Torsion and the second fundamental form for distributions

The second fundamental form of Riemannian geometry is generalised to the case of a manifold with a linear connection and an integrable distribution. This bilinear form is generally not symmetric and its skew part is the torsion. The form itself is closely related to the shape map of the connection. The codimension one case generalises the traditional shape operator of Riemannian geometry

Shape maps for second order partial differential equations

We analyse the singularity formation of congruences of solutions of systems of second order PDEs via the construction of \emph{shape maps}. The trace of such maps represents a congruence volume whose collapse we study through an appropriate evolution equation, akin to Raychaudhuri's equation. We develop the necessary geometric framework on a suitable jet space in which the shape maps appear naturally associated with certain linear connections. Explicit computations are given, along with a nontrivial example

Tangent bundle geometry induced by second order partial differential equations

We show how the tangent bundle decomposition generated by a system of ordinary differential equations may be generalized to the case of a system of second order PDEs `of connection type'. Whereas for ODEs the decomposition is intrinsic, for PDEs it is necessary to specify a closed 1-form on the manifold of independent variables, together with a transverse local vector field. The resulting decomposition provides several natural curvature operators. The harmonic map equation is examined, and in this case both the 1-form and the vector field arise naturally

Integrative multiomics analysis highlights immune-cell regulatory mechanisms and shared genetic architecture for 14 immune-associated diseases and cancer outcomes

Developing functional insight into the causal molecular drivers of immunological disease is a critical challenge in genomic medicine. Here, we systematically apply Mendelian randomization (MR), genetic colocalization, immune-cell-type enrichment, and phenome-wide association methods to investigate the effects of genetically predicted gene expression on ten immune-associated diseases and four cancer outcomes. Using whole blood-derived estimates for regulatory variants from the eQTLGen consortium (n = 31,684), we constructed genetic risk scores for 10,104 genes. Applying the inverse-variance-weighted MR method transcriptome wide while accounting for linkage disequilibrium structure identified 664 unique genes with evidence of a genetically predicted effect on at least one disease outcome (p < 4.81 × 10(−5)). We next undertook genetic colocalization to investigate cell-type-specific effects at these loci by using gene expression data derived from 18 types of immune cells. This highlighted many cell-type-dependent effects, such as PRKCQ expression and asthma risk (posterior probability = 0.998), which was T cell specific. Phenome-wide analyses on 311 complex traits and endpoints allowed us to explore shared genetic architecture and prioritize key drivers of disease risk, such as CASP10, which provided evidence of an effect on seven cancer-related outcomes. Our atlas of results can be used to characterize known and novel loci in immune-associated disease and cancer susceptibility, both in terms of elucidating cell-type-dependent effects as well as dissecting shared disease pathways and pervasive pleiotropy. As an exemplar, we have highlighted several key findings in this study, although similar evaluations can be conducted via our interactive web platform

From rag picking to riches: fashion education meets textile waste

Referred to as the 'Golden Dustman' (Evans 1998) Martin Margiela's approach to sourcing and reworking vintage garments was likened to that of a Victorian ragpicker. Today, the abundance of second hand clothing donated to charity shops presents fashion designers with the opportunity to reprise Margiela's role, by considering textile waste as valuable, raw materials. Donating unwanted garments to charity is a prolific cultural practice, perceived as philanthropic and sustainable. However, donations of unwanted clothing comprise 80% fast fashion, which cannot easily be re-used, re-sold or biodegraded. Emmanuel House, a homeless charity in Nottingham have a three-tier sorting system: 1. To clothe its service users; 2. for re-sale in the charity’s shop to fund its work; 3. to be sold as 'rag' by the kilo (shipped to 3rd world countries). This conversion to cash process raises various ethical concerns. This paper reports on a social/design innovation collaboration between Emmanuel House, and Year One BA Fashion Design students, which is raising awareness of what happens to clothing donated to charities, including: the resource rich sorting process; unwanted clothing versus clothing poverty; the potential for a circular design approach at end of product lifetime; how strategic re-design can lead to innovative, suitable clothes that enhance user experience and self-esteem. Through volunteering for Emmanuel House, the students have acquired insights into homelessness and sorting charitable donations. By using their tacit knowledge of textile quality and performance, garment construction, fit and silhouette, they have identified valuable materials within existing garments. The selected items formed the basis for critically designed solutions, created using a 'deconstruction/ reconstruction' methodology, incorporating upcycling, customization, overdyeing and repair. Outcomes integrate practical details to accommodate rough sleeping and outdoor, nomadic living, including: waterproof/warm outerwear; multifunctionality/ transformability; multiple pockets for carrying/concealing items; e-textile functionality to augment light/ heat

A Group Theoretical Identification of Integrable Equations in the Li\'enard Type Equation x¨+f(x)x˙+g(x)=0\ddot{x}+f(x)\dot{x}+g(x) = 0 : Part II: Equations having Maximal Lie Point Symmetries

In this second of the set of two papers on Lie symmetry analysis of a class of Li\'enard type equation of the form $\ddot {x} + f(x)\dot {x} + g(x)= 0$, where over dot denotes differentiation with respect to time and $f(x)$ and $g(x)$ are smooth functions of their variables, we isolate the equations which possess maximal Lie point symmetries. It is well known that any second order nonlinear ordinary differential equation which admits eight parameter Lie point symmetries is linearizable to free particle equation through point transformation. As a consequence all the identified equations turn out to be linearizable. We also show that one can get maximal Lie point symmetries for the above Li\'enard equation only when $f_{xx} =0$ (subscript denotes differentiation). In addition, we discuss the linearising transformations and solutions for all the nonlinear equations identified in this paper.Comment: Accepted for publication in Journal of Mathematical Physic

Autonomous three dimensional Newtonian systems which admit Lie and Noether point symmetries

We determine the autonomous three dimensional Newtonian systems which admit Lie point symmetries and the three dimensional autonomous Newtonian Hamiltonian systems, which admit Noether point symmetries. We apply the results in order to determine the two dimensional Hamiltonian dynamical systems which move in a space of constant non-vanishing curvature and are integrable via Noether point symmetries. The derivation of the results is geometric and can be extended naturally to higher dimensions.Comment: Accepted for publication in Journal of Physics A: Math. and Theor.,13 page

Two dimensional dynamical systems which admit Lie and Noether symmetries

We prove two theorems which relate the Lie point symmetries and the Noether symmetries of a dynamical system moving in a Riemannian space with the special projective group and the homothetic group of the space respectively. The theorems are applied to classify the two dimensional Newtonian dynamical systems, which admit a Lie point/Noether symmetry. Two cases are considered, the non-conservative and the conservative forces. The use of the results is demonstrated for the Kepler - Ermakov system, which in general is non-conservative and for potentials similar to the H\`enon Heiles potential. Finally it is shown that in a FRW background with no matter present, the only scalar cosmological model which is integrable is the one for which 3-space is flat and the potential function of the scalar field is exponential. It is important to note that in all applications the generators of the symmetry vectors are found by reading the appropriate entry in the relevant tables.Comment: 25 pages, 17 table

Unequal Mass Binary Black Hole Plunges and Gravitational Recoil

We present results from fully nonlinear simulations of unequal mass binary black holes plunging from close separations well inside the innermost stable circular orbit with mass ratios q = M_1/M_2 = {1,0.85,0.78,0.55,0.32}, or equivalently, with reduced mass parameters $\eta=M_1M_2/(M_1+M_2)^2 = {0.25, 0.248, 0.246, 0.229, 0.183}$. For each case, the initial binary orbital parameters are chosen from the Cook-Baumgarte equal-mass ISCO configuration. We show waveforms of the dominant l=2,3 modes and compute estimates of energy and angular momentum radiated. For the plunges from the close separations considered, we measure kick velocities from gravitational radiation recoil in the range 25-82 km/s. Due to the initial close separations our kick velocity estimates should be understood as a lower bound. The close configurations considered are also likely to contain significant eccentricities influencing the recoil velocity.Comment: 12 pages, 5 figures, to appear in "New Frontiers" special issue of CQ