intense-field ionization of the valence isoelectronic molecules and

We report on the photodynamics of the valence isoelectronic molecules carbon dioxide () and carbon disulfide () under intense-field, single-molecule conditions in focused 50 fs, 800 nm laser pulses. Ion mass spectra are recorded as a function of intensity in the range from to . Ion yields are measured in the absence of the focal volume effect without need for data deconvolution. In both targets, formation of the molecular parent ion () is by far the dominant process; the yield of other species is only a few percent. We observe that the yield of this parent ion is strictly proportional to an integer power of the intensity, until saturation sets in. We conclude the parent ion is formed through multiphoton single-electron ionization, with the photoelectron originating from the doubly degenerate, non-bonding highest-occupied molecular orbital (HOMO). The non-bonding nature of this HOMO explains the remarkable stability of the parent ion.

Short Minimal Codes and Covering Codes via Strong Blocking Sets in Projective Spaces

Minimal linear codes are in one-to-one correspondence with special types of blocking sets of projective spaces over a finite field, which are called strong or cutting blocking sets. Minimal linear codes have been studied since decades but their tight connection with cutting blocking sets of finite projective spaces was unfolded only in the past few years, and it has not been fully exploited yet. In this paper we apply finite geometric and probabilistic arguments to contribute to the field of minimal codes. We prove an upper bound on the minimal length of minimal codes of dimension <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula> over the <inline-formula> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula>-element Galois field which is linear in both <inline-formula> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula>, hence improve the previous superlinear bounds. This result determines the minimal length up to a small constant factor. We also improve the lower and upper bounds on the size of so called higgledy-piggledy line sets in projective spaces and apply these results to present improved bounds on the size of covering codes and saturating sets in projective spaces as well.

On cutting blocking sets and their codes

AbstractLetPG⁡(r,q){\operatorname{PG}(r,q)}be ther-dimensional projective space over the finite fieldGF⁡(q){\operatorname{GF}(q)}. A set𝒳{\mathcal{X}}of points ofPG⁡(r,q){\operatorname{PG}(r,q)}is a cutting blocking set if for each hyperplane Π ofPG⁡(r,q){\operatorname{PG}(r,q)}the setΠ∩𝒳{\Pi\cap\mathcal{X}}spans Π. Cutting blocking sets give rise to saturating sets and minimal linear codes, and those having size as small as possible are of particular interest. We observe that from a cutting blocking set obtained in [20], by using a set of pairwise disjoint lines, there arises a minimal linear code whose length grows linearly with respect to its dimension. We also provide two distinct constructions: a cutting blocking set ofPG⁡(3,q3){\operatorname{PG}(3,q^{3})}of size3⁢(q+1)⁢(q2+1){3(q+1)(q^{2}+1)}as a union of three pairwise disjointq-order subgeometries, and a cutting blocking set ofPG⁡(5,q){\operatorname{PG}(5,q)}of size7⁢(q+1){7(q+1)}from seven lines of a Desarguesian line spread ofPG⁡(5,q){\operatorname{PG}(5,q)}. In both cases, the cutting blocking sets obtained are smaller than the known ones. As a byproduct, we further improve on the upper bound of the smallest size of certain saturating sets and on the minimum length of a minimalq-ary linear code having dimension 4 and 6.

Combinatorial and harmonic-analytic methods for integer tilings

AbstractA finite set of integersAtiles the integers by translations if$\mathbb {Z}$can be covered by pairwise disjoint translated copies ofA. Restricting attention to one tiling period, we have$A\oplus B=\mathbb {Z}_M$for some$M\in \mathbb {N}$and$B\subset \mathbb {Z}$. This can also be stated in terms of cyclotomic divisibility of the mask polynomials$A(X)$and$B(X)$associated withAandB.In this article, we introduce a new approach to a systematic study of such tilings. Our main new tools are the box product, multiscale cuboids and saturating sets, developed through a combination of harmonic-analytic and combinatorial methods. We provide new criteria for tiling and cyclotomic divisibility in terms of these concepts. As an application, we can determine whether a setAcontaining certain configurations can tile a cyclic group$\mathbb {Z}_M$, or recover a tiling set based on partial information about it. We also develop tiling reductions where a given tiling can be replaced by one or more tilings with a simpler structure. The tools introduced here are crucial in our proof in [24] that all tilings of period$(pqr)^2$, where$p,q,r$are distinct odd primes, satisfy a tiling condition proposed by Coven and Meyerowitz [2].

Unstable entropies and dimension theory of partially hyperbolic systems

In this paper we define unstable topological entropy for any subsets (not necessarily compact or invariant) in partially hyperbolic systems as a Carathéodory–Pesin dimension characteristic, motivated by the work of Bowen and Pesin etc. We then establish some basic results in dimension theory for Bowen unstable topological entropy, including an entropy distribution principle and a variational principle in general setting. As applications of this new concept, we study unstable topological entropy of saturated sets and extend some results in Bowen (1973 Trans. Am. Math. Soc. 184 125–36); Pfister and Sullivan (2007 Ergod. Theor. Dynam. Syst. 27 929–56). Our results give new insights to the multifractal analysis for partially hyperbolic systems.

Increasing oxygen prescribing during the COVID-19 pandemic

Valid oxygen prescriptions for hospital inpatients have been a long-standing problem and have been described extensively in BMJ Open Quality with numerous quality improvement projects (QIPs) with the aim of...

Constructing saturating sets in projective spaces using subgeometries

A $$\varrho $$ -saturating set of $$\text {PG}(N,q)$$ is a point set $${\mathcal {S}}$$ such that any point of $$\text {PG}(N,q)$$ lies in a subspace of dimension at most $$\varrho $$ spanned by points of $${\mathcal {S}}$$ . It is generally known that a $$\varrho $$ -saturating set of $$\text {PG}(N,q)$$ has size at least $$c\cdot \varrho \,q^\frac{N-\varrho }{\varrho +1}$$ , with $$c>\frac{1}{3}$$ a constant. Our main result is the discovery of a $$\varrho $$ -saturating set of size roughly $$\frac{(\varrho +1)(\varrho +2)}{2}q^\frac{N-\varrho }{\varrho +1}$$ if $$q=(q')^{\varrho +1}$$ , with $$q'$$ an arbitrary prime power. The existence of such a set improves most known upper bounds on the smallest possible size of $$\varrho $$ -saturating sets if $$\varrho <\frac{2N-1}{3}$$ . As saturating sets have a one-to-one correspondence to linear covering codes, this result improves existing upper bounds on the length and covering density of such codes. To prove that this construction is a $$\varrho $$ -saturating set, we observe that the affine parts of $$q'$$ -subgeometries of $$\text {PG}(N,q)$$ having a hyperplane in common, behave as certain lines of $$\text {AG}\big (\varrho +1,(q')^N\big )$$ . More precisely, these affine lines are the lines of the linear representation of a $$q'$$ -subgeometry $$\text {PG}(\varrho ,q')$$ embedded in $$\text {PG}\big (\varrho +1,(q')^N\big )$$ .