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This paper presents a general analytical solution for the problem of locating points in planar regions with an arbitrary geometry at the boundary. The proposed methodology overcomes the traditional solutions used for polygonal regions. The method originated from the explicit evaluation of the contour integral using the Residue and Cauchy theorems, which then evolved toward a technique very similar to the winding number and, finally, simplified into a variant of ray-crossing approach slightly more informed and more universal than the classic approach, which had been used for decades. The very close relation of both techniques also emerges during the derivation of the solution. The resulting algorithm becomes simpler and potentially faster than the current state of the art for point locations in arbitrary polygons because it uses fewer operations. For polygonal regions, it is also applicable without further processing for special cases of degeneracy, and it is possible to use in fully integer arithmetic; it can also be vectorized for parallel computation. The major novelty, however, is the extension of the technique to virtually any shape or segment delimiting a planar domain, be it linear, a circular arc, or a higher order curve. [ABSTRACT FROM AUTHOR]

When a boat disappears over the horizon, does a distant observer detect the last moment in which the boat is visible, or the first moment in which the boat is not visible? This apparently ludicrous way of reasoning, heritage of long-lasting medieval debates on decision limit problems, paves the way to sophisticated contemporary debates concerning the methodological core of mathematics, physics and biology. These ancient, logically-framed conundrums throw us into the realm of bounded objects with fuzzy edges, where our mind fails to provide responses to plain questions such as: given a closed curve with a boundary (say, a cellular membrane) how do you recognize what is internal and what is external? We show how the choice of an alternative instead of another is not arbitrary, rather points towards entirely different ontological, philosophical and physical commitments. This paves the way to novel interpretations and operational approaches to challenging issues such as black hole singularities, continuous time in quantum dynamics, chaotic nonlinear paths, logarithmic plots, demarcation of living beings. In the sceptical reign where judgements seem to be suspended forever, the contemporary scientist stands for a sort of God equipped with infinite power who is utterly free to dictate the rules of the experimental settings. [ABSTRACT FROM AUTHOR]

We consider paths of steepest descent, in the complex plane, for the norm of a non-constant one variable polynomial ƒ. We show that such paths, starting from a zero of the logarithmic derivative of ƒ and ending in a root of ƒ, draw a tree in the complex plane, and we give an upper bound estimate on their lengths. In some cases, we obtain a finer estimate that depends only on the set of roots of ƒ, not on their multiplicity, and we wonder if this can be done in general. We also extend this question to finite Blaschke products for the unit disk. [ABSTRACT FROM AUTHOR]

We introduce and discuss a connectedness induced by n-ary relations ( $$n>1$$ an integer) on their underlying sets. In particular, we focus on certain n-ary relations with the induced connectedness allowing for a definition of digital Jordan curves. For every integer $$n>1$$ , we introduce one such n-ary relation on the digital plane $${\mathbb {Z}}^2$$ and prove a digital analogue of the Jordan curve theorem for the induced connectedness. It follows that these n-ary relations may be used as convenient structures on the digital plane for the study of geometric properties of digital images. For $$n=2$$ , such a structure coincides with the (specialization order of the) Khalimsky topology and, for $$n>2$$ , it allows for a variety of Jordan curves richer than that provided by the Khalimsky topology. [ABSTRACT FROM AUTHOR]

Lagrangian flux through a fixed curve segment naturally gives rise to the notion of a donating region, a compact point set in which each particle will pass through the curve and contribute to the flux. A precise geometric determination of the donating region has not been available until now. The author proposes an explicit, constructive, and analytical definition of the donating region based on two characteristic curves of the flow field, viz. streaklines and timelines. It is also shown that, within a time interval of finite size, each particle in the proposed donating region has a net effect of going across the fixed curve once. The presented analysis might be potentially useful for flow topology studies, Lagrangian flux calculation, and explicit interface tracking. [ABSTRACT FROM AUTHOR]

Abstract: The Inscribed Square Conjecture has been open since 1911. It states that any plane Jordan curve contains four points on a non-degenerate square. In this article two different discrete versions of this conjecture are introduced and proved. The first version is in the field of digital topology: it is proved that the conjecture holds for digital simple closed 4-curves, and that it is false for 8-curves. The second one is in the topological graph theory field: it is proved that any cycle of the grid contains an inscribed square with integer vertices. The proofs are based on a theorem due to Pak. An infinite family of 4-curves in the digital plane containing a single non-degenerate inscribed square is introduced as well as a second infinite family containing one 4-curve with exactly inscribed squares for each positive integer value of . Finally an algorithm with time complexity is given to find inscribed squares in simple digital curves. [Copyright &y& Elsevier]

Abstract: This paper discusses an attempt at identifying a property of circuits in (nonplanar) graphs resembling the separation property of circuits in planar graphs derived from the Jordan Curve Theorem. If is a graph and is a circuit in , we say that two circuits in form a split of if the symmetric difference of their edges sets is equal to the edge set of , and if they are separated in by the intersection of their vertex sets. García Moreno and Jensen, A note on semiextensions of stable circuits, Discrete Math. 309 (2009) 4952–4954, asked whether such a split exists for any circuit whenever is 3-connected. We observe that if true, this implies a strong form of a version of the Cycle Double-Cover Conjecture suggested in the Ph.D. thesis of Luis Goddyn. The main result of the paper shows that the property holds for Hamilton circuits in cubic graphs. [ABSTRACT FROM AUTHOR]

In an undirected simple graph, we define connectedness induced by a set of walks of the same lengths. We show that the connectedness is preserved by the strong product of graphs with walk sets. This result is used to introduce a graph on the vertex set ℤ² with sets of walks that is obtained as the strong product of a pair of copies of a graph on the vertex set ℤ with certain walk sets. It is proved that each of the walk sets in the graph introduced induces connectedness on ℤ² that satisfies a digital analogue of the Jordan curve theorem. It follows that the graph with any of the walk sets provides a convenient structure on the digital plane ℤ² for the study of digital images. [ABSTRACT FROM AUTHOR]