arXiv subjects

Yue Ren

Publications and source records attributed to Yue Ren.

At least 19 recordsLinked to original sources

arXiv2026

Non-Archimedean Polydisc Spaces and Applications to Optimisation

We propose a new framework for optimisation over non-Archimedean spaces inspired by Berkovich geometry. Specifically, we introduce polydisc spaces, which consists of products of closed balls over a non-Archimedean field. These spaces retain the rigid hierarchical structure of the non-Archimedean field whilst acquiring many desirable geometric features absent from it. We show that metric trees embed naturally into these spaces, demonstrating their capacity to represent hierarchical data. We study their metric geometry, establishing properties such as geodesic uniqueness, confirming their comaptibility with classical optimisation techniques. We further propose a class of real-valued functions given by linear combinations of absolute values of polynomials. These functions admit a piecewise polynomial description along geodesics and satisfy a universal approximation property. We formulate a theory of optimisation on polydisc spaces: we prove existence of minimisers and explore algorithms for finding them. We provide an accompanying open-source Julia library implementing the core objects and optimisation procedures introduced.

math.OC↗

arXiv2026

CardioLens: Revealing the Clinical Reality Gap of MLLMs via Multi-Sequence Cardiac MRI Evaluations

Multimodal Large Language Models (MLLMs) have shown strong performance on public medical benchmarks, yet existing evaluations often remain weak proxies for clinical use, relying on isolated inputs and simplified recognition-style tasks. We introduce CardioLens, a leakage-resistant evaluation testbed for multi-sequence Cardiovascular Magnetic Resonance (CMR), constructed from private hospital archives through a rigorous report-to-QA construction and verification pipeline. CardioLens contains 473,896 slices and 13,494 verified QA pairs across 4D Cine, LGE, perfusion, and T2-weighted imaging, and evaluates three stages of CMR interpretation: image understanding, report generation, and disease diagnosis. Across 24 state-of-the-art MLLMs, CardioLens reveals a substantial clinical reality gap: models perform poorly overall, with performance degrading along the real CMR workflow. Confusion analysis further shows a category-collapse failure mode, where models default to frequent abnormal categories rather than distinguishing clinically distinct findings. To rule out MLLM-compatible input construction as the primary cause, we compare random, clinically motivated, and data-driven slice selection protocols under different slice budgets; performance changes only marginally, typically by about 1%. Explicit reasoning prompts also fail to rescue performance, often making models more conservative rather than improving visual evidence use. These results show that current MLLMs remain far from reliable CMR interpretation, where clinical decisions require integrating distributed evidence across sequences, views, and temporal phases. CardioLens provides a clinically grounded testbed for developing next-generation MLLMs toward real-world clinical deployment.

cs.CV↗

arXiv2026

A lifting partition theorem for tropical tritangent classes to smooth space sextic curves

The set of tritangent planes to smooth tropical space sextic curves has 15 connected components, recording continuous displacements of planes preserving the tritangency condition. These 15 tritangent classes are polyhedral complexes in $\mathbf{R}^3$, and each of them contains the tropicalization of precisely eight tritangent planes to any smooth space sextic curve with the given tropicalization. Prior joint work of the authors with Len confirms that each tropical tritangent plane has 0, 1, 2, 4 or 8 lifts to classical tritangent planes defined over the algebraic closure of the field over which the original algebraic curve is defined. Our main theorem states that when the input classical curve is generic, then only six out of the ten possible partitions of 8 into powers of 2 arise from lifting multiplicities of tritangent classes. Furthermore, we show that these partitions are completely determined by the dimension of a suitable connected subcomplex of the class and the existence of a member with a tropical tangency of a predetermined combinatorial type.

math.AG↗

arXiv2026

Root bounds of vertical systems using tropical geometry

Sparse polynomial systems with vertical coefficient dependencies arise naturally when describing the critical points of optimization problems and, when augmented with linear forms, the steady states of chemical reaction networks. Moreover, any polynomial system is the specialization of such a parametrized system. We prove that the generic number of complex zeros of an augmented vertically parametrized system is the tropical intersection number of a tropical linear space and a classical linear space. In the special case when the matroid of the tropical linear space is cotransversal, we express this number as a mixed volume. We also obtain bounds on the maximal number of positive zeros, which is often the significant number in applications. We derive lower bounds from the number of intersections between positive tropicalizations, and when the positive zeros have toric structure, we provide upper bounds that are simpler and in some cases smaller than the generic root count. The resulting algorithms are implemented in Julia.

math.AG↗

arXiv2025

Tropical methods for building real space sextics with totally real tritangent planes

This paper proposes the use of combinatorial techniques from tropical geometry to build the 120 tritangent planes to a given smooth algebraic space sextic. Although the tropical count is infinite, tropical tritangents come in 15 equivalence classes, each containing the tropicalization of exactly eight classical tritangents. Under mild genericity conditions on the tropical side, we show that liftings of tropical tritangents are defined over quadratic extensions of the ground field over which the input sextic curve is defined. When the input curve is real, we prove that every complex liftable member of a given tropical tritangent class either completely lifts to the reals or none of its liftings are defined over the reals. As our main application we use these methods to build examples of real space sextics with 64 and 120 totally real tritangents, respectively. The paper concludes with a discussion of our results in the arithmetic setting.

math.AG↗

arXiv2025

Zero-dimensional tropicalizations in OSCAR

We present algorithms for computing zero-dimensional tropical varieties as implemented in OscarZerodimensionalTropicalization.jl. The algorithms include a mathematical workaround for a common practical issue arising when working with polynomials over inexact fields in existing software systems.

math.AG↗

arXiv2025

The tropical galaxy of a Laman graph

A Laman graph $G$ is a minimally rigid graph in dimension two, and its realization number is its number of distinct embeddings with fixed generic edge lengths. While conjectured to grow exponentially in the number of vertices of $G$, the best proven lower bound is merely $2$. Motivated by the fact that the realization number can be expressed as a tropical intersection product involving $\mathrm{Trop}(G)$, the Bergman fan of the graphic matroid of $G$, and the fact that stars of $\mathrm{Trop}(G)$ naturally lead to lower bounds thereof, we introduce the tropical galaxy of $G$ together with a galactic pairing thereon. We study structural properties of this pairing, such as under which conditions it is non-trivially subadditive, and connect it being non-zero to arboreal pairs. We also present a software package for working with tropical galaxies.

math.CO↗

arXiv2025

A tropical approach to rigidity: counting realisations of frameworks

A realisation of a graph in the plane as a bar-joint framework is rigid if there are finitely many other realisations, up to isometries, with the same edge lengths. Each of these finitely-many realisations can be seen as a solution to a system of quadratic equations prescribing the distances between pairs of points. For generic realisations, the size of the solution set depends only on the underlying graph so long as we allow for complex solutions. We provide a characterisation of the realisation number - that is the cardinality of this complex solution set - of a minimally rigid graph. Our characterisation uses tropical geometry to express the realisation number as an intersection of Bergman fans of the graphic matroid. As a consequence, we derive a combinatorial upper bound on the realisation number involving the Tutte polynomial. Moreover, we provide computational evidence that our upper bound is usually an improvement on the mixed volume bound.

math.CO↗

arXiv2024

Embedding polynomial systems into vertically parametrised families: A case study on ODEbase

Vertically parametrised polynomial systems are a particular nice class of parametrised polynomial systems for which a lot of interesting algebraic information is encoded in its combinatorics. Given a fixed polynomial system, we empirically study what constitutes a good vertically parametrised polynomial system that gives rise to it and how to construct said vertically parametrised polynomial system. For data, we use all polynomial systems in ODEbase, which we have transcribed to an OSCAR readable format, and made available as a Julia package OscarODEbase.

math.AG↗

arXiv2024

A tropical method for solving parametrized polynomial systems

We give a framework for constructing generically optimal homotopies for parametrized polynomial systems from tropical data. Here, generically optimal means that the number of paths tracked is equal to the generic number of solutions. We focus on two types of parametrized systems -- vertically parametrized and horizontally parametrized systems -- and discuss techniques for computing the tropical data efficiently. We end the paper with several case studies, where we analyze systems arising from chemical reaction networks, coupled oscillators, and rigid graphs.

math.AG↗

arXiv2023

Generic root counts of tropically transverse systems -- An invitation to tropical geometry in OSCAR

The main mathematical focus of this paper is a class of parametrised polynomial systems that we refer to as being tropically transverse. We show how their generic number of solutions can be expressed as the mixed volume of a modified system. We then provide an alternate proof of a recent result by Borovik et al, on the number of equilibria of coupled nonlinear oscillators using elementary tropical geometry. The proof draws upon a wide range of concepts across tropical geometry, which we will use as an opportunity to give a first overview over various tropical features in OSCAR.

math.AG↗

arXiv2023

Hands-on Tropical Geometry

This is a guide on how to create 3d printable models of tropical surfaces, curves, and combinations thereof. It uses Polymake to construct bounded models of the tropical objects, and OpenSCAD to thicken and export them to any common 3D printable file format.

math.AG↗

arXiv2023

On intersections and stable intersections of tropical hypersurfaces

We prove that every connected component of an intersection of tropical hypersurfaces contains a point of their stable intersection unless their stable intersection is empty. This is done by studying algebraic hypersurfaces that tropicalize to them and the tropicalization of their intersection.

math.CO↗

arXiv2022

Sharp bounds for the number of regions of maxout networks and vertices of Minkowski sums

We present results on the number of linear regions of the functions that can be represented by artificial feedforward neural networks with maxout units. A rank-k maxout unit is a function computing the maximum of $k$ linear functions. For networks with a single layer of maxout units, the linear regions correspond to the upper vertices of a Minkowski sum of polytopes. We obtain face counting formulas in terms of the intersection posets of tropical hypersurfaces or the number of upper faces of partial Minkowski sums, along with explicit sharp upper bounds for the number of regions for any input dimension, any number of units, and any ranks, in the cases with and without biases. Based on these results we also obtain asymptotically sharp upper bounds for networks with multiple layers.

math.CO↗

arXiv2022

Generic root counts and flatness in tropical geometry

We use tropical and non-archimedean geometry to study the generic number of solutions of families of polynomial equations over a parameter space $Y$. In particular, we are interested in the choices of parameters for which the generic root count is attained. Our families are given as subschemes $X\subseteq T$ where $T$ is a relative torus over $Y$. We generalize Bernstein's theorem from an intersecting family of hypersurfaces $X=V(f_1)\cap\dots\cap V(f_n)$ to an intersecting family of higher-codimensional schemes $X=X_1\cap\dots\cap X_k$, replacing the mixed volume by a tropical intersection product. Central to our work is the notion of tropical flatness of $X$ around a point $P\in Y$, which allows us to transfer tropical properties of the fiber over $P$ to generic properties. We show that tropical flatness holds over a dense open subset of the Berkovich analytification $Y^\text{an}$, and that the tropical intersection number is attained as a root count at all $P\in Y^\text{an}$ around which the $X_i$'s are tropically flat and the tropical prevariety of the fibers $\bigcap_{i=1}^k\text{Trop}(X_{i,P})$ is bounded. We then study the generic root count of a wide class of parametrized square polynomial systems. This in particular gives tropical formulas for the volumes of Newton-Okounkov bodies, and the number of complex steady states of chemical reaction networks.

math.AG↗

arXiv2021

Design and Characterization of Self Lubricating Refractory High Entropy Alloy Based Multilayered Films

Refractory high entropy alloys (RHEA) have been proven to have excellent mechanical properties with a potential use as protective thin films. However, the combination of high hardness with low friction and wear is a major challenge in the design of self lubricating RHEA films. In this study, we show that designing of NbMoWTa/Ag multilayered films give a remarkable reduction in friction and at same time maintain high hardness. Interestingly, it's found that the bcc superlattice dominates in both layers and the interfaces are highly coherent when the individual layer thickness h is reduced below 10 nm. The film properties are then strongly dependent on h ranging from 100 to 2.5 nm, and the most promising properties are obtained when the interface structure transforms from incoherent to coherent ones. Specially, the multilayer with h = 2.5 nm exhibits superior tribological performance over monolithic NbMoWTa, due to the significant coherent strengthening along with the self-lubricating ability in the multilayer. This tailored phase transition and coherent structure between matrix and lubrication phases can also provide an optimal wear rate-coefficient of friction (COF) combination, which is higher than most of the Ag containing self lubricating films. The current work might open a new route towards the development of innovative self lubricating RHEA films with excellent tribological properties in general.

cond-mat.mtrl-sci↗

arXiv2021

Role of a fractal shape of the inclusions on acoustic attenuation in a nanocomposite

Nanophononic materials are promising to control the transport of sound in the GHz range and heat in the THz range. Here we are interested in the influence of a dendritic shape of inclusion on acoustic attenuation. We investigate a Finite Element numerical simulation of the transient propagation of an acoustic wave-packet in 2D nanophononic materials with circular or dendritic inclusions periodically distributed in matrix. By measuring the penetration length, diffusivity, and instantaneous wave velocity, we find that the multi-branching tree-like form of dendrites provides a continuous source of phonon-interface scattering leading to an increasing acoustic attenuation. When the wavelength is far less than the inter-inclusion distance, we report a strong attenuation process in the dendritic case which can be fitted by a compressed exponential function with $β>1$.

cond-mat.mes-hall↗

arXiv2020

Parallel Computation of tropical varieties, their positive part, and tropical Grassmannians

In this article, we present a massively parallel framework for computing tropicalizations of algebraic varieties which can make use of finite symmetries. We compute the tropical Grassmannian TGr$_0(3,8)$, and show that it refines the $15$-dimensional skeleton of the Dressian Dr$(3,8)$ with the exception of $23$ special cones for which we construct explicit obstructions to the realizability of their tropical linear spaces. Moreover, we propose algorithms for identifying maximal-dimensional tropical cones which belong to the positive tropicalization. These algorithms exploit symmetries of the tropical variety even though the positive tropicalization need not be symmetric. We compute the maximal-dimensional cones of the positive Grassmannian TGr$^+(3,8)$ and compare them to the cluster complex of the classical Grassmannian Gr$(3,8)$.

math.AG↗