Papers

This paper provides a data-driven design guaranteeing parameter convergence in model reference adaptive control (MRAC) when the to-be-controlled system is subject to process noise. In the context of MRAC, parameter convergence refers to ensuring convergence of the adaptive gains to a solution of the matching equations, or to an approximate solution when noise is present. In classical MRAC, even small noise may induce parameter drift, thus lacking robustness to noise. Meanwhile, existing robust MRAC methods cannot ensure parameter convergence without imposing excitation conditions on data. A key feature of the proposed framework is to ensure convergence of the adaptive gains to an approximate solution of the matching equations without relying on persistently exciting signals. Furthermore, the matching error can be explicitly characterized as a function of the noise. This explicit characterization allows to establish a necessary and sufficient condition on the noise characteristics under which the limit closed-loop system matrix is Hurwitz. In the noise-free case, the proposed framework results in exact parameter convergence. Notably, as compared to existing methods achieving exact parameter convergence in the noise-free case, the condition on data in the proposed framework is weaker.

Causal mediation analysis decomposes a treatment effect into indirect pathways through mediators and direct pathways not operating through them. Modern biomedical studies often involve high-dimensional covariates and mediators that are noisy proxies for lower-dimensional latent biological processes. Existing methods typically rely on sparsity, linear factor models, or ignore the connection among variables in the learned representations, which can be restrictive when measurements are nonlinear and covariate and mediator factors are structurally dependent. We propose MediEncoder, a representation-learning framework for nonlinear high-dimensional mediation analysis. MediEncoder jointly learns low-dimensional covariate and mediator representations using a coupled encoder-decoder architecture with a cross-factor network that links treatment and covariate representations to mediator representations. The learned features are then used in a cross-fitted efficient influence function-based estimator of natural direct and indirect effects. The resulting estimator is multiply robust and asymptotically normal under suitable regularity conditions. Simulations show that MediEncoder improves estimation accuracy over competing dimension-reduction approaches, and an application to Alzheimer's Disease Neuroimaging Initiative data illustrates its utility in high-dimensional biomedical causal mediation analysis.

This paper introduces a comprehensive two-dimensional analytical model of a toroidal magnetic ring with circular cross-section under sinusoidal excitation. Applying Maxwell's equations in local polar coordinates within a complex permeability, the model derives analytical expressions for the internal magnetic field, magnetic flux, complex impedance, and total losses. It rigorously separates the contributions of eddy current losses, hysteresis losses, and winding losses, while explicitly incorporating the skin effect in the conductive core via Bessel functions. An expression for the apparent permeability is also provided, enabling the nonlinear core behavior to be mapped onto simplified linear material models. The resulting analytical model offers a computationally efficient and accurate foundation for standardized magnetic material characterization, such as Brockhaus and Iwatsu ring measurements, as a powerful alternative to 2D and 3D finite element analysis.

Laplace–King representations combine spherical harmonics with King functions [Wang et al., Chin. Phys. B 34, 065201 (2025)], the radial kernels of shifted isotropic Gaussians. The radial parameters may vary across angular modes. We prove that, for every angular degree, fixed-width kernels with positive real shifts have dense complex linear span in a Gaussian-weighted radial \(L^2\) space. Finite Laplace–King representations are dense in the corresponding three-dimensional weighted space; allowing variable widths preserves density in the same reference norm. A generating identity connects the kernels to generalized Laguerre polynomials. The self-adjoint King operator is unitarily equivalent to the free radial Schrödinger operator; its spectral resolution defines a continuous King mixture model (KMM) through imaginary-shift kernels in a distinct weighted Hilbert space.

Let (\Gamma) be a finite compact connected metric graph and let (A\in L^\infty(\Gamma)) be a real magnetic potential. The magnetic Laplacian (H_A) with standard vertex conditions is defined by the closed quadratic form [ q_A[u]=\sum_e\int_e |(-i\partial_x-A_e)u_e|^2,dx. ] A magnetic Cheeger constant is introduced by adding to the usual boundary term the frustration index of the potential on subgraphs. The first point of the paper is that, on a metric graph, this frustration index is exactly a finite dimensional (\ell^1) flux distance determined by the periods of (A) on cycles. Consequently the Cheeger constant can be written directly in terms of Aharonov Bohm fluxes. We prove a Cheeger type lower bound for the bottom of the spectrum and derive the corresponding explicit lower estimate in terms of the distance of the global cycle flux vector from the integral flux lattice. The estimate also gives (L^2) decay bounds for the magnetic heat semigroup and for the magnetic energy. The constants are not asserted to be sharp; the emphasis is on the flux dependence and on the self contained metric graph formulation.

Expanding upon the work of arXiv:2502.08799, we provide a non-explosion criterion for branched rough differential equations (bRDEs) where we continue to allow for the drift and the rough coefficient to be unbounded in the branched setting. Moreover, by providing a characterization of “pure area” branched rough paths – branched rough paths over the zero path, we provide two different constructions of bRDEs which explode in finite time, and thus demonstrating the sharpness of the criterion. Finally, by realizing a trade-off between the growth of the coefficient of the bRDE and the decay of its higher-order derivatives, we provide a new non-explosion principle for bRDEs which allows for the coefficient to grow even faster than what is provided in arXiv:2502.08799.

We derive the vertical chromaticity $\xi_y$ of the Fermilab Muon g-2 storage ring in closed analytic form. Expanding the Hamiltonian as a Taylor polynomial in the dynamical variables and integrating the equations of motion order by order, we obtain the vertical second-order aberrations of the homogeneous magnetic dipole ($\mathtt{DI}$) and the combined-function dipole-and-electrostatic-quadrupole element ($\mathtt{DIQ}$) used in the muon $g{-}2$ ring. Composing the per-element maps over the periodic dispersion orbit yields a closed-form expression for the vertical chromaticity $\xichromy$ of the continuous-ring $\mathtt{DIQ360}$ model, in direct functional analogy with the horizontal result of our earlier work on the same ring (Ref.~\refcite{ChromCPO11}). Comparison against COSY INFINITY differential-algebra computation shows agreement at the $10^{-11}$ level across all three ring models ($\mathtt{DIQ360}$ closed form and the modular $\mathtt{DIEQ\_ON}$, $\mathtt{DIEQ}$ via per-element composition) for muon $g{-}2$ electrostatic-quadrupole (ESQ) voltages $\Vesq \in [10, 26]\,\mathrm{kV}$.

We begin by presenting the mathematical rationale underlying classical deductive inference. We then introduce the foundational ideas of the Bayesian inference framework. Results lying at the interface of Statistics and Ergodic Theory are outlined, providing a theoretical framework applicable to the prediction and analysis of real-world phenomena from random data. This text is expository in nature - no new results are presented; rather, recently published results are described in a didactic manner. Throughout, we work with Hölder equilibrium measures, which encompass a substantially more general class of processes than i.i.d. ones.

Control Barrier Functions (CBFs) have been proposed to ensure safety of autonomous systems. This paper considers control policies that switch between CBF constraints. Under this approach, we represent a complex non-convex safe region as a union of sets that are computationally tractable to verify. We denote this framework as union-CBFs and make the following contributions. First, considering switching CBF-QP controllers, we propose a sufficient condition that ensures (i) the system undergoes a finite number of switches in any finite time interval and ensures (ii) the forward invariance of the closed-loop system in between switches. Second, we consider two types of switching strategies and propose union-CBFs conditions for each strategy to satisfy (i) and (ii). Third, we formulate Sum-of-Squares (SOS) algorithms to verify the conditions. The experiments show that our union-CBFs framework results in a larger safe region compared to high-degree polynomial CBFs. We also show the efficiency of the verification algorithms using a polynomial system model.

For two subspaces $U,V\subseteq\RR^n$, the circumcentered–reflection method (CRM) of Behling, Bello-Cruz, and Santos~\citeyearpar{BBS2018} computes the projection onto $U\cap V$ using only the reflections across $U$ and $V$, with known linear-convergence rate equal to the cosine of the Friedrichs angle. We prove that, when CRM is initialized in $V$, it contracts at the strictly smaller rate $\rho_V=(\sin^2\theta_p-\sin^2\theta_F)/(\sin^2\theta_p+\sin^2\theta_F)$, where $\theta_F\in(0,\pi/2]$ is the Friedrichs angle, whose cosine we denote by $c_F\coloneqq\cos\theta_F$, and $\theta_p\in[\theta_F,\pi/2]$ is the largest principal angle between $U$ and $V$. The bound is sharp, attained on an explicit ray in $V$, and optimal among parameter-free single-step iterations. The constant itself is not new: Bauschke, Bello-Cruz, Nghia, Phan, and Wang~\citeyearpar{BBNPW2016} identified it as the optimal rate of the relaxed alternating-projection family and of their adaptive linesearch map $B_T$. Our contribution is that the parameter-free geometric circumcenter attains it as well, via Kantorovich's inequality applied to a single self-adjoint operator on $V$. Restricted to $V$, CRM coincides pointwise with the linesearch maps $A_T$ and $B_T$ from the Gubin–Polyak–Raik framework~\citep{GPR1967}. We further prove $\rho_V<c_F^2$ whenever $\theta_F<\pi/2$, with one-step convergence exactly when $\theta_F=\theta_p$. Over-reflecting either or both of $R_U$, $R_V$ inside the circumcenter does not help. Going faster than $\rho_V$ universally requires memory: Chebyshev semi-iteration applied to $P_VP_U$ attains a strictly smaller rate, beating $\rho_V$ by a factor at most $2$, attained in the limit $\theta_F\to\theta_p$.

The classical Rogers–Shephard inequalities were extended to the Firey $L_p$-summation by Bianchini and Colesanti in the plane and by Zvavitch and the second and fourth authors for locally anti-blocking convex bodies in $\mathbb{R}^n$, leaving open the equality cases. We characterize the equality cases of these inequalities: in both cases, for $p>1$, equality holds if and only if the convex body is a simplex with one vertex at the origin.

We establish new polynomial-weighted recursions for Andrews' smallest parts function. Our results use the generating series for the spt function, a harmonic Maass form of weight 3/2, paired with the Dedekind eta function via the Rankin-Cohen bracket. In weights larger than 2, we find a nontrivial quasimodular component, which we determine for the relevant weights. We apply the holomorphic projection operator and the vanishing of cusp form spaces of small enough weight to obtain our results.

We study possible singular endpoints of the three-dimensional incompressible Navier–Stokes equations by parabolic rescaling. Weak limits can lose information through pressure, nonlinear products, concentration, spatial tails, or symmetry, so we keep these quantities as part of one augmented endpoint state and use a fixed whole-space pressure representative throughout the shrinking chain. We prove compact endpoint realization, nonoverlapping assignment of positive defects, two-generation inheritance, rotational alternatives, and a high-critical kinetic normalization centered at the original singular point. We show that positive rotational distance alone does not coerce a productive rotational Reynolds stress, and we give an explicit conditional compactness theorem under which frequency-tight amplitude-fast records pass strongly to a nonzero ancient Euler profile. We then study a possible Euler limit that is invisible under positive heat evolution. Exact heat-orbit nullity forces cancellation separately on each interaction-energy shell and isotropic covariance on every Fourier sphere; independently, any nonzero $L^2$ field normalized along its long heat orbit loses all mass from every fixed spatial ball. Frequency anti-concentration, the accumulating heat-chain passage, and retention of a singular witness remain open; no unconditional regularity theorem is claimed.

We study the axisymmetric Navier–Stokes equations with swirl in the variables $F=u^\theta/r$, $G=\omega^\theta/r$, and $\Gamma=ru^\theta$. Their distinct scaling degrees lead to several critical transition surfaces rather than a single scalar exponent. We prove mesoscopic and temporal critical-face identities, source/contact trace estimates, compact finite-depth preparation obstructions, an affine six-power flattening barrier, a fixed-old-current compression ceiling, and compactness for rescaled strain generators with bounded action and bounded variation. A sharp radial estimate gives exponential control of the time occupied by large swirl-energy records; on parabolic time intervals it yields logarithmic routing from total kinetic records to the meridional component, and an independent cone argument routes non-meridional Type-II records to a scale-critical signed compression action. The source-free circulation also satisfies an exact quadratic dissipation identity, while scalar-weight calculations show that no radial power simultaneously yields a compression-free continuity law and a scale-zero current action. The results exclude several compact and passively prepared recurrence mechanisms under explicit hypotheses; meridional Type-II records, signed critical compression, noncompact high-frequency behavior, and boundary-entry alternatives remain.

One of S.T. Yau's problems asks the following: given a $3$-dimensional asymptotically flat manifold $M$ with non-negative scalar curvature and $L^2$-norm of the curvature tensor at most $1$, if the mass of $M$ is small, is there a bilipschitz diffeomorphism from $M$ to the flat Euclidean space $\mathbb{R}^3$? We provide a strong positive answer to this problem by using our previous work \cite{DS25}.

Path-planning for autonomous vehicles in threat-laden environments is a fundamental challenge because the problem is nonlinear and nonconvex even in simplest scenarios. While traditional optimal control methods can be used to find ideal paths, the computational time is often too slow for real-time decision-making. To solve this challenge, we propose a method based on Deep Deterministic Policy Gradient (DDPG) and model the threat as possibly multiple circular 'no-go' zones. A mission is regarded as a failure if the vehicle enters this restricted zone at any time or does not reach a neighborhood of the destination. The DDPG agent is trained through trial and error in a simulated environment, learning a direct mapping from its current state (position and heading) to a series of feasible actions that guide the agent to safely reach its destination. The reword function has three parts: (a) an attractive field centered at the final destination, (b) some repulsive fields centered at the origins of circular obstacles, and (c) a penalty of control energy consumption (the magnitude of heading change) that indirectly in favor for straight path. The DDPG trains the agent using these incentives to find the largest possible set of starting points wherein a safe path to the destination is guaranteed. This provides critical information for mission planning, showing beforehand whether a task is achievable from a given starting point, assisting pre-mission planning activities. The approach is validated in simulation. A comparison between the DDPG method and a traditional optimal control (pseudo-spectral) method is carried out. The results show that the learning-based agent produces effective paths while being significantly faster, making it a better fit for real-time applications.

Unambiguous state discrimination (USD) measurements are attractive because outcomes are either marked as conclusive (i.e., error free) or inconclusive (i.e., erased). We study affine filtering measurements, a structured variant of USD for decoding classical linear codes over pure-state classical-quantum channels, where a conclusive outcome identifies an affine subspace containing the transmitted codeword and an inconclusive outcome is treated as an erasure. For a group-covariant indexing of pure-state codewords, we show that the optimal design of affine filtering measurements is a semidefinite program that can be reduced to a linear program via character-based diagonalization. We use the resulting measurement to build a quantum decoding framework for local codes, and we demonstrate (via simulations on regular LDPC codes from Gallager ensembles using single parity check local constraints) that affine filtering based decoding can outperform symbol-wise USD and symbol-wise pretty good measurement based decoding methods on i.i.d. pure-state channels. In an independent and concurrent work, Buzet and Chailloux study similar fine-grained USD measurements for symmetric families of states. Their focus is on the code-agnostic setting whereas our focus is on code-aware constructions and decoding.

We present PDE-Agents, a multi-agent ecosystem that automates the full lifecycle of partial differential equation (PDE) and finite element method (FEM) simulations through natural-language interaction. Three large language model agents, Simulation, Analytics, and Database, are orchestrated by a LangGraph supervisor and run locally using Qwen3-Coder-Next and Llama 4 Scout on dual NVIDIA RTX PRO 6000 Blackwell GPUs. The model-agnostic architecture is validated across two generations of open-source LLMs. A GraphRAG knowledge base using Neo4j and 768-dimensional vector embeddings provides material properties, failure patterns, and prior run lineage. We report seven contributions: (i) verification and validation showing second-order spatial convergence, O(h^2), for three heat-equation benchmarks; (ii) a 50-task ablation with a frozen knowledge graph comparing KG On, KG Off, and KG Smart, where KG Smart achieves 100% success and the highest output quality, with physics score 0.933 versus 0.853 and material property fidelity (MPF) 0.926 versus 0.796 for KG Off; (iii) a novel-material study using three fictional materials known only to the knowledge graph, where KG Smart reaches MPF = 1.00 versus 0.34 without the graph; (iv) failure analysis tracing KG On's three failures to budget exhaustion and timeout, identifying warm-start injection as the main reliability factor; (v) an adaptive framework selecting retrieval mode per task; (vi) production metrics from 1,369 runs showing 97.8% overall success and 85.4% first-try success; and (vii) a 100-task knowledge-graph growth study showing an 8.8% MPF gain on hard tasks while easy and novel tasks remain at ceiling. All code, models, and evaluation artifacts are openly released. These results show that integration pattern, rather than knowledge content alone, determines whether GraphRAG helps or hinders LLM agents.

We study orthogonal modular forms associated with moduli spaces of lattice-polarized K3 surfaces whose generic transcendental lattices are of the form $T = H \oplus H \oplus L(-1)$ where $L$ is a root lattice of type $A_n$ or $D_n$. In Picard numbers $10$ through $17$, we use explicit Jacobian elliptic fibrations to construct modular forms on type IV domains associated with orthogonal groups $\mathrm{O}^+(T)$. We show that the coefficients of suitable Weierstrass models naturally realize generators for the corresponding graded rings of orthogonal modular forms.

The Behrens–Fisher problem concerns inference on the difference of two normal means when both variances are unknown and unequal. It is a classical example in which nuisance parameters prevent ordinary exact fixed-sample inference, and it has long served as a benchmark for the foundations of inference. We revisit it through the inferential model (IM) framework of Martin and Liu. After conditioning and regular marginalization, the exact association is two-dimensional, with one coordinate for the standardized mean contrast and one for the variance ratio. Their one-dimensional generalized marginal IM is then best understood as a cylindrical two-dimensional predictive random set: sharp in its mean-contrast projection, by Hsu's stochastic domination, and vacuous in the variance ratio. Our main result is a precise validity-first optimality: among prior-free procedures that retain exact, uniform, finite-sample validity, the IM interval is the shortest. We prove minimaxity and admissibility in the cylindrical class and, by a projection argument, extend this to rectangular and general two-dimensional predictive random sets. A companion tradeoff principle shows that any adaptive procedure can only redistribute interval width across variance-ratio regimes, never shorten it uniformly. A Monte Carlo study bears this out: Welch and the bootstrap under-cover, whereas the conservative fiducial does not dominate the IM interval, being shorter only where the latter over-covers and longer where validity binds.

We introduce two infinite families of 3-regular planar graphs. Both families are conceptual adversaries to the Pohl-Warnsdorf algorithm for finding Hamiltonians. We provide a closed form calculation of the number of Hamiltonians.

We study the large-domain asymptotics of geometric observables in Berry's random wave model on $\mathbb{R}^d$. We show that, in sharp contrast with the behavior of stationary random fields with absolutely continuous spectral measures, any observable whose fluctuations are asymptotically fully correlated with its second Wiener chaos projection belongs to a common universality class governed by a fractional Gaussian field with Hurst index $H=(1-d)/2$. This class also includes the classical stationary Poisson line process in $\mathbb{R}^d$. Our findings show that suitable raw observables of Berry's random wave (such as critical point counts or non-nodal level set volumes) have large-domain fluctuations that become arbitrarily close – in the sense of random tempered distributions – to those generated by a (possibly noisy) Poisson line process. This probabilistic approximation provides evidence that the large-scale filamentary patterns observed in numerical simulations of random waves – often referred to as "scars" or "scarlets" following the numerical investigations of Heller, O'Connor and Gehlen (1987)– may admit a natural probabilistic interpretation. In the second part of our work, we characterize the scaling limit – in a distributional sense – of suitable quadratic transformations of the Radon–Fourier coefficients associated with a large class of stationary fields. We show that random waves are characterized by the property that such a scaling limit is a generalized random field obtained by composing white noise on the affine Grassmannian of lines with a dimension-dependent deterministic operator. As an application of our main results, we derive explicit conditions ensuring that quadratic functionals of pullback monochromatic waves on compact Riemannian manifolds exhibit distributional limits in the fractional Gaussian universality class described above.

We construct and analyze an exactly divergence-free Scott-Vogelius finite element method for the surface Stokes problem. The proposed scheme simultaneously enforces the tangentiality and incompressibility constructs exactly and has the same number of unknowns as the two-dimensional Euclidean discretization. Our construction extends the surface finite element framework of [10,11] to Scott–Vogelius discretizations defined on curved Clough–Tocher triangulations. In contrast to previous isoparametric Scott–Vogelius methods based on macro-element constructions, the present approach defines the finite element spaces directly on the refined surface triangulation, leading to a substantially simpler and more practical implementation. We prove inf-sup stability of the method and derive optimal-order convergence in the isoparametric regime.

This work is motivated by the analysis of temporal difference algorithms, where stability can be guaranteed by bounding the eigenvalues of an associated matrix derived from a, typically non-reversible, Markov kernel. We generalize the existing sufficient conditions for stability and show that the associated eigenvalues can be bounded in terms of the Dirichlet spectral gap of the Markov kernel. We derive a collection of methods for showing that non-reversible Markov chains have positive spectral gaps. We show that if a Markov chain has positive absolute spectral gap, then it has a positive Dirichlet spectral gap. In the case of discrete-time linear Gaussian systems, we give explicit bounds for both Dirichlet and absolute spectral gaps. Additionally, we present an example of a Markov chain which is $V$-uniformly geometrically ergodic but has zero Dirichlet spectral gap.

Public-key cryptosystems eliminate the requirement for pre-shared secret keys by enabling encryption with a publicly disclosed key and decryption with a corresponding private key. In this article we generalize the public-key cryptosystems to ternary algebraic structures, with particular attention to ElGamal as a representative family. We introduce the necessary algebraic background for nonderived ternary structures, including special elements, ternary group rings, and a matrix ternarization procedure that maps binary rings and group rings to antidiagonal symbolic matrices closed under ternary multiplication. Building on these foundations, we formulate a ternary analogue of the ElGamal three-step protocol (key generation, ephemeral encryption, and decryption via querelements) and derive explicit ternary power and querelement formulas that enable correct decryption. Concrete instantiations and numerical examples over a ternary fraction field, a matrix-ternarized finite group ring, and a finite \((6,3)\)-ring (field) validate the construction and illustrate admissible word-length quantization and cycle behaviour of ternary powers. The ternary framework highlights two practical advantages: richer algebraic structure (querelements replace binary inverses) that increases algebraic complexity for attackers, and higher information density (matrix ternarization transfers paired/plaintext vectors). Formal hardness assumptions, optimized parameter choices, and comprehensive security and performance analyses remain necessary future work.

Von Neumann's Hilbert-space formalism of quantum mechanics constitutes a logico-physical theory of observed or measured reality. Imposing the logical constraint of Booleanity, essential for objectively shareable descriptions among observers, reveals the physical meaning of classicality inherently embedded within the formalism itself. Starting from this consideration, the present work reformulates the quantum-classical transition via Hilbert-space classical mechanics (HCM), grounding classicality not in the dynamical limit ($\hbar \to 0$), but in the logical constraint of Booleanity (i.e., the mutual commutativity of preparable states). Within this state-centric framework, applying the Einstein-Podolsky-Rosen (EPR) criterion alongside locality and measurement independence reduces standard quantum mechanics to the HCM model. Thus, the EPR argument reveals not quantum incompleteness, but the implicit classical boundaries of its own premises. To resolve this impasse, we articulate a nuanced quantum ontology grounded in a fundamental structural bipartition between the observational environment and the observed object, which accommodates three categorical distinctions: ontic, processional, and tropos-existential. Building on this, we propose a criterion of objective reality wherein descriptive objectivity is treated as merely a sufficient condition for physical reality. This addresses the historical Bohr-Einstein ambiguity, enabling the quantum formalism to ontologically unify objective measured phenomena and non-objective observed interference within a context-dependent framework.

In this note, we establish an explicit upper bound for the dimension of the rational cohomology for a simple algebraic group over an algebraically closed field of prime characteristic.

We describe new dependent-rounding algorithms for bipartite graphs. Given a fractional matching $x$ of graph $G = (U \cup V, E)$, the algorithms return an integral solution $X$ such that each right-node $v \in V$ has at most one edge, and where the variables $X_e$ also satisfy broad non-positive correlation properties. In particular, for any edges $e_1, e_2$ sharing a left-node $u \in U$, the variables $X_{e_1}, X_{e_2}$ have *strong* negative-correlation, i.e. the expectation of $X_{e_1} X_{e_2}$ is significantly below $x_{e_1} x_{e_2}$. Dependent rounding schemes with these properties have been used for a approximation algorithms for job-scheduling on unrelated machines to minimize weighted completion times, among other applications. Our new algorithm achieves simpler and qualitatively stronger bounds compared to prior algorithms. In particular, we achieve a negative-correlation property $$\mathbb E[X_{e_1} X_{e_2}] \leq 0.79751 \ x_{e_1} x_{e_2}, $$ which is a significant constant-factor improvement over Baveja, Qu & Srinivasan (2023).

Order types are an equivalence relation between point configurations that capture their combinatorial and convexity properties. Let $P$ be a $\kappa$-colored sequence of $n \ge d+1$ points in general position in $\mathbb{R}^d$. Let $\rho$ be a $\kappa$-colored order type on $k \le d+1$ points that has positive density on $P$; that is, for some constant $\delta >0$, there are $\delta \cdot \binom{n}{k}$ $k$-point subsequences of $P$ that have the same order type as $\rho$ and the same color pattern. In this paper we show that there exists a constant $c >0$ (depending only on $d, \delta$, $k$ and $\kappa$) and disjoint subsets $X_1,\dots,X_k$ of $P$, each with at least $c \cdot n$ points, such that for every choice of $k$ points $x_i \in X_i$, $(x_1,\dots,x_k)$ has the same order type and color pattern as $\rho$.

The time-harmonic elastic-electromagnetic interior transmission eigenvalue problem (EEITEP) arises when an elastic body becomes invisible to an incident electromagnetic wave. This spectral problem is typically non-elliptic and non-self-adjoint, making its analysis delicate. In this paper, we study the discreteness of transmission eigenvalues and the boundary localization of the associated eigenfunctions. For a general bounded Lipschitz domain, we prove that the set of positive transmission eigenvalues, if non-empty, is discrete with $\infty$ as its only possible accumulation point. For a radially symmetric domain, we demonstrate the existence of a sequence of transmission eigenvalues and derive their asymptotic behavior. We rigorously show that the corresponding transmission eigenfunctions exhibit boundary localization in their electromagnetic components, whereas the elastic displacement field remains globally distributed throughout the domain. Finally, we derive lower bounds for the $L^{\infty}$-norms of the electromagnetic gradients normalized by their $L^2$-norms, quantifying their blow-up behavior near the boundary along this sequence. These findings reveal a potential spectral mechanism for developing super-resolution imaging methods in elastic-electromagnetic scattering.