Recovering plane curves of low degree from their inflection lines and inflection points

In this paper we consider the following problem: is it possible to recover a smooth plane curve of degree d ≥ 3 from its inflection lines? We answer the posed question positively for a general smooth plane quartic curve, making the additional assumption that also one inflection point is given, and for any smooth plane cubic curve.     

A 3-Slope Theorem for the infinite relaxation in the plane

In this paper we consider the infinite relaxation of the corner polyhedron with 2 rows. For the 1-row case, Gomory and Johnson proved in their seminal paper a sufficient condition for a minimal function to be extreme, the celebrated 2-Slope Theorem. Despite increased interest in understanding the multiple row setting, no generalization of this theorem was known for this case. We present an extension of the 2-Slope Theorem for the case of 2 rows by showing that minimal 3-slope functions satisfying an additional regularity condition are facets (and hence extreme). Moreover, we show that this regularity condition is necessary, unveiling a structure which is only present in the multi-row setting.     

Symplectically degenerate maxima via generating functions

We provide a simple proof of a theorem due to Nancy Hingston, asserting that symplectically degenerate maxima of any Hamiltonian diffeomorphism $\phi $ of the standard symplectic $2d$ -torus are non-isolated contractible periodic points or their action is a non-isolated point of the average-action spectrum of $\phi $ . Our argument is based on generating functions.     

Smoothness of holonomy covers for singular foliations and essential isotropy

Given a singular foliation, we attach an “essential isotropy” group to each of its leaves, and show that its discreteness is the integrability obstruction of a natural Lie algebroid over the leaf. We show that a condition ensuring discreteness is the local surjectivity of a transversal exponential map associated with the maximal ideal of vector fields prescribed to be tangent to the foliation. The essential isotropy group is also shown to control the smoothness of the holonomy cover of the leaf (the associated fiber of the holonomy groupoid), as well as the smoothness of the associated isotropy group. Namely, the (topological) closeness of the essential isotropy group is a necessary and sufficient condition for the holonomy cover to be a smooth (finite-dimensional) manifold and the isotropy group to be a Lie group. These results are useful towards understanding the normal form of a singular foliation around a compact leaf. At the end of this article we briefly outline work of ours on this normal form, to be presented in a subsequent paper.     

On v-Domains and Star Operations

Let * be a star operation on an integral domain D. Let f (D) be the set of all nonzero finitely generated fractional ideals of D. Call D a *-Prüfer (respectively, (*, v)-Prüfer) domain if (FF ^?1)* = D (respectively, (F ^v F ^?1)* = D) for all F ∈  f (D). We establish that *-Prüfer domains (and (*, v)-Prüfer domains) for various star operations * span a major portion of the known generalizations of Prüfer domains inside the class of v-domains. We also use Theorem 6.6 of the Larsen and McCarthy book [30 Larsen , M. D. , McCarthy , P. J. ( 1971 ). Multiplicative Theory of Ideals . New York : Academic Press . [Google Scholar]], which gives several equivalent conditions for an integral domain to be a Prüfer domain, as a model, and we show which statements of that theorem on Prüfer domains can be generalized in a natural way and proved for *-Prüfer domains, and which cannot be. We also show that in a *-Prüfer domain, each pair of *-invertible *-ideals admits a GCD in the set of *-invertible *-ideals, obtaining a remarkable generalization of a property holding for the “classical” class of Prüfer v-multiplication domains. We also link D being *-Prüfer (or (*, v)-Prüfer) with the group Inv*(D) of *-invertible *-ideals (under *-multiplication) being lattice-ordered.     

On Some Types of Rigid Sets in Banach Spaces

One of the properties characterizing Euclidean spaces says - roughly speaking- that their unit sphere has nice invariant properties. More precisely, a finite dimensional normed space has an Euclidean norm if and only if the group of isometries acts transitively on its unit sphere (the norm is “transitive”); such property of the sphere is also called “rigidity”. More recently, another notion of “rigidity” for compact sets, connected with “isometric sequences”, received some attention. Infinite rigid sets are diametral; moreover, under suitable assumptions on the space, they are also contained in the boundary of a sphere. These notions are connected with many problems, in different areas. Here we discuss and compare these two notions of rigid set, trying to indicate new relations among them and with some other properties of sets. Several examples complete the paper.     

On the Mosaic Method for Many-Dimensional Modal Logics: A Case Study Combining Tense and Modal Operators

We present an extension of the mosaic method aimed at capturing many-dimensional modal logics. As a proof-of-concept, we define the method for logics arising from the combination of linear tense operators with an “orthogonal” S5-like modality. We show that the existence of a model for a given set of formulas is equivalent to the existence of a suitable set of partial models, called mosaics, and apply the technique not only in obtaining a proof of decidability and a proof of completeness for the corresponding Hilbert-style axiomatization, but also in the development of a mosaic-based tableau system. We further consider extensions for dealing with the case when interactions between the two dimensions exist, thus covering a wide class of bundled Ockhamist branching-time logics, and present for them some partial results, such as a non-analytic version of the tableau system.     

Topology design for on-demand dual-path routing in wireless networks

One way to achieve reliability with low-latency is through multi-path routing and transport protocols that build redundant delivery channels (or data paths) to reduce end-to-end packet losses and retransmissions. However, the applicability and effectiveness of such protocols are limited by the topological constraints of the underlying communication infrastructure. Multiple data delivery paths can only be constructed over networks that are capable of supporting multiple paths. In mission-critical wireless networks, the underlying network topology is directly affected by the terrain, location and environmental interferences, however the settings of the wireless radios at each node can be properly configured to compensate for these effects for multi-path support. In this work we investigate optimization models for topology designs that enable end-to-end dual-path support on a distributed wireless sensor network. We consider the case of a fixed sensor network with isotropic antennas, where the control variable for topology management is the transmission power on network nodes. For optimization modeling, the network metrics of relevance are coverage, robustness and power utilization. The optimization models proposed in this work eliminate some of the typical assumptions made in the pertinent network design literature that are too strong in this application context.     

Umt-domains and domains with prüfer integral closure

An integral domain R is said to be a UMT-domain if uppers to zero in R[X) are maximal t-ideals. We show that R is a UMT-domain if and only if its localizations at maximal tdeals have Prüfer integral closure. We also prove that the UMT-property is preserved upon passage to polynomial rings. Finally, we characterize the UMT-property in certian pullback constructions; as an application, we show that a domain has Prüfer integral closure if and only if all its overrings are UMT-domains.