Intersection Property Research Articles

A fixed point theorem for twist maps

<p style='text-indent:20px;'>Poincaré's last geometric theorem (Poincaré-Birkhoff Theorem [<xref ref-type="bibr" rid="b2">2</xref>]) states that any area-preserving twist map of annulus has at least two fixed points. We replace the area-preserving condition with a weaker intersection property, which states that any essential simple closed curve intersects its image under <inline-formula><tex-math id="M1">\begin{document}$ f $\end{document}</tex-math></inline-formula> at least at one point. The conclusion is that any such map has at least one fixed point. Besides providing a new proof to Poincaré's geometric theorem, our result also has some applications to reversible systems.</p>

Summand intersection property on the class of exact submodules

A module M is said to have the SIP if intersection of each pair of direct summands is also a direct summand of M. In this article, we define a module M to have the SIPr if and only if intersection of each pair of exact direct summands is also a direct summand of M where r is a left exact preradical for the category of right modules. We investigate structural properties of SIPr-modules and locate the implications between the other summand intersection properties. We deal with decomposition theory as well as direct summands of SIPr-modules. We provide examples by looking at special left exact preradicals.

Injective envelopes and the intersection property

We consider the ideal structure of a reduced crossed product of a unital C∗-algebra equipped with an action of a discrete group. More specifically we find sufficient and necessary conditions for the group action to have the intersection property, meaning that non-zero ideals in the reduced crossed product restrict to non-zero ideals in the underlying C∗-algebra. We show that the intersection property of a group action on a C∗-algebra is equivalent to the intersection property of the action on the equivariant injective envelope. We also show that the centre of the equivariant injective envelope always contains a C∗-algebraic copy of the equivariant injective envelope of the centre of the injective envelope. Finally, we give applications of these results in the case when the group is C∗-simple.

Simulation and Uncertainty Quantification of Electron Beams in Active Spacecraft Charging Scenarios

Novel active sensing methods have been recently proposed to measure the electrostatic potential of noncooperative objects in geosynchronous equatorial orbit and deep space. Such approaches make use of electron beams to excite the emission of secondary electrons and X-rays and infer properties of the emitting surface. However, the detectability of secondary electrons is severely complicated in the presence of complex charged bodies, making computationally efficient simulation frameworks necessary for in situ potential estimation. The purpose of this paper is twofold: firstly, to introduce and test a quasi-analytical, uncoupled, and computationally efficient electron beam expansion and deflection model for active charging applications; and secondly, to characterize the uncertainty in the beam–target intersection properties, which condition the measurement of secondary electrons. The results show that a combination of secondary electrons and X-ray methods is highly desirable to yield a robust and accurate measure of the potential of a target spacecraft.

Democracy and Demography: Intersectional Dimensions of German Politics

AbstractFeminist theory revealed liberal democracy as gendered masculine in a macrointersectional way that privileged racial-ethnic and economic power, enforced heteronormativity, and constructed gender-binary citizenship. Merely reformed to accommodate women, many brotherhood–breadwinner democracies now face deeper challenges. As the second demographic transition undermines the hegemony of binary gender relations, it reorganizes political conflict on an axis of reproductive politics. Germany’s Green and Alternative für Deutschland parties exemplify opposite ends of this axis. The Green clusters of issues reflect intersectional societal ideals that demasculinize democracy, while reactionary populism repoliticizes masculinity to defend the family–state relations of the breadwinner–brotherhood gender system.

Equivariant perverse sheaves and quasi-hereditary algebras

Let X denote a quasi-projective variety over a field on which a connected linear algebraic group G acts with finitely many orbits. Then, the G-orbits define a stratification of X. We establish several key properties of the category of equivariant perverse sheaves on X, which have locally constant cohomology sheaves on each of the orbits. Under the above assumptions, we show that this category comes close to being a highest weight category in the sense of Cline, Parshall and Scott and defines a quasi-hereditary algebra. We observe that the above hypotheses are satisfied by all toric varieties and by all spherical varieties associated to connected reductive groups over any algebraically closed field.Next we show that the odd dimensional intersection cohomology sheaves vanish on all spherical varieties defined over algebraically closed fields of positive characteristics, extending similar results for spherical varieties defined over the field of complex numbers by Michel Brion and the author in prior work. Assuming that the linear algebraic group G and the action of G on X are defined over a finite field Fq, and where the odd dimensional intersection cohomology sheaves on the orbit closures vanish, we also establish several basic properties of the mixed category of mixed equivariant perverse sheaves so that the associated terms in the weight filtration are finite sums of the shifted equivariant intersection cohomology complexes on the orbit closures.

Evaluation of Intersection Properties Using MARS Method for Improving Urban Traffic Performance: Case Study of Tekirdağ, Turkey

Increasing urban traffic performance is a technical problem that has been investigated by many researchers these days. Traffic performance can be increased in many ways, as part of the transportation planning process, on smaller scales, or with different methods and techniques. The determination of traffic intervention areas in urban transportation planning is an intervention type that determines the rate at which the traffic performance will increase. Although transportation planning is an integrated issue, the type of traffic modification and prior intervention on intersections are often determined with partitive paradigms and strategies. It is a significant opportunity for decision makers to be informed in advance of the effects of intersection characteristics on the overall traffic performance. However, it is not an attempted or tested concept to perform a general assessment of the impact of the intersection characteristics on the overall performance of the intersections. In this study, a four-stage integrated analysis including the multivariate adaptive regression splines (MARS) method is proposed for the overall traffic performance evaluation. The traffic characteristics of intersections are first indexed and categorized. The intersection performance results are then obtained using the VISSIM traffic simulation software. Subsequently, the relationship between them is determined with the MARS method, and the effects of the intersection characteristics on the traffic performance are investigated. Tekirdağ, a metropolitan city located in northwestern Turkey, is selected for the case study. According to the obtained simulation results, the suggestions related to the development of intersection performances are made and tested.

Cameron-Liebler $k$-sets in $\mathrm{AG}(n,q)$

We study Cameron-Liebler $k$-sets in the affine geometry, so sets of $k$-spaces in $\mathrm{AG}(n,q)$. This generalizes research on Cameron-Liebler $k$-sets in the projective geometry $\mathrm{PG}(n,q)$. Note that in algebraic combinatorics, Cameron-Liebler $k$-sets of $\mathrm{AG}(n,q)$ correspond to certain equitable bipartitions of the association scheme of $k$-spaces in $\mathrm{AG}(n,q)$, while in the analysis of Boolean functions, they correspond to Boolean degree $1$ functions of $\mathrm{AG}(n,q)$.
 We define Cameron-Liebler $k$-sets in $\mathrm{AG}(n,q)$ by intersection properties with $k$-spreads and show the equivalence of several definitions. In particular, we investigate the relationship between Cameron-Liebler $k$-sets in $\mathrm{AG}(n,q)$ and $\mathrm{PG}(n,q)$. As a by-product, we calculate the character table of the association scheme of affine lines. Furthermore, we characterize the smallest examples of Cameron-Liebler $k$-sets.
 This paper focuses on $\mathrm{AG}(n,q)$ for $n > 3$, while the case for Cameron-Liebler line classes in $\mathrm{AG}(3,q)$ was already treated separately.

Intrinsic characterizations of C-realcompact spaces

<pre>c-realcompact spaces are introduced by Karamzadeh and Keshtkar in Quaest. Math. 41, no. 8 (2018), 1135-1167. We offer a characterization of these spaces X via c-stable family of closed sets in X by showing that X is c-realcompact if and only if each c-stable family of closed sets in X with finite intersection property has nonempty intersection. This last condition which makes sense for an arbitrary topological space can be taken as an alternative definition of a c-realcompact space. We show that each topological space can be extended as a dense subspace to a c-realcompact space with some desired extension properties. An allied class of spaces viz CP-compact spaces akin to that of c-realcompact spaces are introduced. The paper ends after examining how far a known class of c-realcompact spaces could be realized as CP-compact for appropriately chosen ideal P of closed sets in X.</pre>

Free idempotent generated semigroups: The word problem and structure via gain graphs

Building on the previous extensive study of Yang, Gould and the present author, we provide a more precise insight into the group-theoretical ramifications of the word problem for free idempotent generated semigroups over finite biordered sets. We prove that such word problems are in fact equivalent to the problem of computing intersections of cosets of certain subgroups of direct products of maximal subgroups of the free idempotent generated semigroup in question, thus providing decidability of those word problems under group-theoretical assumptions related to the Howson property and the coset intersection property. We also provide a basic sketch of the global semigroup-theoretical structure of an arbitrary free idempotent generated semigroup, including the characterisation of Green’s relations and the key parameters of non-regular \({\cal D}\)-classes. In particular, we prove that all Schützenberger groups of IG(ℇ) for a finite biordered set ℇ must be among the divisors of the maximal subgroups of IG(ℇ).

Hausdorff dimension of intersections with planes and general sets

We give conditions on a general family P_{\lambda}\colon \R^n\to\R^m , \lambda \in \Lambda , of orthogonal projections which guarantee that the Hausdorff dimension formula \dim A\cap P_{\lambda}^{-1}\{u\}=s-m holds generically for measurable sets A\subset\R^n with positive and finite s -dimensional Hausdorff measure, s>m , and with positive lower density. As an application we prove for measurable sets A,B\subset\R^n with positive s - and t -dimensional measures, and with positive lower density that if s + (n-1)t/n > n , then \dim A\cap (g(B)+z) = s+t - n for almost all rotations g and for positively many z\in\R^n .

Highly irregular orbits for subshifts of finite type: large intersections and emergence

In their recent paper (2019 arXiv:1904.03424), the first author, Kiriki and Soma introduced a concept of pointwise emergence to measure the complexity of irregular orbits. They constructed a residual subset of the full shift with high pointwise emergence. In this paper we consider the set of points with high pointwise emergence for topologically mixing subshifts of finite type. We show that this set has full topological entropy, full Hausdorff dimension, and full topological pressure for any Hölder continuous potential. Furthermore, we show that this set belongs to a certain class of sets with large intersection property. This is a natural generalization of Färm and Persson (2011 Nonlinearity 24 1291–1309) to pointwise emergence and Carathéodory dimension.

A genus 4 origami with minimal hitting time and an intersection property

In a minimal flow, the hitting time is the exponent of the power law, as r goes to zero, for the time needed by orbits to become r-dense. We show that on the so-called Ornithorynque origami, the hitting time of the flow in an irrational slope equals the diophantine type of the slope. We give a general criterion for such equality. In general, for genus at least two, hitting time is strictly bigger than diophantine type.

Directed sets and topological spaces definable in o‐minimal structures

We study directed sets definable in o-minimal structures, showing that in expansions of ordered fields these admit cofinal definable curves, as well as a suitable analogue in expansions of ordered groups, and furthermore that no analogue holds in full generality. We use the theory of tame pairs to extend the results in the field case to definable families of sets with the finite intersection property. We then apply our results to the study of definable topologies. We prove that all definable topological spaces display properties akin to first countability, and give several characterizations of a notion of definable compactness due to Peterzil and Steinhorn (J. Lond. Math. Soc. (2) 59 (1999) 769–786) generalized to this setting.

Fractal Intersections and Products via Algorithmic Dimension

Algorithmic fractal dimensions quantify the algorithmic information density of individual points and may be defined in terms of Kolmogorov complexity. This work uses these dimensions to bound the classical Hausdorff and packing dimensions of intersections and Cartesian products of fractals in Euclidean spaces. This approach shows that two prominent, fundamental results about the dimension of Borel or analytic sets also hold for arbitrary sets.

Reconceptualising ‘reasonable adjustments’ for the successful employment of autistic women

Many more autistic women than men do not receive diagnosis in childhood. In addition, they are often socialised from an early age to conceal or ‘camouflage’ their differences in order to fit in. Gendered prejudices often dictate workplace roles which serve to further disadvantage women. Employers are often unaware of how to ensure their workplaces are accommodating for autistic women, and it may seem inconvenient or unnecessary to make adjustments. This article seeks to highlight some issues related to promoting the meaningful recruitment and inclusion of autistic women in the workplace. It draws on qualitative research data to highlight the intersectional dimensions of being an autistic woman at work. The emerging themes are discussed with a view to improving knowledge about how to improve workplace environments and practices thereby support autistic women to succeed. Points of interest Autistic women ‘camouflage’ or ‘mask’ aspects of their autism in order to conceal or compensate for their differences. This may mean they appear to cope and consequently, their needs for support or adjustments at work are overlooked. The research methods use the narratives of autistic women to illuminate the difficulties and advantages of being both autistic and a woman in the workplace. This research suggests that sexist expectations of women to interact as non-autistic person would at work, particularly in relation to emotions, impacts upon their opportunities to progress in their chosen careers. The research discusses barriers and support at work. It contributes to improving awareness of the diverse and complex challenges faced by autistic women.

LARGE INTERSECTION PROPERTIES FOR LIMSUP SETS GENERATED BY RECTANGLES IN COMPACT METRIC SPACES

The large intersection classes in real Euclidean spaces have been introduced by Falconer in 1985. First, we generalize Falconer’s large intersection classes to compact metric spaces and show that these classes are closed under countable intersections. Next, we prove that the limsup sets generated by rectangles in a compact metric space enjoy this intersection property under certain full Hausdorff measure assumptions. We also apply our results to high-dimensional Diophantine approximation on fractals.

Residual intersections and modules with Cohen-Macaulay Rees algebra

In this paper, we consider a finite, torsion-free module E over a Gorenstein local ring. We provide sufficient conditions for E to be of linear type and for the Rees algebra R(E) of E to be Cohen-Macaulay. Our results are obtained by constructing a generic Bourbaki ideal I of E and exploiting properties of the residual intersections of I.

TREASURE: Text Mining Algorithm Based on Affinity Analysis and Set Intersection to Find the Action of Tuberculosis Drugs against Other Pathogens

Tuberculosis (TB) is one of the top causes of death in the world. Though TB is known as the world’s most infectious killer, it can be treated with a combination of TB drugs. Some of these drugs can be active against other infective agents, in addition to TB. We propose a framework called TREASURE (Text mining algoRithm basEd on Affinity analysis and Set intersection to find the action of tUberculosis dRugs against other pathogEns), which particularly focuses on the extraction of various drug–pathogen relationships in eight different TB drugs, namely pyrazinamide, moxifloxacin, ethambutol, isoniazid, rifampicin, linezolid, streptomycin and amikacin. More than 1500 research papers from PubMed are collected for each drug. The data collected for this purpose are first preprocessed, and various relation records are generated for each drug using affinity analysis. These records are then filtered based on the maximum co-occurrence value and set intersection property to obtain the required inferences. The inferences produced by this framework can help the medical researchers in finding cures for other bacterial diseases. Additionally, the analysis presented in this model can be utilized by the medical experts in their disease and drug experiments.

Intersection properties for singular radial solutions of quasilinear elliptic equations with Hardy type potentials

We are interested in singular positive solutions of a quasilinear elliptic equation with a singular coefficient where , , and . The differential operator includes the standard Laplace, m-Laplace and ℓ-Hessian operators. In the case , a solution of the problem gives a radial solution of . This problem has a one-parameter family of singular positive solutions and one exact singular solution. We obtain an intersection number of two singular solutions in the critical and supercritical cases. The main technical tool is a phase plane analysis.