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At least 91 records · Page 5Linked to original sources
arXiv2026

Self-normalized scaled quadratic variation

The concept of a scaled quadratic variation was originally introduced by E. Gladyshev in 1961 for processes with Gaussian increments. Using certain deterministic scaling, arrived at from the covariance of the process, Gladyshev showed that the sum of scaled square increments along the dyadic partition sequence converges almost surely to a finite limit. In this paper, we propose a pathwise counterpart in which the deterministic normalization is replaced by a self-normalizing factor built from the $p$-th variation of the path along a given sequence of partitions. The resulting quantity requires no probabilistic assumption and no knowledge of a covariance structure, and its scale is both path-dependent and sensitive to the partition sequence. Under a mild regularity condition on the limiting $p$-th variation, we show that the self-normalized and the classical deterministic normalizations are comparable, and for fractional Brownian motion the two agree up to a multiplicative constant. We establish a switching behaviour in the index, and prove that for $p \ge 2$ the self-normalized scaled quadratic variation obeys a smooth-transformation formula under $C^2$ maps; at $p=2$ this recovers the known transformation rule for quadratic variation. Since only squared increments are scaled, the construction polarizes, yielding a matrix-valued scaled quadratic variation for every $p \geq 1$ for $\mathbb R^d$ valued paths. We conclude with examples beyond the Gaussian setting.

math.PR↗
arXiv2026

A near-quantum-limited diamond maser amplifier operating at millikelvin temperatures

Current microwave quantum technologies require the amplification of weak signals with minimal added noise at millikelvin temperatures. To date, this stringent requirement has been met exclusively by superconducting technologies, such as Josephson or kinetic-inductance parametric amplifiers. A fundamentally distinct alternative approach could be offered by masers, the microwave counterpart of lasers, which were predicted as early as the 1950s to achieve quantum-limited noise performance under ideal conditions. However, their dependence on cryogenic operation historically limited further advancement. Here we demonstrate the first-ever non-superconducting, near-quantum-limited maser amplifier operating at millikelvin temperatures utilising nitrogen impurity spins (P1 centres) in diamond. Population inversion is achieved via microwave pumping, exploiting a four-spin cross-relaxation mechanism. We realise a maximum power gain exceeding 30 dB, an added noise of approximately 2.55 quanta above the standard quantum limit, and a maximum 1 dB output compression point of -63 dBm at 6.595 GHz. The ability to operate in strong static magnetic fields of arbitrary orientation may offer a complementary, non-superconducting route for applications such as semiconducting spin-qubit readout, magnetic-resonance spectroscopy, and dark-matter axion searches.

quant-ph↗
arXiv2026

Long time behaviour of Mean Field Games with fractional diffusion

In this paper we study the long time behaviour of mean field games systems with fractional diffusion, modeling the case that the individual dynamics of the players is driven by independent jump processes and controlled through the drift term, while being confined by an external field in order to guarantee ergodicity. In the case of globally Lipschitz, locally uniformly convex Hamiltonian, and weakly coupled costs satisfying the Lasry-Lions monotonicity condition, we prove that there is a unique solution $(u_T,m_T)$ to the mean field game problem in $(0,T)$ and we show that, if $T$ is sufficiently large, $(u_T,m_T)$ satisfies the so-called turnpike property, namely it is exponentially close to the (unique) stationary ergodic state for any proportionally long intermediate time.

math.AP↗
arXiv2026

Impartial removing games on grid graphs

A subset of the vertex set of a graph is geodetically convex if it contains every vertex on any shortest path between two elements of the subset. The convex hull of a set of vertices is the smallest convex set containing the set. We study two games in which two players take turns selecting vertices of a graph. The last player to move is the winner. The achievement game ends when the convex hull of the unselected vertices does not contain every vertex in the graph. In the avoidance game, the convex hull of the remaining vertices must contain every vertex. We determine the nim-value of these games for the family of grid graphs. We also provide some results for lattice graphs. Key tools in this analysis are delayed gamegraphs, option preserving maps, and case analysis diagrams.

math.CO↗
arXiv2026

Dissociating performance from compositional feature learning

Out-of-distribution (OOD) generalisation is considered a hallmark of human and animal intelligence. To achieve OOD through composition, a system must discover the environment-invariant properties of experienced input-output mappings and transfer them to novel inputs. This can be realised if an intelligent system can identify appropriate, task-invariant, and composable input features, as well as the composition methods, thus allowing it to act based not on the interpolation between learnt data points but on the task-invariant composition of those features. We propose that in order to confirm that an algorithm does indeed learn compositional structures from data, it is not enough to just test on an OOD setup, but one also needs to confirm that the features identified are indeed compositional. We showcase this by exploring two tasks with clearly defined OOD metrics that are not OOD solvable by three commonly used neural networks: a Multi-Layer Perceptron (MLP), a Convolutional Neural Network (CNN), and a Transformer. In addition, we develop two novel network architectures imbued with biases that allow them to be successful in OOD scenarios. We show that even with correct biases and almost perfect OOD performance, an algorithm can still fail to learn the correct features for compositional generalisation.

cs.LG↗
arXiv2026

Unsupervised Clustering for Fault Analysis in High-Voltage Power Systems Using Voltage and Current Signals

The widespread use of sensors in modern power grids has led to the accumulation of large amounts of voltage and current waveform data, especially during fault events. However, the lack of labeled datasets poses a significant challenge for fault classification and analysis. This paper explores the application of unsupervised clustering techniques for fault diagnosis in high-voltage power systems. A dataset provided by the Réseau de Transport d'Électricité (RTE) is analyzed, with frequency domain features extracted using the Fast Fourier Transform (FFT). The K-Means algorithm is then applied to identify underlying patterns in the data, enabling automated fault categorization without the need for labeled training samples. The resulting clusters are evaluated in collaboration with power system experts to assess their alignment with real-world fault characteristics. The results demonstrate the potential of unsupervised learning for scalable and data-driven fault analysis by structuring large unlabeled waveform datasets and supporting targeted expert review.

cs.LG↗
arXiv2026

On properness of moduli stacks of $D^{\times}$-shtukas over ramified legs

Given a maximal order $\mathcal{D}$ of a central division algebra $D$ over a global function field $F$, we prove an explicit sufficient condition for moduli stacks of $\mathcal{D}^\times$-shtukas to be proper over a finite field (modulo a suitable central action) in terms of the \emph{local invariants} of $D$ and \emph{bounds}. Our proof is a refinement of E.~Lau's result (Duke Math. J. \textbf{140} (2007)), which showed the properness of the \emph{leg morphism} (or \emph{characteristic morphism}) away from the ramification locus of $D$. %, by carefully measuring the contribution of ``ramified legs''. We also establish non-emptiness of Newton and Kottwitz--Rapoport strata for moduli stacks of $\mathcal{B}^\times$-shtukas, where $\mathcal{B}$ is a maximal order of a central simple algebra over $F$.

math.NT↗
arXiv2026

Tensorization is a powerful but underexplored tool for compression and interpretability of neural networks

Tensorizing a neural network involves reshaping some or all of its dense weight matrices into higher-order tensors and approximating them using low-rank tensor network decompositions. This technique has shown promise as a model compression strategy for large-scale neural networks. However, despite encouraging empirical results, tensorized neural networks (TNNs) remain underutilized in mainstream deep learning. In this position paper, we offer a perspective on both the potential and current limitations of TNNs. We argue that TNNs represent a powerful yet underexplored framework for deep learning--one that deserves greater attention from both engineering and theoretical communities. Beyond compression, we highlight the value of TNNs as a flexible class of architectures with distinctive scaling properties and increased interpretability. A central feature of TNNs is the presence of bond indices, which introduce new latent spaces not found in conventional networks. These internal representations may provide deeper insight into the evolution of features across layers, potentially advancing the goals of mechanistic interpretability. We conclude by outlining several key research directions aimed at overcoming the practical barriers to scaling and adopting TNNs in modern deep learning workflows.

cs.LG↗
arXiv2026

Spaces with the maximal projection constant revisited

Let $n \geq 2$ be an integer such that an equiangular set of vectors $w_1, \ldots, w_d$ of the maximal possible cardinality (that is, attaining the classical Gerzon upper bound) exists in $\mathbb{K}^n$, where $\mathbb{K}=\mathbb{R}$ or $\mathbb{K}=\mathbb{C}$ (so that $d=\frac{n(n+1)}{2}$ in the real case and $d=n^2$ in the complex case). We provide a complete characterization of $n$-dimensional normed spaces whose absolute projection constant is maximal among all $n$-dimensional normed spaces over $\mathbb{K}$. The characterization states that $X$ has the maximal projection constant if and only if it is isometric to a space whose dual unit ball is contained between the absolutely convex hull of the vectors $w_1, \ldots, w_d$ and a suitably rescaled zonotope generated by the same vectors. As a consequence, we obtain that, in the considered situations, $n=2$ with $\mathbb{K}=\mathbb{R}$ is the only case in which there is, up to isometry, a unique norm on $\mathbb{K}^n$ with the maximal projection constant. In this case, the unit ball is a linear image of a regular hexagon in $\mathbb{R}^2$.

math.FA↗
arXiv2026

A Divisor-Sum Analogue of the Collatz Map

Let $σ$ denote the sum-of-divisors function and let $\mathcal{R}$ send an odd integer $n$ to $σ(n)$ and an even integer $n$ to $n/2$. We conjecture that every orbit of $\mathcal{R}$ reaches $1$; this implies that there is no odd $2^{k}$-perfect number for any $k \ge 1$. We prove a pointwise descent estimate for the map $T$ induced by $\mathcal{R}$ on the odd integers, and show that a single application of $T$ divides almost every odd $n$ by $(\log n)^{2\log 2-\varepsilon}$, for every fixed $\varepsilon>0$. The estimate does not iterate, and we determine what is missing.

math.NT↗
arXiv2026

ProteinZero: Self-Improving Protein Generation via Online Reinforcement Learning

Protein generative models have shown remarkable promise in protein design, yet their success rates remain constrained by reliance on curated sequence-structure datasets and by misalignment between supervised objectives and real design goals. We present ProteinZero, an online reinforcement learning framework for inverse folding models that enables scalable, automated, and continuous self-improvement with computationally efficient feedback. ProteinZero employs a reward pipeline that combines structural guidance from ESMFold with a novel self-derived ddG predictor, providing stable multi-objective signals while avoiding the prohibitive cost of physics-based methods. To ensure robustness in online RL, we further introduce a novel embedding-level diversity regularizer that mitigates mode collapse and promotes sequence-level diversity among generated designs. Within a general RL formulation balancing multi-reward optimization, KL-divergence from a reference model, and diversity regularization, ProteinZero achieves robust improvements across designability, predicted stability, recovery, and diversity. On the CATH-4.3 benchmark, it consistently outperforms state-of-the-art baselines including ProteinMPNN, ESM-IF, and InstructPLM, reducing design failure rates by 36-48% and achieving success rates above 90% across diverse folds. Importantly, a complete RL run can be executed on a single 8xGPU node within three days, including reward computation and data generation. These results indicate that efficient online RL fine-tuning can complement supervised pretraining by allowing protein generative models to evolve continuously from their own outputs and optimize multiple design objectives without labeled data, opening new possibilities for exploring the vast protein design space. Code and model checkpoints are available at https://github.com/ziwenwang28/ProteinZero.

cs.LG↗
arXiv2026

Brockett cost function for symplectic eigenvalues

The sum of symplectic eigenvalues and corresponding eigenvectors of symmetric positive-definite matrices in the sense of Williamson's theorem can be computed via minimization of a trace cost function under the symplecticity constraint. Optimal solutions to this problem only offer a symplectic basis for the symplectic eigenspace corresponding to the sought symplectic eigenvalues. In this note, we introduce a Brockett cost function and investigate its properties and the connection with the symplectic eigenvalues and eigenvectors of the considered matrix. Specifically, we prove that any stationary point consists of symplectic eigenvectors, characterize the saddle points and global minimizers based on which the trace minimization theorem for the symplectic eigenvalues is re-established, and the nonexistence of local nonglobal minimizers is justified.

math.OC↗
arXiv2026

Broad Validity of the First-Order Approach in Moral Hazard

We consider the standard moral hazard problem with limited liability. The first-order approach (FOA) is the main tool for its solution, but existing sufficient conditions for its validity are restrictive. Our main result shows that the FOA is broadly valid, as long as the agent's reservation utility is sufficiently high. In basic examples, the FOA is valid for almost any positive reservation wage. We establish existence and uniqueness of the optimal contract. We derive closed-form solutions with various functional forms. We show that optimal contracts are either linear or piecewise linear option contracts with log utility and output distributions in an exponential family with a linear sufficient statistic (including Gaussian, exponential, binomial, geometric, and gamma distributions). We provide an algorithm that finds the optimal contracts at negligible computational cost, whether or not the FOA is valid.

econ.TH↗
arXiv2026

Buchsbaumness, Macaulayfication and Castelnuovo-Mumford regularity of monomial curves

Projective monomial curves are associated with rings generated by monomials of equal degree in two variables. In this paper, we give an infinite class of non-smooth, non Cohen-Macaulay $k$-Buchsbaum projective monomial curves for any $k\geq 1$ and find the monomial generators for the respective Macaulayfication. More generally, we demonstrate a method to find the Macaulayfication of a $k$-Buchsbaum monomial curve for any $k\geq 1$. We also discuss Castelnuovo-Mumford regularity of certain curves in terms of $k$-Buchsbaumness.

math.AC↗
arXiv2026

On the Distribution of Age of Information in Time-varying Updating Systems

Age of Information (AoI) is a crucial metric for quantifying information freshness in real-time systems where the sampling rate of data packets is time-varying. Evaluating AoI under such conditions is challenging, as system states become temporally correlated and traditional stationary analysis is inapplicable. We investigate an $M_{t}/G/1/1$ queueing system with a time-varying sampling rate and probabilistic preemption, proposing a novel analytical framework based on multi-dimensional partial differential equations (PDEs) to capture the time evolution of the system's status distribution. To solve the PDEs, we develop a decomposition technique that breaks the high-dimensional PDE into lower-dimensional subsystems. Solving these subsystems allows us to derive the Aol distribution at arbitrary time instances. We show AoI does not exhibit a memoryless property, even with negligible processing times, due to its dependence on the historical sampling process. Our framework extends to the stationary setting, where we derive a closed-form expression for the Laplace-Stieltjes Transform (LST) of the steady-state AoI. Numerical experiments reveal AoI exhibits a non-trivial lag in response to sampling rate changes. Our results also show that no single preemption probability or processing time distribution can minimize Aol violation probability across all thresholds in either time-varying or stationary scenarios. Finally, we formulate an optimization problem and propose a heuristic method to find sampling rates that reduce costs while satisfying AoI constraints.

cs.IT↗
arXiv2026

Properties of Random Code Ensembles over Classical-Quantum Channels

We show two properties of i.i.d. code ensembles over classical-quantum (CQ) channels with arbitrary output states. The first property is that the probability distribution of the error exponent across the ensemble accumulates above a threshold that is strictly larger than the CQ random coding exponent (RCE) at low rates, while coinciding with it at rates close to the mutual information of the channel. This result, combined with the works by Dalai, Renes, Li and Yang and Cheng and Liu, implies that the ensemble distribution of error exponents concentrates around the CQ RCE in the high rate regime. Moreover, in the same rate region the threshold we derive coincides with the ensemble-average of the exponent, that is, the CQ typical random coding (TRC) exponent. The second property we derive is that the probability that a randomly selected code from an ensemble contains a nested code of the same rate such that each of its codewords achieves the CQ expurgated exponent goes to one asymptotically in the codeword length. Such nested code can be obtained by expurgating a vanishingly small fraction of codewords as the codeword length increases. This result refines Holevo's work [5], which proved that the expurgated exponent can be achieved by discarding the worst half of the codewords from at least one code in the ensemble.

cs.IT↗
arXiv2026

Galois Rings: Ring-Theoretic Properties and Applications to Coulomb Branches and Affine Hecke Algebras

Galois rings and Galois orders, introduced by Futorny and Ovsienko, are realized as subrings of fixed subrings of skew group (or monoid) rings and have numerous applications in the structure and representation theory of associative algebras. This paper consists of two parts. The first parte investigates ring-theoretic properties that follow from the Galois ring structure alone. In particular, we estabilish natural conditions under which Galois are Ore domains or (semi)prime Goldie rings. We also study several ring-theoretic dimensions and combine the theories of Galois rings and PI-algebras to obtain new structural results. In the second part, we apply these results, together with general techniques from ring theory, to affine Hecke algebras in the sense of Ginzburg, Kapranov, and Vasserot, as well as to spherical Coulomb branch algebras. In particular, we prove that these algebras are Jacobson semiprimitive, satisfy the Nullstellensatz, and determine several of their ring-theoretic dimensions. For affine Hecke algebras, we further prove that they satisfy the maximal Nullstellensatz, are integral over their centers, and determine their T-ideals of polynomial identities, PI-degree, and PI-exponents. For spherical Coulomb branch algebras, we compute the Krull dimension, estabilish that they satisfy the Gelfand-Kirillov conjecture, and prove that they are not PI-algebras

math.RA↗