arXiv subjects

Donghan Kim

Publications and source records attributed to Donghan Kim.

At least 19 recordsLinked to original sources

arXiv2026

Partition invariance of variation indices and classical $p$-variation spaces

We study the variation index of a continuous path along a refining partition sequence, defined as the infimum of exponents $p\ge1$ for which the corresponding $p$-th variation sums are uniformly bounded. Under natural geometric assumptions on the partitions, we prove that this index coincides with the classical variation index, defined using all finite partitions, for every path with positive H\"older regularity. The result extends to paths with an all-orders Dini modulus and yields a characterization of the classical variation index in terms of dyadic Faber--Schauder coefficients. Explicit counterexamples show that the partition assumptions cannot in general be omitted. Even under these assumptions, however, membership in the corresponding function spaces at the critical exponent may still depend on the partition sequence. We further obtain compact subcritical embeddings into the space of paths with vanishing classical $p$-variation and establish a strict hierarchy of coefficient and variation spaces. Finally, we show that the continuous embedding of the classical $p$-variation space into the space of continuous paths with uniformly bounded dyadic $p$-th variation sums has nonclosed range.

math.FA↗

arXiv2026

The Gromov-Hausdorff Distance Between Consecutive Spheres

We determine the Gromov-Hausdorff distance between consecutive unit round spheres equipped with their geodesic metrics. Put $\zeta_n:=\arccos(-\tfrac{1}{n+1}),$ the common geodesic distance between distinct vertices of a regular simplex with $n+2$ vertices inscribed in $\mathbb{S}^n$. We prove that $$ d_{\mathrm{GH}}(\mathbb{S}^n,\mathbb{S}^{n+1})=\frac{\zeta_n}{2} \qquad(n\geq1), $$ resolving a conjecture of Lim, M\'emoli, and Smith. All cases $n\geq4$ were previously open. This equality is established by explicitly constructing a family of correspondences $\mathcal R_n\subseteq \mathbb{S}^{n+1}\times \mathbb{S}^n$, whose distortion matches the known quantitative Borsuk-Ulam lower bound $\zeta_n$. We also introduce synchronized spherical joins and suspensions of correspondences and prove that the distortion of a join is exactly the maximum of the distortions of its factors. In particular, suspension preserves distortion. Applying these join and suspension operations to the optimal correspondences $\mathcal R_n$ yields new bounds for spheres of nonconsecutive dimensions, including $$ \lim_{m\to\infty} d_{\mathrm{GH}}\bigl(\mathbb{S}^m,\mathbb{S}^{m+d(m)}\bigr) = \frac{\pi}{4} \qquad\text{whenever } d(m)\geq1,\ \text{and }d(m)=o(m).$$

math.MG↗

arXiv2026

Pathwise Portfolio Theory and Market Viability

The theory of portfolios, and its allied notions and fundamental results concerning growth optimality, the num\'eraire property, and ``market viability'' -- which rules out the possibility of financing nontrivial future liability streams starting with arbitrarily small initial capital -- is developed in a pathwise setting, completely devoid of probabilistic considerations. The approach replaces the familiar semimartingale decomposition of stochastic analysis for assets' returns, by decompositions generated through suitable trend extractors and their associated residual paths; then deploys F\"ollmer's celebrated pathwise version of classical It\^o integration and calculus. The resulting growth-num\'eraire and viability-boundedness equivalences bear considerable similarities to their semimartingale counterparts, but need not collapse into a single equivalence class in the pathwise setting; this separation is illustrated by two examples.

q-fin.MF↗

arXiv2026

Proper Homotopy Nonrigidity of Open Contractible Manifolds

We study whether proper homotopy equivalence classifies open contractible manifolds arising as interiors of compact contractible manifolds. In dimensions three and four, it does: any two such interiors that are properly homotopy equivalent are homeomorphic. In every even dimension $N\geq 6$ and every odd dimension $N\geq 9$, however, it does not: a single proper homotopy type contains infinitely many pairwise nonhomeomorphic smooth open contractible $N$-manifolds. The cases of dimensions five and seven remain unresolved.

math.GT↗

arXiv2026

Topological Signatures of Diffusive Release in Porous Media

We used persistent homology to quantify the multiscale topological and geometric organization of porous media, including solid connectivity and the formation of loop-like and cavity-like structures across spatial scales. Through statistical analysis, we show that these topological and geometric features are closely associated with diffusion-driven release behavior in porous media. In particular, even within each target-porosity level, samples with richer topological features tend to exhibit long-tailed release, indicating that release behavior depends not only on the amount of pore space but also on the multiscale organization of the solid phase. We further show that persistent homology-based features can classify release-curve regimes using a simple classification model. Notably, feature extraction is substantially faster than finite element diffusion simulations. Together, these results suggest that persistent homology provides a lightweight, interpretable, and geometry-based descriptor for screening diffusive release behavior in porous media.

cs.CE↗

arXiv2026

Pathwise integration beyond Young via Faber--Schauder energy spaces

We develop a pathwise integration theory based on Faber--Schauder energy spaces. The approach replaces the classical H\"older--Young and finite-variation Young conditions by dyadic summability conditions expressed in terms of Faber--Schauder coefficients. On the normalized interval $[0,1]$, these conditions define Banach spaces $\mathcal{E}^p$, which we call Faber--Schauder energy spaces. For $p,q>1$ satisfying $1/p+1/q\ge1$, we prove that every pair $f\in\mathcal{E}^p$ and $g\in\mathcal {E}^q$ admits a continuous pathwise integral $I_{f,g}$, constructed from dyadic left Riemann sums. We call $I_{f,g}$ the Faber--Schauder integral, and show that it depends boundedly and bilinearly on $(f,g)$ in the corresponding energy norms. The integral satisfies additivity, integration by parts, and a dyadic Young--Lo\`eve estimate. It is also the uniform limit of classical Riemann--Stieltjes integrals of finite Faber--Schauder approximations. The Faber--Schauder integral agrees with the classical Young integral whenever the latter is available, but also applies to deterministic and Gaussian examples for which neither the H\"older--Young condition nor the finite-variation Young condition can be verified. In this sense, it provides a Faber--Schauder coefficient-based extension of Young's framework.

math.CA↗

arXiv2026

Banach spaces of continuous paths with finite $p$-th variation

We study pathwise $p$-th variation of continuous paths on a compact interval along a fixed partition sequence. Although the class of continuous paths with finite $p$-th variation is generally not linear, we develop a coefficient-based approach via Faber-Schauder expansions that, for any $p>1$, enables the construction of paths with prescribed $p$-th variation while preserving useful linear structures and H\"older regularity. We first construct continuous paths with linear $p$-th variation from suitable conditions on their Faber-Schauder coefficients. We then prescribe nonlinear $p$-th variation through a multiplicative transformation and show that, whenever nonempty, the class of H\"older continuous paths with a given $p$-th variation is dense in $C([0,1])$. Next, we introduce a transport procedure that turns a Banach subspace of continuous functions into a Banach subspace of paths with explicitly controlled $p$-th variation. We also prove stability of the associated pathwise F\"ollmer-It\^o map on these transported subspaces. Finally, via time-changes, we show that this constructive framework extends from $q$-adic partition sequences to broader classes of dense $q$-refining partition sequences.

math.PR↗

arXiv2026

Rethinking Forward Processes for Score-Based Nonlinear Data Assimilation in High Dimensions

Data assimilation is the process of estimating the state of a dynamical system over time by combining model predictions with measurements. This task becomes challenging when the system is nonlinear and high-dimensional. To address this, score-based Bayesian filters have recently emerged. However, these methods still show unsatisfactory performance in certain cases, particularly under spatially sparse measurements. Such degradation stems from heuristic approximations of the likelihood score, whose errors can accumulate over time. This limitation arises because the methods simply adopt a classical forward process for generative modeling that transforms a data distribution toward a Gaussian distribution, which is independent of the measurement equation. Here, we propose a forward process tailored for filtering that transforms the system state toward the measurement space, enabling a theoretically sound formulation of the likelihood score. Based on this, we develop the Measurement-Aware Score-based Filter (MASF). We evaluate MASF on Kolmogorov flow, a high-dimensional fluid benchmark with up to $\mathcal{O}(10^5)$ dimensions, under diverse measurement operators, including nonlinear cases with a dimensional mismatch between the state and the measurements. MASF shows improved performance over existing score-based filters and ensemble-type Kalman filters. Notably, MASF achieves up to a $28.2\times$ wall-clock speedup compared with the baselines when using amortized pretraining. Our implementation is available at \texttt{https://github.com/tcnllab-oss/masf}.

stat.ML↗

arXiv2026

Superconductivity in Isolated Single Copper Oxygen Plane

One of the central questions in cuprate superconductivity is if superconductivity can exist in an isolated single CuO$_2$ plane without any interlayer coupling. There have been numerous experimental efforts to answer this question, but it still has not been clearly resolved. Here we present a heterostructure system with an isolated half-unit-cell La$_{2-x}$Sr$_x$CuO$_4$ which has a single CuO$_2$ plane. Using in-situ angle-resolved photoemission spectroscopy, we measured the electronic and gap structures of a single CuO$_2$ plane. We observed a \textit{d}-wave-like gap which closes somewhat above the bulk T$_c$. Moreover, almost identical gap properties are seen for both single CuO$_2$ plane and bulk. These observations lead us to the conclusion that the d-wave superconductivity of cuprates also exists in a single CuO$_2$ plane. Our results demonstrate that cuprate superconductivity is essentially a two-dimensional phenomenon and provide a platform to study cuprate superconductivity in a purely two-dimensional system.

cond-mat.supr-con↗

arXiv2025

Low-dimensionality-induced tunable ferromagnetism in SrRuO$_3$ ultrathin films

Quantum materials near electronic or magnetic phase boundaries exhibit enhanced tunability, as their emergent properties become highly sensitive to external perturbations. Here, we demonstrate precise control of ferromagnetism in a SrRuO$_3$ ultrathin film, where a high density of states (DOS), arising from low-dimensional quantum states, places the system at the crossover between a non-magnetic and bulk ferromagnetic state. Using spin- and angle-resolved photoemission spectroscopy (SRPES/ARPES), transport measurements, and theoretical calculations, we systematically tune the Fermi level via electron doping across the high-DOS point. We directly visualize the spin-split band structure and reveal its influence on both magnetic and transport properties. Our findings provide compelling evidence that magnetism can be engineered through DOS control at a phase crossover, establishing a pathway for the rational design of tunable quantum materials.

cond-mat.str-el↗

arXiv2025

Ligand Engineering for Precise Control of Ultrathin CsPbI3 Nanoplatelet Superlattices for Efficient Light-Emitting Diodes

Strongly-confined perovskite nanoplatelets (PeNPLs) offer opportunities not found in conventional isotropic nanocubes, especially in producing linearly polarized light, as well as enhancing outcoupling through control over the transition dipole moment. But this requires ultrathin nanoplatelets with three or fewer monolayers of PbI6 octahedra across the thickness, which are challenging to synthesise uniformly, and their luminescence is strongly affected by surface defects. Together, these limit the performance of ultrathin PeNPLs in light-emitting diodes (LEDs). Here, we address these challenges with an ancillary ligand engineering strategy. We demonstrate that ligands with phosphoryl functional groups strongly bind to the perovskite surface, while having an organic backbone that is not sterically bulky ensures high ligand density. By modulating nucleation and growth, these ancillary ligands lead to monodisperse PeNPLs that stack more uniformly when self-assembled into superlattices, with suppressed agglomeration. As a result, from edge-up PeNPL superlattices, we achieve enhanced degree of polarization, while from face-down PeNPL superlattices, we achieve enhanced outcoupling that results in LEDs with 13.1% external quantum efficiency, the highest reported for ultrathin PeNPL LEDs. This work establishes ancillary ligand-induced synthesis as a decisive route to achieve uniform nanoplatelets with robust orientation control, enabling full utilization of the multifunctionality of anisotropic PeNPLs.

physics.optics↗

arXiv2025

Metric Topologies on Multiset Spaces as Topological Monoids and Their Group Completion

We construct a multiset space $\mathbb{N}[X]$ over a metric space $X$ that simultaneously enjoys desirable topological properties and admits a natural matching metric $d_{\mathbb{N}[X]}$, making it a metrizable abelian topological monoid whose structure is compatible with the original metric on $X$. This framework extends naturally to the free abelian group $\mathbb{Z}[X]$, where a metric $d_{\mathbb{Z}[X]}$ induces a metrizable abelian topological group structure. We further identify the metric completion of $\mathbb{N}[X]$, showing that it carries a canonical extension of the matching metric.

math.MG↗

arXiv2025

Graphon particle systems with common noise

We study a nonlinear graphon particle system driven by both idiosyncratic and common noise, where interactions are governed by a graphon and represented as positive finite measures. Each particle evolves via a McKean-Vlasov-type SDE with graphon-weighted conditional laws. We prove a law of large numbers for the empirical and interaction measures, using generalized Wasserstein metrics and weak convergence techniques suited for the non-Markovian structure induced by common noise.

math.PR↗

arXiv2025

Rapid low-temperature synthesis of graphene-coated SiC substrates for remote and van der Waals epitaxy

Non-conventional epitaxial techniques, such as van der Waals epitaxy (vdWE) and remote epitaxy, have attracted substantial attention in the semiconductor research community for their capability to repeatedly produce high-quality free-standing films from a single mother wafer. Successful implementation of these epitaxial techniques depends on creating a robust, uniform two-dimensional (2D) material surface. The conventional method for fabricating graphene on silicon carbide (SiC) is high-temperature graphitization. However, the extremely high temperature required for silicon sublimation (typically above 1500 {\deg}C) causes step-bunching of the SiC surface, forming non-uniform multilayer graphene stripes and an unfavorable surface morphology for epitaxial growth. Here, we developed a wafer-scale graphitization technique that allows fast synthesis of single-crystalline graphene at ultra-low temperatures by metal-assisted graphitization (MAG). We found annealing conditions that enable SiC dissociation while avoiding silicide formation, producing uniform single-crystalline graphene while maintaining the surface morphology of the substrate. The graphene thickness can be controlled by varying the metal thickness or annealing temperature, enabling remote epitaxy or vdWE. We successfully produced freestanding single-crystalline III-N (AlN, GaN) films on graphene/SiC via the 2D material-based layer transfer technique. Our results show that low-temperature graphene synthesis via MAG offers a promising route to producing large-scale ultra-wide bandgap free-standing crystalline membranes.

cond-mat.mtrl-sci↗

arXiv2025

Equichordal Points of Convex Bodies

The equichordal point problem is a classical question in geometry, asking whether there exist multiple equichordal points within a single convex body. An equichordal point is defined as a point through which all chords of the convex body have the same length. This problem, initially posed by Fujiwara and further investigated by Blaschke, Rothe, and Weitzenb\"ock, has remained an intriguing challenge, particularly in higher dimensions. In this paper, we rigorously prove the nonexistence of multiple equichordal points in $n$-dimensional convex bodies for $n \geq 2$. By utilizing topological tools such as the Borsuk-Ulam theorem and analyzing the properties of continuous functions and mappings on convex bodies, we resolve this long-standing question.

math.MG↗

arXiv2025

Expensive Homeomorphism of Convex Bodies

In this paper, we address the longstanding question of whether expansive homeomorphisms can exist within convex bodies in Euclidean spaces. Utilizing fundamental tools from topology, including the Borsuk-Ulam theorem and Brouwer's fixed-point theorem, we establish the nonexistence of such mappings. Through an inductive approach based on dimension and the extension of boundary homeomorphisms, we demonstrate that expansive homeomorphisms are incompatible with the compact and convex structure of these bodies. This work highlights the interplay between topological principles and metric geometry, offering new insights into the constraints imposed by convexity.

math.MG↗

arXiv2024

Super-Polynomial Growth of the Generalized Persistence Diagram

The Generalized Persistence Diagram (GPD) for multi-parameter persistence naturally extends the classical notion of persistence diagram for one-parameter persistence. However, unlike its classical counterpart, computing the GPD remains a significant challenge. The main hurdle is that, while the GPD is defined as the M\"obius inversion of the Generalized Rank Invariant (GRI), computing the GRI is intractable due to the formidable size of its domain, i.e., the set of all connected and convex subsets in a finite grid in $\mathbb{R}^d$ with $d \geq 2$. This computational intractability suggests seeking alternative approaches to computing the GPD. In order to study the complexity associated to computing the GPD, it is useful to consider its classical one-parameter counterpart, where for a filtration of a simplicial complex with $n$ simplices, its persistence diagram contains at most $n$ points. This observation leads to the question: 'Given a $d$-parameter simplicial filtration, could the cardinality of its GPD (specifically, the support of the GPD) also be bounded by a polynomial in the number of simplices in the filtration?' This is the case for $d=1$, where we compute the persistence diagram directly at the simplicial filtration level. If this were also the case for $d\geq2$, it might be possible to compute the GPD directly and much more efficiently without relying on the GRI. We show that the answer to the question above is negative, demonstrating the inherent difficulty of computing the GPD. More specifically, we construct a sequence of $d$-parameter simplicial filtrations where the cardinalities of their GPDs are not bounded by any polynomial in the the number of simplices. Furthermore, we show that several commonly used methods for constructing multi-parameter filtrations can give rise to such "wild" filtrations.

math.AT↗

arXiv2024

Robot Metabolism: Towards machines that can grow by consuming other machines

Biological lifeforms can heal, grow, adapt, and reproduce -- abilities essential for sustained survival and development. In contrast, robots today are primarily monolithic machines with limited ability to self-repair, physically develop, or incorporate material from their environments. While robot minds rapidly evolve new behaviors through AI, their bodies remain closed systems, unable to systematically integrate material to grow or heal. We argue that open-ended physical adaptation is only possible when robots are designed using a small repertoire of simple modules. This allows machines to mechanically adapt by consuming parts from other machines or their surroundings and shed broken components. We demonstrate this principle on a truss modular robot platform. We show how robots can grow bigger, faster, and more capable by consuming materials from their environment and other robots. We suggest that machine metabolic processes like those demonstrated here will be an essential part of any sustained future robot ecology.

cs.RO↗