Transverse Theory Research Articles

Selection of appropriate location of turbines using score function in transversals of an intuitionistic fuzzy soft hypergraphs

The concept of intuitionistic fuzzy soft set is applied to generalize the theory of transversals in hypergraphs. The notion of transversals of an Intuitionistic Fuzzy Soft Hypergraphs (IFSHGs) and locally minimal transversals of IFSHGs are pioneered with some of its specifications. It is also proved that H ˜ is (μ, ν)-tempered IFSHGs if H ˜ is support simple, elementary and simply ordered. Then, an algorithm is developed and proposed to find the minimal transversals of IFSHGs. An application is also identified in selecting appropriate location for the installation of wind turbines. Finally the proposed algorithm works in finding the suitable place for wind turbine installation. As a result the proposed algorithm is helpful in making decisions.

Projection theorems for linear-fractional families of projections

AbstractMarstrand’s theorem states that applying a generic rotation to a planar setAbefore projecting it orthogonally to thex-axis almost surely gives an image with the maximal possible dimension$\min(1, \dim A)$. We first prove, using the transversality theory of Peres–Schlag locally, that the same result holds when applying a generic complex linear-fractional transformation in$PSL(2,\mathbb{C})$or a generic real linear-fractional transformation in$PGL(3,\mathbb{R})$. We next show that, under some necessary technical assumptions, transversality locally holds for restricted families of projections corresponding to one-dimensional subgroups of$PSL(2,\mathbb{C})$or$PGL(3,\mathbb{R})$. Third, we demonstrate, in any dimension, local transversality and resulting projection statements for the families of closest-point projections to totally-geodesic subspaces of hyperbolic and spherical geometries.

Large Fronts in Nonlocally Coupled Systems Using Conley–Floer Homology

In this paper, we study travelling front solutions for nonlocal equations of the type ∂tu=N∗S(u)+∇F(u),u(t,x)∈Rd.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\partial _t u = N * S(u) + \ abla F(u), \\qquad u(t,x) \\in {{\ extbf {R}}}^d. \\end{aligned}$$\\end{document}Here, N * denotes a convolution-type operator in the spatial variable x in {{textbf {R}}}, either continuous or discrete. We develop a Morse-type theory, the Conley–Floer homology, which captures travelling front solutions in a topologically robust manner, by encoding fronts in the boundary operator of a chain complex. The equations describing the travelling fronts involve both forward and backward delay terms, possibly of infinite range. Consequently, these equations lack a natural phase space, so that classic dynamical systems tools are not at our disposal. We therefore develop, from scratch, a general transversality theory, and a classification of bounded solutions, in the absence of a phase space. In various cases the resulting Conley–Floer homology can be interpreted as a homological Conley index for multivalued vector fields. Using the Conley–Floer homology, we derive existence and multiplicity results on travelling front solutions.

Decoherent Excitation of Transverse Free Currents in Dielectric Liquids via Inter-Molecular Interactions

We present a theoretical model for a class of optical scattering experiments in which short-duration, linearly-polarized electromagnetic pulses scatter off dielectric liquids. The pattern of scattering, particularly in the transverse direction, indicates that significant free currents are generated in the direction orthogonal to the polarization of the incident light. Modelling the target as a dense cluster of two-level systems, we show that transverse free currents are produced by short duration, electric-dipole interactions between proximate molecules, and result in scattering patterns similar to those observed in the experiments. Calculations provide a rationale for why these scattering patterns are not observed in the same molecules at lower densities or with lower field intensities. These features make this model a relevant alternative to proposed transverse optical magnetism theories.

Carnot’s theory of transversals and its applications by Servois and Brianchon: the awakening of synthetic geometry in France

Carnot’s theory of transversals and its applications by Servois and Brianchon: the awakening of synthetic geometry in France

Certified numerical algorithm for isolating the singularities of the plane projection of generic smooth space curves

Isolating the singularities of a plane curve is the first step towards computing its topology. For this, numerical methods are efficient but not certified in general. We are interested in developing certified numerical algorithms for isolating the singularities. In order to do so, we restrict our attention to the special case of plane curves that are projections of smooth curves in higher dimensions. This type of curve appears naturally in robotics applications and scientific visualization. In this setting, we show that the singularities can be encoded by a regular square system whose solutions can be isolated with certified numerical methods. Our analysis is conditioned by assumptions that we prove to be generic using transversality theory. We also provide a semi-algorithm to check their validity. Finally, we present experiments, some of which are not reachable by other methods, and discuss the efficiency of our method.

Inherent characteristic analysis of a precast concrete vibrating table based on the Rayleigh-Ritz method

To assess a precast concrete vibrating table, a system mechanics model was established by using the transverse vibration theory of thin plates. The model was then simplified as a stiffened plate structure according to the structural characteristics of a vibration platen. Specifically, the vibration mode functions of the vibration platen were established by utilizing a two-dimensional beam function method in which the maximum kinetic energy and maximum strain energy of the vibration platen were derived. Additionally, the Rayleigh-Ritz method based on the displacement variation principle was utilized to analyze the natural frequency and mode shapes of the vibration platen. A comparison between the theoretical calculation results and the finite element simulation results indicated that the established vibration mode functions can appropriately reflect the mechanical behavior of the vibration platen, which verified the correctness of the analysis method for the inherent characteristic of the vibration platen and provided a preliminarily theoretical basis for the performance improvement of a precast concrete vibrating table.

Fair-by-design matching

Matching algorithms are used routinely to match donors to recipients for solid organs transplantation, for the assignment of medical residents to hospitals, record linkage in databases, scheduling jobs on machines, network switching, online advertising, and image recognition, among others. Although many optimal solutions may exist to a given matching problem, when the elements that shall or not be included in a solution correspond to individuals, it becomes of paramount importance that the solution is selected fairly. In this paper we study individual fairness in matching problems. Given that many maximum matchings may exist, each one satisfying a different set of individuals, the only way to guarantee fairness is through randomization. Hence we introduce the distributional maxmin fairness framework which provides, for any given input instance, the strongest guarantee possible simultaneously for all individuals in terms of satisfaction probability (the probability of being matched in the solution). Specifically, a probability distribution over feasible solutions is maxmin-fair if it is not possible to improve the satisfaction probability of any individual without decreasing it for some other individual which is no better off. Our main contribution is a polynomial-time algorithm building on techniques from minimum cuts, and edge-coloring algorithms for regular bipartite graphs, and transversal theory. In the special case of bipartite matching, our algorithm runs in $$O((|V|^2 + |E| |V|^{2/3})\cdot (\log |V|)^2)$$ expected time. An experimental evaluation of our fair-matching algorithm shows its ability to scale to graphs with tens of millions of vertices and hundreds of millions of edges, taking only a few minutes on a simple architecture. To the best of our knowledge, this yields the first large-scale implementation of the egalitarian mechanism of Bogomolnaia and Moulin (Econometrica 72(1):257–279, 2004). Our analysis confirms that our method provides stronger satisfaction probability guarantees than non-trivial baselines.

Query-oriented text summarization based on hypergraph transversals

The rise in the amount of textual resources available on the Internet has created the need for tools of automatic document summarization. The main challenges of query-oriented extractive summarization are (1) to identify the topics of the documents and (2) to recover query-relevant sentences of the documents that together cover these topics. Existing graph- or hypergraph-based summarizers use graph-based ranking algorithms to produce individual scores of relevance for the sentences. Hence, these systems fail to measure the topics jointly covered by the sentences forming the summary, which tends to produce redundant summaries. To address the issue of selecting non-redundant sentences jointly covering the main query-relevant topics of a corpus, we propose a new method using the powerful theory of hypergraph transversals. First, we introduce a new topic model based on the semantic clustering of terms in order to discover the topics present in a corpus. Second, these topics are modeled as the hyperedges of a hypergraph in which the nodes are the sentences. A summary is then produced by generating a transversal of nodes in the hypergraph. Algorithms based on the theory of submodular functions are proposed to generate the transversals and to build the summaries. The proposed summarizer outperforms existing graph- or hypergraph-based summarizers by at least 6% of ROUGE-SU4 F-measure on DUC 2007 dataset. It is moreover cheaper than existing hypergraph-based summarizers in terms of computational time complexity.

About Intrinsic Transversality of Pairs of Sets

The article continues the study of the 'regular' arrangement of a collection of sets near a point in their intersection. Such regular intersection or, in other words, transversality properties are crucial for the validity of qualification conditions in optimisation as well as subdifferential, normal cone and coderivative calculus, and convergence analysis of computational algorithms. One of the main motivations for the development of the transversality theory of collections of sets comes from the convergence analysis of alternating projections for solving feasibility problems. This article targets infinite dimensional extensions of the intrinsic transversality property introduced recently by Drusvyatskiy, Ioffe and Lewis as a sufficient condition for local linear convergence of alternating projections. Several characterisations of this property are established involving new limiting objects defined for pairs of sets. Special attention is given to the convex case.

Transversality in Variational Analysis

We discuss various aspects of newly developed extension of the classical transversality theory to variational analysis and optimization theory. In particular, we give interpretations in transversality terms of some key results (relating to subdifferential calculus, necessary optimality conditions and linear convergence of alternating projections) and prove a set-valued version of the Thom transversality theorem for semi-algebraic objects.

A fast algorithm to construct a representation for transversal matroids

Transversal matroids were introduced in the mid 1960s and unified many results in transversal theory. Piff and Welsh proved that a transversal matroid is representable over all sufficiently large fields. To date, their merge algorithm is the only known algorithm to construct a representation matrix for a given transversal matroid. In this paper, a new algorithm to construct a representation matrix for a given transversal matroid is proposed that is faster than the Piff–Welsh algorithm. Let $$G=(V^{(1)}\dot{\cup }V^{(2)},E)$$ be the bipartite graph representing a transversal matroid and $${\mathcal {M}}$$ the set of all complete matchings of G. The time complexity of the proposed algorithm is $$O\left( |V^{(1)}|^{1/2}|E| + |V^{(1)}| |{\mathcal {M}}|\right) $$ . This algorithm makes use of complete matchings of bipartite graphs instead of matrix determinants, and an enumeration algorithm is used to find these matchings.

A method for selective excitation of Ince-Gaussian modes in an end-pumped solid-state laser

A method for selective excitation of Ince-Gaussian modes is presented. The method is based on the spatial distributions of Ince-Gaussian modes as well as the transverse mode selection theory. Significant diffraction loss is introduced in a resonator by using opaque lines at zero-intensity positions, and this loss allows to excite a specific mode; we call this method “loss control.” We study the method by means of numerical simulation of a half-symmetric laser resonator. The simulated field is represented by angular spectrum of the plane waves representation, and its changes are calculated by the two-dimensional fast Fourier transform algorithm when it passes through the optical elements and propagates back and forth in the resonator. The output lasing modes of our method have an overlap of over 90 % with the target Ince-Gaussian modes. The method will be beneficial to the further study of properties and potential applications of Ince-Gaussian modes.

Helly numbers of acyclic families

The Helly number of a family of sets with empty intersection is the size of its largest inclusion-wise minimal sub-family with empty intersection. Let F be a finite family of open subsets of an arbitrary locally arc-wise connected topological space Γ. Assume that for every sub-family G⊆F the intersection of the elements of G has at most r connected components, each of which is a Q-homology cell. We show that the Helly number of F is at most r(dΓ+1), where dΓ is the smallest integer j such that every open set of Γ has trivial Q-homology in dimension j and higher. (In particular dRd=d.) This bound is best possible. We also prove a stronger theorem where the intersection of small sub-families may have more than r connected components, each possibly with nontrivial homology in low dimension. As an application, we obtain several explicit bounds on Helly numbers in geometric transversal theory for which only ad hoc geometric proofs were previously known; in certain cases, the bound we obtain is better than what was previously known. In fact, our proof bounds the Leray number of the nerves of the families under consideration and thus also yields, under similar assumptions, a fractional Helly theorem, a (p,q)-theorem and the existence of small weak ϵ-nets.

A formação em Terapia Ocupacional: entre o ideal e o real

This theoretical study aims at discussing the process of idealization of the profession, by students of the Federal University of Espírito Santo, in confrontation with the reality of practical experiences, relating this question with the formation of the occupational therapist. For this discussion were used as reference studies on the identity and training in Occupational Therapy, transversality theory and production of subjectivity. Indicates that the idealization of Occupational Therapy, how will be built in the training process, can be determinant of the quality of the actions in the reality of care. It proposes to consider training as an open space, multivector, which can produce differentiations, trials, state natural modes of existence, emphasizing the importance of the teacher-student relationship in the process.

How humans fly

This paper is devoted to the general problem of reconstructing the cost from the observation of trajectories, in a problem of optimal control. It is motivated by the following applied problem, concerning HALE drones: one would like them to decide by themselves for their trajectories, and to behave at least as a good human pilot. This applied question is very similar to the problem of determining what is minimized in human locomotion. These starting points are the reasons for the particular classes of control systems and of costs under consideration. To summarize, our conclusion is that in general, inside these classes, three experiments visiting the same values of the control are needed to reconstruct the cost, and two experiments are in general not enough. The method is constructive. The proof of these results is mostly based upon the Thom's transversality theory.

Two Results on Infinite Transversal Theory

By the assistance of an independence space on a set S with finite rank, this paper presents a way to determine a family of subsets of S to possess an independent transversal. Afterwards, it provides another result on infinite transversal relative to partial transversals of a family of subsets of S. The two results are in the field of transversal theory. Keywords: transversal; independence space; finite rank; finite partial transversal. 2010 Mathematics Subject Classification: 05B35; 52A35

The weight of matter

Einstein's traceless 1919 gravitational theory is analyzed from a variational viewpoint. It is shown to be equivalent to a transverse (invariant only under diffeomorphisms that preserve the Lebesgue measure) theory, with an additional Weyl symmetry, in which the gauge is partially fixed so that the metric becomes unimodular. In spite of the fact that this symmetry forbids direct coupling of the potential energy with the gravitational sector, the equivalence principle is recovered in the unimodular gauge owing to Bianchi's identities.

SuperDARN observations of high-m ULF waves with curved phase fronts and their interpretation in terms of transverse resonator theory

[1] The Hankasalmi SuperDARN radar in Finland, while operating in a high spatial and temporal resolution mode, has measured the ionospheric signature of a naturally occurring ULF wave in scatter artificially induced by the Tromso Heater. The wave had a period of 100 s and exhibited curved phase fronts across the heated volume (about 180 km along a single radar beam). Spatial information provided by the radar has enabled an m-number for the wave of about 38 to be determined. It is demonstrated here that the curved phase fronts are a generic feature of nonstationary poloidal waves in a transverse resonator, caused by the common action of the field line curvature, the plasma pressure, and the equilibrium current. Some features of the observed event agree with the resonator in the vicinity of the ring current, where it is proposed that the wave is excited by a moving source in the form of a proton cloud drifting in the magnetosphere in the azimuthal direction.

A biomechanical inactivation principle

This paper develops the mathematical side of a theory of inactivations in human biomechanics. This theory has been validated by practical experiments, including zero-gravity experiments. The theory mostly relies on Pontryagin’s maximum principle on the one side and on transversality theory on the other side. It turns out that the periods of silence in the activation of muscles that are observed in practice during the motions of the arm can appear only if “something like the energy expenditure” is minimized. Conversely, minimization of a criterion taking into account the “energy expenditure” guaranties the presence of these periods of silence, for sufficiently short movements.