Miracle-dz/UniFAR-FTUs huggingface dataset:
Quantum K-theoretic geometric Satake: the $\operatorname{SL}_{n}$ case

The geometric Satake correspondence gives an equivalence of categories between the representations of a semisimple group $G$ and the spherical perverse sheaves on the affine Grassmannian $Gr$ of its Langlands dual group. Bezrukavnikov and Finkelberg developed a derived version of this equivalence which relates the derived category of $G^{\vee }$ -equivariant constructible sheaves on $Gr$ with the category of $G$ -equivariant ${\mathcal{O}}(\mathfrak{g})$ -modules. In this paper, we develop a K-theoretic version of the derived geometric Satake which involves the quantum group $U_{q}\mathfrak{g}$ . We define a convolution category $K\operatorname{Conv}(Gr)$ whose morphism spaces are given by the $G^{\vee }\times \mathbb{C}^{\times }$ -equivariant algebraic K-theory of certain fibre products. We conjecture that $K\operatorname{Conv}(Gr)$ is equivalent to a full subcategory of the category of $U_{q}\mathfrak{g}$ -equivariant ${\mathcal{O}}_{q}(G)$ -modules. We prove this conjecture when $G=\operatorname{SL}_{n}$ . A key tool in our proof is the $\operatorname{SL}_{n}$ spider, which is a combinatorial description of the category of $U_{q}\mathfrak{sl}_{n}$ representations. By applying horizontal trace, we show that the annular $\operatorname{SL}_{n}$ spider describes the category of $U_{q}\mathfrak{sl}_{n}$ -equivariant ${\mathcal{O}}_{q}(\operatorname{SL}_{n})$ -modules. Then we use quantum loop algebras to relate the annular $\operatorname{SL}_{n}$ spider to $K\operatorname{Conv}(Gr)$ . This gives a combinatorial/diagrammatic description of both categories and proves our conjecture.
Distribution of Resonances in Scattering by
 Thin Barriers

We study high energy resonances for the operators $-\Delta +\delta_{\partial\Omega}\otimes V$ and $-\Delta+\delta_{\partial\Omega}'\otimes V\partial_\nu$ where $\Omega$ is strictly convex with smooth boundary, $V:L^2(\partial\Omega)\to L^2(\partial\Omega)$ may depend on frequency, and $\delta_{\partial\Omega}$ is the surface measure on $\partial\Omega$. These operators are model Hamiltonians for quantum corrals and leaky quantum graphs. We give a quantum version of the Sabine Law from the study of acoustics for both the $\delta$ and $\delta'$ interactions. It characterizes the decay rates (imaginary parts of resonances) in terms of the system's ray dynamics. In particular, the decay rates are controlled by the average reflectivity and chord length of the barrier. For the $\delta$ interaction we show that generically there are infinitely many resonances arbitrarily close to the resonance free region found by our theorem. In the case of the $\delta'$ interaction, the quantum Sabine law gives the existence of a resonance free region that converges to the real axis at a fixed polynomial rate and is optimal in the case of the unit disk in the plane. As far as the author is aware, this is the only class of examples that is known to have resonances converging to the real axis at a fixed polynomial rate but no faster. The proof of our theorem requires several new technical tools. We adapt intersecting Lagrangian distributions to the semiclassical setting and give a description of the kernel of the free resolvent as such a distribution. We also construct a semiclassical version of the Melrose--Taylor parametrix for complex energies. We use these constructions to give a complete microlocal description of boundary layer operators and to prove sharp high energy estimates on the boundary layer operators in the case that $\partial\Omega$ is smooth and strictly convex.
Lecture notes on Liouville theory and the DOZZ formula

The purpose of these notes, based on a series of 4 lectures given by the author at IHES, is to explain the recent proof of the DOZZ formula for the three point correlation functions of Liouville conformal field theory (LCFT). We first review the probabilistic construction of the $N$ point correlation functions of LCFT on the Riemann sphere for $N \geq 3$ (based on a series of works of the author with David, Kupiainen and Rhodes). We then explain the construction of the two point correlation functions of LCFT, also called reflection coefficient. These probabilistic constructions will be justified from the point of view of Polyakov's path integral formulation of LCFT. Finally, we explain the proof of the DOZZ formula for the three point correlation functions of LCFT (based on 2 papers by the author with Kupiainen and Rhodes).
A Study of Deep Learning Networks for Motion Compensation in Cardiac Gated Spect Images

Motion compensation is an effective approach for noise suppression and motion blur reduction in cardiac gated SPECT imaging. In this work, we investigate the potential benefit of using a deep learning network for motion compensation in a sequence of gated images throughout the cardiac cycle in the presence of large inter-subject variability and imaging degrading factors. We make use a set of clinical acquisitions from 130 subjects and quantify the motion compensation accuracy by variants of two known cascaded learning networks (namely VTN and VoxelMorph). The results in the experiments show that both networks can yield accurate compensation results in both standard dose and half dose studies. Specifically, VTN achieved a relative MSE of 0.0312 (full dose) and 0.0561 (half dose), compared to 0.0340 (full dose) and 0.0356 (half dose) for VoxelMorph. Both networks also outperformed the classical optical flow equation (OFE) method.
A reliable numerical method for solving a certain class of singular initial value problems using reproducing kernel algorithm

The aim of this study is to present a good modernistic strategy for solving some well-known classes of Lane-Emden type singular differential equations. The proposed approach is based on the reproducing kernel Hilbert space (RKHS) and introducing the reproducing kernel properties in which the initial conditions of the problem are satisfied. The analytical solution that obtained involves in the form of a convergent series with easily computable terms in its reproducing kernel space. The approximation solution is expressed by n-term summation of reproducing kernel functions and it is converge to the analytical solution. Our investigations indicate that there is excellent agreement between the numerical results and the RKHS method, which is applied to some examples to illustrate the accuracy, efficiency, and applicability of the method. The present work shows the potential of the RKHS technique in solving such nonlinear singular initial value problems.
Tilting Bundles on Hypertoric Varieties

Recently McBreen and Webster constructed a tilting bundle on a smooth hypertoric variety (using reduction to finite characteristic) and showed that its endomorphism ring is Koszul. In this short note we present alternative proofs for these results. We simply observe that the tilting bundle constructed by Halpern-Leistner and Sam on a generic open Geometric Invariant Theory substack of the ambient linear space restricts to a tilting bundle on the hypertoric variety. The fact that the hypertoric variety is defined by a quadratic regular sequence then also yields an easy proof of Koszulity.
Local Marchenko-Pastur law at the hard edge of sample covariance matrices

Let XN be a N × N matrix whose entries are independent identically distributed complex random variables with mean zero and variance 1N. We study the asymptotic spectral distribution of the eigenvalues of the covariance matrix XN*XN for N → ∞. We prove that the empirical density of eigenvalues in an interval [E, E + η] converges to the Marchenko-Pastur law locally on the optimal scale, Nη/E≫(logN)b, and in any interval up to the hard edge, (logN)bN2≲E≤4−κ, for any κ > 0. As a consequence, we show the complete delocalization of the eigenvectors.
BMR Freeness for Icosahedral Family

ABSTRACT We verify the Broué–Malle–Rouquier (BMR) freeness for cyclotomic Hecke algebras associated with complex reflection groups G17, G18, G19 in the Shephard–Todd classification. Together with results of Ariki, Ariki–Koike, Broué–Malle, Marin, Marin–Pfeiffer, and Chavli, this settled affirmatively the BMR freeness conjecture. Our verification is inspired by Bergman’s diamond lemma and requires 1GB memory and 5 days calculation on a PC.
Completed Iwahori-Hecke algebras and parahoric Hecke algebras for Kac-Moody groups over local fields

Let G be a split Kac-Moody group over a non-archimedean local field. We define a completion of the Iwahori-Hecke algebra of G. We determine its center and prove that it is isomorphic to the spherical Hecke algebra of G using the Satake isomorphism. This is thus similar to the situation of reductive groups. Our main tool is the masure I associated to this setting, which is the analogue of the Bruhat-Tits building for reductive groups. Then, for each special and spherical facet F, we associate a Hecke algebra. In the Kac-Moody setting, this construction was known only for the spherical subgroup and for the Iwahori subgroup.
A Deep Learning Model for Green Algae Detection on SAR Images

This study developed a textural-enhanced deep learning (DL) model based on the classic U-net framework for green algae detection in Sentinel-1 synthetic aperture radar (SAR) imagery. Four special modifications are made in the framework: texture-fused input dataset, texture concatenation to effectively use the texture information, weighted loss function to settle the imbalance of algae–seawater samples, and an attention module to facilitate model focus on the discriminative features efficiently. To build the proposed model, we collected 119 Sentinel-1 SAR images acquired in the Yellow Sea and manually labeled 8441 samples, among which 4421/1896/2124 were used as the training/validation/testing dataset, respectively. Experiments show that the classification achieves the mean intersection over union (mIOU) of 86.31%, outperforming previous DL methods. Furthermore, each modification is effective, and the weighted loss function plays the most critical role. Moreover, we monitored green tide in the Yellow Sea from 2019 to 2021 using the proposed model and analyzed the relationship between green tide interannual variation and two primary environmental factors: nitrate concentration and sea surface temperature (SST). The interannual variation is characterized via three crucial indexes: bloom duration, coverage area, and nearshore damage. The detection results reveal that the bloom duration is the longest (shortest) in 2019 (2020), corresponding to the biggest (smallest) coverage area in 2019 (2020). In addition, the nearshore damage is the heaviest (lightest) in 2021 (2020). We also found that the interannual variation of green tide scales is partly related to the available nitrate concentration and SST variation in algae-distributed regions.
Thermodynamics of Quantum Fields in Nonstationary Spacetimes

Quantum field theory (QFT) on non-stationary spacetimes is well understood from the side of the algebra of observables. The state space, however, is largely unexplored, due to the non-existence of distinguished states (vacuum, scattering states, thermal states). Project C7 of the SFB 676 was focused on characterisations of states by asymptotic conditions, e.g. holography (in case the boundary has sufficiently many symmetries), on a precise version of an approximate particle interpretation (for instance in Robertson--Walker spacetimes) and on the determination in terms of expectation values of locally covariant fields. Additionally, the backreaction of quantum matter fields on the curvature as well as the perturbative quantisation of the Einstein--Klein--Gordon system in the case of a cosmological background have been investigated. Finally, a detailed analysis and construction of equilibrium states for interacting field theories has been performed.
Wild Character Varieties, Meromorphic Hitchin Systems and Dynkin Diagrams

The theory of Hitchin systems is something like a ‘global theory of Lie groups’ where one works over a Riemann surface rather than just at a point. This chapter describes how one can take this analogy a few steps further by attempting to make precise the class of rich geometric objects that appear in this story (including the non-compact case) and discusses their classification, outlining a theory of ‘Dynkin diagrams’ as a step towards classifying some examples of such objects.
Isometric embeddings of a class of separable metric spaces into Banach spaces

Let $(M,d)$ be a bounded countable metric space and $c>0$ a constant, such that $d(x,y)+d(y,z)-d(x,z) \ge c$, for any pairwise distinct points $x,y,z$ of $M$. For such metric spaces we prove that they can be isometrically embedded into any Banach space containing an isomorphic copy of $\ell_\infty$.
The volume of singular Kähler–Einstein Fano varieties

We show that the anti-canonical volume of an $n$ -dimensional Kähler–Einstein $\mathbb{Q}$ -Fano variety is bounded from above by certain invariants of the local singularities, namely $\operatorname{lct}^{n}\cdot \operatorname{mult}$ for ideals and the normalized volume function for real valuations. This refines a recent result by Fujita. As an application, we get sharp volume upper bounds for Kähler–Einstein Fano varieties with quotient singularities. Based on very recent results by Li and the author, we show that a Fano manifold is K-semistable if and only if a de Fernex–Ein–Mustaţă type inequality holds on its affine cone.
On row sequences of Pad\'e and Hermite-Pad\'e approximation

A survey of direct and inverse type results for row sequences of Pad\'e and Hermite-Pad\'e approximation is given. A conjecture is posed on an inverse type result for type II Hermite-Pad\'e approximation when it is known that the sequence of common denominators of the approximating vector rational functions has a limit. Some inverse type results are proved for the so called incomplete Pad\'e approximants which may lead to the proof of the conjecture and the connection is discussed.
Conformal Scattering of Maxwell fields on Reissner–Nordström–de Sitter Black Hole Spacetimes

We construct a complete conformal scattering theory for Maxwell fields in the static exterior region of a Reissner-Nordstr{\o}m-de Sitter black bole spacetime. This is done using uniform energy decay results that we obtain in a separate paper, to show that the trace operators are injective and have closed ranges. We then solve the Goursat problem (characteristic Cauchy problem) for Maxwell fields on the null boundaries showing that the trace operators are also surjective.
Existence of weak solution for mean curvature flow with transport term and forcing term

We study the mean curvature flow with given non-smooth transport term and forcing term, in suitable Sobolev spaces. We prove a global-in-time existence of the weak solutions for the mean curvature flow with the terms, by using the modified Allen-Cahn equation that holds useful properties such as the monotonicity formula.
Symmetry Results for Fractional Elliptic Systems and Related Problems

We study elliptic gradient systems with fractional laplacian operators on the whole space where u: R n → R m , H ∈ C 2, γ(R m ) for γ > max (0, 1 − 2 min {s i }), s = (s 1,…, s m ) for 0 < s i < 1 and ∇H(u) = (H u i (u 1, u 2,…, u m )) i . We prove De Giorgi type results for this system for certain values of s and in lower dimensions, i.e. n = 2, 3. Just like the local case, the concepts of orientable systems and H − monotone solutions play the key role in proving symmetry results. In addition, we provide optimal energy estimates, a monotonicity formula, a Hamiltonian identity and various Liouville theorems.
Effect of Narrowband UV-B Irradiation on the Growth Performance of House Crickets

Indoor co-cultivation systems can answer to the need for sustainable and resilient food production systems. Rearing organisms under light-emitting diodes (LEDs) irradiation provides the possibility to control and shape the emitted light spectra. UV-B-irradiation (280–315 nm) can positively affect the nutritional composition of different plants and other organisms, whereas information on edible insects is scarce. To evaluate the potential effect of the photosynthetically active radiation (PAR) and LED-emitting LEDs on the rearing and nutritional quality of edible insects, house crickets (Acheta domesticus) were reared from the age of 21 days under controlled LED spectra, with an additional UV-B (0.08 W/m2) dose of 1.15 KJm2 d−1 (illuminated over a period for 4 h per day) for 34 days. UV-B exposure showed no harm to the weight of the crickets and significantly increased their survival by ca. 10% under narrowband UV-B treatment. The nutritional composition including proteins, fat and chitin contents of the insects was not affected by the UV-B light and reached values of 60.03 ± 10.41, 22.38 ± 2.12 and 9.33 ± 1.21%, respectively, under the LED irradiation. Therefore, house crickets can grow under LED irradiation with a positive effect of narrowband UV-B application on their survival.
Fair Multi-Resource Allocation in Heterogeneous Servers With an External Resource Type

This paper considers the problem of fair allocation of multiple types of resources in heterogeneous servers, along with a resource type external to those servers. Our work is motivated by the need for fair multi-resource allocation in mobile edge computing (MEC), where the users must upload their tasks over a single dedicated wireless communication link that exists outside the computing servers. We propose a fair multi-resource allocation mechanism for this environment, termed Task Share Fairness with External Resource (TSF-ER), which finds the Kalai-Smorodinsky bargaining solution satisfying important fairness properties. We show that TSF-ER is envy-free, Pareto optimal, and strategy-proof, and it satisfies the property of sharing incentive. Large-scale simulation driven by Google and Alibaba cluster trace further shows that TSF-ER significantly outperforms the existing utilitarian, Nash social welfare maximizer, and egalitarian solutions, leading to fairer resource allocation while maintaining a high level of resource utilization.
REALIZABILITY PROBLEM FOR COMMUTING GRAPHS

We investigate properties which ensure that a given finite graph is the commuting graph of a group or semigroup. We show that all graphs on at least two vertices such that no vertex is adjacent to all other vertices is the commuting graph of some semigroup. Moreover, we obtain complete classifications of the graphs with an isolated vertex or edge that are the commuting graph of a group and the cycles that are the commuting graph of a centrefree semigroup.
Strong sequential completeness of the natural domain of a conditional expectation operator in Riesz spaces

Strong convergence and convergence in probability were generalized to the setting of a Riesz space with conditional expectation operator, $T$, in [{{\sc Y. Azouzi, W. -C. Kuo, K. Ramdane, B. A. Watson}, {Convergence in Riesz spaces with conditional expectation operators}, {\em Positivity}, {\bf 19} {(2015), 647-657}}] as $T$-strong convergence and convergence in $T$-conditional probability, respectively. Generalized $L^{p}$ spaces for the cases of $p=1,2,\infty$, were discussed in the setting of Riesz spaces as $\mathcal{L}^{p}(T)$ spaces in [{{\sc C. C. A. Labuschagne, B. A. Watson}, {Discrete stochastic integration in Riesz spaces}, {\em Positivity}, {\bf 14} {(2010), 859-875}}]. An $R(T)$ valued norm, for the cases of $p=1,\infty,$ was introduced on these spaces in [{{\sc W. Kuo, M. Rogans, B.A. Watson}, {Mixing processes in Riesz spaces}, {\em Journal of Mathematical Analysis and Application}, {\bf 456} {(2017), 992-1004}}] where it was also shown that $R(T)$ is a universally complete $f$-algebra and that these spaces are $R(T)$-modules. In [{{\sc Y. Azouzi, M. Trabelsi}, {$L^p$-spaces with respect to conditional expectation on Riesz spaces}, {\em Journal of Mathematical Analysis and Application}, {\bf 447} {(2017), 798-816}}] functional calculus was used to consider $\mathcal{L}^{p}(T)$ for $p\in (1,\infty)$. In this paper we prove the strong sequential completeness of the space $\mathcal{L}^{1}(T)$, the natural domain of the conditional expectation operator $T$, and the strong completeness of $\mathcal{L}^{\infty}(T)$.
Hopf Bifurcation for an SIS model with age structure

An SIS model is investigated in which the infective individuals are assumed to have an infection-age structure. The model is formulated as an abstract non-densely defined Cauchy problem. We study some dynamical properties of the model by using the theory of integrated semigroup, the Hopf bifurcation theory and the normal form theory for semilinear equations with non-dense domain. Qualitative analysis indicates that there exist some parameter values such that this SIS model has a non-trivial periodic solution which bifurcates from the positive equilibrium. Furthermore, the explicit formulae are given to determine the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions. Numerical simulations are also carried out to support our theoretical results.
Language Tests, Motivation, and Demotivation: A Complex Relationship

Tests are an integral part of most formal language learning processes, exerting far-reaching effects on many aspects of language learning. Among other things, tests affect language learning motivation (LLM), an element that is fundamental to many learning contexts. However, little attention has been paid to how tests affect LLM and what the pedagogical implications may be. Using qualitative semi-structured interviews with 20 adolescent Arab learners, this study attempts to explore how and when tests can motivate or demotivate language learners. It also examines the main strategies learners deploy to cope with the demotivating experiences associated with test taking. The data suggest that tests have the potential to both motivate and demotivate learners before, during, and after taking a test. Additionally, the results reveal several strategies that learners employ to cope with demotivation. This leads to a discussion regarding pedagogical implications on how to effectively employ tests to increase learners’ motivation and decrease their demotivation.
Prevalence of musculoskeletal symptoms among Portuguese call center operators: associations with gender, Body Mass Index, hours of work and sitting time

According to previous studies, the prevalence of musculoskeletal symptoms among call center operators is high. However, although this is a developing occupation in Portugal, there is a lack of studies assessing this issue as well as its associations with different risk factors. Therefore, the aim of this study is to investigate the 7-day and 12-month prevalence of musculoskeletal symptoms among Portuguese call center operators and their associations with gender, body mass index, hours of work and sitting time. The study was conducted in a call center company in Portugal. One-hundred and forty-eight workers agreed to participate, and filled out questionnaires to evaluate sociodemographic, anthropometric, and occupational variables, as well as the Nordic Musculoskeletal Questionnaire (NMQ) to assess musculoskeletal symptoms in the last 7 days and 12 months of 9 body regions. NMQ revealed the 12-month prevalence of musculoskeletal symptoms, the 3 most affected body regions were the neck (56.1%), the low back (54.7%) and the shoulders (43.9%). And the 7-day prevalence, the three most affected body region were the same, but in different percentages: low back (31.8%), neck (23.6%) and shoulders (21.6%). Significant associations were found between gender and reporting symptoms in the wrist/hands (p=0.033) and the knees (p=0.031), with females reporting significantly more symptoms than males; and between body mass index and reporting symptoms in the thighs/hips, with overweight operators reporting more symptoms (p=0.010). No significant associations were found for the hours of work, neither in sitting time. Conclusion: Workplace interventions are needed to decrease the prevalence of musculoskeletal complaints among call center workers.