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Incidence Geometry

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lightbulbAbout this topic
Incidence geometry is a branch of mathematics that studies the properties and relationships of points, lines, and other geometric objects based on their incidence relations, which describe how these objects intersect or are positioned relative to one another, without necessarily relying on the concepts of distance or angle.
lightbulbAbout this topic
Incidence geometry is a branch of mathematics that studies the properties and relationships of points, lines, and other geometric objects based on their incidence relations, which describe how these objects intersect or are positioned relative to one another, without necessarily relying on the concepts of distance or angle.

Key research themes

1. What are the combinatorial and incidence structures underlying Archimedean solids, specifically the rhombicosidodecahedron, and how can induced graphs reflect their properties?

This theme focuses on the combinatorial formalization of incidence relationships among vertices, edges, and faces in highly symmetric polyhedra. By axiomatizing local incidence structures inspired by the rhombicosidodecahedron, researchers can define nodal points with specific incidence counts and induce graphs reflecting shared region types. Understanding these structures provides insight into their geometric and graph-theoretic properties and offers frameworks to generalize to other uniform polyhedra.

Key finding: The paper introduces R-Faccial Systems as an abstract combinatorial framework that axiomatizes the incidences of regions, points, and types observed in the uniform rhombicosidodecahedron. It formally defines R-Faccial Nodal... Read more

2. How do the loci of triangle centers behave in Poncelet triangle families between conics, and under what conditions do these loci become conic (elliptic) curves?

Research in this area investigates the geometric properties of families of Poncelet triangles inscribed and circumscribed between two conics, especially the confocal pair of ellipses. A key question is which notable triangle centers sweep loci that are conics, typically ellipses, as the triangles vary, and which conditions or affine relationships characterize this behavior. Understanding these locus ellipticity phenomena sheds light on deep connections between classical geometry, complex analysis, and algebraic geometry.

Key finding: The authors develop a comprehensive theory predicting whether the locus of a triangle center over specific Poncelet triangle families is a conic curve. For families embedded between confocal ellipses, they prove that if a... Read more

3. Under what geometric and combinatorial conditions can polyhedra be inscribed or circumscribed to convex bodies in Euclidean spaces, and what are the existence results and counterexamples in various dimensions?

This theme examines the classical problem of inscribing or circumscribing polyhedra to convex bodies in Euclidean spaces of various dimensions. It addresses questions such as: Given a convex body in R^n, is there always a polyhedron with vertices on the boundary (inscribed) or one containing the body with each face touching the boundary (circumscribed)? The theme incorporates techniques from geometry, topology, and combinatorics to establish existence theorems, identify obstructions, and analyze special cases like squares inscribed in planar Jordan curves or cubes circumscribed around convex bodies. These fundamental results inform both geometric measure theory and combinatorial geometry.

Key finding: The paper surveys classical and recent results on the existence of polyhedra inscribed in or circumscribed about convex bodies in R^n. It proves that any planar convex body admits an inscribed square or rectangle with... Read more

All papers in Incidence Geometry

Dementia is an increasing focus for policymakers, civil organizations and multidisciplinary researchers. The most recent descriptive epidemiological research into dementia is enabling investigation into how the prevalence and incidence... more
Primary biliary cholangitis (PBC) has been regarded as female-predominant without evidence of gender difference in survival. We aimed to compare the overall survival, incidence and prevalence of PBC in two well defined population-based... more
Background In South Africa, large increases in early adult mortality during the 1990s and early 2000s have reversed since public HIV treatment rollout in 2004. In a rural population in KwaZulu-Natal, we investigate trends in parental... more
Lactation may protect women with previous gestational diabetes mellitus (GDM) from developing type 2 diabetes mellitus, but the results of existing studies are inconsistent, ranging from null to beneficial. We aimed to conduct a... more
Objective. To compile an injury profile of 46 fast bowlers aged 11 - 18 years, and to identify the associated risk factors for injury during one academy cricket season. Methods. The fast bowlers selected were tested and observed for one... more
Laryngeal complications (LCs) following cardiac operations contribute to increased morbidity and resource utilization. Using a nationally representative cohort of cardiac surgical patients, we characterized the incidence of LC as well as... more
Background: The incidence of kidney cancer from different areas of Iran was reported. Nevertheless, there is no available systematic reviews in this regard. Therefore, the present systematic review carried out to estimate the incidence... more
Introduction. The incidence and risk factors for ventilator-related pneumonia (VAP) in patients with delirium are deficient, and there is a lack of in-depth knowledge of the impact of VAP on outcomes in this population. We investigated... more
To estimate incidence of cleft lip and cleft palate among Jordanian infants and to identify associated anomalies, we carried out a prospective, hospital-based study over a 5-year period, January 2000-January 2005. The total number of... more
Introduction: Post-anaesthesia care unit (PACU) delirium is a potentially preventable condition that results in a significant long-term effect. In a multicentre prospective cohort study, we investigate the incidence and risk factors of... more
Results: Age-standardized incidence rate (ASR) (per 100,000 population) of breast cancer among women increased dramatically from 11.8 in 2001 to 22.7 in 2014, indicating a 92.4% increase over the decade. Colorectal cancer incidence was... more
The importance of circadian factors in managing patients is poorly understood. We present two retrospective cohort studies showing that lungs reperfused between 4 and 8 AM have a higher incidence (OR 1.12; 95% CI 1.03 to 1.21; p=0.01) of... more
Background: Esophageal Cancer (EC) is the eighth most common cancer in terms of incidence. Despite various studies, there is no general estimate of EC incidence in Iran. Therefore, the present study is conducted to evaluate the incidence... more
In the literature, there are but a few incidence geometries on which the McLaughlin sporadic group McL acts as a flag-transitive automorphism group. Their highest rank is four. In the present paper, we construct a geometry of rank six on... more
Eleven-year retrospective study was carried out on small ruminants submitted for necropsy in the Department of Veterinary Pathology, University of Ibadan between 1999 and 2010 to determine the disease pattern, incidence and type of... more
Objective: Assessing changes in breast cancer (BC) incidence and stage distribution in the District of Modena, Italy, during the period 1992-1998, and their relationship to a mammographic screening program launched in 1995. Methods:... more
We study the properties (2T) 1 , (IP) 2 , PRI, RPRI, QPRI and RWPRI for inductively minimal pairs (Γ, G) consisting of a finite geometry and a group acting flag-transitively on it. For each of these properties, we characterize which... more
We show that a generalized hexagon of order 3 which contains a subhexagon of order (1, 3) must be the split Cayley hexagon H(3).
Our purpose is to elaborate a theory of planar nets or unfoldings for polyhedra, its generalization and extension to polytopes and to combinatorial polytopes, in terms of morphisms of geometries and the adjacency graph of facets.
Results: There were a total of 5909 childhood cancer cases; 2959 aged 0-14 years (1659 males, 1300 females) and 2950 aged 15-24 years (1590 males, 1358 females). For 0-14-year-old boys, there was statistically significant sinusoidal... more
Inductively minimal geometries form an inÿnite family of incidence geometries on which ÿnite symmetric groups act ag-transitively. They were introduced in Buekenhout et al.
Low Birth Weight (LBW) is an indicator of public health. The morbidity and mortality risk of LBW is 4 times greater than non-LBW. Low birth weight (LBW) both in pre term and term gestational age infants often occurs jaundice. This study... more
Many fungal species cause infection in the nose and paranasal sinuses. Almost half of them are capable of causing systemic and CNS infections. Clinical presentation, imaging features, and treatment differ based on type of fungal... more
Wasting is an acute malnutrition which could indirectly increase infant mortality and morbidity. Currently, wasting was serious public health problem in Indonesia with 11,1% of prevalence. This study aimed to determine the incidence of... more
Background : The burden of surgical site infections (SSI) in Brazilian hospitals is high. Both international and local guidelines from governmental agencies advise preventive interventions against SSI. Objective: To measure the combined... more
by H Vyas
prevalence of coeliac disease (CD). Recent meta-analysis estimates prevalence at 5.8%. The DS Medical Interest Group (DSMIG), does not recommend routine screening, despite guidance from the European Society of Paediatric Gastroenterology... more
492 Recent Change in the Incidence of Cervical Cerclage Maria Bisulli, MD; Neil S. Seligman, MD; Komal Soin, MS; Anju Suhag, MD; Vincenzo Berghella, MD Div. of Maternal-Fetal Medicine, Dept. of Ob/Gyn, Jefferson Medical College of Thomas... more
Background Coccidioidomycosis is an emerging infection in the southwestern United States. We examined the effects of precipitation and temperature on the incidence of coccidioidomycosis in California during 2000-2020, and estimated... more
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