Fully-Dynamic and Kinetic Conflict-Free Coloring of Intervals with Respect to Points

We introduce the fully-dynamic conflict-free coloring problem for a set S of intervals in R^1 with respect to points, where the goal is to maintain a conflict-free coloring for S under insertions and deletions. A coloring is conflict-free if for each point p contained in some interval, p is contained in an interval whose color is not shared with any other interval containing p. We investigate trade-offs between the number of colors used and the number of intervals that are recolored upon insertion or deletion of an interval. Our results include: - a lower bound on the number of recolorings as a function of the number of colors, which implies that with O(1) recolorings per update the worst-case number of colors is Omega(log n/log log n), and that any strategy using O(1/epsilon) colors needs Omega(epsilon n^epsilon) recolorings; - a coloring strategy that uses O(log n) colors at the cost of O(log n) recolorings, and another strategy that uses O(1/epsilon) colors at the cost of O(n^epsilon/epsilon) recolorings; - stronger upper and lower bounds for special cases. We also consider the kinetic setting where the intervals move continuously (but there are no insertions or deletions); here we show how to maintain a coloring with only four colors at the cost of three recolorings per event and show this is tight

Dynamic Conflict-Free Colorings in the Plane

We study dynamic conflict-free colorings in the plane, where the goal is to maintain a conflict-free coloring (CF-coloring for short) under insertions and deletions. - First we consider CF-colorings of a set S of unit squares with respect to points. Our method maintains a CF-coloring that uses O(log n) colors at any time, where n is the current number of squares in S, at the cost of only O(log n) recolorings per insertion or deletion We generalize the method to rectangles whose sides have lengths in the range [1, c], where c is a fixed constant. Here the number of used colors becomes O(log^2 n). The method also extends to arbitrary rectangles whose coordinates come from a fixed universe of size N, yielding O(log^2 N log^2 n) colors. The number of recolorings for both methods stays in O(log n). - We then present a general framework to maintain a CF-coloring under insertions for sets of objects that admit a unimax coloring with a small number of colors in the static case. As an application we show how to maintain a CF-coloring with O(log^3 n) colors for disks (or other objects with linear union complexity) with respect to points at the cost of O(log n) recolorings per insertion. We extend the framework to the fully-dynamic case when the static unimax coloring admits weak deletions. As an application we show how to maintain a CF-coloring with O(sqrt(n) log^2 n) colors for points with respect to rectangles, at the cost of O(log n) recolorings per insertion and O(1) recolorings per deletion. These are the first results on fully-dynamic CF-colorings in the plane, and the first results for semi-dynamic CF-colorings for non-congruent objects

Conflict-Free Coloring of Points and Simple Regions in the Plane

We study conflict-free colorings, where the underlying set systems arise in geometry. Our main result is a general framework for conflict-free coloring of regions with low union complexity. A coloring of regions is conflict-free if for any covered point in the plane, there exists a region that covers it with a unique color (i.e., no other region covering this point has the same color). For example, we show that we can conflict-free color any family of n pseudo-discs with O(log n) color

High Efficiency and New Potential of RSLDE: A Green Technique for the Extraction of Bioactive Molecules from Not Completely Exhausted Plant Biomass and Organic Industrial Processing Waste

A product is characterized by low environmental impact if, during the whole process (from extraction of raw materials from solid natural matter to disposal), its negative contribution to environment modification is significantly reduced or eliminated. According to circular economy, it is important to take into consideration other aspects, such as the possibility to improve the efficiency of extraction process by modifying the principle on which it is based and allowing the recovery of not completely exhausted waste, obtaining other active ingredients, and favoring the recycling of normally eliminated materials. The purpose of this work was to propose more efficient and greener alternatives to conventional solid–liquid extraction processes. Major features are the rapidity of the process, extraction at room temperature and high yields. Rapid Solid–Liquid Dynamic Extraction (RSLDE) represents an innovative solid–liquid extraction technology that allows the solid matrices containing extractable substances in an organic or inorganic solvent and their mixtures to be exhausted in shorter time than current techniques. The principle at the basis of this novel process consists of the generation of a negative pressure gradient between the inside and the outside of the solid matrix, which induces the extraction of compounds not chemically linked to the solid matter, being insoluble in the extractant liquid. Therefore, this work focuses on how RSLDE can potentially bring several improvements in the field of solid–liquid extraction, especially for industrial applications

Analyzing Harmonic Polarities: A Tonal Narrative Approach

This dissertation aims to develop an approach to analyzing common-practice repertoire based on the dynamic interplay of centripetal and centrifugal forces. It aims at interpreting various kinds of chromaticism and modulation in terms of the interaction of forces moving away from the tonic or principal key (centrifugal) and those returning to it (centripetal). Centripetal forces also correspond to the force of cadential substantiation of keys, not only the principal key, which I call temporal-centripetal force; temporal-centrifugal forces correspond to the phenomena of tonal instability, of motion through multiple regions. The dynamic interplay and counterbalancing of these forces is a core concern of the dissertation. In chapter 1 I build upon Arnold Schoenberg\u27s visionary metaphor for modulation as a struggle and a competition between the tonic and its regions, imagined as a sovereign ruler and his subjects. The tendencies of the regions generate centrifugal forces; the ruling tonic\u27s desire to subjugate them correspond to centripetal forces. What I call temporal-centripetal forces correspond to the possibility for temporary centripetal forces to be generated by a region, which becomes the ruler of its own realm. The crucial application to analyzing harmonic motion is that centripetal and centrifugal forces are dynamically interdependent; each requires the other in order to stimulate a living tonal narrative. In order to measure centrifugal forces, the tonal narrative approach draws on Gottfried Weber and Schoenberg\u27s maps of tonal space and Weber\u27s elucidation of degrees of relatedness between keys, which are explored in chapter 2. Centrifugal forces divide into sharp and flat sides or types, which correspond to very general character and coloristic shades: sharp-centrifugal forces associate with brightening and greater activity and flat-centrifugal forces with darkening and sometimes passivity. In chapter 3, I explore how the harmonic motion of entire movements can be divided into distinctive functions that define their overall shape, such as intensifications (accumulations of dissonance or centrifugal force or both), culminations, counterbalancing of sharp- and flat-CF forces, and the attainment a complete tour of keys or regions in the tonal spectrum. This chapter also offers a hierarchy of key-area substantiation, determining the structural significance of regions appearing in tonal narratives. Chapter 4 interprets sonata form in terms of centripetal and centrifugal forces unfolding in broad stages, which often correspond with formal parts but sometimes cross their boundaries. It examines the expansion of centrifugal trajectories in piano sonata development sections by Mozart and Beethoven; the culmination of this expansion is the complete traversal of the enharmonic circle in the Waldstein sonata (first movement). I analyze this development section in terms of the number of fundamental steps travelled from the subordinate key to the point of furthest remove. Chapter 5 develops a hermeneutic reading of Schubert\u27s sonata D. 894/i; centrifugal and centripetal forces are matched to pastoral/epic expressive modes identified in this movement by Robert Hatten. This work also features an immense modulatory trajectory around the enharmonic circle, corresponding to an epic narrative journey into the tonal underworld. A remarkable aspect of the development section is the transient recurrences of passages returning to the pastoral mode and centripetal forces; these provide a welcome contrast and respite from the inexorable flatward trajectory. The conclusion of the dissertation briefly offers some applications of analyzing harmonic polarities to piano performance, drawing particularly on aspects of touch and technique discussed by Boris Berman

Large bichromatic point sets admit empty monochromatic 4-gons

We consider a variation of a problem stated by Erd˝os and Szekeres in 1935 about the existence of a number fES(k) such that any set S of at least fES(k) points in general position in the plane has a subset of k points that are the vertices of a convex k-gon. In our setting the points of S are colored, and we say that a (not necessarily convex) spanned polygon is monochromatic if all its vertices have the same color. Moreover, a polygon is called empty if it does not contain any points of S in its interior. We show that any bichromatic set of n ≥ 5044 points in R2 in general position determines at least one empty, monochromatic quadrilateral (and thus linearly many).Postprint (published version

LIPIcs, Volume 258, SoCG 2023, Complete Volume

LIPIcs, Volume 258, SoCG 2023, Complete Volum