0000000000552130

AUTHOR

Richard Montgomery

showing 3 related works from this author

Bicycle paths, elasticae and sub-Riemannian geometry

2020

We relate the sub-Riemannian geometry on the group of rigid motions of the plane to `bicycling mathematics'. We show that this geometry's geodesics correspond to bike paths whose front tracks are either non-inflectional Euler elasticae or straight lines, and that its infinite minimizing geodesics (or `metric lines') correspond to bike paths whose front tracks are either straight lines or `Euler's solitons' (also known as Syntractrix or Convicts' curves).

Mathematics - Differential GeometryGeodesicGeneral Physics and AstronomyGeometryRiemannian geometry01 natural sciencessymbols.namesakeMathematics - Metric GeometryClassical Analysis and ODEs (math.CA)FOS: Mathematics0101 mathematicsMathematical PhysicsMathematics53C17 (Primary) 53A17 53A04 (Secondary)Group (mathematics)Plane (geometry)Applied Mathematics010102 general mathematicsMetric Geometry (math.MG)Statistical and Nonlinear Physics010101 applied mathematicsDifferential Geometry (math.DG)Mathematics - Classical Analysis and ODEsMetric (mathematics)Euler's formulasymbolsNonlinearity
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Efficacy and Safety of Degludec versus Glargine in Type 2 Diabetes.

2017

BACKGROUND Degludec is an ultralong-Acting, once-daily basal insulin that is approved for use in adults, adolescents, and children with diabetes. Previous open-label studies have shown lower day-To-day variability in the glucose-lowering effect and lower rates of hypoglycemia among patients who received degludec than among those who received basal insulin glargine. However, data are lacking on the cardiovascular safety of degludec. METHODS We randomly assigned 7637 patients with type 2 diabetes to receive either insulin degludec (3818 patients) or insulin glargine U100 (3819 patients) once daily between dinner and bedtime in a double-blind, treat-To-Target, event-driven cardiovascular outco…

Insulin degludecBlood GlucoseMalemedicine.medical_treatmentDEVOTE Study GroupInsulin GlargineType 2 diabetesKaplan-Meier Estimate030204 cardiovascular system & hematologylaw.inventiondiabetes ; insulin0302 clinical medicineRandomized controlled triallawCardiovascular DiseaseGLUCOSE CONTROL11 Medical and Health SciencesRISKCOMPLICATIONSOUTCOMESIncidenceGeneral MedicineMiddle AgedInsulin Long-ActingVARIABILITYCardiovascular Diseasesdiabetes mellitusFemaleLife Sciences & BiomedicineHumanmedicine.drugmedicine.medical_specialty030209 endocrinology & metabolismAged; Blood Glucose; Cardiovascular Diseases; Diabetes Mellitus Type 2; Double-Blind Method; Female; Humans; Hypoglycemia; Hypoglycemic Agents; Incidence; Insulin Glargine; Insulin Long-Acting; Kaplan-Meier Estimate; Male; Middle Aged; Medicine (all)HypoglycemiaBedtimeArticleEVENTS03 medical and health sciencesHYPOGLYCEMIAMedicine General & InternalDouble-Blind MethodInternal medicineDiabetes mellitusGeneral & Internal MedicinemedicineHumansHypoglycemic AgentsIntensive care medicineMETAANALYSISAgedScience & TechnologyHypoglycemic AgentInsulin glarginebusiness.industryInsulinmedicine.diseaseDiabetes Mellitus Type 2businessBASAL INSULIN
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Sard property for the endpoint map on some Carnot groups

2016

In Carnot-Caratheodory or sub-Riemannian geometry, one of the major open problems is whether the conclusions of Sard's theorem holds for the endpoint map, a canonical map from an infinite-dimensional path space to the underlying finite-dimensional manifold. The set of critical values for the endpoint map is also known as abnormal set, being the set of endpoints of abnormal extremals leaving the base point. We prove that a strong version of Sard's property holds for all step-2 Carnot groups and several other classes of Lie groups endowed with left-invariant distributions. Namely, we prove that the abnormal set lies in a proper analytic subvariety. In doing so we examine several characterizat…

Mathematics - Differential Geometry0209 industrial biotechnologyPure mathematics53C17 22F50 22E25 14M17SubvarietyGroup Theory (math.GR)02 engineering and technologySard's property01 natural sciencesSet (abstract data type)020901 industrial engineering & automationAbnormal curves; Carnot groups; Endpoint map; Polarized groups; Sard's property; Sub-Riemannian geometry; Analysis; Mathematical PhysicsMathematics - Metric GeometryFOS: MathematicsPoint (geometry)Canonical mapAbnormal curves; Carnot groups Endpoint map Polarized groups Sard's property Sub-Riemannian geometry Analysis0101 mathematicsMathematics - Optimization and ControlMathematical PhysicsMathematicsApplied Mathematics010102 general mathematicsta111Polarized groupsCarnot groupLie groupEndpoint mapMetric Geometry (math.MG)Base (topology)ManifoldSub-Riemannian geometryDifferential Geometry (math.DG)Optimization and Control (math.OC)Carnot groupsAbnormal curvesMathematics - Group TheoryAnalysis
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