Separation equations for 2D superintegrable systems on constant curvature spaces

Second-order conformal quantum superintegrable systems in two dimensions are Laplace equations on a manifold with an added scalar potential and three independent 2nd order conformal symmetry operators. They encode all the information about 2D Helmholtz or time-independent Schrödinger superintegrable systems in an efficient manner: each of these systems admits a quadratic symmetry algebra (not usually a Lie algebra) and is multiseparable. We study the separation equations for the systems as a family rather than separate cases. We show that the separation equations comprise all of the various types of hypergeometric and Heun equations in full generality. In particular, they yield all of the 1D Schrödinger exactly solvable (ES) and quasi-exactly solvable (QES) systems related to the Heun operator. We focus on complex constant curvature spaces and show explicitly that there are eight pairs of Laplace separation types and these types account for all separable coordinates on the 20 flat space and 9 2-sphere Helmholtz superintegrable systems, including those for the constant potential case. The different systems are related by Stäckel transforms, by the symmetry algebras and by Böcher contractions of the conformal algebra so(4, C) to itself, which enables all systems to be derived from a single one: the generic potential on the complex 2-sphere. This approach facilitates a unified view of special function theory, incorporating hypergeometric and Heun functions in full generality

Classical and Quantum Integrable Systems in \wt{\gr{gl}}(2)^{+*} and Separation of Variables

Classical integrable Hamiltonian systems generated by elements of the Poisson commuting ring of spectral invariants on rational coadjoint orbits of the loop algebra \wt{\gr{gl}}^{+*}(2,{\bf R}) are integrated by separation of variables in the Hamilton-Jacobi equation in hyperellipsoidal coordinates. The canonically quantized systems are then shown to also be completely integrable and separable within the same coordinates. Pairs of second class constraints defining reduced phase spaces are implemented in the quantized systems by choosing one constraint as an invariant, and interpreting the other as determining a quotient (i.e., by treating one as a first class constraint and the other as a gauge condition). Completely integrable, separable systems on spheres and ellipsoids result, but those on ellipsoids require a further modification of order \OO(\hbar^2) in the commuting invariants in order to assure self-adjointness and to recover the Laplacian for the case of free motion. For each case - in the ambient space ${\bf R}^{n}$, the sphere and the ellipsoid - the Schr\"odinger equations are completely separated in hyperellipsoidal coordinates, giving equations of generalized Lam\'e type.Comment: 28 page

SYNGAP1 encephalopathy:A distinctive generalized developmental and epileptic encephalopathy

Objective To delineate the epileptology, a key part of the SYNGAP1 phenotypic spectrum, in a large patient cohort. Methods Patients were recruited via investigators' practices or social media. We included patients with (likely) pathogenic SYNGAP1 variants or chromosome 6p21.32 microdeletions incorporating SYNGAP1. We analyzed patients' phenotypes using a standardized epilepsy questionnaire, medical records, EEG, MRI, and seizure videos. Results We included 57 patients (53% male, median age 8 years) with SYNGAP1 mutations (n = 53) or microdeletions (n = 4). Of the 57 patients, 56 had epilepsy: generalized in 55, with focal seizures in 7 and infantile spasms in 1. Median seizure onset age was 2 years. A novel type of drop attack was identified comprising eyelid myoclonia evolving to a myoclonic-atonic (n = 5) or atonic (n = 8) seizure. Seizure types included eyelid myoclonia with absences (65%), myoclonic seizures (34%), atypical (20%) and typical (18%) absences, and atonic seizures (14%), triggered by eating in 25%. Developmental delay preceded seizure onset in 54 of 56 (96%) patients for whom early developmental history was available. Developmental plateauing or regression occurred with seizures in 56 in the context of a developmental and epileptic encephalopathy (DEE). Fifty-five of 57 patients had intellectual disability, which was moderate to severe in 50. Other common features included behavioral problems (73%); high pain threshold (72%); eating problems, including oral aversion (68%); hypotonia (67%); sleeping problems (62%); autism spectrum disorder (54%); and ataxia or gait abnormalities (51%). Conclusions SYNGAP1 mutations cause a generalized DEE with a distinctive syndrome combining epilepsy with eyelid myoclonia with absences and myoclonic-atonic seizures, as well as a predilection to seizures triggered by eating.</p

Building on Progress Towards Fair Access : annual report 2019

Supplementary Table 3: Anti-epileptic drug use in patients with SYNGAP1 mutations or deletion

Bôcher and Abstract Contractions of 2nd Order Quadratic Algebras

Quadratic algebras are generalizations of Lie algebras which include the symmetry algebras of 2nd order superintegrable systems in 2 dimensions as special cases. The superintegrable systems are exactly solvable physical systems in classical and quantum mechanics. Distinct superintegrable systems and their quadratic algebras can be related by geometric contractions, induced by Bôcher contractions of the conformal Lie algebra so(4,C) to itself. In this paper we give a precise definition of Bôcher contractions and show how they can be classified. They subsume well known contractions of e(2,C) and so(3,C) and have important physical and geometric meanings, such as the derivation of the Askey scheme for obtaining all hypergeometric orthogonal polynomials as limits of Racah/Wilson polynomials. We also classify abstract nondegenerate quadratic algebras in terms of an invariant that we call a canonical form. We describe an algorithm for finding the canonical form of such algebras. We calculate explicitly all canonical forms arising from quadratic algebras of 2D nondegenerate superintegrable systems on constant curvature spaces and Darboux spaces. We further discuss contraction of quadratic algebras, focusing on those coming from superintegrable systems

Data from: SYNGAP1 encephalopathy: a distinctive generalized developmental and epileptic encephalopathy

Objective. To delineate the epileptology, a key part of the SYNGAP1 phenotypic spectrum, in a large patient cohort. Methods. Patients were recruited via investigators’ practices or social media. We included patients with (likely) pathogenic SYNGAP1 variants or chromosome 6p21.32 microdeletions incorporating SYNGAP1. We analysed patients’ phenotypes using a standardized epilepsy questionnaire, medical records, EEG, MRI and seizure videos. Results. We included 57 patients (53% male, median age 8 years) with SYNGAP1 mutations (n=53) or microdeletions (n=4). 56/57 patients had epilepsy: generalized in 55, with focal seizures in seven and infantile spasms in one. Median seizure onset age was 2 years. A novel type of drop attack was identified comprising eyelid myoclonia evolving to a myoclonic-atonic (n=5) or atonic (n=8) seizure. Seizure types included eyelid myoclonia with absences (65%), myoclonic seizures (34%), atypical (20%) and typical (18%) absences and atonic seizures (14%), triggered by eating in 25%. Developmental delay preceded seizure onset in 54/56 (96%) patients of whom early developmental history was available. Developmental plateauing or regression occurred with seizures in 56 in the context of developmental and epileptic encephalopathies (DEEs). 55/57 patients had intellectual disability, which was moderate to severe in 50. Other common features included behavioural problems (73%), high pain threshold (72%), eating problems including oral aversion (68%), hypotonia (67%), sleeping problems (62%), autism spectrum disorder (54%) and ataxia or gait abnormalities (51%). Conclusions. SYNGAP1 mutations cause a generalized DEE with a distinctive syndrome combining epilepsy with eyelid myoclonia and myoclonic-atonic seizures, and predilection to seizures triggered by eating

Data from: SYNGAP1 encephalopathy: a distinctive generalized developmental and epileptic encephalopathy

Objective. To delineate the epileptology, a key part of the SYNGAP1 phenotypic spectrum, in a large patient cohort. Methods. Patients were recruited via investigators’ practices or social media. We included patients with (likely) pathogenic SYNGAP1 variants or chromosome 6p21.32 microdeletions incorporating SYNGAP1. We analysed patients’ phenotypes using a standardized epilepsy questionnaire, medical records, EEG, MRI and seizure videos. Results. We included 57 patients (53% male, median age 8 years) with SYNGAP1 mutations (n=53) or microdeletions (n=4). 56/57 patients had epilepsy: generalized in 55, with focal seizures in seven and infantile spasms in one. Median seizure onset age was 2 years. A novel type of drop attack was identified comprising eyelid myoclonia evolving to a myoclonic-atonic (n=5) or atonic (n=8) seizure. Seizure types included eyelid myoclonia with absences (65%), myoclonic seizures (34%), atypical (20%) and typical (18%) absences and atonic seizures (14%), triggered by eating in 25%. Developmental delay preceded seizure onset in 54/56 (96%) patients of whom early developmental history was available. Developmental plateauing or regression occurred with seizures in 56 in the context of developmental and epileptic encephalopathies (DEEs). 55/57 patients had intellectual disability, which was moderate to severe in 50. Other common features included behavioural problems (73%), high pain threshold (72%), eating problems including oral aversion (68%), hypotonia (67%), sleeping problems (62%), autism spectrum disorder (54%) and ataxia or gait abnormalities (51%). Conclusions. SYNGAP1 mutations cause a generalized DEE with a distinctive syndrome combining epilepsy with eyelid myoclonia and myoclonic-atonic seizures, and predilection to seizures triggered by eating

Drivers of political parties’ climate policy preferences: lessons from Denmark and Ireland

Political parties are important actors in domestic climate politics. What drives variation in parties’ climate policy preferences? To contribute to a growing literature on the party politics of climate change, we focus on the roles of public opinion, party competition, and parties’ traditional policy preferences in shaping parties’ climate policy preferences in Denmark and Ireland. In case studies that draw on in-depth interviews with policy practitioners, we show how parties respond to public opinion, accommodate issue-owners, and are powerfully constrained and enabled by their existing preferences. These mechanisms also help to explain different responses on climate policy across the left-right spectrum. Competition between mainstream parties is particularly powerful, but can constrain as much as it enables ‘greener’ climate policy preferences. While climate change may be a distinctive problem, the party politics of climate change features similar incentives and constraints as other domains