Papers

Using cubic cyclotomic classes, character theory, and product constructions motivated by generalized Denniston partial difference sets, we construct negative Latin-square-type partial difference sets in $\mathbb{F}_{2^6}^{+}\times\mathbb{Z}_4^4$, $\mathbb{Z}_4^4\times\mathbb{F}_{2^{10}}^{+}$, $\mathbb{F}_{2^{10}}^{+}\times\mathbb{Z}_8^4$, $\mathbb{Z}_8^4\times\mathbb{F}_{2^{14}}^{+}$, and $\mathbb{Z}_4^4\times\mathbb{Z}_{16}^2$. The first four constructions replace cubic cyclotomic partitions by partitions of non-elementary abelian 2-groups having the same character-value patterns. To the best of our knowledge, these are the first partial difference sets with the stated parameters in the indicated groups.

We prove a non-arithmeticity property for the complex eigenvalues of a Zariski dense subsemigroups of real reductive algebraic groups. This property will be used in [7] in order to establish a local limit theorem for the operator norm of products of random matrices.

Martingale posteriors quantify uncertainty by forward-imputing observations from one-step-ahead predictive distributions, but implementations stop after finitely many imputations. For the empirical Pólya-urn posterior of a quantile the law of the stopped state is derived. The quantile of the stopped urn measure keeps the familiar martingale tail-sum variance fraction; the deployed stochastic-approximation tracker with frozen gain $c$ does not. Its variance carries an explicit factor $G_a$ with $a=cf_0(q_\tau)$, which may fall below or exceed the tail fraction, and a density-adapted gain restores calibration through a density-free inflation. Shared urn innovations yield the joint law of finitely many quantile levels. For conditional quantile regression, a smoothed martingale posterior started at the ordinary quantile-regression estimator with a full inverse-Jacobian matrix gain satisfies a process Bernstein–von Mises theorem with calibrated finite-horizon bands; scalar or diagonal gains cannot match the sandwich covariance process.

We study the Liouville equation $-\Delta u = e^u$ on a complete, connected, non-compact, boundaryless Riemannian surface $(M, g)$ with non-negative Ricci curvature. Assuming only some asymptotic lower bound on the solution, we establish classification results for both the solutions and the ambient manifold, discussing also their optimality. Our results reveal a close connection between the volume growth of the manifold and the classification of both the solutions and the underlying manifold.

We develop a non-parametric approach to refine forecast-error histories for safety stock estimation without relying on error magnitude or recency alone. Using forecast-error data from a fast-moving consumer goods environment, we show that one year of errors is insufficient to characterize service-level risk, while pooling several years is problematic because error distributions change over time. Non-parametric annual comparisons indicate a progressive reduction in bias and dispersion, consistent with learning in the forecasting process. We propose LOWDII (Leave-One-Out Wasserstein Distributional Influence Index), a diagnostic that evaluates the distributional influence of each historical forecast error on the empirical uncertainty distribution used for safety stock calibration. LOWDII identifies observations whose influence is disproportionate to their representativeness, separating transient distortions from persistent tail behaviour without imposing parametric assumptions. We evaluate the method in a discrete-event simulation using subsequent-year demand realizations as validation. The results show that LOWDII achieves the target service level while reducing average stock by 5.1% to 22.6% relative to benchmark methods. From a managerial perspective, LOWDII helps firms translate improvements in the forecasting process into safety stock decisions, avoiding the projection of obsolete historical errors into the future and capturing the economic benefit of lower uncertainty requirements earlier.

L. Babai introduced a method for constructing a cubic graph whose automorphism group is isomorphic to a given finite group $G$, obtained by modifying a corresponding Cayley graph of $G$. Building on this approach, we construct a cubic graph that admits a polyhedral map whose automorphism group, as well as the automorphism group of the polyhedral map itself, is isomorphic to $G$.

We develop a theory of nonlinear shrinkage covariance estimation for nonparanormal (Gaussian-copula) models, in which each observed coordinate is an unknown strictly increasing transformation of a latent Gaussian vector. This model accommodates arbitrary marginal skewness and heavy marginal tails while retaining a Gaussian dependence structure, and it is the natural semiparametric setting for heavy-tailed, asymmetric financial returns. Our estimator, marginal-free nonlinear shrinkage (MENS), applies an oracle nonlinear shrinkage function to the eigenvalues of the normal-scores rank-covariance matrix. We give the almost-sure convergence of the empirical spectral distribution of the normal-scores covariance to the generalized Marchenko-Pastur law of Sigma, and asymptotic optimality of MENS among rotation-equivariant estimators under Frobenius loss. We establish a Baik-Ben Arous-Peche phase transition for spiked latent correlations. The MENS attains the robustness of rank-based estimation and the efficiency of nonlinear shrinkage at once within this class. We corroborate the theory with a simulation study that isolates the marginal-invariance property and the spiked transition. In an out-of-sample minimum-variance backtest on S&P 500 stocks, MENS delivers a better-conditioned covariance estimate, lower realized portfolio volatility, and lower turnover than linear shrinkage, illustrating its practical value for high-dimensional allocation and decision-making.

For a graph $G$ of order $n$, with adjacency eigenvalues $\lambda_1(G) \geq \cdots \geq \lambda_n(G)$, the energy of $G$ is defined to be \[\mathcal{E}(G)=\sum_{i=1}^{n} |\lambda_i(G)|.\] A well-known conjecture from the 1980s by Fajtlowicz states that for any graph $G$, \[\mathcal{E}(G) \ge 2\left(n-\alpha(G)\right),\] where $\alpha(G)$ denotes the independence number. We prove this conjecture.

Let $E\subset \mathbb{R}$ be a self-similar set generated by a homogeneous iterated function system $\Phi$ with contraction ratio $\rho\in (0,1)$. Assume that $\Phi$ satisfies the open set condition and $\dim_{\rm H}E<1$. Let $f$ be a $C^1$-diffeomorphism on $\mathbb{R}$. We prove that if $\log|f'(x)|/\log\rho\not\in\mathbb{Q}$ for every $x\in E\cap f^{-1}(E)$, then the upper Minkowski dimension of $f(E)\cap E$ is strictly less than the Hausdorff dimension of $E$. We also establish a quantitative dimension drop result when $E$ is a missing-digit set and $f$ is an affine map with rational slope satisfying a certain arithmetic condition. Based on these results and a result of Shmerkin [Ann. of Math., 2019], we obtain characterizations of $\gamma$ in various contexts such that $\overline{\dim}_{\rm M}\left((\gamma E+\alpha)\cap E\right)<\dim_{\rm H}E$ for every $\alpha\in\mathbb{R}$.

We provide a direct injection for the well-known strong log-convexity of the Bell numbers $B_n$, that is $B_mB_n\le B_{m-1}B_{n+1}$ for every $1\le m\le n$. Our injection $\Pi_m\times\Pi_n\to \Pi_{m-1}\times\Pi_{n+1}$, where $\Pi_n$ denotes the set of all partitions of $[n]$, preserves the total number of blocks in the pair of partitions. In other words, it is also an injection for the strong $q$-log-convexity of Touchard polynomials, a result established by Chen, Wang, and Yang using analytical arguments. As an application of the injection, we also recover a related result of Chern, Diaconis, Kane, and Rhoades.

We study the simultaneous recovery of the initial source and sound speed for the scalar wave equation from a single boundary measurement. Although the recovery of either parameter separately is well understood under suitable geometric hypotheses, simultaneous recovery remains open in general, and stable recovery is further obstructed by the inherent instability of the linearized problem. We show that both uniqueness and stability can be obtained when the two unknowns are coupled through a prescribed constitutive relation arising from a common underlying material. Under a quantitative nondegeneracy condition, motivated by calibrated material regimes, the coupled inverse problem reduces to a single inverse source problem. Applying the microlocal and Carleman framework of Stefanov and Uhlmann to this reduced problem, we establish global uniqueness and Lipschitz stability under geometric conditions involving strictly convex foliations and geodesic visibility. We also obtain partial-data results, proving local uniqueness and Lipschitz stability in compact visible regions where the constitutive nondegeneracy condition holds.

We study two classes of oscillatory Jacobi-type weight functions $x^c(1-x^2)^{\lambda-1/2}\exp(i\zeta x)$, $x\in[-1,1], \lambda>-1/2, c\in\{0,1\}$. Other restrictions are imposed on $\lambda$ and $\zeta$ to guarantee the existence of the associated orthogonal polynomials. By using the ladder operators established in the recent literature for monic orthogonal polynomials associated with Jacobi-type weight functions and three compatibility conditions, we derive two coupled difference equations satisfied by the three-term recurrence coefficients. Compared with the existing results, these equations are structurally simpler and of lower order. Once the initial values are determined, the recurrence coefficients can be computed at any stage through the difference equations. The obtained expressions enable us to conjecture the symbolic forms for the recurrence coefficients.

This paper concerns neutral inclusions for imperfect bonding of high-conductivity type. An inclusion, which is a bounded domain, is said to be of imperfect bonding of high-conductivity type if the flux is discontinuous along its boundary while the potential is continuous. The inclusion is neutral to a uniform field if the presence of the inclusion does not perturb the field outside the inclusion. It is known that ellipses and ellipsoids can be neutral to all uniform fields by introducing a proper imperfect bonding coefficient on boundaries. The purpose of this paper is to prove the converse. We prove that if an inclusion of imperfect bonding of high-conductivity type is neutral to all uniform fields, then it is an ellipse or an ellipsoid. The neutrality condition is given by existence of the solution to a certain differential equation on the boundary surface and the main result is proved by converting the neutrality condition into an algebraic boundary identity characterizing ellipses and ellipsoids.

Random-feature methods reduce high-dimensional elliptic PDE collocation to linear coefficient problems, but full-dimensional trial spaces overlook lower-dimensional structure. We introduce the Hierarchical Analysis-of-Variance Random Feature Method (HA-RFM), which selects coordinate blocks using closed Sobol indices of the PDE residual, identifies oblique low-rank features from fitted-predictor gradients, and couples all retained features in one regularized least-squares solve. Under structural and stability hypotheses, we establish an $L^2$ error bound that links solution and residual truncation to finite-width approximation and regularized finite-sample fitting, and we derive guarantees for width and structure recovery. The resulting width is polynomial in the dimension at fixed interaction order, with dimension-independent higher-order contributions under uniform structural control. Residual screening achieves exact recovery of the prescribed three-pair support, while fitted-predictor gradients recover oblique directions through dimension $50$. In random-ridge tests, less than $1\%$ additional width reduces errors by factors of $14$-$39$ over coordinate blocks and $34$-$100$ over equal-width full-dimensional RFM. Semilinear computations extend HA-RFM through dimension $100$, while dense and distributed interactions delineate the coordinate families required for broader structure.

Quantum teleportation is a key protocol for transmitting quantum information using entanglement and classical communication. Its reliability is constrained by both the availability and fidelity of shared entangled pairs, which are affected by stochastic generation and memory decoherence. In this work, we focus on encoded teleportation, in which quantum information is encoded using a quantum error-correcting code and transmitted as a codeword. We evaluate reliability in terms of logical error probability, considering latency-constrained settings where entanglement is accumulated over time and degrades while in memory. We develop a unified framework that captures the interaction between entanglement availability, decoherence, and coding decisions. Our results show that the benefits of longer codes depend on the availability and fidelity of entangled pairs, as acquiring additional resources introduces delays that can reduce their quality. To address this latency-reliability tradeoff, we leverage code puncturing to enable flexible encoded teleportation, allowing the effective code length to adapt across different latency regimes while preserving a common stabilizer structure. Numerical results show that encoded teleportation can provide substantial reliability gains over uncoded transmission under a common entanglement-acquisition latency constraint, and that selecting appropriate punctured codes improves performance across varying latency budgets. Overall, our results highlight the importance of resource-aware adaptation for reliable quantum networking.

Weissler's conjectured characterization of complex hypercontractivity on the Hamming cube remained open in the ranges $2<p\le q<3$ and $\frac32<p\le q<2$. Ivanisvili–Nazarov established the diagonal cases. We prove the remaining strict off-diagonal cases through the corresponding two-point inequality. Our argument gives a self-contained proof for every $2<p\le q<\infty$ and, by duality, for every $1<p\le q<2$, including the previously known diagonal cases. The proof is based on a flow argument.

We give a spectral description of the self-similar collapse profile of the Constantin-Lax-Majda (CLM) equation, the $a=0$ anchor of the generalized family $w_t + a\,u\,w_x = u_x\,w$, $u_x = Hw$. Linearizing about the exact profile $\Omega(y) = -y/(y^2+1/4)$ and realizing $L_0$ as a closed operator on the origin-$H^2$ space, we prove three things at $a=0$. Its essential spectrum meets the closed half-plane $\{\mathrm{Re}\,\lambda \ge -1/2\}$ in the single vertical line $\{\mathrm{Re}\,\lambda = -1/2\}$: the line is placed by a log-widening Weyl sequence, and an explicit Hardy-Mellin resolvent bound constructively empties the rest of the half-plane apart from $0$ and $1$. Its full point spectrum over $\mathbb{C}$, on the odd realization, is exactly $\{0,1\}$, the scaling and time-shift symmetry modes, with no embedded eigenvalues; removing these by the standard modulation leaves a spectral gap of $1/2$ on $X$. The linear semigroup and its exact decay rate $e^{-\tau/2}$ are computed in closed form, but on a weighted space of the conjugated variable reached from $X$ by a bounded transfer map; we keep the two separate, since $L_0$ is non-normal and a spectral gap does not by itself give a decay rate in the $X$ norm. A realization dichotomy identifies the in-strip smear of generic discretizations as the faithful spectrum of the maximal $L^2$ realization, which origin-$H^2$ removes. For $a>0$ we prove a conditional two-line inclusion for each admissible smooth focusing profile, recompute the branch $c_l(a)$ of Lushnikov, Silantyev, and Siegel as a cross-check, and record the formal scaling-relevance exponent $s^*(a) = 1/c_l(a)$, below which fractional dissipation is asymptotically subdominant in self-similar variables for fixed sufficiently regular data. The contribution is the realization-dependent spectral picture of the collapse profile itself.

Conditional optimization tailors decisions to contextual or event information, but its practical use is often limited by the difficulty of learning the relevant conditional distribution from finite samples of a target joint distribution. This challenge is especially acute when target joint data are scarce or unavailable, or when few observations fall in the conditioning region of interest. Related joint data may be available from multiple sources, such as different stores, markets, populations, or operating environments, but these sources may be biased relative to the target distribution and cannot be pooled naively. We develop a distributionally robust framework based on optimal transport (OT) for harnessing such heterogeneous data in conditional optimization. The framework constructs ambiguity sets over joint distributions using OT distances to empirical source distributions and optimizes worst-case conditional performance over plausible target laws. We propose three OT ambiguity sets that capture different ways of using heterogeneous sources: enforcing simultaneous source consistency, aggregating source discrepancies through weights, and centering the ambiguity set at an OT barycenter. We derive tractable reformulations, establish feasibility conditions, discuss parameter choices, and characterize the relationships among the formulations, revealing trade-offs between robustness, information aggregation, and computational complexity. We demonstrate the value of the framework through a conditional assortment problem using demand and product-feature data from multiple stores.

Let \(f(k)\) be the minimum integer \(N\) such that any red–blue edge-coloring of the ordered complete graph on \(N\) vertices contains a set of \(k\) vertices whose induced coloring is admissible. In this note, we obtain the exact value of $f(k)$ for $k\ge 3$, which confirms a conjecture posed by Bradač, Liu, Wu and Xu.

Federated learning (FL) over wireless networks suffers from significant training latency and degraded convergence due to unreliable wireless transmission, especially under blocked propagation environments. Although reconfigurable intelligent surfaces (RISs) can improve communication reliability, existing wireless FL studies rarely characterize the trade-off between learning convergence and communication delay under modulation-dependent transmission errors. In this paper, we consider a wireless FL system operating under RIS-assisted blocked-link propagation scenarios, and focus on adaptive modulation and sub-channel allocation for convergence-latency aware communication design. By characterizing the effect of symbol errors on uploaded local gradients, we derive a convergence-related upper bound that reveals the impact of symbol error rate (SER) on FL loss decay. Based on this result, we formulate a joint convergence-latency optimization problem, which is cast as a mixed-integer nonlinear programming (MINLP) problem, and solve it using a low-complexity hybrid alternating optimization framework. Extensive experiments on MNIST, CIFAR-10, and Speech Commands show that the proposed scheme consistently achieves faster convergence and higher test accuracy than existing adaptive communication schemes, especially in complex tasks and challenging wireless scenarios.

Let $A$ be the self-adjoint operator in $L^{2}(-1,1)$, generated by the second-order classical Legendre differential equation% \[ \ell\lbrack y](t)=-\left( (1-t^{2})y^{\prime}(t)\right) ^{\prime}+ky(t)=\lambda y(t)\quad(t\in(-1,1)), \] which has the Legendre polynomials $\{P_{m}\}_{m=0}^{\infty}$ as a complete sequence of eigenfunctions; here $k$ is a fixed, non-negative real number. This is the Friedrichs extension of the minimal operator associated with $\ell[\cdot]$ in $L^2(-1,1)$. For each $n \in \mathbb{N}$, we show that $\mathcal{D}(A^{n})$ is characterized by one integrability condition instead of $2n$ boundary conditions as dictated by the classical Glazman-Krein-Naimark theory. We also prove that if $f\in\mathcal{D}(A^{n})$ then $f^{(n)}\in L^{2}(-1,1).$ This smoothness result extends known results when $n=1$ and $n=2.$ Furthermore, this result is optimal in the sense that there exists $g\in\mathcal{D}(A^{n})$ with $g^{(n+1)}\notin L^{2}(-1,1)$.

We construct weak equilibrium measures and weak capacities for the hypoelliptic third-order Langevin diffusion motivated by an accelerated sampling algorithm (Mou et al. (2021) J. Mach. Learn. Res., 22(42), 1–41). In this process, the Brownian noise acts only in the highest-order auxiliary variable and reaches the physical variables through a step-three Hörmander chain, so the standard uniformly elliptic boundary-flux theory is not directly applicable at characteristic points of phase-space balls. We prove an elliptic-regularization stability theorem for the corresponding hitting laws and then define the weak equilibrium measure and weak capacity. The proof combines the boundary-hitting stability strategy of Lee–Ramil–Seo (2026, arXiv:2503.12610v2) with localized hypoelliptic heat-kernel estimates (Pigato (2022) Stoch. Process. Appl., 145, 117–142) adapted to the third-order chain. We obtain the bounded-domain weak capacity–hitting identity and a Lyapunov drift argument in the spirit of Lee–Ramil–Seo that yields positive Harris recurrence and extends the construction to a whole-space weak equilibrium measure, and whole-space capacity–hitting identity.

We show that if all the even exponents of an odd perfect number N are equal to 2 except for one, then $3^{23,000,000,000}$ divides N .

High-dimensional hyperinterpolation is severely hampered by the curse of dimensionality, as its coefficient tensors grow exponentially with the ambient dimension. Existing research predominantly focuses on heuristic algorithmic optimizations, often overlooking the inherent structural properties of these tensors. This paper establishes a rigorous theory of intrinsic low-$\epsilon$-Tucker-rank for hyperinterpolation coefficient tensors, delivering near-optimal low-rank approximations with error bounds that are nearly independent of the dimension. We further construct a unified, Tucker-compatible theoretical framework that integrates both Chidori-type and Fiber-type tensor CUR (TCUR) decompositions, deriving tight and stable Frobenius-norm error estimates that depend exclusively on tensor spectral properties and index set geometry. We mathematically verify the convergence and numerical stability of greedy adaptive index selection schemes and prove their near-optimality, enabling a fully tensor-free hyperinterpolation workflow that avoids constructing the full coefficient array. Three practical greedy TCUR algorithms and a lightweight TCUR-to-Tucker recompression pipeline are proposed as direct corollaries of our structural theory. Numerical experiments across three distinct families of high-dimensional test functions validate all theoretical predictions and confirm the intrinsic low-rank compressibility of hyperinterpolation coefficients. In contrast to prior algorithm-centric studies, this work prioritizes rigorous theoretical characterization over implementation tricks, establishing a unified structural and mathematical foundation for high-dimensional hyperinterpolation.

A transmission scheme for secure communications, called secret-message transmission by echoing encrypted probes (STEEP), can deliver a positive secrecy rate, even when eavesdropping channels are much stronger than that between users, subject to sufficient asymmetric power allocations from users. This paper considers an optimization problem for STEEP applied to MIMO (or MIMOME) channels between multi-antenna users and multi-antenna Eve. We reveal for the first time a significant increase of the achievable secrecy rate of MIMO STEEP using optimized precoders.

We establish a quantitative adiabatic estimate for a class of finite-dimensional non-Hermitian Schrödinger dynamics. The original non-Hermitian Hamiltonian is assumed to be diagonalizable with real spectrum and non-crossing eigenvalues. We then construct a dynamically compatible time-dependent metric operator, and its positive square root defines a Dyson map. The associated Dyson-transformed Hamiltonian is Hermitian. When this Dyson-transformed Hamiltonian satisfies the Hermitian adiabatic assumption, the standard resolvent-projection method gives an explicit adiabatic estimate in the Hermitian representation. Pulling this estimate back through the Dyson map gives an approximation in the original non-Hermitian representation. The resulting Dyson-pulled-back projections are then compared with the spectral projections of the original non-Hermitian Hamiltonian by using a contour-resolvent estimate. The final bound contains two contributions: the pulled-back Hermitian adiabatic error and the projection-comparison error. A two-level non-Hermitian model is presented to illustrate the hypotheses of the theorem and the two contributions appearing in the final estimate.

We study the linearized motion of a spherical rigid body immersed in an incompressible viscous fluid occupying the whole space, under Navier slip boundary conditions with friction. While semigroup regularity and decay estimates are well understood for the no-slip case, such results have been limited to bounded domains in the slip setting. By exploiting the geometric regularity of the sphere, we establish strong $L^q$-regularity, bounded analyticity, and sharp $ L^q-L^r$ estimates for the corresponding fluid-structure operator. These results would provide a foundation for constructing strong $W^{2,1}_q$ solutions and Kato-type solutions to the corresponding nonlinear system.

Secure communications over SIMO and MISO channels between wireless nodes are commonly encountered in applications and widely considered in the literature. This paper investigates such a problem by considering a newly proposed scheme called secrecy-message transmission by echoing encrypted probes (STEEP). We focus on a type of heavily eavesdropped channels where an aggressive eavesdropper (Eve) has multiple antennas and is capable of jamming and eavesdropping in the full-duplex mode. Assuming optimal jamming and eavesdropping by Eve, we present the secrecy rates achievable by STEEP in two forms: using SIMO channel in phase 1 and using MISO channel in phase 2, and vice versa. With any given channel condition between every pair of nodes and any finite jamming power by Eve, the secrecy rate of STEEP in either form is shown to be positive subject to a positive power in phase 1 and a sufficiently large power in phase 2. The connections between STEEP in either form and the conventional SIMO and MISO schemes are provided, and their secrecy rates are compared comprehensively subject to random fading channels and finite transmit powers.

We introduce an open-source library for machine-checked control theory in the interactive proof assistant Lean to lay foundations for the verification of cyber-physical systems. To this end, as representative theorems, we present formalizations of Lyapunov stability theory and the small-gain theorem. First, the machinery employed for formalizing Lyapunov stability, i.e., neighborhood filters, allows stating a Lyapunov theorem that covers both points and sets and applies to continuous, discrete, and hybrid systems. Second, the small-gain theorem is proved via stating input-output systems as relations without the usual well-posedness assumption. The Lean formalization of each of these theorems is then presented. We conclude by discussing the library architecture and mentioning some of the other system theoretic results that are formalized in the library along with future plans.

This paper aims to establish the global regularity and large time behavior of solutions to the three-dimensional (3D) compressible magnetohydrodynamics (MHD) equations with vanishing vertical magnetic resistivity in the upper half-space with no-slip boundary condition on velocity and perfectly conducting boundary condition on magnetic field. By exploiting anisotropic Sobolev inequalities and elaborated estimates, we are able to achieve global-in-time uniform regularity estimates of solutions, which are independent of small vertical resistivity coefficient $\varepsilon$. These uniform global regularity estimates allow us to pass to the limit as \(\varepsilon \to 0\) and obtain the convergence to the corresponding MHD system without vertical magnetic resistivity globally in time. Moreover, the $H^{1}\left(\mathbb{R}_{+}^{3}\right)$ decay rates of solutions to the original system are also derived based on the detailed analysis on semigroup of the related linearized operator in half space and the uniform energy estimates achieved. In contrast to the decay estimates for the limit system obtained by Gao and Xie \cite{JDE}, the present decay estimate exhibits a slower rate, attributable to the absence of higher-order normal derivative estimates, which results from the occurrence of boundary layers. Finally, we combine both sets of regularity and decay results to rigorously prove an explicit time-uniform $L^2$ convergence rate of order $\varepsilon^{\frac{1}{4}}$ for the vanishing vertical magnetic resistivity limit process.