Papers

We investigate the lifting of quantum Gromov-Hausdorff convergence through Toeplitz type $C^*$-algebra extensions by stable ideals in the framework of noncommutative metric geometry. Working with the spectral metric space construction of Hawkins and Zacharias (Comm. Math. Phys. 350 (2017), 475-506), we consider a sequence of complete sub-operator systems of the quotient or the unital $C^*$-algebra underlying the stable ideal, converging in the quantum Gromov-Hausdorff distance. We study whether this induces a corresponding convergent sequence of complete sub-operator systems of the extension. To address this problem, we construct complete sub-operator systems of the extension associated with those of the quotient and the unital $C^*$-algebra underlying the stable ideal. We also introduce the notion of unital $2$-contractive approximation together with its Toeplitz type refinement to provide the compatibility required by the commutator structure of the Dirac operator on the extension. We prove that, under this approximation hypothesis on the convergent sequence in the quotient or the unital $C^*$-algebra underlying the stable ideal, quantum Gromov-Hausdorff convergence lifts to the extension.

Generalization remains a pivotal challenge in deep learning, where traditional optimizers like Stochastic Gradient Descent (SGD) often converge to sharp minima, leading to overfitting and reduced performance on unseen data. Building on Sharpness-Aware Minimization (SAM), for seeking flat minima associated with improved generalization, we propose the Extragradient-Inspired Sharpness-Aware Minimization (EISAM), a novel optimizer that enhances generalization via the extragradient technique. EISAM uses a two-step update process: a prediction step investigating the geometry of the loss landscape and a perturbation step that refines updates with a base optimizer. This approach achieves better generalization performance than SAM. Crucially, EISAM reduces sensitivity to the perturbation radius, enhancing robustness, and simplifying the tuning across diverse settings. Extensive experiments on benchmark datasets demonstrate that EISAM consistently outperforms SGD, Adaptive Moment Estimation (Adam), and SAM in test accuracy and training efficiency across various architectures. Theoretical analysis further confirms that EISAM tightens the generalization bound by steering parameters toward flatter minima with reduced curvature. Accompanied by a thorough hyperparameter analysis, EISAM offers practical tuning guidance, establishing it as a robust, scalable, and broadly applicable optimization solution that advances both the theory and practice in deep learning.

In this review we discuss semi-classical methods that are traditionally used to describe many-body systems in physics, but may also be used to describe partitions of integers in analytic number theory. Specifically, we explore the connection between the methods of statistical mechanics and number partitions. Though the two fields appear very different, their fundamental issues bear a close resemblance. In the former case it is the distribution of a given amount of energy among the particles in an ensemble at a given temperature with well defined properties, while in the latter case it is the way an integer is partitioned into other integers, with or without restrictions. We begin with a discussion of the single-particle quantum density of states, also called the level density, in which we illustrate the connection between the density of states and the classical periodic orbits through the semiclassical trace formula. This is then extended to many particle systems. We show that the asymptotic number partition is reproduced by the average (smooth) part of the level density at discrete integer values of the argument. In the especially interesting case of distinct square partitions, pronounced oscillations are well reproduced by the periodic orbit theory in terms of a few orbits characterised by Pythagorean number triples. We speculate on the connection to Fermat's theorem as to why such regular oscillations (though vanishing asymptotically) exist only in this special case. Finally, we discuss some new results for integer partitions of primes, both unrestricted and distinct.

In this paper, we study the optimal control problems for stochastic Volterra integral equations driven by G-Brownian motion under Volatility Ambiguity. With the help of G-stochastic analysis techniques and the weak convergence methods, we obtain the variation of the cost functional and the variational inequality. Under the convexity assumptions, we establish the stochastic maximum principle, which serves as both a necessary and sufficient condition for optimal control.

This article established the existence results for the $m$-harmonic equation involving critical Choquard nonlinearity and subcritical perturbation. We first explore the minimizers of the $m$-harmonic operator with the critical Choquard equation. Then, using these minimizers, we establish delicate estimates to show the energy below the threshold level, which helps to recover the compactness. Further, we prove the existence of a nontrivial solution for our problem with different kinds of local and nonlocal subcritical perturbations. To the best of our knowledge, this is the first article dealing with the polyharmonic equation and critical Choquard type nonlinearity. The results obtained are even new for $m\geq 2$.

In next-generation networks, communication systems will no longer be limited to data transmission and will be expected to acquire awareness of the surrounding environment. This leads to the concept of integrated sensing and communication (ISAC), where the same wireless infrastructure is used for both communication and environmental sensing. Thus, ISAC enables the system to transmit information efficiently and observe and interpret channel variations and user behavior. Motivated by this capability, this work focuses on detecting an active attacker in an urban environment scenario, where the attacker intentionally manipulates beamforming directions to increase interference and mislead the transmitter into allocating the main lobe of beam toward itself instead of legitimate users. We apply game-theoretic approaches to model the interaction between legitimate users and the attacker, and integrate the resulting utility-based formulation into a reinforcement learning (RL) framework. Simulation results demonstrate that the proposed method effectively addresses security challenges in dynamic 6G ISAC systems.

Fix a positive integer $n$, and put $B_d=\{d,2d,\ldots,nd\}$. Let $M_k(n)$ be the least integer $m$ for which one translate of each of $B_1,\ldots,B_k$ can be placed pairwise disjointly in $[m]$. We prove that, for every $\eps\in(0,1)$ and all sufficiently large $k$, one has $M_k(n)\le n\lceil(1+\eps)k\rceil$. Since the trivial counting bound gives $M_k(n)\ge nk$, it follows that $M_k(n)=(1+o(1))nk$ for every fixed $n$. This confirms a conjecture of Alon, Dębski, Grytczuk and Przybyło on prescribed-difference packings of fixed-cardinality arithmetic progressions.

We address the question of how small a non-vanishing sum of $N$-th roots of unity with $k$ terms can be. We show upper bounds of the shape $N^{-\alpha_k}$, where $\alpha_k\rightarrow\infty$ with $k$. We also address the question of improving these bounds for a positive proportion of $N$.

Let $S(G)$ denote the Seidel matrix of a simple graph $G$, and let $E_S(G)$ be the Seidel energy of $G$, defined as the sum of the absolute values of the eigenvalues of $S(G)$. In this paper, we study the change of Seidel energy under edge deletion. For an independent-set blow-up graph $G=H[n_1,\ldots,n_p]$, we establish a general structural criterion within the framework of independent-set blow-up graphs. More precisely, if the endpoints of the deleted edge $e$ belong to blow-up parts of sizes $n_a$ and $n_b$, respectively, then $E_S(G-e)>E_S(G)$ whenever both $n_a,n_b$ are at least $4$, or one is $3$ and the other is at least $6$, or one is $2$ and the other is at least $15$. As applications, we obtain the following consequences. First, for every Turán graph $T(n,r)$ with $r\geq4$ and $n\geq4r$, deleting any edge strictly increases the Seidel energy. Second, for complete multipartite graphs, we derive an exact reduced-order spectral criterion for the remaining cases not covered by the structural result. This criterion determines whether the Seidel energy increases, decreases, or remains unchanged after deleting an edge, by using matrices whose orders depend only on the number of partite sets. These results provide affirmative answers to two problems proposed by Tian et al. [Linear and Multilinear Algebra 70 (19) (2022), 4597–4614].

G.\ D.\ Baker formulated a forcing method to interpret integer optimisation problem into $2$-adic linear regression, and proved the NP-hardness of $2$-adic linear regression. We generalise the forcing method to a wider class of $p$-adic optimisation for the case where $p$ is not necessarily $2$, and prove the NP-hardness of $p$-adic linear regression, the NP-hardness of $2$-adic dynamic neural network by S.\ Albeverio, A.\ Khrennikov, and B.\ Tirrozi, and the NP-hardness of a partial generalisation of the $p$-adic optimisation problem associated to van der Put neural network by G.\ L.\ R.\ N'guessan.

In this paper, we initiate a systematic study of $(\infty : \infty)$ holomorphic correspondences that naturally arise as conformal combinations (matings) of transcendental entire maps with Fuchsian groups. This construction parallels the recent theory of finite-degree algebraic correspondences associated with rational maps. Our correspondence combines the dynamics of a transcendental entire function outside a distinguished attracting/parabolic basin with the action of a compatible Fuchsian group within it. We show that the resulting correspondence is the composition of a Möbius involution and the deleted covering correspondence of a meromorphic function having exactly one simple pole. When the transcendental entire function has finitely many singular values, so does this meromorphic function, and its line complex can be described explicitly.

We study binary classification when the two classes are elliptical with a shared radial generator $g$. The log-likelihood ratio is then an additive function of the two squared Mahalanobis radii, with radial link $\varphi=\log g$. For Gaussian classes $\varphi$ is affine and the rule reduces to quadratic discriminant analysis (QDA); otherwise the Bayes link is not the identity. Instead of fitting this link with a spline smoother, we derive its family from the within-class law of the radius and estimate a finite fractional-power projection. We prove that the link is identifiable from the radius law, that the plug-in estimator is $\sqrt{n}$-consistent and asymptotically normal under finite-moment conditions, and that the induced classifier is asymptotically Bayes-optimal in an iterated sieve limit. For $d\ge2$ and non-nested class parameters, the identity link suffices only for affine (Gaussian) generators. The bridge, GAM membership and the affine direction of this dichotomy are verified in Lean 4. We compare against the global Mahalanobis-GAM of Ghosh et al. (2025), re-implemented with mgcv REML splines at equal input budget. On two UCI benchmarks the derived link ties with the fitted GAM. On six financial series from three asset classes, with validation that respects temporal dependence, it is never significantly worse than the fitted GAM: better on CAD/USD and tied on the other five. It beats QDA on the three heaviest-tailed series (oil $[+0.023,+0.069]$, S&P 500 $[+0.039,+0.125]$, JPY/USD $[+0.009,+0.047]$) and ties elsewhere. A rate simulation with a closed-form ground truth reproduces the $\sqrt{n}$ rate and, under covariance heterogeneity, QDA's risk floor against the derived link's vanishing excess risk. The derived link thus matches a tuned GAM without smoothing-parameter selection and improves on QDA where the generator departs from the Gaussian.

This paper presents an approach and application of optimization of spatial packaging of interconnected systems with physical interactions (SPI2) in three-dimensional component placement problems. To enable its application for an automotive use case, SPI2 must support both initial design generation, including component alignment, and robust system-level coordination, requiring improved solution reliability and tractable computational cost. To address these requirements, the proposed methodology improves convergence rate and solution quality by enhancing numerical robustness in gradient-based optimization while reducing computational load. Existing SPI2 approaches are extended through the addition of alignment capabilities, enabling the representation of port-to-port alignments between components. Furthermore, the applicability of SPI2 is expanded by treating component placement locations as design variables, allowing for penalty-based coordination to ensure design feasibility and enabling integration within system-level optimization. The approach is validated using a multi-objective optimization framework based on Nondominated Sorting Genetic Algorithm II (NSGA-II), applied to a combined powertrain optimization and battery chassis integration problem. This demonstrates the effectiveness of the SPI2 in a system-level design context. The results show a twofold application of SPI2 in an automotive use case: first, as a tool for initial design generation, and second, as part of a system-level design coordinator that outperforms a discretized exhaustive search while requiring lower computational cost.

This study harnesses topological analysis in an attempt to reveal structure in the dynamics of a crowd. Topology and in particular persistent homology characterizes relational structures in data through the number of connected components and holes, that is, a loop of pairwise connection with no connections across it. We apply this universal data analysis method to a simulated time series of individual pedestrian positions of a crowd moving through a wide corridor – either uni- or bidirectional. We consider two pedestrians to be connected, when they are sufficiently close. This approach leads to two matrices containing the persistence signatures for the whole time series, so-called CROCKERs. Despite the high level of data abstraction, the CROCKERs' first two principal components on time-delayed positional data show a clear separation of the different parameter configurations. This holds up to symmetry. Our results support our claim that persistent homology is a useful tool to characterize crowd dynamics without introducing any prior assumptions about the detectable spatio-temporal patterns.

The multiobjective optimization of electric machines always involves a trade-off caused by various competing objectives such as performance and cost. A suitable design is usually determined by comparing variants from the Pareto front, which has been generated by a large number of simulation runs. This paper addresses the efficient generation of the Pareto front using a continuation method based on a homotopy method that exploits second-order derivative information to achieve superlinear convergence, enabling the fast generation of new Pareto-optimal points within only a few iterations. A key contribution is the derivation of formulas to compute the Hessian with respect to geometry parameters and shape, thus enabling direct modifications of the motor geometry in the context of Isogeometric Analysis. We apply our method to nonlinear 2D magnetostatic simulations of a permanent magnet synchronous motor and demonstrate its effectiveness by optimizing the cost, mean torque and torque ripple of the motor. Compared to a first-order optimization method, this approach reduces the number of iterations and function evaluations needed, making the pareto optimization fast and efficient.

We study a generalization of the star coloring problem on hypergraphs. For a family of connected subhypergraphs $\mathcal{F}$, we define an $\mathcal{F}$-coloring of a hypergraph as a coloring avoiding monochromatic hyperedges and any 2-colored member of $\mathcal{F}$. We let $\chi^r_{\mathcal{F}}(d)$ be the maximum of the minimum number of colors needed for an $\mathcal{F}$-coloring of an $r$-uniform hypergraph with maximum degree $d.$ We show bounds for $\chi^r_{\mathcal{F}}(d)$, that also yield results on star and acyclic coloring problem on hypergraphs.

The third problem in Ben Green's collection of 100 open problems asks whether an open subset of $(0,1)$ that does not contain $x,y,z$ with $xy=z$ must have measure at most 1/3. We give an affirmative answer to this question. As part of the proof we obtain a result of independent interest that gives a lower bound for the size of the sumset and the difference set of a set of reals in terms not just of its size but also of a parameter that measures how far it is from being an interval.

Energy consumption remains a dominant operational challenge for current and future cellular systems, especially in dense urban deployments. This paper investigates a novel role for non terrestrial network (NTN) high-altitude platform station (HAPS) as an enabler of energy-efficient operation rather than only coverage extension. We define the HAPS-Hypercell as a wide-area non-terrestrial layer that can assume the coverage role of multiple terrestrial macro-cells, enabling, for the first time, the shutdown of both capacity and coverage macro-cells. We develop a comprehensive third generation partnership project (3GPP)-compliant system model, along with two HAPS-Hypercell pairing architectures that capture the interplay among multiple layers, realistic channel conditions, and distributed carrier shutdown (CS) mechanisms. Our results show that the HAPS-Hypercell can effectively reduce overall network power consumption. We then identify key limitations of a straightforward HAPS integration, laying the groundwork for future optimization and providing key insights for next-generation CS operations.

We derive and analyze a time-domain pressure–interface evolution model for a compressible gas bubble in a compressible liquid with surface tension. Starting from the nonlinear two-phase Euler free-boundary problem and linearizing about a spherical Young–Laplace equilibrium, we obtain a coupled bulk–surface hyperbolic system for the liquid pressure, the gas pressure, and the normal displacement of the interface. The time-domain analysis is complicated by the fact that the surface-tension quadratic form is indefinite: the \(Y_0^0\) component is the volume-changing breathing mode, the \(Y_1^m (m = \{-1,0,1\})\) components are neutral translations, and only the higher spherical harmonics give a coercive shape-mode energy. We exploit this decomposition. The coercive sector is treated by a \(C^0\)-semigroup argument, while the breathing mode is analyzed separately by Fourier–Laplace methods. This yields well-posedness for admissible finite-energy data and classical solutions under the natural compatibility conditions. We then justify two limiting descriptions with quantitative error estimates. In an acoustic quasi-static regime, the model reduces to the linearized Rayleigh–Plesset equation for the breathing mode and to the linearized Rayleigh–Lamb equations for the shape modes. In a different regime, it reduces to the frozen-interface acoustic transmission model. Finally, a frequency-domain analysis identifies the corresponding Minnaert, Rayleigh–Lamb, and Fabry–Pérot-type resonance mechanisms as different components and limits of the same pressure–interface formulation.

For a free $\mathbb{Z}_2$-space $X$, the co-index $\mathrm{coind}(X)$ is the largest integer $m$ for which there exists a $\mathbb{Z}_2$-equivariant map $S^m \to X$, where $S^m$ carries the antipodal action. Since suspension sends such a map to a $\mathbb{Z}_2$-equivariant map $S^{m+1}\to S(X),$ one always has $$\mathrm{coind}(S(X)) \geq \mathrm{coind}(X)+1.$$ We prove that the excess over this lower bound can be arbitrarily large. More precisely, for every $n \geq 2$, we construct a finite free $n$-dimensional simplicial $\mathbb{Z}_2$-complex $\mathcal{K}$ such that $\mathrm{coind}(\mathcal{K})=1$ and $\mathrm{coind}(S(\mathcal{K}))=n+1$. This answers a question of Simonyi, Tardos, and Vrécica on the possible growth of co-index under suspension and, equivalently, shows that the co-index lower bound on the chromatic number of a graph $G$ obtained from $B_0(G)$ can exceed the corresponding bound obtained from the box complex $B(G)$ by an arbitrarily large amount.

We develop an integration theory for functions taking values in a metric space. Following a Bochner-type construction, we define the concept of free integral as an element of the Lipschitz-free space $\mathcal{F}(M)$. We establish the main properties of this integral, including duality formulas, and the study of the resulting space of free integrable functions. We also cover when the metric space is a Banach space: in this setting, the free integral has an interpretable decomposition generalising the Bochner integral. We then connect the free integral with the geometry of $\mathcal{F}(M)$ by showing that it always produces convex integrals of molecules. This allows to study extremal properties within the unit ball of $\mathcal{F}(M)$. Finally, we provide a detailed example to illustrate the framework we develop.

We aim to identify scattering network architectures that maximize the separation capacity on data with low intrinsic dimension. The networks we consider employ a fixed monomial nonlinearity and no pooling, so that the only design variable is the frame generated by the network filters. For data modeled as rectifiable sets, we first characterize and bound the separation capacity of general feature extractors in terms of the geometry of the dataset. We then particularize to scattering networks and obtain two design criteria: (i) the filters should meet the data on sufficiently many frequencies, and (ii) the matrices coupling the frame to the geometry of the data should be well-conditioned.

We present an invariant-domain-preserving (IDP) treatment of nonconforming interfaces for Legendre–Gauss–Lobatto Discontinuous Galerkin Spectral Element Methods (LGL-DGSEM) with adaptive mesh refinement (AMR) on Cartesian meshes. The proposed methodology extends recently developed convex limiting and graph-viscosity frameworks for DGSEM to meshes containing hanging nodes. Starting from a conservative mortar formulation, we derive low-order interface fluxes that satisfy the requirements of invariant-domain-preserving discretizations. To avoid the excessive diffusion associated with fully connected mortar couplings, a sparsification strategy based on LGL subcell characteristic functions is introduced, yielding compact interface stencils. The resulting mortar fluxes remain conservative, reduce to the standard conforming formulation on matching interfaces, and naturally fit into graph-viscosity-based low-order schemes used for convex limiting. The proposed construction provides the missing ingredient required to combine high-order DGSEM discretizations, invariant-domain-preserving limiting, and adaptive mesh refinement within a unified framework for nonlinear hyperbolic conservation laws. We provide numerical verifications of the properties of the proposed scheme and run challenging simulations that require positivity limiting and shock-capturing.

We introduce a unified framework to establish the high-dimensional asymptotic behavior of maximal functions associated with radial log-concave probability densities, encompassing the maximal heat semigroup, Hardy-Littlewood maximal function over Euclidean balls, and, additionally, maximal spherical means. Namely, for any $p \in (1, \infty)$, we prove that the $L^p(\mathbb{R}^d)$ operator norms of these maximal operators all converge as the dimension $d \to \infty$ to a single, universal limit $\lambda(p)$. Furthermore, by proving that the $L^p$ operator norms for the heat semigroup $\mathcal G_*^d$ are monotonically non-decreasing in the dimension, we provide explicit quantitative bounds on the universal limit, showing that $\frac{2}{5}\frac{p}{p-1} \le \|\mathcal{G}_*^1\|_{L^p(\mathbb{R}) \to L^p(\mathbb{R})} \le \lambda(p) \le \frac{p}{p-1}$. We also prove an extremality property: among all symmetric convex bodies in high dimensions, the maximal operator associated with the Euclidean ball achieves the asymptotically minimal $L^p$ operator norm. Our main results are established via a general transference principle that allows us to control maximal functions via Fourier multiplier symbols. To estimate these symbols uniformly across log-concave densities, we import variance type bounds and thin-shell type concentration of measure results, which are novel tools in the study of maximal functions. In particular, to prove the extremality property, we require a variance type bound for general log concave measures established in a recent series of breakthroughs in high dimensional convex geometry.

We construct smooth, uniformly convex planar domains that admit minimal, strictly stable solutions of a semilinear Dirichlet problem whose superlevel sets are nonetheless nonconvex. The class of admissible nonlinearities includes, in particular, two prototypical cases: the Gelfand-type nonlinearity $e^u$ and the family of shifted power-type nonlinearities $(a+u)^p$, where $a>0$ and $p>1$. By applying the elementary scaling properties of the Dirichlet problem, we also show that the same lack of convexity of superlevel sets holds for the corresponding parameter-dependent equations. These results provide a negative answer to a question posed by Brezis, who inquired whether the stability of a solution necessarily entails quasiconcavity for these prototypical stable configurations.

We present a novel isogeometric deep learning method, termed SplineNet, for the seamless design and analysis of shell structures with complex geometries. The proposed approach is built upon watertight spline representations, e.g., analysis-suitable unstructured T-splines, and features exact geometric descriptions of Computer-Aided Design (CAD) models in neural networks. Bézier extraction is used to build the network architecture, where Bernstein polynomials serve as the nonlinear activation functions. SplineNet can be applied in a data-free or data-driven way. In the data-free case, energy-based formulations can be naturally incorporated as loss terms, which fulfill the need of Computer-Aided Engineering (CAE) and can be accurately calculated. In particular, the Kirchhoff–Love (KL) model is adopted to solve for the mechanical behaviors of shell structures. This way, CAD and CAE can be tightly integrated in a deep neural network without the time-consuming model/data exchange process. In the data-driven case, SplineNet can be used as the trunk net of Deep Operator Networks (DeepONet) to provide interpretability. Given such a trained network and unseen input data, results can be immediately obtained without retraining the network or repeatedly performing the traditional workflow for analysis. In the end, a variety of numerical examples are studied to demonstrate the effectiveness of the proposed method, especially when real-world complex geometries are involved.

In this paper, we adapt the framework developed by Mason and Snaith to investigate the $n$-level density of zeros in the context of function fields. Specifically, we derive explicit formulas for the $n$-level density of zeros in families of quadratic Dirichlet $L$-functions associated with hyperelliptic curves of genus $g$ over the finite field $\mathbb{F}_{q}$. Employing Mason and Snaith's method, we obtain precise expressions for the $1$-level density in these families and extend the approach to higher-level densities. Furthermore, we apply the method to derive formulas for the $n$-level density of zeros in families of $L$-functions associated with prime characters. Our results are consistent with the findings of Andrade, Jung, and Shamesaldeen in the case $n=1$.

In this paper we explore a fast Poisson solver for problems with a solution that is known to be low-rank. We use an adaptive and warm started cross approximation called Cross-DEIM that iterates between index selection and and cross approximation to generate a low-rank solution. This paper focuses on leveraging a modern machine learning framework, PyTorch, as a general purpose array language to implement low-rank solvers based on Cross-DEIM. PyTorch enables native access to GPUs and accelerators but with a user-friendly high-level interface. We investigate statistical leverage scores for the index selection for the cross approximation due to the cost associated with the pivoted algorithms used with the discrete empirical interpolation methods (DEIM and QDEIM) which are historically preferred. The cross approximation is naturally paired with a Discrete Sine Transform (DST) Poisson solver. This allows the Fast Fourier Transform (FFT) to be evaluated in batches along dimensions independently without any global transpose even in higher dimensions. We present performance results running on a A100 GPU and AMD EPYC CPU demonstrating the usefulness of the approach that enables problems sizes that previously were not feasible.

We study a human-AI service system in which tasks arrive sequentially and are processed through a two-stage architecture: an automated chatbot followed, when necessary, by a human agent. We consider $T$ sequentially arriving tasks, each belonging to one of $K$ heterogeneous types. For each task the decision maker chooses how many resources to allocate to the chatbot, whose type-dependent success probabilities are initially unknown. Tasks not resolved by the chatbot enter type-dependent human-service queues, where they are processed by a human agent with unknown service rates. This model captures a central tradeoff in hybrid service systems: relying more on automation reduces human congestion but increases chatbot costs, while insufficient automation may overload the human agent. We propose the UCB-DPP policy, which combines Upper Confidence Bounds with Drift-Plus-Penalty control to learn the unknown parameters of the system while making queue-aware decisions. We prove that UCB-DPP achieves regret $\widetilde{\mathcal{O}}(K\sqrt{T})$ and guarantees mean-rate stability of the human-service queues. Simulations on synthetic instances show that the proposed policy outperforms natural baselines.

Adaptive gradient methods can favor max-margin separators that differ from gradient descent, yet a fixed positive numerical stability constant eventually changes the update geometry again. This paper studies the rate-controlled middle case for full-batch linear classification on separable data. For memoryless stability-annealed smoothed-sign descent with weighted exponential loss, we prove that the normalized iterates converge to the minimizer of a convex Burg-type barrier over a margin slice. The proof rewrites the dynamics exactly as entropic mirror ascent on a concave dual objective, controls the dual gap by a KL recursion, and yields an explicit S_t^{-1/2} normalized-iterate envelope. The static barrier geometry is fully characterized, including KKT conditions and both endpoint limits. Experiments validate the exact dual identities to floating-point error, illustrate the predicted path and rate diagram, and show an empirical fixed-epsilon crossover scaling in cumulative time. We further report robustness and boundary diagnostics for logistic tails, fixed-epsilon crossover, and adaptive-method variants, delineating the scope of the proved smoothed-sign theory.