Compatible Affine Research Articles

Relativistic cosmology and intrinsic spin of matter: Results and theorems in Einstein-Cartan theory

We start by presenting the general set of structure equations for the 1+3 threading spacetime decomposition in 4 spacetime dimensions, valid for any theory of gravitation based on a metric compatible affine connection. We then apply these equations to the study of cosmological solutions of the Einstein-Cartan theory in which the matter is modeled by a perfect fluid with intrinsic spin. It is shown that the metric tensor can be described by a generic FLRW solution. However, due to the presence of torsion the Weyl tensors might not vanish. The coupling between the torsion and Weyl tensors leads to the conclusion that, in this cosmological model, the universe must either be flat or open, excluding definitely the possibility of a closed universe. In the open case, we derive a wave equation for the traceless part of the magnetic part of the Weyl tensor and show how the intrinsic spin of matter in a dynamic universe leads to the generation and emission of gravitational waves. Lastly, in this cosmological model, it is found that the torsion tensor, which has an intrinsic spin as its source, contributes to a positive accelerated expansion of the universe. Comparing the theoretical predictions of the model with the current experimental data, we conclude that torsion cannot completely replace the role of a cosmological constant.

A Small Molecule Fluorogenic Probe for the Detection of Sphingosine in Living Cells.

The single-chained sphingolipid sphingosine is an essential structural lipid and signaling molecule. Abnormal sphingosine metabolism is observed in several diseases, including cancer, diabetes, and Alzheimer's. Despite its biological importance, there is a lack of tools for detecting sphingosine in living cells. This is likely due to the broader challenge of developing highly selective and live-cell compatible affinity probes for hydrophobic lipid species. In this work, we have developed a small molecule fluorescent turn-on probe for labeling sphingosine in living cells. We demonstrate that this probe exhibits a dose-dependent response to sphingosine and is able to detect endogenous pools of sphingosine. Using our probe, we successfully detected sphingosine accumulation in cells from patients with Niemann-Pick type C1 (NPC1), a lipid transport disorder in which increased sphingosine mediates disease progression. This work provides a simple and accessible method for the detection of sphingosine and should facilitate study of this critical signaling lipid in biology and disease.

P-brane Newton–Cartan geometry

We provide a formal definition of p-brane Newton–Cartan (pNC) geometry and establish some foundational results. Our approach is the same followed in the literature for foundations of Newton–Cartan gravity. Our results provide control of aspects of pNC geometry that are otherwise unclear when using the usual gauge language of nonrelativistic theories of gravity. In particular, we obtain a set of necessary and sufficient conditions that a pNC structure must satisfy in order to admit torsion-free, compatible affine connections and determine the space formed by the latter. This is summarized in Theorem 3.1. Since pNC structures interpolate between Leibnizian structures for p = 0 and Lorentzian structures for p = d − 1 (with d being the dimension of the spacetime manifold), the present work also constitutes a generalization of results of Newton–Cartan and (pseudo-)Riemannian geometry.

Dispersionless integrable hierarchies andGL(2, ℝ) geometry

AbstractParaconformal orGL(2, ℝ) geometry on ann-dimensional manifoldMis defined by a field of rational normal curves of degreen– 1 in the projectivised cotangent bundle ℙT*M. Such geometry is known to arise on solution spaces of ODEs with vanishing Wünschmann (Doubrov–Wilczynski) invariants. In this paper we discuss yet another natural source ofGL(2, ℝ) structures, namely dispersionless integrable hierarchies of PDEs such as the dispersionless Kadomtsev–Petviashvili (dKP) hierarchy. In the latter context,GL(2, ℝ) structures coincide with the characteristic variety (principal symbol) of the hierarchy.Dispersionless hierarchies provide explicit examples of particularly interesting classes of involutiveGL(2, ℝ) structures studied in the literature. Thus, we obtain torsion-freeGL(2, ℝ) structures of Bryant [5] that appeared in the context of exotic holonomy in dimension four, as well as totally geodesicGL(2, ℝ) structures of Krynski [33]. The latter possess a compatible affine connection (with torsion) and a two-parameter family of totally geodesicα-manifolds (coming from the dispersionless Lax equations), which makes them a natural generalisation of the Einstein–Weyl geometry.Our main result states that involutiveGL(2, ℝ) structures are governed by a dispersionless integrable system whose general local solution depends on 2n– 4 arbitrary functions of 3 variables. This establishes integrability of the system of Wünschmann conditions.

The role of conformal symmetry in gravity and the standard model

In this paper we consider conformal symmetry in the context of manifolds with general affine connection. We extend the conformal transformation law of the metric to a general metric compatible affine connection, and find that it is a symmetry of both the geodesic equation and the Riemann tensor. We derive the generalised Jacobi equation and Raychaudhuri equation and show that they are both conformally invariant. Using the geodesic deviation (Jacobi) equation we analyse the behaviour of geodesics in different conformal frames. Since we find that our version of conformal symmetry is exact in classical pure Einstein's gravity, we ask whether one can extend it to the standard model. We find that it is possible to write conformal invariant Lagrangians in any dimensions for vector, fermion and scalar fields, but that such Lagrangians are only gauge invariant in four dimensions. Provided one introduces a dilaton field, gravity can be conformally coupled to matter.

Technetium-99m-labeled rituximab for use as a specific tracer of sentinel lymph node biopsy: a translational research study.

PurposeWe aimed to develop and translate a CD20-antigen-targeted radiopharmaceutical, Technetium-99 m-labeled (99mTc) rituximab, for sentinel lymph node (SLN) detection.Methods99mTc-rituximab was synthesized and tested for stability in human serum. The binding affinity to CD20 was evaluated in Raji cells by flow cytometric analysis. Biodistribution and sentinel node mapping were carried out in bal b/c mice. Eighty-five patients with breast cancer participated in this study. Dynamic sentinel lymphoscintigraphy was first assessed in 12 patients before planar lymphoscintigraphy was assessed in a larger cohort. All patients underwent sentinel lymph node biopsy (SLNB), followed by axillary lymph node dissection.ResultsThe cell-binding study showed that 99mTc-rituximab possessed compatible affinity to human CD20. In the mechanism study, 99mTc-labeled anti-mouse CD20 monoclonal antibodies could bind to mouse CD20 and accumulate in the SLN with 2.62±1.25 % of the percentage of injected activity, which could be blocked by excessive unlabeled antibody. Low uptake of non-sentinel nodes and fast clearance from the injection site were observed in the mice. Sentinel nodes were identified in 82 of 85 breast cancer patients (96.5%) by lymphoscintigraphy and SLNB. The sensitivity, specificity, and accuracy were 96.8% (30/31), 100% (51/51), and 98.8% (81/82), respectively.Conclusion99mTc-rituximab, specifically binding to CD20, met most of the requirements of an ideal sentinel mapping agent for use in clinical settings.

Phage display selection of EGFR-specific antibodies by capture-sandwich panning

Sandwich ELISA experiment requires two mutually compatible affinity reagents, typically antibodies, that are highly sensitive and specific to the target analyte in its native conformation, and whose epitopes do not overlap each other’s. Finding or developing a pair of antibodies that meet these requirements can be a challenge and many otherwise useful antibodies fail in sandwich ELISA-format analysis. In order to discover sandwich immunoassay-compatible antibodies for epidermal growth factor receptor (EGFR), we first immobilized cetuximab, a high-affinity monoclonal antibody against EGFR, on a plastic surface and affinity-captured EGFR in A431 cell lysate. Panning of a phage antibody library on the surface-captured antigen yielded a couple of antibody clones that can be paired with cetuximab and also with each other in the sandwich-format immunoassay. This provides a potentially useful strategy for the development of sandwich ELISA reagents or antibodies against antigens that are hard to purify.

Spectral averaging for trace compatible operators

In this note the notions of trace compatible operators and infin- itesimal spectral flow are introduced. We define the spectral shift function as the integral of infinitesimal spectral flow. It is proved that the spectral shift function thus defined is absolutely continuous and Krein's formula is es- tablished. Some examples of trace compatible affine spaces ofoperators are given.

Generalized Tensor Structures of Kawaguchi Spaces

We prove that, in a generalized Kawaguchi space, there exist intrinsic almost product and almost complex structures associated to a metric of this space. We derive conditions for the integrability of these structures and find compatible affine connections.

Dirac Equation in Space–Time with Torsion

The Dirac equation in a curved space–time endowed with compatible affine connection is reconsidered. After a detailed decomposition of the total action, the equation is obtained by varying with respect to the Dirac spinor and the torsion field. The result is a known Dirac-like equation with constraints that can be interpreted as the equation of a self-interacting spin 1/2 particle in curved space–time. The scheme is then translated into the language of the 2-spinor formalism of curved space–time based on the choice of a null tetrad frame. The spinorial equation so obtained coincides with the standard one in case of no torsion, while in general it remains a nonlinear equation describing a self-interacting spin 1/2 particle. The nonlinearity is produced by the interaction of the particle with its own current that remains conserved as in the free torsion case.

Affine connections and frames of reference in classical mechanics

In recent formulations of Classical Mechanics the concept of affine has been used for the representation of frames of reference, in the sense of measuring accelerations as well as other time derivatives, especially in Continuum Mechanics. In this work we give a mathematical insight into this representation. We define on the space-time manifold a set of compatible affine connections and a set of frames, and we classify these sets in a natural way. Then we establish a map between these sets and study its properties in relation to the classes defined in these sets.