Conformal Geometry Research Articles

Omnidirectional conformal microstrip array antenna with electronically beam switching capabilities for 5G applications

This article presents an omnidirectional and electronically beam switching conformal microstrip array antenna at 3.5 GHz frequency for 5G applications. The 1×2 array microstrip antenna is used as a radiating element. It is placed on a hexagonal shape-holding structure to form the conformal geometry. The radiating elements are excited by a six-faced feeding network. It consists of one 3-way power divider and two 2-way power dividers. These power dividers are serially connected and form a six-faced regular structure with 12 output ports. The output ports of the uniform power divider feed network are connected to the conformal array and provide the omnidirectional radiation pattern. To obtain the beam switching mechanism, the uniform power divider is then loaded with an RF switched power divider and connected to the conformal array. Six angular separation beams (Sector I to VI) are obtained (60°) with RF switches. The response of the feed network along with the radiating element is verified experimentally. The impedance bandwidth is well below <−10 dB and measured isolation among different output ports are below −20 dB. The simple microstrip design technique makes the antenna suitable for omnidirectional and beam switching in a large array configuration for 5G applications.

213Bi]Bi3+/[111In]In3+-neunpa-cycMSH: Theranostic Radiopharmaceutical Targeting Melanoma─Structural, Radiochemical, and Biological Evaluation.

With the emergence of [225Ac]Ac3+ as a therapeutic radionuclide for targeted α therapy (TAT), access to clinical quantities of the potent, short-lived α-emitter [213Bi]Bi3+ (t1/2 = 45.6 min) will increase over the next decade. With this in mind, the nonadentate chelator, H4neunpa-NH2, has been investigated as a ligand for chelation of [213Bi]Bi3+ in combination with [111In]In3+ as a suitable radionuclidic pair for TAT and single photon emission computed tomography (SPECT) diagnostics. Nuclear magnetic resonance (NMR) spectroscopy was utilized to assess the coordination characteristics of H4neunpa-NH2 on complexation of [natBi]Bi3+, while the solid-state structure of [natBi][Bi(neunpa-NH3)] was characterized via X-ray diffraction (XRD) studies, and density functional theory (DFT) calculations were performed to elucidate the conformational geometries of the metal complex in solution. H4neunpa-NH2 exhibited fast complexation kinetics with [213Bi]Bi3+ at RT achieving quantitative radiolabeling within 5 min at 10-8 M ligand concentration, which was accompanied by the formation of a kinetically inert complex. Two bioconjugates incorporating the melanocortin 1 receptor (MC1R) targeting peptide Nle-CycMSHhex were synthesized featuring two different covalent linkers for in vivo evaluation with [213Bi]Bi3+ and [111In]In3+. High molar activities of 7.47 and 21.0 GBq/μmol were achieved for each of the bioconjugates with [213Bi]Bi3+. SPECT/CT scans of the [111In]In3+-labeled tracer showed accumulation in the tumor over time, which was accompanied by high liver uptake and clearance via the hepatic pathway due to the high lipophilicity of the covalent linker. In vivo biodistribution studies in C57Bl/6J mice bearing B16-F10 tumor xenografts showed good tumor uptake (5.91% ID/g) at 1 h post-administration with [213Bi][Bi(neunpa-Ph-Pip-Nle-CycMSHhex)]. This study demonstrates H4neunpa-NH2 to be an effective chelating ligand for [213Bi]Bi3+ and [111In]In3+, with promising characteristics for further development toward theranostic applications.

Continuum dynamics and statistical correction of compositional heterogeneity in multivalent IDP oligomers resolved by single-particle EM

Multivalent intrinsically disordered protein (IDP) complexes are prevalent in biology and act in regulation of diverse processes, including transcription, signaling events, and the assembly and disassembly of complex macromolecular architectures. These systems pose significant challenges to structural investigation, due to continuum dynamics imparted by the IDP and compositional heterogeneity resulting from characteristic low-affinity interactions. Here, we developed a modular pipeline for automated single-particle electron microscopy (EM) distribution analysis of common but relatively understudied semi-ordered systems: ‘beads-on-a-string’ assemblies, composed of IDPs bound at multivalent sites to the ubiquitous ∼20 kDa cross-linking hub protein LC8. This approach quantifies conformational geometries and compositional heterogeneity on a single-particle basis, and statistically corrects spurious observations arising from random proximity of bound and unbound LC8. The statistical correction is generically applicable to oligomer characterization and not specific to our pipeline. Following validation, the approach was applied to the nuclear pore IDP Nup159 and the transcription factor ASCIZ. This analysis unveiled significant compositional and conformational diversity in both systems that could not be obtained from ensemble single particle EM class-averaging strategies, and new insights for exploring how these architectural properties might contribute to their physiological roles in supramolecular assembly and transcriptional regulation. We expect that this approach may be adopted to many other intrinsically disordered systems that have evaded traditional methods of structural characterization.

Conformal geometry on a class of embedded hypersurfaces in spacetimes

In this work, we study various geometric properties of embedded spacelike hypersurfaces in 1 + 1 + 2 decomposed spacetimes with a preferred spatial direction, denoted e μ , which are orthogonal to the fluid flow velocity of the spacetime and admit a proper conformal transformation. To ensure non-vanishing and positivity of the scalar curvature of the induced metric on the hypersurface, we impose that the scalar curvature of the conformal metric is non-negative and that the associated conformal factor φ satisfies , where denotes derivative along the preferred spatial direction. Firstly, it is demonstrated that such hypersurface is either of Einstein type or the spatial twist vanishes on them, and that the scalar curvature of the induced metric is constant. It is then proved that if the hypersurface is compact and of Einstein type and admits a proper conformal transformation, then these hypersurfaces must be isomorphic to the three-sphere, where we make use of some well known results on Riemannian manifolds admitting conformal transformations. If the hypersurface is not of Einstein type and have nowhere vanishing sheet expansion, we show that this conclusion fails. However, with the additional conditions that the scalar curvatures of the induced metric and the conformal metric coincide, the associated conformal factor is strictly negative and the third and higher order derivatives of the conformal factor vanish, the conclusion that the hypersurface is isomorphic to the three-sphere follows. Furthermore, additional results are obtained under the conditions that the scalar curvature of a metric conformal to the induced metric is also constant. Finally, we consider some of our results in context of locally rotationally symmetric spacetimes and show that, if the hypersurfaces are compact and not of Einstein type, then under specified conditions the hypersurface is isomorphic to the three-sphere, where we constructed explicit examples of proper conformal Killing vector fields along e μ .

Effects of the Nature of the Metal Ion, Protein and Substrate on the Catalytic Center in Matrix Metalloproteinase-1: Insights from Multilevel MD, QM/MM and QM Studies.

Matrix metalloproteinase-1 (MMP-1) is a Zn(II) dependent endopeptidase involved in the degradation of collagen, the most abundant structural protein in the extracellular matrix of connective tissues and the human body. Herein we performed a multilevel computational analysis including molecular dynamics (MD), combined quantum mechanics/molecular mechanics (QM/MM), and quantum mechanics (QM) calculations to characterize the structure and geometry of the catalytic Zn(II) within the MMP-1 protein environment in comparison to crystallographic and spectroscopic data. The substrate's removal fine-tuned impact on the conformational dynamics and geometry of the catalytic Zn(II) center was also explored. Finally, the study examined the effect of substituting catalytic Zn(II) by Co(II) on the overall structure and dynamics of the MMP-1 THP complex and specifically on the geometry of the catalytic metal center. Overall our QM/MM and QM studies were in good agreement with the MM description of the Zn(II) centers in the MD simulations.

Plasmid DNA Undergoes Two Compaction Regimes under Macromolecular Crowding.

The laser light scattering experiments were performed to explore the role of dextran (size (d): 2.6, 6.9, and 17.0 nm) in compacting the plasmids (pBS: 2.9 kbps; pCMV-Tag2B: 4.3 kbps; and pET28a: 5.3 kbps) in vitro in the volume fraction (ϕ) range 0.01 to 0.15 of the macromolecular crowder. Two compaction regimes were observed in terms of the radius of gyration (Rg) for plasmid-dextran combinations, wherein the plasmid diffusivity is governed by normal diffusion and subdiffusion, respectively. Generalized scaling, Rg ∼ ϕ-1/(1+x), where x represents the conformational geometry of plasmids, is reported. The plasmid conformation depends on the crowder's size, with larger conformational changes observed in the presence of smaller crowders. The second virial coefficient (A2) and translational diffusion coefficient (Dt) indicate that entropically driven depletion of crowders, excluded volume, and interplasmid repulsive interactions govern plasmids' conformational changes, validated herein from the scaling of Dt with molecular weight.

Einstein gravity as a gauge theory for the conformal group

General relativity in dimension n = p + q can be formulated as a gauge theory for the conformal group , along with an additional field reducing the structure group down to the Poincaré group . In this paper, we propose a new variational principle for Einstein geometry which realizes this fact. Importantly, as opposed to previous treatments in the literature, our action functional gives first order field equations and does not require supplementary constraints on gauge fields, such as torsion-freeness. Our approach is based on the ‘first order formulation’ of conformal tractor geometry. Accordingly, it provides a straightforward variational derivation of the tractor version of the Einstein equation. To achieve this, we review the standard theory of tractor geometry with a gauge theory perspective, defining the tractor bundle a priori in terms of an abstract principal bundle and providing an identification with the standard conformal tractor bundle via a dynamical soldering form. This can also be seen as a generalization of the so called Cartan–Palatini formulation of general relativity in which the ‘internal’ orthogonal group is extended to an appropriate parabolic subgroup of the conformal group.

Linear fractional group as Galois group

We compute all signatures of [Formula: see text] and [Formula: see text] which classify all orientation preserving actions of the groups [Formula: see text] and [Formula: see text] on compact, connected, orientable surfaces with orbifold genus [Formula: see text]. This classification is well-grounded in the other branches of Mathematics like topology, smooth and conformal geometry, algebraic categories, and it is also directly related to the inverse Galois problem.

PECD study of a single-conformer molecule: a critical comparison of experiment and theory.

In this work we address a specific experimental and theoretical question regarding the influence of a conformational population in the modeling of photoelectron circular dichroism (PECD) spectroscopy. In the past two decades, PECD has revealed a rich and complex phenomenology in molecular processes with unprecedented insight, especially in molecular geometry sensitivity. Since the early development of this spectroscopy, theory has pointed out the importance of conformer influence on PECD; in particular, the rotation of methyl groups was surprisingly found to be responsible for strong modulation of the PECD signal. Here, to advance understanding of the effect of rotations, we have chosen to study norcamphor, a single-conformer molecule, as a benchmark for a PECD comparison between experiment and theory at the density functional theory (DFT) and time-dependent density functional theory (TDDFT) levels. The excellent agreement between experimental data and theory for norcamphor sheds light on the influence of rotations and gives a solid explanation for the reasonable qualitative agreement in the PECD of camphor, where three methyl groups are added to the same molecular structure.

Quantitative characterization of the human retinotopic map based on quasiconformal mapping.

The retinotopic map depicts the cortical neurons' response to visual stimuli on the retina and has contributed significantly to our understanding of human visual system. Although recent advances in high field functional magnetic resonance imaging (fMRI) have made it possible to generate the in vivo retinotopic map with great detail, quantifying the map remains challenging. Existing quantification methods do not preserve surface topology and often introduce large geometric distortions to the map. In this study, we developed a new framework based on computational conformal geometry and quasiconformal Teichmüller theory to quantify the retinotopic map. Specifically, we introduced a general pipeline, consisting of cortical surface conformal parameterization, surface-spline-based cortical activation signal smoothing, and vertex-wise Beltrami coefficient-based map description. After correcting most of the violations of the topological conditions, the result was a "Beltrami coefficient map" (BCM) that rigorously and completely characterizes the retinotopic map by quantifying the local quasiconformal mapping distortion at each visual field location. The BCM provided topological and fully reconstructable retinotopic maps. We successfully applied the new framework to analyze the V1 retinotopic maps from the Human Connectome Project (n=181), the largest state of the art retinotopy dataset currently available. With unprecedented precision, we found that the V1 retinotopic map was quasiconformal and the local mapping distortions were similar across observers. The new framework can be applied to other visual areas and retinotopic maps of individuals with and without eye diseases, and improve our understanding of visual cortical organization in normal and clinical populations.

Standard Model in Weyl conformal geometry

We study the Standard Model (SM) in Weyl conformal geometry. This embedding is truly minimal with no new fields beyond the SM spectrum and Weyl geometry. The action inherits a gauged scale symmetry D(1) (known as Weyl gauge symmetry) from the underlying geometry. The associated Weyl quadratic gravity undergoes spontaneous breaking of D(1) by a geometric Stueckelberg mechanism in which the Weyl gauge field (omega _mu ) acquires mass by “absorbing” the spin-zero mode of the {tilde{R}}^2 term in the action. This mode also generates the Planck scale and the cosmological constant. The Einstein-Proca action emerges in the broken phase. In the presence of the SM, this mechanism receives corrections (from the Higgs) and it can induce electroweak (EW) symmetry breaking. The EW scale is proportional to the vev of the Stueckelberg field. The Higgs field (sigma ) has direct couplings to the Weyl gauge field (sigma ^2omega _mu omega ^mu ). The SM fermions only acquire such couplings for non-vanishing kinetic mixing of the gauge fields of D(1)times U(1)_Y. If this mixing is present, part of the mass of Z boson is not due to the usual Higgs mechanism, but to its mixing with massive omega _mu . Precision measurements of Z mass then set lower bounds on the mass of omega _mu which can be light (few TeV). In the early Universe the Higgs field can have a geometric origin, by Weyl vector fusion, and the Higgs potential can drive inflation. The dependence of the tensor-to-scalar ratio r on the spectral index n_s is similar to that in Starobinsky inflation but mildly shifted to lower r by the Higgs non-minimal coupling to Weyl geometry.

Liouville conformal field theory on even-dimensional spheres

Initiated by Polyakov in his 1981 seminal work, the study of two-dimensional Liouville conformal field theory has drawn considerable attention over the past few decades. Recent progress in the understanding of conformal geometry in dimension higher than two has naturally led to a generalization of the Polyakov formalism to higher dimensions based on conformally invariant operators: Graham–Jenne–Mason–Sparling operators and the Q-curvature. This article is dedicated to providing a rigorous construction of Liouville conformal field theory on even-dimensional spheres. This is done at the classical level in terms of a generalized uniformization problem and at the quantum level, thanks to a probabilistic construction based on log-correlated fields and Gaussian multiplicative chaos. The properties of the objects thus defined are in agreement with the ones expected in the physics literature.

Conformal boundary condition and massive gravitons in AdS/BCFT

According to Witten [1], the conformal boundary condition of gravity, which specifies the conformal geometry of the boundary and the trace of the extrinsic curvature, is elliptic and leads to well-defined perturbation theory of gravity about any classical solution. The conformal boundary condition was previously considered in [2, 3] in the context of AdS/BCFT, wherein the equation of motion of the end-of-the-world was derived and emphasized. In this paper, we investigate further other consequences of the conformal boundary condition in AdS/BCFT. We derive the boundary central charges of the holographic Weyl anomaly and show that they are exactly the same for conformal boundary condition and Dirichlet boundary condition. We analysis the metric perturbation with conformal boundary condition (CBC), Dirichlet boundary condition (DBC) and Neumann boundary condition (NBC) imposed on the end-of-the-world brane and show that they admit an interpretation as the fluctuation of the extrinsic curvature (case of CBC and DBC) and the induced metric (case of NBC) of Q respectively. In all cases, the fluctuation modes are massive, which are closely relevant to the massive island formation in the literature. Our results reveal that there are non-trivial gravitational dynamics from extrinsic curvatures on the conformal and Dirichlet branes, which may have interesting applications to the island. We also discuss, in passing, the localization of gravitons in brane world theory. We find that, contrary to NBC, the graviton for CBC/DBC is located on the brane with non-positive tension instead of non-negative tension.

Vibrational Spectroscopy of Homo- and Heterochiral Amino Acid Dimers: Conformational Landscapes.

The homo- and heterochiral protonated dimers of asparagine with serine and with valine were investigated using infrared multiple-photon dissociation (IRMPD) spectroscopy. Extensive quantum-chemical calculations were used in a three-tiered strategy to screen the conformational spaces of all four dimer species. The resulting binary structures were further grouped into five different types based on their intermolecular binding topologies and subunit configurations. For each dimer species, there are eight to fourteen final conformational geometries within a 10 kJ mol−1 window of the global minimum structure for each species. The comparison between the experimental IRMPD spectra and the simulated harmonic IR features allowed us to clearly identify the types of structures responsible for the observation. The monomeric subunits of the observed homo- and heterochiral dimers are compared to the corresponding protonated/neutral amino acid monomers observed experimentally in previous IRMDP/rotational spectroscopic studies. Possible chirality and kinetic influences on the experimental IRMPD spectra are discussed.

On Huber-type theorems in general dimensions

In this paper we present some extensions of the celebrated finite point conformal compactification theorem of Huber [20] for complete open surfaces to general dimensions based on the n-Laplace equation in conformal geometry. We are able to conclude a domain in the round sphere has to be the sphere deleted finitely many points if it can be endowed with a complete conformal metric with the negative part of the smallest Ricci curvature satisfying some integrable conditions. Our proof is based on the improvements of the Arsove-Huber-type theorem on n-superharmonic functions in our earlier work [26]. Moreover, using p-parabolicity, we push the injectivity theorem of Schoen-Yau to allow some negative curvature and therefore establish the finite point conformal compactification theorem for manifolds which have a conformal immersion into the round sphere. As a side product we establish the injectivity of conformal immersions from n-parabolicity alone, which is interesting by itself in conformal geometry.

Conformal topology optimization of multi-material ferromagnetic soft active structures using an extended level set method

Ferromagnetic soft active structures using embedded ferromagnetic particles in the soft polymer matrix can generate flexible locomotion and change configurations remotely, rapidly, and biologically friendly with an applied magnetic field. To achieve the desired motion, these soft active structures can be designed by tailoring the layouts of the ferromagnetic soft polymer. Although many magnetic soft active structures have been designed and fabricated, they are limited by the developer’s intuition and experience. Structural topology optimization has become a promising method to achieve innovative structures by optimizing the material layout, opening a new path for architecting ferromagnetic-driven active structures. Given the widespread adoption of thin-shell structures for soft robots, the extended level set method (X-LSM) and conformal geometry theory are employed to perform topology optimization of the ferromagnetic soft active structures on manifolds. The boundary evolution on a free-form 3D surface can be transferred into a 2D rectangular plane by solving a modified Hamilton–Jacobi equation weighted by conformal factors. The reconciled level set (RLS) method is firstly implemented within the X-LSM framework in this paper to enable the design of multi-material ferromagnetic soft active structures on free-from surfaces. The design objective consists of a subobjective function for kinematic requirement and a subobjective function for minimum compliance. The shape sensitivity was analyzed using the material time derivative and the adjoint variable approach. The proposed method was applied to design several single and multi-material ferromagnetic soft active structures. Two topologically optimized designs have been printed using functional 3D printing technology, or the so-called 4D printing, to physically realize soft active structures with built-in functionalities. The results of the numerical verification and experimental validation demonstrate the effectiveness of the proposed design and fabrication framework.

Unfolding conformal geometry

Conformal geometry is studied using the unfolded formulation à la Vasiliev. Analyzing the first-order consistency of the unfolded equations, we identify the content of zero-forms as the spin-two off-shell Fradkin-Tseytlin module of mathfrak{so}left(2,dright) . We sketch the nonlinear structure of the equations and explain how Weyl invariant densities, which Type-B Weyl anomaly consist of, could be systematically computed within the unfolded formulation. The unfolded equation for conformal geometry is also shown to be reduced to various on-shell gravitational systems by requiring additional algebraic constraints.

Regularized Integrals on Riemann Surfaces and Modular Forms

We introduce a simple procedure to integrate differential forms with arbitrary holomorphic poles on Riemann surfaces. It gives rise to an intrinsic regularization of such singular integrals in terms of the underlying conformal geometry. Applied to products of Riemann surfaces, this regularization scheme establishes an analytic theory for integrals over configuration spaces, including Feynman graph integrals arising from two dimensional chiral quantum field theories. Specializing to elliptic curves, we show such regularized graph integrals are almost-holomorphic modular forms that geometrically provide modular completions of the corresponding ordered A-cycle integrals.

Experimental study on the flow and heat transfer behavior of polymer solutions in the closed liquid ring vacuum pump system

The closed liquid ring vacuum pump system, which mainly includes the liquid ring vacuum pump and the heat exchanger, is widely used for gas pumping in various industries. In this paper, the energy saving effect in the liquid ring vacuum pump and the flow and heat transfer behavior in the shell-and-tube heat exchanger are investigated for aqueous solutions of rigid xanthan gum (XG) and flexible Polyacrylamide (PAM), and the effects of the concentration, Reynolds number, polymer conformation and flow geometry on the flow behavior of polymer solutions are studied. Results show that, despite the significant differences in molecular conformation and rheological properties between XG and PAM, the similar energy saving rate, about 1.5 ∼ 17.5%, is obtained in the liquid ring vacuum pump. However, in the shell-and-tube heat exchanger, XG solutions exhibit the more obvious pressure drop enhancement and heat transfer reduction than that of PAM solutions for the same concentration. Furthermore, due to the polymer shear degradation, the shaft power in the liquid ring vacuum pump decreases first and then stabilizes, conversely, the flow resistance decreases and the flow rate increases in the heat exchanger. The great difference in the flow condition between the intense rotating gas–liquid two-phase flow field and the gentle liquid pipe flow causes the different flow behavior for XG and PAM solutions.

A Tolman-like Compact Model with Conformal Geometry.

In this investigation, we study a model of a charged anisotropic compact star by assuming a relationship between the metric functions arising from a conformal symmetry. This mechanism leads to a first-order differential equation containing pressure anisotropy and the electric field. Particular forms of the electric field intensity, combined with the Tolman VII metric, are used to solve the Einstein–Maxwell field equations. New classes of exact solutions generated are expressed in terms of elementary functions. For specific parameter values based on the physical requirements, it is shown that the model satisfies the causality, stability and energy conditions. Numerical values generated for masses, radii, central densities, surface redshifts and compactness factors are consistent with compact objects such as PSR J1614-2230 and SMC X-1.