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Let $\mathbb H$ denote the three-dimensional Heisenberg group. In this paper, we study vertical curves in $\mathbb H$ and fibers of maps $\mathbb H \to \mathbb R^2$ from a metric perspective. We say that a set in $\mathbb H$ is a vertical curve if it satisfies a cone condition with respect to a homogeneous cone with axis $\langle Z \rangle$, the center of $\mathbb H$. This is analogous to the cone condition used to define intrinsic Lipschitz graphs. In the first part of the paper, we prove that connected vertical curves are locally biH\"older equivalent to intervals. We also show that the class of vertical curves coincides with the class of intersections of intrinsic Lipschitz graphs satisfying a transversality condition. Unlike intrinsic Lipschitz graphs, the Hausdorff dimension of a vertical curve can vary; we construct vertical curves with Hausdorff dimension either strictly larger or strictly smaller than 2. Consequently, there are intersections of intrinsic Lipschitz graphs with Hausdorff dimension either strictly larger or strictly smaller than 2. In the second part of the paper, we consider smooth functions $\beta$ from the unit ball $B$ in $\mathbb H$ to $\mathbb R^2$. We show that, in contrast to the situation in Euclidean space, there are maps such that $\beta$ is arbitrarily close to the projection $\pi$ from $\mathbb H$ to the horizontal plane, but the average $\mathcal{H}^2$ measure of a fiber of $\beta$ in $B$ is arbitrarily small., Comment: 29 pages. 2 figures. Comments are welcome!

The presence of defects in transition metal dichalcogenides (TMDs) can lead to dramatic local changes in their properties which are of interest for a range of technologies including quantum security devices, hydrogen production, and energy storage. It is therefore essential to be able to study these materials in their native environments, including ambient conditions. Here we report single atom resolution imaging of atomic defects in MoS2, WSe2 and WS2 monolayers carried out in ambient conditions using conductive atomic force microscopy (C-AFM). By comparing measurements from a range of TMDs we use C-AFM to chemically identify the most likely atomic species for the defects observed and quantify their prevalence on each material, identifying oxygen chalcogen substitutions and transition metal substitutions as the most likely, and most common, defect types. Moreover, we demonstrate that C-AFM operated in ambient environments can resolve subtle changes in electronic structure with atomic resolution, which we apply to WSe2 monolayers doped using a nitrogen plasma, demonstrating the capability of C-AFM to resolve electronic, and chemical-specific, details at the atomic scale.

In this paper, we study how the cohomology of nilpotent groups is affected by Lipschitz maps. We show that, given a smooth Lipschitz map $f$ between two simply-connected nilpotent Lie groups $G$ and $H$, there is a map $\psi$ that induces an ergodic measure on the space of functions from $G$ to $H$. We call such maps ergodic maps. We show that when $\psi$ is an ergodic map, the pullback $\psi^*\omega$ of a differential form $\omega$ admits a well-defined amenable average $\overline{\psi^{*}}\omega$, and $\overline{\psi^*}$ is a homomorphism of cohomology algebras. In the case that $f$ is a quasi-isometry, the ergodic map $\psi$ is also a quasi-isometry, and $\overline{\psi^*}$ is an isomorphism. This lets us generalize and provide a simplified, self-contained proof of the theorem due to Shalom, Sauer, and Gotfredsen-Kyed that quasi-isometric nilpotent groups have isomorphic cohomology algebras., Comment: 20 pages. Minor changes in the title, abstract, introduction, and acknowledgment

We construct $d$-dimensional polyhedral chains such that the distribution of tangent planes is close to a prescribed measure on the Grassmannian and the chains are either cycles (if the prescribed measure is centered) or their boundary is the boundary of a unit $d$-cube (if the barycenter of the prescribed measure, considered as a measure on $\bigwedge^d \mathbb{R}^n$, is a simple $d$-vector). Such fillings were first proved to exist by Burago and Ivanov [Geom. funct. anal., 2004]; our work gives an explicit construction. Furthermore, in the case that the measure on the Grassmannian is supported on the set of positively oriented $d$-planes, we can construct fillings that are Lipschitz multigraphs. We apply this construction to prove that for anisotropic surface energies, ellipticity for Lipschitz multivalued functions is equivalent to polyconvexity and to show that strict polyconvexity is necessary for the atomic condition to hold.

The translation of AI-generated brain metastases (BM) segmentation into clinical practice relies heavily on diverse, high-quality annotated medical imaging datasets. The BraTS-METS 2023 challenge has gained momentum for testing and benchmarking algorithms using rigorously annotated internationally compiled real-world datasets. This study presents the results of the segmentation challenge and characterizes the challenging cases that impacted the performance of the winning algorithms. Untreated brain metastases on standard anatomic MRI sequences (T1, T2, FLAIR, T1PG) from eight contributed international datasets were annotated in stepwise method: published UNET algorithms, student, neuroradiologist, final approver neuroradiologist. Segmentations were ranked based on lesion-wise Dice and Hausdorff distance (HD95) scores. False positives (FP) and false negatives (FN) were rigorously penalized, receiving a score of 0 for Dice and a fixed penalty of 374 for HD95. Eight datasets comprising 1303 studies were annotated, with 402 studies (3076 lesions) released on Synapse as publicly available datasets to challenge competitors. Additionally, 31 studies (139 lesions) were held out for validation, and 59 studies (218 lesions) were used for testing. Segmentation accuracy was measured as rank across subjects, with the winning team achieving a LesionWise mean score of 7.9. Common errors among the leading teams included false negatives for small lesions and misregistration of masks in space.The BraTS-METS 2023 challenge successfully curated well-annotated, diverse datasets and identified common errors, facilitating the translation of BM segmentation across varied clinical environments and providing personalized volumetric reports to patients undergoing BM treatment.

We prove that in the first Heisenberg group, unlike Euclidean spaces and higher dimensional Heisenberg groups, the best possible exponent for the strong geometric lemma for intrinsic Lipschitz graphs is $4$ instead of $2$. Combined with earlier work from arXiv:2004.11447 and arXiv:2207.03013, our result completes the proof of the strong geometric lemma in Heisenberg groups. One key tool in our proof, and possibly of independent interest, is a suitable refinement of the foliated coronizations which first appeared in arXiv:2004.12522., Comment: 29 pages

Understanding the stress transfer mechanisms from a polymer matrix to two-dimensional reinforcements is essential for the preparation of high performance nanocomposites. In this study, the interfacial stress transfer from a flexible polymer substrate to monolayer and few-layer molybdenum disulfide MoS2 under tension has been investigated. Layer-dependent and strain-dependent photoluminescence (PL) spectroscopy were used to examine the stress transfer efficiency., Comment: 29 pages

We prove that the Heisenberg Riesz transform is $L_2$--unbounded on a family of intrinsic Lipschitz graphs in the first Heisenberg group $\mathbb{H}$. We construct this family by combining a method from \cite{NY2} with a stopping time argument, and we establish the $L_2$--unboundedness of the Riesz transform by introducing several new techniques to analyze singular integrals on intrinsic Lipschitz graphs. These include a formula for the Riesz transform in terms of a singular integral on a vertical plane and bounds on the flow of singular integrals that arises from a perturbation of a graph. On the way, we use our construction to show that the strong geometric lemma fails in $\mathbb{H}$ for all exponents in $[2,4)$. Our results are in stark contrast to two fundamental results in Euclidean harmonic analysis and geometric measure theory: Lipschitz graphs in $\mathbb{R}^n$ satisfy the strong geometric lemma, and the $m$--Riesz transform is $L_2$--bounded on $m$--dimensional Lipschitz graphs in $\mathbb{R}^n$ for $m\in (0,n)$.

This paper describes the use of the Murchison Widefield Array, a low-frequency radio telescope at a radio-quiet Western Australian site, as a radar receiver forming part of a continent-spanning multistatic radar network for the surveillance of space. This paper details the system geometry employed, the orbit-specific radar signal processing, and the orbit determination algorithms necessary to ensure resident space objects are detected, tracked, and propagated. Finally, the paper includes the results processed after a short collection campaign utilising several FM radio transmitters across the country, up to a maximum baseline distance of over 2500 km. The results demonstrate the Murchison Widefield Array is able to provide widefield and persistent coverage of objects in low Earth orbit.

Purpose: A core component of advancing cancer treatment research is assessing response to therapy. Doing so by hand, for example as per RECIST or RANO criteria, is tedious, time-consuming, and can miss important tumor response information; most notably, they exclude non-target lesions. We wish to assess change in a holistic fashion that includes all lesions, obtaining simple, informative, and automated assessments of tumor progression or regression. Due to often low patient enrolments in clinical trials, we wish to make response assessments with small training sets. Deep neuroevolution (DNE) can produce radiology artificial intelligence (AI) that performs well on small training sets. Here we use DNE for function approximation that predicts progression versus regression of metastatic brain disease. Methods: We analyzed 50 pairs of MRI contrast-enhanced images as our training set. Half of these pairs, separated in time, qualified as disease progression, while the other 25 images constituted regression. We trained the parameters of a relatively small CNN via mutations that consisted of random CNN weight adjustments and mutation fitness. We then incorporated the best mutations into the next generations CNN, repeating this process for approximately 50,000 generations. We applied the CNNs to our training set, as well as a separate testing set with the same class balance of 25 progression and 25 regression images. Results: DNE achieved monotonic convergence to 100% training set accuracy. DNE also converged monotonically to 100% testing set accuracy. Conclusion: DNE can accurately classify brain-metastatic disease progression versus regression. Future work will extend the input from 2D image slices to full 3D volumes, and include the category of no change. We believe that an approach such as our could ultimately provide a useful adjunct to RANO/RECIST assessment.

We prove that if a quasiconvex subset $X$ of a metric space $Y$ has finite Nagata dimension and is Lipschitz $k$-connected or admits Euclidean isoperimetric inequalities up to dimension $k$ for some $k$ then $X$ is isoperimetrically undistorted in $Y$ up to dimension $k+1$. This generalizes and strengthens a recent result of the third named author and has several consequences and applications. It yields for example that in spaces of finite Nagata dimension, Lipschitz connectedness implies Euclidean isoperimetric inequalities, and Euclidean isoperimetric inequalities imply coning inequalities. It furthermore allows us to prove an analog of the Federer-Fleming deformation theorem in spaces of finite Nagata dimension admitting Euclidean isoperimetric inequalities.

$\textbf{Purpose}$: To develop a clinical CEST MR fingerprinting (CEST-MRF) method for brain tumor quantification using EPI acquisition and deep learning reconstruction. $\textbf{Methods}$: A CEST-MRF pulse sequence originally designed for animal imaging was modified to conform to hardware limits on clinical scanners while keeping scan time $\leq$ 2 minutes. Quantitative MRF reconstruction was performed using a deep reconstruction network (DRONE) to yield the water relaxation and chemical exchange parameters. The feasibility of the 6 parameter DRONE reconstruction was tested in simulations in a digital brain phantom. A healthy subject was scanned with the CEST-MRF sequence, conventional MRF and CEST sequences for comparison. Reproducibility was assessed via test-retest experiments and the concordance correlation coefficient (CCC) calculated for white matter (WM) and grey matter (GM). The clinical utility of CEST-MRF was demonstrated in 4 patients with brain metastases in comparison to standard clinical imaging sequences. Tumors were segmented into edema, solid core and necrotic core regions and the CEST-MRF values compared to the contra-lateral side. $\textbf{Results}$: The DRONE reconstruction of the digital phantom yielded a normalized RMS error of $\leq$ 7% for all parameters. The CEST-MRF parameters were in good agreement with those from conventional MRF and CEST sequences and previous studies. The mean CCC for all 6 parameters was 0.98$\pm$0.01 in WM and 0.98$\pm$0.02 in GM. The CEST-MRF values in nearly all tumor regions were significantly different (P=0.05) from each other and the contra-lateral side. $\textbf{Conclusion}$: Combination of EPI readout and deep learning reconstruction enabled fast, accurate and reproducible CEST-MRF in brain tumors., Comment: 9 figures, 1 table

Silicon-based tracking detectors have been used in several important applications, such as in cancer therapy using particle beams, and for the discovery of new elementary particles at the Large Hadron Collider at CERN. III-V semiconductor materials are an attractive alternative to silicon for this application, as they have some superior physical properties. They could meet the demands for fast timing detectors allowing time-of-flight measurements with ps resolution while being radiation tolerant and cost-efficient. As a material with a larger density, higher atomic number Z and much higher electron mobility than silicon, GaAs exhibits faster signal collection and a larger signal per {\mu}m of sensor thickness. In this work, we report on the fabrication of n-in-n GaAs thin-film devices intended to serve next-generation high-energy particle tracking detectors. Molecular beam epitaxy (MBE) was used to grow high-quality GaAs films with doping levels sufficiently low to achieve full depletion for detectors with an active thickness of 10 {\mu}m. The signal collection speed of the detector structures was assessed using the transient current technique (TCT). To elucidate the structural properties of the detector, Kelvin probe force microscopy (KPFM) was used, which confirmed the formation of the junction in the detector and revealed residual doping in the intrinsic layer. Our results suggest that GaAs thin films are suitable candidates to achieve thin and radiation-tolerant tracking detectors., Comment: 15 pages, 12 figures

The BraTS 2021 challenge celebrates its 10th anniversary and is jointly organized by the Radiological Society of North America (RSNA), the American Society of Neuroradiology (ASNR), and the Medical Image Computing and Computer Assisted Interventions (MICCAI) society. Since its inception, BraTS has been focusing on being a common benchmarking venue for brain glioma segmentation algorithms, with well-curated multi-institutional multi-parametric magnetic resonance imaging (mpMRI) data. Gliomas are the most common primary malignancies of the central nervous system, with varying degrees of aggressiveness and prognosis. The RSNA-ASNR-MICCAI BraTS 2021 challenge targets the evaluation of computational algorithms assessing the same tumor compartmentalization, as well as the underlying tumor's molecular characterization, in pre-operative baseline mpMRI data from 2,040 patients. Specifically, the two tasks that BraTS 2021 focuses on are: a) the segmentation of the histologically distinct brain tumor sub-regions, and b) the classification of the tumor's O[6]-methylguanine-DNA methyltransferase (MGMT) promoter methylation status. The performance evaluation of all participating algorithms in BraTS 2021 will be conducted through the Sage Bionetworks Synapse platform (Task 1) and Kaggle (Task 2), concluding in distributing to the top ranked participants monetary awards of $60,000 collectively., Comment: 19 pages, 2 figures, 1 table

In this paper, we give a necessary and sufficient condition for a graphical strip in the Heisenberg group $\mathbb{H}$ to be area-minimizing in the slab $\{-1

The resistance to failure through tearing is a crucial mechanical property for the application of different elastomers. In this work, graphene nanoplatelets (GNPs) were introduced into a fluoroelastomer (FKM) matrix with the aim of improving its tear resistance. The fracture energy through tearing was evaluated using the pure shear test. It was found that the tearing energy increased linearly with the volume fraction of the GNPs. At the maximum GNP content, the tearing resistance was 3 times higher, suggesting efficient toughening from the GNPs. Theoretical analysis of the micromechanics was conducted by considering debonding and pull-out of the nanoplatelets as possible toughening mechanisms. It was determined quantitatively that the main toughening mechanism was debonding of the interface rather than pull-out. The formation of cavities at flake ends during the deformation, as confirmed by scanning electron microscopy, was found to contribute to the remarkably high interfacial debonding energy (~1 kJ/m2).

Minimal surfaces in $\mathbb{R}^n$ can be locally approximated by graphs of harmonic functions, i.e., functions that are critical points of the Dirichlet energy, but no analogous theorem is known for $H$-minimal surfaces in the three-dimensional Heisenberg group $\mathbb{H}$, which are known to have singularities. In this paper, we introduce a definition of intrinsic Dirichlet energy for surfaces in $\mathbb{H}$ and study the critical points of this energy, which we call contact harmonic graphs. Nearly flat regions of $H$-minimal surfaces can often be approximated by such graphs. We give a calibration condition for an intrinsic Lipschitz graph to be energy-minimizing, construct energy-minimizing graphs with a variety of singularities, and prove a first variation formula for the energy of intrinsic Lipschitz graphs and piecewise smooth intrinsic graphs., Comment: 37 pages, 3 figures

High-performance sealing materials that can guard key components against high pressure gases and liquid chemicals while withstanding mechanical deformation are of utmost importance in a number of industries. In this present work, graphene nanoplatelets (GNPs) were introduced into a fluoroelastomer (FKM) matrix to improve its mechanical and barrier properties and test its suitability for sealing applications. Nanocomposites filled with different loadings of GNPs were prepared and compared with their counterparts loaded with carbon black (CB). GNPs were dispersed homogeneously with a high degree of in-plane alignment. The tensile and barrier properties of the FKM were improved significantly by the addition of GNPs. Micromechanical modelling based on the shear-lag/rule-of-mixtures theory was used to analyse the reinforcing efficiency of the GNPs. Upon the addition of the GNPs, the elastomer was able to swell anisotropically in liquids, a fact that can be used to tune the swelling properties for sealing applications. In terms of gas permeability, a modification of the well-established Nielsen's theory was introduced to analyse the CO2 permeability for the bulk composite samples. The significantly improved mechanical, thermal and barrier properties at relatively low filler loadings, reveal that the FKM/GNP nanocomposites produced are very promising for use in advanced sealing applications.

There has been a massive growth in the study of transition metal dichalcogenides (TMDs) over the past decade, based upon their interesting and unusual electronic, optical and mechanical properties, such as tuneable and strain-dependent bandgaps. Tungsten disulfide (WS2), as a typical example of TMDs, has considerable potential in applications such as strain engineered devices and the next generation multifunctional polymer nanocomposites. However, controlling the strain, or more practically, monitoring the strain in WS2 and the associated micromechanics have not been so well studied. Both photoluminescence spectroscopy (PL) and Raman spectroscopy have been proved to be effective but PL cannot be employed to characterise multilayer TMDs while it is difficult for Raman spectroscopy to reveal the band structure. In this present study, photoluminescence and Raman spectroscopy have been combined to monitor the strain distribution and stress transfer of monolayer WS2 on a flexible polymer substrate and in polymer nanocomposites. It is demonstrated that WS2 still follows continuum mechanics on the microscale and that strain generates a non-uniform bandgap distribution even in a single WS2 flake through a simple strain engineering. It is shown that these flakes could be useful in optoelectonic applications as they become micron-sized PL emitters with a band gap that can be tuned by the application of external strain to the substrate. The analysis of strain distributions using Raman spectroscopy is further extended to thin-film few-layer WS2 polymer nanocomposites where it is demonstrated that the stress can be transferred effectively to WS2 flakes. The relationship between the mechanical behaviour of single monolayer WS2 flakes and that of few-layer flakes in bulk composites is investigated.

Most polymers are long-lasting and produced from monomers derived from fossil fuel sources. Bio-based and/or biodegradable plastics have been proposed as a sustainable alternative. Amongst those available, polyhydroxyalkanoate (PHA) shows great potential across a large variety of applications but is currently limited to packaging, cosmetics and tissue engineering due to its relatively poor physical properties. An expansion of its uses can be accomplished by developing nanocomposites where PHAs are used as the polymer matrix. Herein, a PHA biopolyester was melt blended with graphene nanoplatelets (GNPs) or with a 1:1 hybrid mixture of GNPs and carbon nanofibers (CNFs). The resulting nanocomposites exhibited enhanced thermal stability while their Young's modulus roughly doubled compared to pure PHA. The hybrid nanocomposites percolated electrically at lower nanofiller loadings compared to the GNP-PHA system. The electrical conductivity at 15 wt.% loading was ~ 6 times higher than the GNP-based sample. As a result, the electromagnetic interference shielding performance of the hybrid material was around 50% better than the pure GNPs nanocomposites, exhibiting shielding effectiveness above 20 dB, which is the threshold for common commercial applications. The thermal conductivity increased significantly for both types of bio-nanocomposites and reached values around 5 W K-1 m-1 with the hybrid-based material displaying the best performance. Considering the solvent-free and industrially compatible production method, the proposed multifunctional materials are promising to expand the range of application of PHAs and increase the environmental sustainability of the plastic and plastic electronics industry., Comment: 26 pages

We prove that the $L_4$ norm of the vertical perimeter of any measurable subset of the $3$-dimensional Heisenberg group $\mathbb{H}$ is at most a universal constant multiple of the (Heisenberg) perimeter of the subset. We show that this isoperimetric-type inequality is optimal in the sense that there are sets for which it fails to hold with the $L_4$ norm replaced by the $L_q$ norm for any $q<4$. This is in contrast to the $5$-dimensional setting, where the above result holds with the $L_4$ norm replaced by the $L_2$ norm. The proof of the aforementioned isoperimetric inequality introduces a new structural methodology for understanding the geometry of surfaces in $\mathbb{H}$. In previous work (2017) we showed how to obtain a hierarchical decomposition of Ahlfors-regular surfaces into pieces that are approximately intrinsic Lipschitz graphs. Here we prove that any such graph admits a foliated corona decomposition, which is a family of nested partitions into pieces that are close to ruled surfaces. Apart from the intrinsic geometric and analytic significance of these results, which settle questions posed by Cheeger-Kleiner-Naor (2009) and Lafforgue-Naor (2012), they have several noteworthy implications, including the fact that the $L_1$ distortion of a word-ball of radius $n\ge 2$ in the discrete $3$-dimensional Heisenberg group is bounded above and below by universal constant multiples of $\sqrt[4]{\log n}$; this is in contrast to higher dimensional Heisenberg groups, where our previous work showed that the distortion of a word-ball of radius $n\ge 2$ is of order $\sqrt{\log n}$., Comment: 118 pages, 6 figures, corrected version with some additional remarks

Electrically conductive, polymeric materials that maintain their conductivity even when under significant mechanical deformation are needed for actuator electrodes, conformable electromagnetic shielding, stretchable tactile sensors and flexible energy storage. The challenge for these materials is that the percolated, electrically conductive networks tend to separate even at low strains, leading to significant piezoresistance. Herein, deformable conductors were fabricated by spray-coating a nitrile substrate with a graphene-elastomer solution. The coatings showed only slight increase in electrical resistance after thousands of bending cycles and repeated folding-unfolding events. The deformable conductors doubled their electrical resistance at 12% strain and were washable without changing their electrical properties. The conductivity-strain behaviour was modelled by considering the nanofiller separation upon deformation. To boost the conductivity at higher strains, the production process was adapted by stretching the nitrile substrate before spraying, after which it was released. This adaption meant that the electrical resistance doubled at 25 % strain. The electrical resistance was found sufficiently low to give a 1.9 dB/{\mu}m shielding in the 8-12 GHz electromagnetic band. The physical and electrical properties, including the EM screening, of the flexible conductors, were found to deteriorate upon cycling but could be recovered through reheating the coating., Comment: 29 pages, 6 figures

We show that the $\beta$--numbers of intrinsic Lipschitz graphs of Heisenberg groups $\mathbb{H}_n$ are locally Carleson integrable when $n \geq 2$. Our technique relies on a recent Dorronsoro inequality \cite{FO} as well as a novel slicing argument. A key ingredient in our proof is a Euclidean inequality bounding the $\beta$--number of a function on a cube of $\mathbb{R}^n$ using the $\beta$--number of the restriction of the function to codimension--1 slices of the cube.

A comprehensive study has been undertaken on the dimensional swelling of graphene-reinforced elastomers in liquids. Anisotropic swelling was observed for samples reinforced with graphene nanoplatelets (GNPs), induced by the in-plane orientation of the GNPs. Upon the addition of the GNPs, the diameter swelling ratio of the nanocomposites was significantly reduced, whereas the thickness swelling ratio increased and was even greater than that of the unfilled elastomers. The swelling phenomenon has been analyzed in terms of a modification of the Flory-Rhener theory. The newly-derived equations proposed herein, can accurately predict the dependence of dimensional swelling (diameter and thickness) on volume swelling, independent of the type of elastomer and solvent. The anisotropic swelling of the samples was also studied in combination with the evaluation of the tensile properties of the filled elastomers. A novel theory that enables the assessment of volume swelling for GNP-reinforced elastomers, based on the filler geometry and volume fraction has been developed. It was found that the swelling of rubber nanocomposites induces a biaxial constraint from the graphene flakes.