Miracle-dz/UniFAR-FTUs huggingface dataset:
Finite 2-groups with odd number of conjugacy classes

In this paper we consider finite 2-groups with odd number of real conjugacy classes. On one hand we show that if $k$ is an odd natural number less than 24, then there are only finitely many finite 2-groups with exactly $k$ real conjugacy classes. On the other hand we construct infinitely many finite 2-groups with exactly 25 real conjugacy classes. Both resuls are proven using pro-$p$ techniques and, in particular, we use the Kneser classification of semi-simple $p$-adic algebraic groups.
On the Geometry of Some Equivariantly Related Manifolds

We provide geometric realizations of both classical and “exotic” $G$-manifolds such as spheres, Kervaire manifolds, bundles over spheres, homogeneous spaces, and connected sums among them. As an application, we apply the process known as Cheeger deformation to produce new metrics of both positive Ricci and almost non-negative curvature on such objects.
Singular conformally invariant trilinear forms, I Multiplicity one results

A normalized holomorphic family (depending on $\boldsymbol \lambda \in \mathbb C^3$) of conformally invariant trilinear forms on the sphere is studied. Its zero set $Z$ is described. For $\boldsymbol \lambda\notin Z$, the multiplicity of the space of conformally invariant trilinear forms is shown to be 1, except perhaps for a denumerable subset.
Microlocal Morse theory of wrapped Fukaya categories

Consider on the one hand the partially wrapped Fukaya category of a cotangent bundle stopped at an appropriately stratifiable singular isotropic. Consider on the other hand the derived category of sheaves with microsupport along the singular isotropic. We show here that the former is equivalent to the compact objects of the latter. Applications include a sheaf-theoretic description of the wrapped Fukaya categories of plumbings and of codimension one Weinstein hypersurfaces of cosphere bundles. While inspired by the Nadler-Zaslow correspondence, our results do not depend on it.
Global well-posedness to the subcritical Oldroyd-B type models in 2D

We prove the global well-posedness to the 2D Oldroyd-B type models with $\nu \Lambda^{2\alpha}u$ and $\eta\Lambda^{2\beta}\tau$ satisfying $(i)\ \alpha>1, \eta=0$ or $(ii)\ \alpha=1,\ \beta>0$. By establishing the gradient estimate of $u$, $\tau$ and $L^\infty$ bound of ${\rm curl u+\Lambda^{-2}curldiv \tau}$, Elgidi-Rousset (Commun. Pure Appl. Math. online, 2015) obtained the global well-posedness for the case $\nu=0$, $\beta=1$. However, for the cases $(i)$ and $(ii)$, it is difficult to improve the regularity of $u$ and $\tau$ directly, especially when $\alpha\rightarrow 1^{+}$ in case $(i)$ and $\beta\rightarrow 0^{+}$ in case $(ii)$. To overcome this difficulty, we exploit a new structure of the equations coming from the dissipation and coupled term. Then we prove the global well-posedness to these cases by energy method which brings us closer to the more interesting case $\alpha=1$, $\eta=0$.
On the Compensator in the Doob-Meyer Decomposition of the Snell Envelope

Let $G$ be a semimartingale, and $S$ its Snell envelope. Under the assumption that $G\in\mathcal{H}^1$, we show that the finite-variation part of $S$ is absolutely continuous with respect to the decreasing part of the finite-variation part of $G$. In the Markovian setting, this enables us to identify sufficient conditions for the value function of the optimal stopping problem to belong to the domain of the extended (martingale) generator of the underlying Markov process. We then show that the \textit{dual} of the optimal stopping problem is a stochastic control problem for a controlled Markov process, and the optimal control is characterised by a function belonging to the domain of the martingale generator. Finally, we give an application to the smooth pasting condition.
The facets of the matroid polytope and the independent set polytope of a positroid

A positroid is a special case of a realizable matroid, that arose from the study of totally nonnegative part of the Grassmannian by Postnikov. Postnikov demonstrated that positroids are in bijection with certain interesting classes of combinatorial objects, such as Grassmann necklaces and decorated permutations. The bases of a positroid can be described directly in terms of the Grassmann necklace and decorated permutation. In this paper, we show how to describe the bases and independent sets directly from the decorated permutation, bypassing the use of the Grassmann necklace.
On the HOMFLY polynomial of 4-plat presentations of knots

In this paper, I give a method to calculate the HOMFLY polynomials of knots by using a representation of the braid group B4 into a group of 3 ? 3 matrices. Also, I will give examples of a 2-bridge knot and a 3-bridge knot that have the same Jone polynomial, but different HOMFLY polynomials.
Phase Off-Axis Modulation Metasurface for Orbital Angular Momentum Mode Multiplexing/Demultiplexing

Orbital angular momentum (OAM) mode multiplexing provides a promising solution for enlarging communication capacity density. Although various OAM mode multiplexing technologies have been investigated, (de)multiplexers possessing a wide spectral response and polarization insensitivity to combine wavelength division multiplexing (WDM) and polarization division multiplexing (PDM) remain elusive. Herein, we propose a phase off-axis modulation strategy for OAM mode (de)multiplexing by encoding the binary Dammann optical vortex grating (DOVG) phase into a geometric phase metasurface that consists of metallic nanoantennas with spatially varying orientations on a dielectric-metal substrate. We show that four OAM modes (−2, −1, 1, 2) were successfully multiplexed and demultiplexed by virtue of the multilevel diffraction of DOVG and its reverse process. The working wavelength of this metasurface ranges from 1260 nm to 1675 nm due to the dispersion-free geometric phase, moreover, the mode symmetry of DOVG and the conjugate modulation of the geometric phase yield polarization insensitivity. As a proof of concept, an OAM mode multiplexing communication system combining WDM and PDM was constructed, and 1.56 Tbit/s QPSK signals were transmitted with the bit-error-rate approaching 10−6. These results demonstrate that this phase off-axis modulation metasurface can effectively (de)multiplex OAM modes and shows good compatibility with WDM and PDM, which may promote the practical application of OAM communication and all-optical interconnection.
On the optimal paving over MASAs in von Neumann algebras

We prove that if A is a singular MASA in a II1 factor M and ! is a free ultralter, then for any x2 M A, withkxk 1, and any n 2, there exists a partition of 1 with projections p1;p2;:::;pn2 A ! (i.e. a paving) such thatk n=1 pixpik 2 p n 1=n, and give examples where this is sharp. Some open problems on optimal pavings are discussed.
JSNet++: Dynamic Filters and Pointwise Correlation for 3D Point Cloud Instance and Semantic Segmentation

In this paper, we propose a novel joint instance and semantic segmentation approach, called JSNet++, to address the instance and semantic segmentation tasks of 3D point clouds simultaneously. We first introduce a basic joint segmentation framework (JSNet). It fuses features from different layers of the backbone network to obtain more discriminative features and makes the two tasks take advantage of each other with a joint instance and semantic segmentation (JISS) module. Specifically, the JISS transforms semantic features into instance embedding space, and then the transformed features are fused with instance features to facilitate instance segmentation. Meanwhile, the JISS module also makes semantic segmentation benefit from instance segmentation by aggregating instance features to semantic feature space. To further reduce the memory consumption of JSNet, we design a dynamic filters for convolution (DFConv) on point clouds. Specifically, we exploit the geometry and density information to generate the dynamic filters, which are used to perform depthwise convolution with the input features. Afterwards, we unify the spatial correlation and channel correlation into a module to fully explore the pointwise correlation in point clouds, and we develop an improved JISS module (JISS*) by using the pointwise correlation module to further improve the accuracy of segmentation. Finally, based on the JSNet, DFConv and JISS*, we propose a new joint segmentation network, termed JSNet++. Experimental results on the benchmarks S3DIS and ScanNet v2 datasets demonstrate the effectiveness of our approach, and our method achieves significant performance improvements over baseline on both instance and semantic segmentation.
Operadic categories and their skew monoidal categories of collections

I describe a generalization of the notion of operadic category due to Batanin and Markl. For each such operadic category I describe a skew monoidal category of collections, such that a monoid in this skew monoidal category is precisely an operad over the operadic category. In fact I describe two skew monoidal categories with this property. The first has the feature that the operadic category can be recovered from the skew monoidal category of collections; the second has the feature that the right unit constraint is invertible. In the case of the operadic category S of finite sets and functions, for which an operad is just a symmetric operad in the usual sense, the first skew monoidal category has underlying category [N, Set], and the second is the usual monoidal category of collections [P, Set] with the substitution monoidal structure.
Deformations and elements of deformation theory

This article consisted of an elementary introduction to deformation theory of varieties, schemes and manifolds, with some applications to local and global shtukas and fever to Newton polygons of $p$-divisible groups . Soft problems and results mainly are considered. In the framework we give review of some novel results in the theory of local shtukas, Anderson-modules, global shtukas, Newton polygons of $p$-divisible groups and on deformations of $p$-divisible groups with given Newton polygons.
Certain logarithmic integrals, including solution of Monthly problem #tbd, zeta values, and expressions for the Stieltjes constants

We solve problem x proposed by O. Oloa, AMM xxx 2012 {\bf 119? } (to appear), p. yyy for certain definite logarithmic integrals. A number of generating functions are developed with certain coefficients $p_n$, and some extensions are presented. The explicit relation of $p_n$ to N\"{o}rlund numbers $B_n^{(n)}$ is discussed. Certain inequalities are conjectured for the $\{p_n\}$ sequence of coefficients, including its convexity, and an upper bound is demonstrated. It is shown that $p_n$ values may be used to express the Stieltjes constants for the Hurwitz and Riemann zeta functions, as well as values of these zeta functions at integer argument. Other summations with the $p_n$ coefficients are presented.
A Novel Application of Deep Learning (Convolutional Neural Network) for Traumatic Spinal Cord Injury Classification Using Automatically Learned Features of EMG Signal

In this study, a traumatic spinal cord injury (TSCI) classification system is proposed using a convolutional neural network (CNN) technique with automatically learned features from electromyography (EMG) signals for a non-human primate (NHP) model. A comparison between the proposed classification system and a classical classification method (k-nearest neighbors, kNN) is also presented. Developing such an NHP model with a suitable assessment tool (i.e., classifier) is a crucial step in detecting the effect of TSCI using EMG, which is expected to be essential in the evaluation of the efficacy of new TSCI treatments. Intramuscular EMG data were collected from an agonist/antagonist tail muscle pair for the pre- and post-spinal cord lesion from five Macaca fasicularis monkeys. The proposed classifier is based on a CNN using filtered segmented EMG signals from the pre- and post-lesion periods as inputs, while the kNN is designed using four hand-crafted EMG features. The results suggest that the CNN provides a promising classification technique for TSCI, compared to conventional machine learning classification. The kNN with hand-crafted EMG features classified the pre- and post-lesion EMG data with an F-measure of 89.7% and 92.7% for the left- and right-side muscles, respectively, while the CNN with the EMG segments classified the data with an F-measure of 89.8% and 96.9% for the left- and right-side muscles, respectively. Finally, the proposed deep learning classification model (CNN), with its learning ability of high-level features using EMG segments as inputs, shows high potential and promising results for use as a TSCI classification system. Future studies can confirm this finding by considering more subjects.
Rational cuspidal curves in projective surfaces. Topological versus algebraic obstructions

We study rational cuspidal curves in projective surfaces. We specify two criteria obstructing possible configurations of singular points that may occur on such curves. One criterion generalizes the result of Fernandez de Bobadilla, Luengo, Melle--Hernandez and Nemethi and is based on the Bezout theorem. The other one is a generalization of the result obtained by Livingston and the author and relies on Ozsvath--Szabo inequalities for $d$-invariants in Heegaard Floer homology. We show by means of explicit calculations that the two approaches give very similar obstructions.
Invasion waves in the presence of a mutualist

This paper studies invasion waves in the diffusive competitor–competitor–mutualist model generalizing the two‐species Lotka–Volterra model studied by Weinberger et al. The mutualist may benefit the invading or the resident species producing two different types of invasions. Sufficient conditions for linear determinacy are derived in both cases, and when they hold, explicit formulas for linear spreading speeds of the invasions are obtained by linearizing the model. While in the first case the linear speed is increased by the mutualist, it is unaffected in the second case. Mathematical methods are based on converting the model into a cooperative reaction–diffusion system. Copyright © 2013 John Wiley & Sons, Ltd.
$W^{\sigma,\epsilon}$-estimates for nonlocal elliptic equations

We prove that the fractional derivatives of solutions to a class of nonlocal fully nonlinear elliptic equations are epsilon-integrable. We follow Fanghua Lin's original approach to the analogous problem for second order equations, by first proving a potential estimate, then combining this estimate with the ABP-type estimate by N. Guillen and R. Schwab to control the size of the superlevel sets of the fractional derivatives of solutions.
A note on a smooth projective surface with Picard number 2

We characterize the integral Zariski decomposition of a smooth projective surface with Picard number 2 to partially solve a problem of B. Harbourne, P. Pokora, and H. Tutaj‐Gasinska [Electron. Res. Announc. Math. Sci. 22 (2015), 103–108].
The Endocannabinoid Analgesic Entourage Effect: Investigations in Cultured DRG Neurons

Background The endocannabinoid 2-Arachidonyl glycerol (2-AG) exerts dose-related anti-nociceptive effects, which are potentiated by the related but inactive 2-palmitoyl glycerol (2-PG) and 2-linoleoyl glycerol (2-LG). This potentiation of analgesia and other in vivo measures was described as the “entourage effect”. We investigated this effect on TRPV1 signalling in cultured dorsal root ganglion (DRG) nociceptors. Methods Adult rat DRG neurons were cultured in medium containing NGF and GDNF at 37°C. 48 h later cultures were loaded with 2 µM Fura2AM for calcium imaging, and treated with 2-AG, 2-PG and 2-LG, individually or combined, for 5 min, followed by 1 µMol capsaicin. The amplitude and latency of capsaicin responses were measured (N=3–7 rats, controls N=16), and analysed. Results In controls, 1 µMol capsaicin elicited immediate calcium influx in a subset of neurons, with average latency of 1.27 ± 0.2 s and amplitude of 0.15 ± 0.01 Units. 2-AG (10–100 µMol) elicited calcium influx in some neurons. In the presence of 2-AG (0.001–100 µMol), capsaicin responses were markedly delayed in 64% neurons by up to 320 s (P<0.001). 2-PG increased capsaicin response latency at 0.1 nMol-100 µMol (P<0.001), in 60% neurons, as did 2-LG at 0.1–100 µMol (P<0.001), in 76% neurons. Increased capsaicin response latency due to 2-AG and 2-PG was sensitive to the CB2 but not to the CB1 receptor antagonist. Combined application of 1 µMol 2-AG, 5 µMol 2-PG and 10 µMol 2-LG, also resulted in significantly increased capsaicin response latency up to 281.5 ± 41.5 s (P<0.001), in 96% neurons, that was partially restored by the CB2, but not the CB1 antagonist. Conclusion 2-AG, 2-LG and 2-PG significantly delayed TRPV1 signalling in the majority of capsaicin-sensitive DRG neurons, that was markedly increased following combined application. Further studies of these endocannabinoids are required to identify the underlying mechanisms.
Generalization of Deep Learning in Digital Pathology: Experience in Breast Cancer Metastasis Detection

Simple Summary Pathology is a cornerstone in cancer diagnostics, and digital pathology and artificial intelligence-driven image analysis could potentially save time and enhance diagnostic accuracy. For clinical implementation of artificial intelligence, a major question is whether the computer models maintain high performance when applied to new settings. We tested the generalizability of a highly accurate deep learning model for breast cancer metastasis detection in sentinel lymph nodes from, firstly, unseen sentinel node data and, secondly, data with a small change in surgical indication, in this case lymph nodes from axillary dissections. Model performance dropped in both settings, particularly on axillary dissection nodes. Retraining of the model was needed to mitigate the performance drop. The study highlights the generalization challenge of clinical implementation of AI models, and the possibility that retraining might be necessary. Abstract Poor generalizability is a major barrier to clinical implementation of artificial intelligence in digital pathology. The aim of this study was to test the generalizability of a pretrained deep learning model to a new diagnostic setting and to a small change in surgical indication. A deep learning model for breast cancer metastases detection in sentinel lymph nodes, trained on CAMELYON multicenter data, was used as a base model, and achieved an AUC of 0.969 (95% CI 0.926–0.998) and FROC of 0.838 (95% CI 0.757–0.913) on CAMELYON16 test data. On local sentinel node data, the base model performance dropped to AUC 0.929 (95% CI 0.800–0.998) and FROC 0.744 (95% CI 0.566–0.912). On data with a change in surgical indication (axillary dissections) the base model performance indicated an even larger drop with a FROC of 0.503 (95%CI 0.201–0.911). The model was retrained with addition of local data, resulting in about a 4% increase for both AUC and FROC for sentinel nodes, and an increase of 11% in AUC and 49% in FROC for axillary nodes. Pathologist qualitative evaluation of the retrained model´s output showed no missed positive slides. False positives, false negatives and one previously undetected micro-metastasis were observed. The study highlights the generalization challenge even when using a multicenter trained model, and that a small change in indication can considerably impact the model´s performance.
Characterizations of ideal cluster points

Abstract Given an ideal ℐ {\mathcal{I}} on ω, we prove that a sequence in a topological space X is ℐ {\mathcal{I}} -convergent if and only if there exists a “big” ℐ {\mathcal{I}} -convergent subsequence. Then we study several properties and show two characterizations of the set of ℐ {\mathcal{I}} -cluster points as classical cluster points of a filter on X and as the smallest closed set containing “almost all” the sequence. As a consequence, we obtain that the underlying topology τ coincides with the topology generated by the pair ( τ , ℐ ) {(\tau,\mathcal{I})} .
Identification and Evolutionary Analysis of User Collusion Behavior in Blockchain Online Social Media

Blockchain technology has given rise to a series of new blockchain online social media (BOSMs), of which Steemit is representative. Such communities are based on a token reward system and attempt to engross users in the knowledge activities of the community through knowledge payment. Studies have found that the reward system of such communities has been abused (e.g., collusion for profit), but few studies have performed an in-depth analysis for this phenomenon. Consequently, real data for Steemit are used as a case study herein to examine the collusion of users in BOSMs. Two user collusion behaviors (group-voting and vote-buying) are defined and measured. On this basis, an identification and evolutionary survival analysis of the two collusion behaviors are conducted for colluding users and colluding groups, and the behavior patterns of user collusion under the token system are deconstructed. The results of this study improve stakeholders’ understanding of user participation behavior in new online communities, and serve as a reference for decision-making in community governance and token design.
Green manure for agricultural sustainability and improvement of soil fertility

Many farmers are incorporating green manuring into their practices as a result of the growing issues facing agriculture, such as climate change, extreme weather events, soil deterioration, and land contamination due to the excessive use of chemical fertilizers. Growing green manure is a useful and affordable way to ensure the long-term productivity of agricultural soils. Given the significant loss of soil fertility caused by improper agronomic practices, human-induced land degradation is extremely serious. Due to their positive effects on the physical, chemical, and biological quality of the soil as well as the proper justification of soil fertility, green manures can play a significant role. Utilizing green manure is one of the trends in environmental protection that aids in the agricultural sector's transition to sustainability by conserving resources and meeting consumer demand for food. Modern farmers are faced with a number of challenges as they try to transition to productive and ecologically friendly practices. To preserve soil fertility while utilizing less chemicals is one of them. Green manure crops could be a potential answer to this issue. Back in ancient times, farmers learned that this method had a positive effect on the soil. Thus, for growers who want to reduce the use of hazardous chemicals for soil fertilization, green manure is indispensable. In terms of operations, many farmers must employ green manure to reduce the overuse of chemical fertilizers in agriculture.
On the summablility of truncated double Fourier series

We estimate the truncated double trigonometric series $\sum_{n=0}^{N}\sum_{m=0}^{M}a_{mn} {e}^{2\pi \imath \left(m x+n y\right)}$, $a_{mn} \in\mathbb{C}$, in Lebesgue spaces with mixed norms in terms of the $p^{th}-q^{th}$ power finite double sums of its coefficients. We obtain these estimates for all possible values of the exponents involved then we provide examples of matrices in ${\mathbb{C}}^{M\times N}$ that maximize some of them up to a constant independent of $M$ and $N$.
Existence of nodal solutions for quasilinear elliptic problems in $\mathbb{R}^N$

We prove the existence of one positive, one negative, and one sign-changing solution of a $p$-Laplacian equation on $\mathbb{R}^N$, with a $p$-superlinear subcritical term. Sign-changing solutions of quasilinear elliptic equations set on the whole of $\mathbb{R}^N$ have only been scarcely investigated in the literature. Our assumptions here are similar to those previously used by some authors in bounded domains, and our proof uses fairly elementary critical point theory, based on constraint minimization on the nodal Nehari set. The lack of compactness due to the unbounded domain is overcome by working in a suitable weighted Sobolev space.