We show that an analogue of the Ball-Box Theorem for step 2, completely non-integrable bundles fr... more We show that an analogue of the Ball-Box Theorem for step 2, completely non-integrable bundles from smooth sub-Riemannian geometry hold true for a class of non-differentiable tangent subbundles that satisfy a geometric condition. In the final section of the paper we give examples of such bundles and an application to dynamical systems.
We derive some new conditions for integrability of dynamically defined C 1 invariant splittings i... more We derive some new conditions for integrability of dynamically defined C 1 invariant splittings in arbitrary dimension and co-dimension. In particular we prove that every 2-dimensional C 1 invariant decomposition on a 3-dimensional manifold satisfying a volume domination condition is uniquely integrable. In the special case of volume preserving diffeomorphisms we show that standard dynamical domination is already sufficient to guarantee unique integrability.
The rapid spread of the Omicron SARS-CoV-2 variant (B.1.1.529) resulted in international efforts ... more The rapid spread of the Omicron SARS-CoV-2 variant (B.1.1.529) resulted in international efforts to quickly assess its escape from immunity generated by vaccines and previous infections. Numerous laboratories published Omicron neutralization data as preprints and reports. The understandable limitations and variability in such rapid reporting of early results however made it difficult to make definitive statements about the data. Here, we aggregate and analyze Omicron neutralization data from 23 reporting laboratories up to 2021-12-22. There are enough data to identify multiple trends and make two definitive points. First, in twice-vaccinated individuals, titer fold drop of Omicron relative to wild type is more than 19x, likely substantially more given the number of measurements below the limit of detection of the assay. Second, out to one month post third vaccination with an mRNA vaccine, or twice vaccinated after an earlier infection, the titer fold drop to Omicron is substantially...
We give new sufficient conditions for the integrability and unique integrability of continuous ta... more We give new sufficient conditions for the integrability and unique integrability of continuous tangent sub-bundles on manifolds of arbitrary dimension, generalizing Frobenius' classical Theorem for C^1 sub-bundles. Using these conditions we derive new criteria for uniqueness of solutions to ODE's and PDE's and for the integrability of invariant bundles in dynamical systems. In particular we give a novel proof of the Stable Manifold Theorem and prove some integrability results for dynamically defined dominated splittings.
We derive some new conditions for integrability of dynamically defined C invariant splittings in ... more We derive some new conditions for integrability of dynamically defined C invariant splittings in arbitrary dimension and co-dimension. In particular we prove that every 2-dimensional C invariant decomposition on a 3-dimensional manifold satisfying a volume domination condition is uniquely integrable. In the special case of volume preserving diffeomorphisms we show that standard dynamical domination is already sufficient to guarantee unique integrability.
We show that an analogue of the Ball-Box Theorem holds true for a class of corank 1, non-differen... more We show that an analogue of the Ball-Box Theorem holds true for a class of corank 1, non-differentiable tangent subbundles that satisfy a geometric condition. In the final section of the paper we give examples of such bundles and an application to dynamical systems.
We investigate the integrability of two-dimensional invariant distributions (tangent sub-bundles)... more We investigate the integrability of two-dimensional invariant distributions (tangent sub-bundles) which arise naturally in the context of dynamical systems on 3-manifolds. In particular, we prove unique integrability of dynamically dominated and volume-dominated Lipschitz continuous invariant decompositions as well as distributions with some other regularity conditions.
We show that an analogue of the Ball-Box Theorem for step 2, completely non-integrable bundles fr... more We show that an analogue of the Ball-Box Theorem for step 2, completely non-integrable bundles from smooth sub-Riemannian geometry hold true for a class of non-differentiable tangent subbundles that satisfy a geometric condition. In the final section of the paper we give examples of such bundles and an application to dynamical systems.
We derive some new conditions for integrability of dynamically defined C 1 invariant splittings i... more We derive some new conditions for integrability of dynamically defined C 1 invariant splittings in arbitrary dimension and co-dimension. In particular we prove that every 2-dimensional C 1 invariant decomposition on a 3-dimensional manifold satisfying a volume domination condition is uniquely integrable. In the special case of volume preserving diffeomorphisms we show that standard dynamical domination is already sufficient to guarantee unique integrability.
The rapid spread of the Omicron SARS-CoV-2 variant (B.1.1.529) resulted in international efforts ... more The rapid spread of the Omicron SARS-CoV-2 variant (B.1.1.529) resulted in international efforts to quickly assess its escape from immunity generated by vaccines and previous infections. Numerous laboratories published Omicron neutralization data as preprints and reports. The understandable limitations and variability in such rapid reporting of early results however made it difficult to make definitive statements about the data. Here, we aggregate and analyze Omicron neutralization data from 23 reporting laboratories up to 2021-12-22. There are enough data to identify multiple trends and make two definitive points. First, in twice-vaccinated individuals, titer fold drop of Omicron relative to wild type is more than 19x, likely substantially more given the number of measurements below the limit of detection of the assay. Second, out to one month post third vaccination with an mRNA vaccine, or twice vaccinated after an earlier infection, the titer fold drop to Omicron is substantially...
We give new sufficient conditions for the integrability and unique integrability of continuous ta... more We give new sufficient conditions for the integrability and unique integrability of continuous tangent sub-bundles on manifolds of arbitrary dimension, generalizing Frobenius' classical Theorem for C^1 sub-bundles. Using these conditions we derive new criteria for uniqueness of solutions to ODE's and PDE's and for the integrability of invariant bundles in dynamical systems. In particular we give a novel proof of the Stable Manifold Theorem and prove some integrability results for dynamically defined dominated splittings.
We derive some new conditions for integrability of dynamically defined C invariant splittings in ... more We derive some new conditions for integrability of dynamically defined C invariant splittings in arbitrary dimension and co-dimension. In particular we prove that every 2-dimensional C invariant decomposition on a 3-dimensional manifold satisfying a volume domination condition is uniquely integrable. In the special case of volume preserving diffeomorphisms we show that standard dynamical domination is already sufficient to guarantee unique integrability.
We show that an analogue of the Ball-Box Theorem holds true for a class of corank 1, non-differen... more We show that an analogue of the Ball-Box Theorem holds true for a class of corank 1, non-differentiable tangent subbundles that satisfy a geometric condition. In the final section of the paper we give examples of such bundles and an application to dynamical systems.
We investigate the integrability of two-dimensional invariant distributions (tangent sub-bundles)... more We investigate the integrability of two-dimensional invariant distributions (tangent sub-bundles) which arise naturally in the context of dynamical systems on 3-manifolds. In particular, we prove unique integrability of dynamically dominated and volume-dominated Lipschitz continuous invariant decompositions as well as distributions with some other regularity conditions.
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