In this paper, we study tropical Weierstrass points. These are the analogues for tropical curves of ramification points of line bundles on algebraic curves. For a divisor on a tropical curve, we associate intrinsic weights to the connected components of the locus of tropical Weierstrass points. These are obtained by analyzing the slopes of rational functions in the complete linear series of the divisor. We prove that for a divisor $D$ of degree $d$ and rank $r$ on a genus $g$ tropical curve, the sum of weights is equal to $d - r + rg$. We establish analogous statements for tropical linear series. In the case $D$ comes from the tropicalization of a divisor, these weights control the number of Weierstrass points that are tropicalized to each component. Our results provide answers to open questions originating from the work of Baker on specialization of divisors from curves to graphs. We conclude with multiple examples that illustrate interesting features appearing in the study of tropical Weierstrass points, and raise several open questions.

Inspired by work of Besson-Courtois-Gallot, we construct a flow called the natural flow on a non-positively curved Riemannian manifold $M$. As with the natural map, the $k$-Jacobian of the natural flow is directly related to the critical exponent $δ$ of the fundamental group. There are several applications of the natural flow that connect dynamical, geometrical, and topological invariants of the manifold. First, we give $k$-dimensional linear isoperimetric inequalities when $k > δ$. This, in turn, produces lower bounds on the Cheeger constant. We resolve a recent conjecture of Dey-Kapovich on the non-existence of $k$-dimensional compact, complex subvarieties of complex hyperbolic manifolds with $2k > δ$. We also provide upper bounds on the homological dimension, generalizing work of Kapovich and work of Farb with the first two authors. Using the natural flow together with Morse theory, we also give upper bounds on the cohomological dimension, which partially resolve a conjecture of Kapovich. Finally, we introduce a new growth condition on the Bowen-Margulis measure that we call uniformly exponentially bounded that we connect to the cohomological dimension and which could be of independent interest.

We introduce a new compactness principle which we call the gluing property. For a measurable cardinal $κ$ and a cardinal $λ$, we say that $κ$ has the $λ$-gluing property if every sequence of $λ$-many $κ$-complete ultrafilters on $κ$ can be glued into a $κ$-complete extender. We show that every $κ$-compact cardinal has the $2^κ$-gluing property, yet non-necessarily the $(2^κ)^+$-gluing property. Finally, we compute the exact consistency-strength for $κ$ to have the $ω$-gluing property; this being $o(κ)=ω_1$.

We formulate a uniform tail bound for empirical processes indexed by a class of functions, in terms of the individual deviations of the functions rather than the worst-case deviation in the considered class. The tail bound is established by introducing an initial ``deflation'' step to the standard generic chaining argument. The resulting tail bound is the sum of the complexity of the ``deflated function class'' in terms of a generalization of Talagrand's $γ$ functional, and the deviation of the function instance, both of which are formulated based on the natural seminorm induced by the corresponding Cramér functions. Leveraging another less demanding natural seminorm, we also show similar bounds, though with implicit dependence on the sample size, in the more general case where finite exponential moments cannot be assumed. We also provide approximations of the tail bounds in terms of the more prevalent Orlicz norms or their ``incomplete'' versions under suitable moment conditions.

We consider the CY orbifolds connected with the products of N = (2, 2) supersymmetric minimal models. We use the spectral flow construction of states, as well as its mirror one, to explicitly show that these CY orbifolds are canonically combined into mirror pairs of isomorphic models, associated to the mutually dual pairs of admissible groups of Berglund- Hubsh-Krawitz.

In this work we present a new approach to constructing Calabi-Yau orbifold models required for compactification in superstring theory. We use the connection of CY orbifolds with the class of exactly solvable N=2 SCFT models to explicitly construct a complete set of fields in these models using the twisting of the spectral flow and the requirement of mutual locality of the fields.

Given a fibration over a perfect field of positive characteristic, we study an Iitaka-type inequality for the anticanonical divisors. We conclude that it holds when the source of the fibration is a threefold or when the target is a curve, the general fibre is regular and the pair induced on it from the ambient space is strongly F-regular. We then give counterexamples in characteristics 2 and 3 for fibrations with non-normal fibres, constructed from Tango--Raynaud surfaces.

In the frequency power spectral density, periodic oscillations appear as a Dirac comb at integer multiples of the frequency of the period. In weakly nonlinear systems or systems close to the primary instability threshold, the periodicity may be perturbed, resulting in deviations from the Dirac comb. We review and discuss a stochastic model of such semiperiodic fluctuations, while also providing several new results which widen the applicability of the model. The fluctuations are described as a superposition of pulses with a fixed shape. Closed form expressions are derived for the frequency power spectral density in the case of periodic pulse arrivals and a random distribution of pulse amplitudes. In general, the spectrum is a Dirac comb located at multiples of the inverse periodicity time and modulated by the pulse spectrum. Deviations from strict periodicity in the arrivals are considered in two ways: either as a random offset to each periodic arrival (jitter) or as independently distributed waiting times between arrivals (renewal). In this contribution, we show that both ways of including deviations from periodicity remove the Dirac comb with remarkable efficiency, leaving mainly the spectrum of the pulse function. Where the jitter process modulates the mass of the higher harmonics, the renewal process leads to spectral broadening. We clarify the effects of random pulse amplitudes on the frequency power spectrum and demonstrate the applicability of normally distributed waiting times to modeling. Contrary to the previous literature, we argue that negative waiting times do not pose problems for the theory, broadening the applicability of the normal approximation. Randomness in the pulse arrival times is investigated by numerical realizations of the process, and the model is used to describe time series of kinetic energy of fluctuating motions in two-dimensional thermal convection.

The Mordell--Lang conjecture for abelian varieties states that the intersection of an algebraic subvariety $X$ with a subgroup of finite rank is contained in a finite union of cosets contained in $X$. In this article, we prove a uniform version of this conjecture, meaning that that the number of cosets necessary does not depend on the ambient abelian variety. To achieve this, we prove a general gap principle on algebraic points that extends the gap principle for curves embedded into their Jacobians, previously obtained by Dimitrov--Gao--Habegger and Kühne. Our new gap principle also implies the full uniform Bogomolov conjecture in abelian varieties.

Cryo-Electron Tomography (cryo-ET) is a new 3D imaging technique with unprecedented potential for resolving submicron structural detail. Existing volume visualization methods, however, cannot cope with its very low signal-to-noise ratio. In order to design more powerful transfer functions, we propose to leverage soft segmentation as an explicit component of visualization for noisy volumes. Our technical realization is based on semi-supervised learning where we combine the advantages of two segmentation algorithms. A first weak segmentation algorithm provides good results for propagating sparse user provided labels to other voxels in the same volume. This weak segmentation algorithm is used to generate dense pseudo labels. A second powerful deep-learning based segmentation algorithm can learn from these pseudo labels to generalize the segmentation to other unseen volumes, a task that the weak segmentation algorithm fails at completely. The proposed volume visualization uses the deep-learning based segmentation as a component for segmentation-aware transfer function design. Appropriate ramp parameters can be suggested automatically through histogram analysis. Finally, our visualization uses gradient-free ambient occlusion shading to further suppress visual presence of noise, and to give structural detail desired prominence. The cryo-ET data studied throughout our technical experiments is based on the highest-quality tilted series of intact SARS-CoV-2 virions. Our technique shows the high impact in target sciences for visual data analysis of very noisy volumes that cannot be visualized with existing techniques.

Contemporary software systems (CSS), such as the internet of things (IoT) based software systems, incorporate new concerns and characteristics inherent to the network, software, hardware, context awareness, interoperability, and others, compared to conventional software systems. In this sense, requirements engineering (RE) plays a fundamental role in ensuring these software systems' correct development looking for the business and end-user needs. Several software technologies supporting RE are available in the literature, but many do not cover all CSS specificities, notably those based on IoT. This research article presents RETIoT (Requirements Engineering Technology for the Internet of Things based software systems), aiming to provide methodological, technical, and tooling support to produce IoT software system requirements document. It is composed of an IoT scenario description technique, a checklist to verify IoT scenarios, construction processes, and templates for IoT software systems. A feasibility study was carried out in IoT system projects to observe its templates and identify improvement opportunities. The results indicate the feasibility of RETIoT templates' when used to capture IoT characteristics. However, further experimental studies represent research opportunities, strengthen confidence in its elements (construction process, techniques, and templates), and capture end-user perception.

Skew completable unimodular rows of odd length are completable over polynomial extension of a local ring if dimension of local ring and length of unimodular rows are same.

Geodesics in the space of positive Lagrangian submanifolds are solutions of a fully non-linear degenerate elliptic PDE. We show that a geodesic segment in the space of positive Lagrangians corresponds to a one parameter family of special Lagrangian cylinders, called the cylindrical transform. The boundaries of the cylinders are contained in the positive Lagrangians at the ends of the geodesic. The special Lagrangian equation with positive Lagrangian boundary conditions is elliptic and the solution space is a smooth manifold, which is one dimensional in the case of cylinders. A geodesic can be recovered from its cylindrical transform by solving the Dirichlet problem for the Laplace operator on each cylinder. Using the cylindrical transform, we show the space of pairs of positive Lagrangian spheres connected by a geodesic is open. Thus, we obtain the first examples of strong solutions to the geodesic equation in arbitrary dimension not invariant under isometries. In fact, the solutions we obtain are smooth away from a finite set of points.

In this article, we prove that if $R$ is a finitely generated ring over $\mathbb{Z}$ of dimension $d, d\geq2, \frac{1}{d!}\in R$, then any unimodular row over $R[X]$ of length $d+1$ can be mapped to a factorial row by elementary transformations.

It is shown that Kazama-Suzuki conditions for the denominator subgroup of N=2 superconformal $G/H$ coset model determine Generalized K$\ddot{a}$hler geometry on the target space of the corresponding N=2 supersymmetric $σ$-model.

A smoothness-increasing accuracy conserving filtering approach to the regularization of discontinuities is presented for single domain spectral collocation approximations of hyperbolic conservation laws. The filter is based on convolution of a polynomial kernel that approximates a delta-sequence. The kernel combines a $k^{th}$ order smoothness with an arbitrary number of ${m}$ zero moments. The zero moments ensure a $m^{th}$ order accurate approximation of the delta-sequence to the delta function. Through exact quadrature the projection error of the polynomial kernel on the spectral basis is ensured to be less than the moment error. A number of test cases on the advection equation, Burger's equation and Euler equations in 1D and 2D shown that the filter regularizes discontinuities while preserving high-order resolution

For any $s \in \mathbb{C}$ with $\Re(s)>0$, denote by $η_{n-1}(s)$ the $(n-1)^{th}$ partial sum of the Dirichlet series for the eta function $η(s)=1-2^{-s}+3^{-s}-\cdots \;$, and by $R_n(s)$ the corresponding remainder. Denoting by $u_n(s)$ the segment starting at $η_{n-1}(s)$ and ending at $η_n(s)$, we first show how, for sufficiently large $n$ values, the circle of diameter $u_{n+2}(s)$ lies strictly inside the circle of diameter $u_n(s)$, to then derive the asymptotic relationship $R_n(s) \sim (-1)^{n-1}/n^s$, as $n \rightarrow \infty$. Denoting by $D=\left\{s \in \mathbb{C}: \; 0< \Re(s) < \frac{1}{2}\right\}$ the open left half of the critical strip, define for all $s\in D$ the ratio $χ_n^{\pm}(s) = η_n(1-s) / η_n(s)$. We then prove that the limit $L(s)=\lim_{N(s)<n\to\infty} χ_n^{\pm}(s)$ exists at every point $s$ of the domain $D$. The function $L(s)$ is continuous on $D$ if and only if the Riemann Hypothesis is true. Finally, we remark how the asymptotic behaviour of $R_n(s)$ can also provide insights substantiating the so called Simple Zeros Conjecture.

This is a review article of my recent papers on free field construction of D-branes in N = 2 superconformal minimal models and Gepner models.

Let $k$ be a nonarchimedean local field. For any $n\geq 3$, we construct the first examples of robust quasi-isometric embeddings of non-elementary free groups into $\mathsf{GL}_n(k)$ which are not limits of Anosov representations. If $\bf{K}=\mathbb{R},\mathbb{C}$, we exhibit examples of non-locally rigid, robust quasi-isometric embeddings of virtually free groups into $\mathsf{GL}_n(\bf{K})$, $n\geq 3$, which are not limits of Anosov representations. Moreover, we exhibit a non-Anosov robust quasi-isometric embedding of the free semigroup $\mathbb{Z}\ast \mathbb{Z}^{+}$ into $\mathsf{GL}_3(\mathbb{C})$, which is a limit of Anosov representations.

We prove that for any bounded convex domain $Ω\subset \mathbb{R}^n$, the function \begin{equation*} ψ_Ω(ξ) = \int_{\mathbb{R}^n\setminusΩ} \frac{\mathrm{d}x}{|x-ξ|^{2n}}, \quad ξ\inΩ, \end{equation*} has exactly one critical point. This confirms an conjecture proposed by Clapp, Pistoia and Saldaña in [J. Math. Pures Appl. 205 (2026), 103783]. The proof uses a spherical coordinates representation to write $ψ_Ω$ as an integral of the distance function $ρ(ξ,ω)$. This approach is not limited to $ψ_Ω$. Instead, it provides a general framework for analyzing a broad class of functionals involving the boundary distance. We also examine non-convex domains. In particular, a single annulus exhibits a full circle of critical points, while multiple concentric annuli produce finitely many critical spheres. These examples show that the convexity hypothesis is essential for the uniqueness conclusion. The method developed here for handling spherical integrals involving the distance function is likely to be useful in other geometric and analytic contexts.

Random subsampling of edges is a commonly employed technique in graph algorithms, underlying a vast array of modern algorithmic breakthroughs. Unfortunately, using this technique often leads to randomized algorithms with no clear path to derandomization because the analyses rely on a union bound on exponentially many events. In this work, we revisit this goal of derandomizing randomized sampling in graphs. We give several results related to bounded-independence edge subsampling, and in the process of doing so, generalize several of the results of Alon and Nussboim (FOCS 2008), who studied bounded-independence analogues of random graphs (which can be viewed as edge subsamples of the complete graph). Most notably, we show: 1. $O(\log(m))$-wise independence suffices for preserving connectivity when sampling at rate $1/2$ in a graph with minimum cut $\geq κ\log(m)$ with probability $1 - \frac{1}{\mathrm{poly}(m)}$ (for a sufficiently large constant $κ$). 2. $O(\log(m))$-wise $\frac{1}{\mathrm{poly}(m)}$-almost independence suffices for ensuring cycle-freeness when sampling at rate $1/2$ in a graph with minimum cycle length $\geq κ\log(m)$ with probability $1 - \frac{1}{\mathrm{poly}(m)}$ (for a sufficiently large constant $κ$). To demonstrate the utility of our results, we revisit the classic problem of using parallel algorithms to find graphic matroid bases, first studied in the work of Karp, Upfal, and Wigderson (FOCS 1985). In this regime, we show that the optimal algorithms of Khanna, Putterman, and Song (arxiv 2025) can be explicitly derandomized while maintaining near-optimality.

Retrieval-Augmented Generation (RAG) enhances Large Language Models (LLMs) with external knowledge but remains vulnerable to low-authority sources that can propagate misinformation. We investigate whether LLMs can perceive information authority - a capability extending beyond semantic understanding. To address this, we introduce AuthorityBench, a comprehensive benchmark for evaluating LLM authority perception comprising three datasets: DomainAuth (10K web domains with PageRank-based authority), EntityAuth (22K entities with popularity-based authority), and RAGAuth (120 queries with documents of varying authority for downstream evaluation). We evaluate five LLMs using three judging methods (PointJudge, PairJudge, ListJudge) across multiple output formats. Results show that ListJudge and PairJudge with PointScore output achieve the strongest correlation with ground-truth authority, while ListJudge offers optimal cost-effectiveness. Notably, incorporating webpage text consistently degrades judgment performance, suggesting authority is distinct from textual style. Downstream experiments on RAG demonstrate that authority-guided filtering largely improves answer accuracy, validating the practical importance of authority perception for reliable knowledge retrieval. Code and benchmark are available at: https://github.com/Trustworthy-Information-Access/AuthorityBench.

Beamsplitters represent fundamental components in both classical and quantum optical systems, enabling the distribution of light, as well as the generation of interference, superposition and entanglement. However, optical networks constructed from conventional bulk 2x2-beamsplitters encounter inherent scalability issues, as the number of required beamsplitters scales quadratically with the number of optical modes for a fully connected network. Metasurfaces offer a promising route to overcome these constraints. By manipulating light at the wavelength scale compact optical components with advanced functionalities can be constructed, which address several modes simultaneously. In this work, we design and experimentally utilize a metasurface as a multiport beamsplitter. Furthermore, we realize a multimode interferometer composed of two cascaded metasurfaces. We characterize the individual and cascaded metasurfaces using classical light, showing controllable splitting ratios through tunable phase relations. We then expand the approach to quantum light, employing single photons to demonstrate second- and third-order photon correlations, as well as single photon interference across multiple spatial paths. These results establish metasurface-based multiport beamsplitters as a scalable and reconfigurable platform bridging classical and quantum photonics.

For the de Rham mapping cone cochain complex induced by a smooth closed 2-form, we explicitly write down the associated mapping cone Thom form in the sense of Mathai-Quillen. Our construction uses the mapping cone covariant derivative, carrying the extra information brought by the 2-form. Our main tool is the Berezin integral. As the main result, we show that this Thom form is closed with respect to the mapping cone differentiation, its integration along the fiber is 1, and it satisfies the transgression formula.

We develop a Euclidean path-integral control to characterize optimal firm behavior in an economy governed by Walrasian equilibrium, Pareto efficiency, and non-cooperative Markovian feedback Nash equilibrium. The approach recasts the problem as a Lagrangian stochastic control system with forward-looking dynamics, thereby avoiding the explicit construction of a value function. Instead, optimal policies are obtained from a continuously differentiable Ito process generated through integrating factors, which yields a tractable alternative to conventional solution methods for complex market environments. This construction is useful in settings with nonlinear stochastic differential equations where standard Hamilton-Jacobi-Bellman (HJB) formulations are difficult to implement. Consistent with Feynman-Kac-type representations, the resulting solutions need not be unique. In economies with a large number of firms, the analysis admits a natural comparison with mean-field game formulations. Our main contribution is to derive a noncooperative feedback Nash equilibrium within this path-integral setting and to contrast it with outcomes implied by mean-field interactions. Several examples illustrate the method's applicability and highlight differences relative to solutions based on the Pontryagin maximum principle generated by HJB.

The AMS-02 experiment plans to install a new silicon microstrip tracker layer (Layer-0) on top of the existing detector, increasing the cosmic-ray acceptance by a factor of 3. Layer-0 employs a design in which multiple silicon microstrip detectors (SSDs) are connected in series to form long detector ladders. We present a detailed performance study of the flight-model ladders using a 350~GeV mixed hadron beam at the CERN SPS. The study focuses on the following aspects: (i) the performance of ladders with different numbers of SSDs, for which the intrinsic spatial resolution at normal incidence varies from $9.5~μ\mathrm{m}$ to $11.4~μ\mathrm{m}$ for ladders composed of 8 to 12 SSDs; (ii) the response consistency for particles impacting on the \emph{Head} and \emph{Tail} regions of the ladder; and (iii) the dependence of the detector performance on the particle incidence angle.

We construct an example of an asymptotically conical (AC) non-Kähler expanding gradient Ricci soliton that has a Kähler tangent cone at infinity. This yields an example of a Kähler cone that can be desingularised by a smooth AC expanding gradient Ricci soliton but not by a smooth AC expanding gradient Kähler--Ricci soliton.

A high-granularity telescope system with a large sensitive area and low material budget has been developed for high-energy heavy ion beam tests. The telescope consists of nine layers of silicon microstrip detectors (SSDs), whose performance was validated through a heavy ion beam test at the CERN SPS. A hybrid machine learning algorithm is proposed to address the challenges of nuclear charge measurement with SSDs. The system achieves a spatial resolution of $\mathcal{O}(1) \,$\SI{}{\micro\metre} and a charge resolution better than 0.16 charge units for nuclei from $Z = 1$ to $Z = 29$, with a sensitive area of $8 \times 8 \, \mathrm{cm}^2$. To the best of our knowledge, this represents the most precise charge and spatial resolution simultaneously achieved by a silicon telescope to date.

The notion of $δ$-Novikov algebras was introduced recently as a generalization of Novikov and bicommutative algebras. It looks like $δ$-Novikov algebras have a richer structure than Novikov algebras. So, unlike Novikov algebras, they have a $2$-dimensional simple algebra for $δ=-1.$ The present paper is dedicated to the study of $3$-dimensional $δ$-Novikov algebras for $δ\notin \big\{0,1\big\}.$ The algebraic and geometric classifications of complex $3$-dimensional $δ$-Novikov algebras are given. As a corollary, we prove that there are no simple $3$-dimensional $δ$-Novikov algebras.

The scientific potential of X-ray polarimetry has long been recognized, but the challenges in measuring polarization have left it largely unexplored, particularly in the hard X-ray regime. While tremendous advancement has been made in soft X-ray polarimetery, the lack of sensitive hard X-ray polarimeters and polarisation measurements continues to limit our understanding of high-energy astrophysical processes. With the development of hard X-ray mirrors, it is now possible to develop a sensitive focal plane hard X-ray polarimeter. One such effort is CXPOL, a prototype developed at PRL, India, which consists of a plastic scintillator as active scatterer readout by PMT surrounded by CsI(Tl) scintillators in cylindrical array with SiPM readout from one side. First results of the prototype have been demonstrated in 20 to 80 keV energy range. The sensitivity of the instrument can be significantly enhanced using faster and better light yield scintillator like NaI as absorbers. Further, the use of a position-sensitive scatterer and absorbers, can also provide spectroscopic information by measuring the interaction position along the length and from the known energy depositions in the detectors. Position sensitive detectors are also helpful in mitigating the systematic effects introduced by the off-axis events in the polarisation measurements. Here, we demonstrate the detection sensitivity in the 100x20x5 mm^3 NaI(Tl) scintillator absorber readout on both ends by SiPM arrays operating in co-incidence. In this work, we characterize the first prototype of this detector system and investigate the variation in energy and position resolution, and light output with irradiation position along the length of the detector. The two end readout in co-incidence also reduces the overall SiPM background per absorber by an order of magnitude, further enhancing the polarimetric sensitivity of the instrument.