Miracle-dz/UniFAR-FTUs huggingface dataset:
Inverse problems and uncertainty quantification

In a Bayesian setting, inverse problems and uncertainty quantification (UQ)—the propagation of uncertainty through a computational (forward) model—are strongly connected. In the form of conditional expectation the Bayesian update becomes computationally attractive. This is especially the case as together with a functional or spectral approach for the forward UQ there is no need for time-consuming and slowly convergent Monte Carlo sampling. The developed sampling-free non-linear Bayesian update is derived from the variational problem associated with conditional expectation. This formulation in general calls for further discretisation to make the computation possible, and we choose a polynomial approximation. After giving details on the actual computation in the framework of functional or spectral approximations, we demonstrate the workings of the algorithm on a number of examples of increasing complexity. At last, we compare the linear and quadratic Bayesian update on the small but taxing example of the chaotic Lorenz 84 model, where we experiment with the influence of dierent observation or measurement operators on the update.
CD meets CAT

Abstract We show that if a noncollapsed CD ⁢ ( K , n ) {\mathrm{CD}(K,n)} space X with n ≥ 2 {n\geq 2} has curvature bounded above by κ in the sense of Alexandrov, then K ≤ ( n - 1 ) ⁢ κ {K\leq(n-1)\kappa} and X is an Alexandrov space of curvature bounded below by K - κ ⁢ ( n - 2 ) {{K}-\kappa(n-2)} . We also show that if a CD ⁢ ( K , n ) {\mathrm{CD}(K,n)} space Y with finite n has curvature bounded above, then it is infinitesimally Hilbertian.
Finite generation and continuity of topological Hochschild and cyclic homology

The goal of this paper is to establish fundamental properties of the Hochschild, topological Hochschild, and topological cyclic homologies of commutative, Noetherian rings, which are assumed only to be F-finite in the majority of our results. This mild hypothesis is satisfied in all cases of interest in finite and mixed characteristic algebraic geometry. We prove firstly that the topological Hochschild homology groups, and the homotopy groups of the fixed point spectra $TR^r$, are finitely generated modules. We use this to establish the continuity of these homology theories for any given ideal. A consequence of such continuity results is the pro Hochschild-Kostant-Rosenberg theorem for topological Hochschild and cyclic homology. Finally, we show more generally that the aforementioned finite generation and continuity properties remain true for any proper scheme over such a ring.
The strong maximal rank conjecture and moduli spaces of curves

Building on recent work of the authors, we use degenerations to chains of elliptic curves to prove two cases of the Aprodu-Farkas strong maximal rank conjecture, in genus $22$ and $23$. This constitutes a major step forward in Farkas' program to prove that the moduli spaces of curves of genus $22$ and $23$ are of general type. Our techniques involve a combination of the Eisenbud-Harris theory of limit linear series, and the notion of linked linear series developed by the second author.
Localization and Compactness in Bergman and Fock spaces

In this paper we study the compactness of operators on the Bergman space of the unit ball and on very generally weighted Bargmann-Fock spaces in terms of the behavior of their Berezin transforms and the norms of the operators acting on reproducing kernels. In particular, in the Bergman space setting we show how a vanishing Berezin transform combined with certain (integral) growth conditions on an operator $T$ are sufficient to imply that the operator is compact. In the weighted Bargmann-Fock space setting we show that the reproducing kernel thesis for compactness holds for operators satisfying similar growth conditions. The main results extend the results of Xia and Zheng to the case of the Bergman space when $1 < p < \infty$, and in the weighted Bargmann-Fock space setting, our results provide new, more general conditions that imply the work of Xia and Zheng via a more familiar approach that can also handle the $1 < p < \infty$ case.
Dirac operators with $W^{1,\infty}$-potential under codimension one collapse

We study the behavior of the spectrum of the Dirac operator together with a symmetric $W^{1, \infty}$-potential on spin manifolds under a collapse of codimension one with bounded sectional curvature and diameter. If there is an induced spin structure on the limit space $N$ then there are convergent eigenvalues which converge to the spectrum of a first order differential operator $D$ on $N$ together with a symmetric $W^{1,\infty}$-potential. If $N$ is orientable and the dimension of the limit space is even then $D$ is the Dirac operator $D^N$ on $N$ and if the dimension of the limit space is odd, then $D = D^N \oplus -D^N$.
Partition C*-algebras

We give a definition of partition C*-algebras: To any partition of a finite set, we assign algebraic relations for a matrix of generators of a universal C*-algebra. We then prove how certain relations may be deduced from others and we explain a partition calculus for simplifying such computations. This article is a small note for C*-algebraists having no background in compact quantum groups, although our partition C*-algebras are motivated from those underlying Banica-Speicher quantum groups (also called easy quantum groups). We list many open questions about partition C*-algebras that may be tackled by purely C*-algebraic means, ranging from ideal structures and representations on Hilbert spaces to K-theory and isomorphism questions. In a follow up article, we deal with the quantum algebraic structure associated to partition C*-algebras.
Metagenomics exploring the effect of recombinant rice based on lotus seed starch-broken rice flour on intestinal flora in rats

High fat diet may cause obesity, diabetes, atherosclerosis, hypertension, hyperlipidemia and other diseases, and even mortality. The recombinant rice based on mixing lotus seed starch and broken rice flour was used to feed high-fat diet rats, and its effects on body weight and organ index were determined. The changes of intestinal flora in rats were studied by macrogenomic technology, and the effects of lotus seed starch-broken rice flour recombinant rice (LSBR) on intestinal flora in rats were discussed. The results showed that adding enough LSBR could effectively inhibit the weight gain of rats, reduce the liver index from 0.3498 to 0.2836, and the kidney index from 0.0079 to 0.0072. The reduction of intestinal flora abundance and diversity caused by high-fat diet were improved, the relative abundance ratio of Firmicutes/Bacteroidete was reduced from 11.04 to 2.17, while the relative abundance of beneficial bacteria was increased.
Response to First-Line Treatment with Immune-Checkpoint Inhibitors in Patients with Advanced Cutaneous Squamous Cell Carcinoma: A Multicenter, Retrospective Analysis from the German ADOReg Registry

Simple Summary Cutaneous squamous cell carcinoma (cSCC) is one of the most common malignancies of the skin with poor survival outcomes in advanced stages of the disease. Recent clinical trials demonstrated the efficacy of checkpoint-inhibitors (CPI) therapy for advanced stage disease, but there is a lack of data from real-world cohorts and trial-ineligible patients. In this retrospective, real-world cohort study, we investigated the efficacy of first-line checkpoint-inhibitor treatment in 39 patients with advanced cSCC from eight different German cancer centers and stratified outcomes by the immune status of the patients. Our data demonstrate that patients receiving CPI achieved high response rates with durable remissions in about 20% of patients. CPI also evoked tumor responses in patients with active autoimmune diseases and lymphoproliferative disorders, although these responses were often short-lived, resulting in a significantly shorter overall survival. Notably, CPI therapy was safe with only 15% of patients discontinuing for toxicity. Abstract Cutaneous squamous cell carcinoma (cSCC) is a common malignancy of the skin and has an overall favorable outcome, except for patients with an advanced stage of the disease. The efficacy of checkpoint inhibitors (CPI) for advanced cSCC has been demonstrated in recent clinical studies, but data from real-world cohorts and trial-ineligible cSCC patients are limited. We retrospectively investigated patients with advanced cSCC who have been treated with CPI in a first-line setting at eight German skin cancer centers registered within the multicenter registry ADOReg. Clinical outcome parameters including response, progression-free (PFS) and overall survival (OS), time-to-next-treatment (TTNT), and toxicity were analyzed and have been stratified by the individual immune status. Among 39 evaluable patients, the tumor response rate (rwTRR) was 48.6%, the median PFS was 29.0 months, and the median OS was not reached. In addition, 9 patients showed an impaired immune status due to immunosuppressive medication or hematological diseases. Our data demonstrated that CPI also evoked tumor responses among immunocompromised patients (rwTRR: 48.1 vs. 50.0%), although these responses less often resulted in durable remissions. In line with this, the median PFS (11 vs. 40 months, p = 0.059), TTNT (12 months vs. NR, p = 0.016), and OS (29 months vs. NR, p < 0.001) were significantly shorter for this patient cohort. CPI therapy was well tolerated in both subcohorts with 15% discontinuing therapy due to toxicity. Our real-world data show that first-line CPI therapy produced strong and durable responses among patients with advanced cSCC. Immunocompromised patients were less likely to achieve long-term benefit from anti-PD1 treatment, despite similar tumor response rates.
Solution of Cassels' Problem on a Diophantine Constant over Function Fields

This paper deals with the analogue of Inhomogeneous Diophantine Approximation in function fields. The inhomogeneous approximation constant of a Laurent series $\theta\in\mathbb F_q\left(\left(\frac{1}{t}\right)\right)$ with respect to $\gamma\in\mathbb F_q\left(\left(\frac{1}{t}\right)\right)$ is defined to be $c(\theta,\gamma)=\inf_{0\neq N\in\mathbb F_q\left[t\right]}|N|\cdot|\langle N\theta - \gamma \rangle|$. We show that for every $\theta$ there exists $\gamma$ such that $c(\theta,\gamma)\geq q^{-2}$, and find a sufficient condition on $\theta$ which forces $c(\theta,\gamma) \leq q^{-2}$ for every $\gamma$. Given $\theta$, we prove that the set $BA_{\theta}=\left\{\gamma\in\mathbb F_q\left(\left(\frac{1}{t}\right)\right)\;:\; c(\theta,\gamma)>0\right\}$ has full Hausdorff dimension. Our methods allow us to solve the case of vectors in $\mathbb F_q\left(\left(\frac{1}{t}\right)\right)^d$ as well. Our results offer a strengthening to analogues of results for real inhomogeneous approximation.
Conformally Kähler, Einstein–Maxwell geometry

On a given compact complex manifold or orbifold $(M,J)$, we study the existence of Hermitian metrics $\tilde g$ in the conformal classes of K\"ahler metrics on $(M,J)$, such that the Ricci tensor of $\tilde g$ is of type $(1,1)$ with respect to the complex structure, and the scalar curvature of $\tilde g$ is constant. In real dimension $4$, such Hermitian metrics provide a Riemannian counter-part of the Einstein--Maxwell (EM) equations in general relativity, and have been recently studied in \cite{ambitoric1, LeB0, LeB, KTF}. We show how the existence problem of such Hermitian metrics (which we call in any dimension {\it conformally K\"ahler, EM} metrics) fits into a formal momentum map interpretation, analogous to results by Donaldson and Fujiki~\cite{donaldson, fujiki} in the cscK case. This leads to a suitable notion of a Futaki invariant which provides an obstruction to the existence of conformally K\"ahler, EM metrics invariant under a certain group of automorphisms which are associated to a given K\"ahler class, a real holomorphic vector field on $(M,J)$, and a positive normalization constant. Specializing to the toric case, we further define a suitable notion of $K$-polystability and show it provides a (stronger) necessary condition for the existence of toric, conformally K\"ahler, EM metrics. We use the methods of \cite{ambitoric2} to show that on a compact symplectic toric $4$-orbifold with second Betti number equal to $2$, $K$-polystability is also a sufficient condition for the existence of (toric) conformally K\"ahler, EM metrics, and the latter are explicitly described as ambitoric in the sense of \cite{ambitoric1}. As an application, we exhibit many new examples of conformally K\"ahler, EM metrics defined on compact $4$-orbifolds, and obtain a uniqueness result for the construction in \cite{LeB0}.
Absolute continuity and weak uniform mixing of random walk in dynamic random environment

We prove results for random walks in dynamic random environments which do not require the strong uniform mixing assumptions present in the literature. We focus on the "environment seen from the walker"-process and in particular its invariant law. Under general conditions it exists and is mutually absolutely continuous to the environment law. With stronger assumptions we obtain for example uniform control on the density or a quenched CLT. The general conditions are made more explicit by looking at hidden Markov models or Markov chains as environment and by providing simple examples.
Simple weak modules for the fixed point subalgebra of the Heisenberg vertex operator algebra of rank $1$ by an automorphism of order $2$ and Whittaker vectors

Let $M(1)$ be the vertex operator algebra with the Virasoro element $\omega$ associated to the Heisenberg algebra of rank $1$ and let $M(1)^{+}$ be the subalgebra of $M(1)$ consisting of the fixed points of an automorphism of $M(1)$ of order $2$. We classify the simple weak $M(1)^{+}$-modules with a non-zero element $w$ such that for some integer $s\geq 2$, $\omega_i w\in{\mathbb C}w$ ($i=\lfloor s/2\rfloor+1,\lfloor s/2\rfloor+2,\ldots,s-1$), $\omega_{s}w\in{\mathbb C}^{\times}w$, and $\omega_i w=0$ for all $i>s$. The result says that any such simple weak $M(1)^{+}$-module is isomorphic to some simple weak $M(1)$-module or to some $\theta$-twisted simple weak $M(1)$-module.
Minimal Surfaces in Minimally Convex Domains

In this paper, we prove that every conformal minimal immersion of a compact bordered Riemann surface $M$ into a minimally convex domain $D\subset \mathbb{R}^3$ can be approximated, uniformly on compacts in $\mathring M=M\setminus bM$, by proper complete conformal minimal immersions $\mathring M\to D$. We also obtain a rigidity theorem for complete immersed minimal surfaces of finite total curvature contained in a minimally convex domain in $\mathbb{R}^3$, and we characterize the minimal surface hull of a compact set $K$ in $\mathbb{R}^n$ for any $n\ge 3$ by sequences of conformal minimal disks whose boundaries converge to $K$ in the measure theoretic sense.
Fourier transforms on $SL_2(\mathbb{Z}/p^n\mathbb{Z}) and related numerical experiments

We detail an explicit construction of ordinary irreducible representations for the family of finite groups $SL_2({\mathbb Z} /p^n {\mathbb Z})$ for odd primes $p$ and $n\geq 2$. For $n=2$, the construction is a complete set of irreducible complex representations, while for $n>2$, all but a handful are obtained. We also produce an algorithm for the computation of a Fourier transform for a function on $SL_2({\mathbb Z} /p^2 {\mathbb Z})$. With this in hand we explore the spectrum of a collection of Cayley graphs on these groups, extending analogous computations for Cayley graphs on $SL_2({\mathbb Z}/p {\mathbb Z})$ and suggesting conjectures for the expansion properties of such graphs.
Selective decay in fluids with advected quantities: MHD and Hall MHD

Modifications of the equations of ideal fluid dynamics with advected quantities are introduced for either the energy h or the Casimir quantities C in the Lie-Poisson formulation. The dissipated quantity (energy or Casimir, respectively) is shown to decrease in time until the modified system reaches a state that holds for ideal energy-Casimir equilibria, namely δ(h+C) = 0. The result holds for Lie-Poisson equations in general, independently of the Lie algebra and the choice of Casimir. This process is illustrated with a number of selected decay examples that pass to stable energy-Casimir equilibria for magnetohydrodynamics (MHD) and Hall MHD in 2D and 3D by decay of either the energy or the Casimirs.
City, Consumption and Interculturalism: How Cities Can Facilitate Consumer Acculturation

With the rise of interculturalism as an alternative paradigm to the dominant multicultural integration policies in immigration countries, the importance of cities, as landscapes of intercultural interactions and consumption has become more and more important. This study aims to investigate how cities and city-related consumption practices play a role in consumer acculturation, an area that is largely overlooked in previous research. A hermeneutic approach is used to analyse and interpret the data collected through semi-structured and unstructured go-along interviews with 18 Iranian immigrants who live in Dortmund, Germany. Beyond the dichotomy of the home and host countries, the findings of this study show how city-related activities and interactions can lead to the construction of a sense of belonging to the hosting society. We show how such a sense of belonging can be constructed through immigrant consumers’ involvement in city-related rituals, private appropriation of public space and reterritorialisation.
A classification of isomorphism-invariant random digraphs

We classify isomorphism-invariant random digraphs according to where randomness resides, namely, arcs, vertices, and vertices and arcs together which in turn yield arc random digraphs (ARD), vertex random digraphs (VRD) and vertex-arc random digraphs (VARD), respectively. This digraph classification can be viewed as an extension of the classification of isomorphism-invariant random graphs. We introduce randomness in the direction of the edges of a given graph and obtain direction random digraphs (DRD) as well. We classify DRDs according to which component is random in addition to the direction and study the relations of DRDs with VARDs, VRDs and ARDs. We also consider random nearest neighbor digraphs and determine their membership with respect to these digraph families.
Topological obstructions to nonnegative scalar curvature and mean convex boundary

We study topological obstructions to the existence of a Riemannian metric on manifolds with boundary such that the scalar curvature is non-negative and the boundary is mean convex. We construct many compact manifolds with boundary which admit no Riemannian metric with non-negative scalar curvature and mean convex boundary. For example, we show that the manifold $(T^{n-2}\times \Sigma )\# N$, where $\Sigma$ is a compact, connected and orientable surface which is not a disk or a cylinder and $N$ is a closed $n$-dimensional manifold, does not admit a metric of non-negative scalar curvature and mean convex boundary, and the manifold $(I\times T^{n-1})\#N$, where $I=[a,b]$, does not admit a metric of positive scalar curvature and mean convex boundary.
Bi-amalgamations subject to the arithmetical property

This paper establishes necessary and sufficient conditions for a bi-amalgamation to inherit the arithmetical property, with applications on the weak global dimension and transfer of the semihereditary property. The new results compare to previous works carried on various settings of duplications and amalgamations, and capitalize on recent results on bi-amalgamations. All results are backed with new and illustrative examples arising as bi-amalgamations.
A diffusive logistic problem with a free boundary in time-periodic environment: favorable habitat or unfavorable habitat

We study the diffusive logistic equation with a free boundary in timeperiodic environment. To understand the effect of the dispersal rate $d$, the original habitat radius $h_0$, the spreading capability $\mu$, and the initial density $u_0$ on the dynamics of the problem, we divide the time-periodic habitat into two cases: favorable habitat and unfavorable habitat. By choosing $d$, $h_0$, $\mu$ and $u_0$ as variable parameters, we obtain a spreading-vanishing dichotomy and sharp criteria for the spreading and vanishing in time-periodic environment. We introduce the principal eigenvalue $\lambda_1(d, \alpha, \gamma, h(t), T)$ to determine the spreading and vanishing of the invasive species. We prove that if $\lambda_1(d, \alpha, \gamma, h_0, T)\leq 0$, the spreading must happen; while if $\lambda_1(d, \alpha, \gamma, h_0, T)> 0$, the spreading is also possible. Our results show that the species in the favorable habitat can establish itself if the dispersal rate is slow or the occupying habitat is large. In an unfavorable habitat, the species vanishes if the initial density of the species is small, while survive successfully if the initial value is big. Moreover, when spreading occurs, the asymptotic spreading speed of the free boundary is determined.
Mechanical Constraint Effect on DNA Persistence Length

Persistence length is a significant criterion to characterize the semi-flexibility of DNA molecules. The mechanical constraints applied on DNA chains in new single-molecule experiments play a complex role in measuring DNA persistence length; however, there is a difficulty in quantitatively characterizing the mechanical constraint effects due to their complex interactions with electrostatic repulsions and thermal fluctuations. In this work, the classical buckling theory of Euler beam and Manning’s statistical theories of electrostatic force and thermal fluctuation force are combined for an isolated DNA fragment to formulate a quantitative model, which interprets the relationship between DNA persistence length and critical buckling length. Moreover, this relationship is further applied to identify the mechanical constraints in different DNA experiments by fitting the effective length factors of buckled fragments. Then, the mechanical constraint effects on DNA persistence lengths are explored. A good agreement among the results by theoretical models, previous experiments, and present molecular dynamics simulations demonstrates that the new superposition relationship including three constraint-dependent terms can effectively characterize changes in DNA persistence lengths with environmental conditions, and the strong constraint-environment coupling term dominates the significant changes of persistence lengths; via fitting effective length factors, the weakest mechanical constraints on DNAs in bulk experiments and stronger constraints on DNAs in single-molecule experiments are identified, respectively. Moreover, the consideration of DNA buckling provides a new perspective to examine the bendability of short-length DNA.