While discrete-time Stochastic Portfolio Theory (SPT) provides a robust framework for market analysis, existing work on functional generation has predominantly focused on long-only portfolios defined on the entire unit simplex. This paper extends the geometric framework of functional generation to the broader class of bankruptcy-proof long-short portfolios defined on local market state spaces. We establish that, within this admissible setting, pseudo-arbitrage is fully characterized by the concavity of the generating function on the market state space, thereby relaxing the usual global domain requirement. A central contribution of this work is a geometric characterization of the short-selling mechanism. We prove that the presence of short selling is equivalent to the negativity of the maximal concave extension of the generating potential. This phenomenon is linked to the steepness of the logarithmic gradient as the market approaches a zero boundary nested inside the simplex. To systematically exploit this mechanism, we introduce the barycentric scaling transformation, a constructive methodology that maps classical long-only generating functions onto restricted domains to engineer admissible strategies with controlled short-selling exposure. Finally, through the analysis of specific shrunken portfolios, we identify a geometric phase transition: under suitable boundary conditions, admissible strategies exhibit a long-only core and a short-selling region in a qualitative sense (without asserting an exact partition of the state space). This provides a unified geometric perspective on relative arbitrage beyond the long-only constraint.

Homomorphic encryption (HE) enables privacy-preserving analytics but remains hindered by high computational overhead. We find that the inverse square root-a key primitive in many statistical and machine learning workloads-exhibits inconsistent and often suboptimal performance across HE libraries and hardware. This stems from a core trade-off between two costly HE operations: evaluating high-degree Chebyshev polynomials to speed up Newton's method versus performing bootstrapping to manage ciphertext noise. Because their relative costs vary by up to 6x across environments, any fixed configuration proves inherently inefficient. To address this challenge, we present HE-DAP, a cross-platform optimization framework that automatically navigates this trade-off. By profiling an environment's unique performance characteristics, HE-DAP finds the optimal balance between polynomial degree and iteration count to accelerate the encrypted inverse square root computation for a given accuracy target. Our evaluation on Lattigo, HEaaN-CPU, and HEaaN-GPU shows that HE-DAP's adaptive approach yields significant performance gains. It accelerates the core inverse square root computation by up to 2.35x over the fixed configuration in PP-STAT while maintaining high numerical accuracy (MRE <= 3.1 x 10^-8). We further demonstrate that optimizing this fundamental building block directly enhances the end-to-end performance of complex statistical analyses, confirming the practical benefits of our environment-aware approach. By automatically adapting to heterogeneous execution environments, HE-DAP demonstrates that principled parameter optimization can make privacy-preserving statistical analytics practical at scale.

A measurement of the ratio of semileptonic branching fractions $\mathcal{R}(Jψ)$, defined as $\mathcal{R}(J/ψ) \equiv \mathcal{B}(B_c^+ \to J/ψτ^+ ν_τ)/\mathcal{B}(B_c^+ \to J/ψμ^+ ν_μ)$, is reported using a sample of proton-proton collision data corresponding to an integrated luminosity of 5.4 fb$^{-1}$ recorded by the LHCb experiment in 2016--2018 at a center-of-mass energy of 13 TeV. The measured value is found to be $\mathcal{R}(J/ψ) = 0.51 \pm 0.12\text{(stat)} \pm 0.08\text{(syst)}$, which is within 1.8 standard deviations of the predictions from the Standard Model assuming lepton flavor universality.

In this work, we investigate a cosmic inflation model based on a cosine-type potential with a negative cosmological constant. This model originates from a classical solution of the Wheeler-DeWitt equation. The equation of motion for the inflaton field can be solved analytically without relying on approximation schemes, such as the slow-roll conditions. The predictions of the spectral index, the tensor-to-scalar ratio, and the running spectral index are calculated and compared with experimental constraints from Planck Collaboration, Atacama Cosmology Telescope Collaboration (ACT), and Dark Energy Spectroscopic Instrument (DESI).

We establish a universal scaling law for collective charging of quantum batteries, independent of microscopic details. We prove that the ergotropy-to-energy ratio approaches unity at least as fast as $\sim N^{-1}$ with the number of batteries $N$, implying generic asymptotic freedom. We further show how the universal $1/N$ scaling can be overcome: when the battery state becomes asymptotically pure, the convergence can be substantially faster, including $\sim N^{-b}$ with $b>1$ and even exponential scaling in $N^2$. Rigorous finite-$N$ upper and lower bounds on the ergotropy-to-energy ratio are further derived, providing nonasymptotic guarantees for the universal $1/N$ scaling.

We consider the $SO(d)$ and $O(d)$ invariant sectors of the bosonic $d$-matrix harmonic oscillator with $U(N)$ gauge symmetry. The micro-canonical degeneracy $\mathcal{Z}( N , d , k )$ for fixed energy $k$ is expressed as a pairing between an $N$-dependent vector and a $d$-dependent vector in the space of partitions of the integer $k$. This pairing formula is derived by counting invariant words in multi-matrix variables $X^i_{j,a}$, using properties of Clebsch-Gordan multiplicities (Kronecker coefficients) for the symmetric group $S_k$, Schur-Weyl duality and harmonic analysis on the homogeneous space $U(d)/SO(d)$. Analytic formulae for large $N$ and $k$ with $ k \le N $ are obtained using group integrals over $U(N)$ and $SO(d)$ (or $ O(d)$). The micro-canonical heat capacity in this regime is negative and turns positive, at a critical value $k_{\rm crit}$, due to finite $N$ modifications to the counting, thus forming what we denote as a characteristic caloric fold in the $ E $ versus $T$ curve. Data from the pairing formula is well fitted by $k_{\rm crit} \sim { N^2 \over 4 }$ for small values of $d$. A derivation of this large $N$ formula is given using a matrix model approximation and semi-classical analysis of the eigenvalue density. The large $N,d$ limit of the degeneracies reveals a key role for ribbon graph combinatorics. The caloric fold is also notably a property of black hole thermodynamics in anti-de-Sitter spaces. We propose the spherically symmetric \(SO(d)\) and \(O(d)\) invariant sectors of \(d\)-matrix quantum mechanics as tractable matrix systems for capturing key features of dual descriptions of black-hole thermodynamics.

Denial-of-service (DoS) and distributed denial-of-service (DDoS) mitigation requires separating malicious traffic from benign traffic while minimizing disruption to legitimate users. Prior work proposed mapping honeypot traffic partitioning to a weighted MaxCut problem and solving the resulting graphs with variational quantum algorithms. We extend this proof of principle direction with a reproducible event-level honeypot-to-QUBO pipeline, labeled temporal bipartite benchmark graphs with 16, 32, 66, and 110 event nodes, QAOA executions on IBM quantum hardware, classical heuristic baselines, a noiseless matrix product state reference, and a routing overhead analysis across quantum processor architectures. The largest benchmark is a 110-node, 181-edge instance executed on three IBM backends. Our results show that a shallow QAOA can execute real traffic partitioning workloads at the utility scale, while backend architecture and routing overhead affect objective quality, security metrics, and observed runtime. Because simple classical heuristics can solve the current labeled benchmark graphs, these experiments are not a quantum advantage claim. Instead, we deliberately use a fixed, shallow QAOA implementation to enable controlled comparisons across problem sizes and hardware architectures. This work establishes a hardware feasibility and architecture benchmark framework, and demonstrates that MaxCut cost, security quality, routing overhead, and runtime must be reported as separate metrics for cybersecurity relevant quantum optimization.

The high-frequency gravitational waves emitted by a binary neutron star merger remnant carry information on matter at densities and temperatures beyond those reached in isolated neutron stars. We quantify how tightly current multi-messenger constraints already determine the dominant post-merger frequency $f_{2,\rm mean}$. Adopting a set of cold equations of state (EOSs) constrained jointly by gravitational-wave tidal deformability, NICER mass--radius measurements, massive-pulsar masses, chiral effective field theory at low density, and perturbative QCD at asymptotically high density, for each binary mass we select the softest and stiffest models of the multi-messenger posterior and follow their coalescence with fully general-relativistic hydrodynamics simulations. Together with a broad set of EOSs drawn from the literature ($82$ models in total), these simulations show that, once the binary mass and a single measure of the stellar compactness ($Λ$ or $R$) are held fixed, the residual spread of $f_{2,\rm mean}$ is only $\sim 100\,{\rm Hz}$, a factor of several below the $\gtrsim 500\,{\rm Hz}$ range spanned by an EOSs set including those already disfavored by the data. This tight calibration of the cold-matter prediction implies that a future high-frequency detection departing from it would point directly to additional physics, such as a hadron--quark transition occurring at finite temperature. We further confirm the quasi-universal relation $(f_1+f_3)/2 \approx f_{2,\rm mean}$ to within $\sim 116\,{\rm Hz}$, which provides a model-independent estimate of $f_{2,\rm mean}$ from the secondary spectral peaks.

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.

Motivated in part by anomaly cancellation for integral charge spectra in chiral gauge theory, we study the symmetric locus in the ideal degree-three Prouhet-Tarry-Escott problem. A symmetric integer solution is one whose entries are paired about a common center $c\in \frac12\mathbb Z$. This symmetry reduces the problem to a sum-of-two-squares equation, $x^2+y^2=u^2+v^2$, in integer variables, subject to the appropriate parity conditions. Thus the problem is governed by representations as sums of two squares. For the full symmetric locus, let $N_{\mathrm{sym}}(H)$ denote the number of nontrivial symmetric integer solutions of height at most $H$, counted with unordered multiset conventions and summed over the admissible centers. Then \begin{align*} N_{\mathrm{sym}}(H) = \frac{4\log 2}{3π^2}H^3\log H+O(H^3). \end{align*} The logarithmic enhancement comes from the second moment of the sum-of-two-squares representation function. In particular, the symmetric locus is larger than one would expect from the naive $H^3$ degree-weighted box-counting scale alone. This asymptotic identifies a large arithmetically structured subfamily of the ideal degree-three solution space, and suggests that paired anomaly-free integral charge spectra reflect a fundamental number-theoretic structure.

Homomorphic quantum error correction aims to protect quantum data against both unauthorized access and environmental noise during server-based processing. We investigate the algebraic compatibility between quantum homomorphic encryption and quantum error correction, determining precise conditions under which encrypted encoded states remain inside the relevant code space during storage and computation. Our work establishes a necessary and sufficient criterion for an $[[n,1,d]]$ stabilizer code to remain compatible with the restricted transversal block-Pauli masking $U_{\rm enc}(a,b)=(X^aZ^b)^{\otimes n}$, stated explicitly for $[[n,1,d]]$ codes and extending directly to code-space preservation for $[[n,k,d]]$ codes. We verify this condition for standard examples (bit-flip and Shor codes, with the phase-flip repetition code following analogously), derive a practical criterion for Calderbank-Shor-Steane codes, and extend the analysis to three-dimensional color codes. A critical challenge emerges for non-Clifford gate implementation: the Shor code lacks a naive transversal $T$-gate implementation of the desired logical operation on encrypted encoded data. We present two routes around this obstruction. First, suitable triorthogonal codes admit transversal $T$-type logical implementations, up to Clifford corrections. Second, logical-gate masking gives code-space compatibility for arbitrary stabilizer codes, provided that suitable unitary representatives of the required logical gates are available. These results separate code-space compatibility from a full cryptographic security proof and provide explicit criteria for combining error correction with homomorphic processing in cloud quantum computing.

Recent high-resolution imaging observations of strong lens systems reveal dense low-mass perturbers. We propose a soliton core, whose central density is boosted by a supermassive black hole (SMBH), in the fuzzy dark matter (FDM) model as an efficient perturber in strong gravitational lensing. The higher central density makes it less efficient in the tidal mass loss, and leads to the higher impact in gravitational lensing. We show that the mass profile of a $\sim 10^6M_\odot$ perturber in JVAS B1938+666, which does not resemble any known astronomical object, can be well explained by a soliton core in the FDM model with the mass of $4\times 10^{-21}$eV hosting an SMBH with the mass of $4\times 10^5M_\odot$. The high mass of the SMBH may be explained by several scenarios that predict heavy SMBH seeds such as the direct collapse black hole formation and primordial black holes.

Wilson coefficients in dimension-six effective field theory are constrained in a combined fit to several ATLAS measurements. These inputs probe Higgs-boson processes across multiple production and decay modes, di-Higgs signatures in the $b\bar{b}γγ$ and $b\bar{b}ττ$ final states, $WW$ and $WZ$ diboson signatures, electroweak $Zjj$ final states, high-mass Drell-Yan interactions, and $t \bar t$ events in both resolved and boosted topologies. Precision electroweak observables from LEP, SLD, and ATLAS are also included. A total of 48 parameters, including individual Wilson coefficients in the Warsaw basis and linear combinations of Wilson coefficients, are constrained simultaneously. Constraints on two-Higgs-doublet models and heavy-vector-boson models are also obtained by matching a relevant sub-set of the results with their parameters. This combined fit provides the most comprehensive effective field theory interpretation of experimental data by the ATLAS Collaboration to date. No significant deviations from the Standard Model are observed.

Channel estimation in vehicular communication is a crucial element in the advancement of intelligent transportation systems. However, the use of pilot signals in the IEEE 802.11p standard is insufficient for accurate channel estimation in high-mobility scenarios. Data pilot-aided (DPA) estimation helps address this, but suffers from demapping errors. We propose a simplified Temporal Convolutional Network-based estimator (DPA-TCN) trained on a mixed signal-to-noise ratio dataset to improve estimation performance and reduce computational complexity. Our DPA-TCN estimator achieves a bit error rate comparable to a state-of-the-art long-short-term memory network with DPA and temporal averaging (LSTM-DPA-TA) while reducing the complexity of the model by approximately 65%.

In academic publishing, names connect scholars to their work. When scholars change their names, including for marriage, academic recognition, or gender transition, they may lose credit for past publications. However, despite significant impacts on citation accuracy and researcher well-being, no existing studies examine how naming policies in computer science serve researchers who change their names. We use a mixed-methods approach combining surveys, interviews, and large-scale citation analysis of papers from eight major computer science venues from 2019-2025. We document the multi-year advocacy effort that established the first name change policies, identify implementation barriers including incomplete publisher updates and months-long processing delays. Researchers continue being cited with misparsed and incorrect names despite publisher updates. When these citation errors happen, interviewees report significant mental health impacts, including stress, anxiety, and safety risks. Empirically, we find that venues with accessible and visible name change policies have significantly fewer citation errors compared to inaccessible policies (899 vs. 996 errors per 1,000 papers). Our annotation analysis shows that deadnaming of transgender researchers in citations decreased by 92% from 2019 to 2024. Our findings demonstrate the importance of inclusive publishing policies, for which name change policy advocacy led by trans researchers has been a significant driver. We recommend that venues adopt proactive visible name change policies, support queer advocacy groups, and improve publication infrastructure to build an inclusive publishing landscape. The accompanied toolkit to check errors in bibliographic latex file is available here https://github.com/pranav-ust/cite-updater.

Combining asynchronous Byzantine Fault Tolerant (BFT) consensus with Proof-of-Stake (PoS) creates a trilemma between Sybil resistance, reward distribution fairness, and protection against persistent plutocracy. Existing DAG-BFT approaches (Narwhal+Tusk, Bullshark, and Mysticeti) prioritize liveness over the fairness implications of stake-based selection, resulting in persistent longitudinal centralization. FairWave is a dual-channel DAG BFT protocol that separates anchor selection from reward distribution. The selection channel is super-linear in stake, guaranteeing Sybil gain < 1 for all split factors K > 1. The reward channel is sub-linear, using square-root stake normalization to mitigate rich-get-richer dynamics. The finalized DAG structure provides deterministic uptime and latency factors, allowing honest validators to agree on operational quality without any external oracle. To avoid circular dependency between selection outcomes and selection weights, reputation is used in a lagged form: the active value at epoch e equals the prior epoch's final value. We derive closed-form constraints for both channels and validate them through nine empirical analyses (approximately 550,000 Monte Carlo rounds) against eight baselines. FairWave achieves a Gini coefficient of 0.149 (vs. Pure-PoS's 0.488), a monotone HHI reduction from 0.039 to 0.021 over 50,000 epochs, an optimal-adversary Sybil split of K* = 1, and a success-rate coefficient of variation of 5.2% under +/-25% input perturbation. Safety (agreement and validity) is a formal consequence of the 2f+1 strong-support commit rule, holding unconditionally for f < n/3; the empirical differential is the monotone-continuous liveness-degradation curve, which decreases from 99.6% commit rate at b=0.20 to 71.1% at the theoretical bound b=1/3 without the discontinuous cliff characteristic of view-change-driven leader-BFT.

Collaborative computation across organizations is often constrained by the need to process sensitive data and proprietary code without exposing them to untrusted infrastructure or participants. Cryptographic approaches such as fully homomorphic encryption and secure multi-party computation provide strong confidentiality but remain impractical for general workloads due to their extreme computational cost. We present the Two-Way Confidential Virtual Machine (2cVM), a two-layer architecture that pairs a hardware trusted execution environment with an intra-workload isolation layer. Unlike regular Confidential Virtual Machines, 2cVM enforces mutual isolation between co-resident workloads, ensuring that participants retain control over their data and code. All computation in 2cVM is governed by a Commitment Manifest that enumerates participants, component composition, permitted data channels, and authorized outputs; the manifest is locked to the VM and incorporated into attestation evidence, making the policy immutable and independently verifiable throughout the VM's lifetime. A proof-of-concept realization combines AMD SEV-SNP for hardware protection with the WebAssembly Component Model for fine-grained sandboxing of participant code. Evaluation on commodity hardware across four benchmark classes shows that the two isolation layers do not accumulate linearly: once a workload executes inside the WebAssembly sandbox, the marginal cost of enabling hardware memory protection is small. Overhead is workload-dependent, governed primarily by memory access pattern, ranging from negligible for sequential workloads to approximately 2x for irregular, pointer-chasing access patterns. These results indicate that 2cVM provides a practical and verifiable foundation for privacy-preserving collaborative computation.

Skill documents, structured natural-language instructions that guide Large Language Model (LLM) agents, are critical to modern agent frameworks, yet LLMs struggle to write skills that actually work. On SkillsBench, human-authored skills improve pass rates by 16.2 percentage points, while LLM-authored skills provide no measurable gain. We introduce SkillAxe, a fully unsupervised framework that enables LLMs to iteratively diagnose and refine their own skills. SkillAxe decomposes skill quality into four interpretable dimensions (quality impact, trigger precision, instruction compliance with fault attribution, and solution-path coverage), producing structured improvement briefs that require no ground-truth labels, test suites, or environment rewards. On SkillsBench, SkillAxe improves pass rates by 28\% relative over unimproved LLM skills and closes 47--67\% of the gap to human-authored skills. We validate the approach as a continuous improvement engine in the wild on SpreadsheetBench, where a SkillAxe-built skill library learns from past agent trajectories and raises pass rate from 16.0\% to 52.0\% using only 22 skills.

Internet of Things (IoT) deployments combine heterogeneous, resource-constrained devices with weak security configurations, exposed services, limited logging, patching constraints, and long lifecycles. Signature- and threshold-based controls remain useful baselines, but they are insufficient as standalone mechanisms in dynamic IoT networks. Likewise, offline artificial intelligence (AI) benchmark performance alone does not establish operational deployability. This article presents a conceptual framework and research agenda for a Linux-based IoT edge gateway that combines resource-aware flow-level AI-assisted risk scoring, event-level explainability, and bounded mitigation through eBPF/XDP. The controller applies reversible, time-limited actions subject to critical-device safeguards, updates packet-level enforcement state, and records structured logs. The architecture separates complex reasoning and policy control in user space from concise packet-handling decisions in the kernel. It also defines a future hardware-aware evaluation pathway covering detection quality, resource cost, response timing, rollback behaviour, and legitimate-traffic preservation. The paper does not report new experimental measurements or claim measured superiority or completed real-time performance.

The Graph Burning Problem (GBP) is a NP-Hard combinatorial optimization problem that models the propagation of influence or contagion in a network. The propagation is represented through the metaphor of a fire spreading through the vertices of a graph. A burning process takes place in a series of discrete time-steps. At each time step, the burning process is characterized by a propagation (where burned nodes spread the fire to their neighbors), and an ignition (where one additional unburned node is chosen to become burned). The minimum number of steps required to burn all vertices of a graph defines its burning number. Literature provides integer linear programming formulations to solve the problem, but with no surprise, such approaches struggle to converge as the graph size increases. Therefore, reducing the search space explored by these formulations becomes a key point to improve performances. In this work, we study the similarities of the Graph Burning Problem with the well-known Dominating Set Problem. We propose a new formulation based on this study and apply dominance rules and symmetry-breaking techniques to reduce the search space and consequently speed up resolution time. We also introduce a perturbation of the proposed objective function, as well as a pruning rule for the perturbed model in order to further accelerate its resolution.

Let $Σ$ be a closed embedded minimal hypersurface in $\mathbb{S}^{n+1}$. We establish an improved lower bound for the first non-zero eigenvalue of the induced Laplace-Beltrami operator on $Σ$. It is slightly better than the bound of Duncan-Sire-Spruck.

In this paper, we develop a quiver approach to coquasi-Hopf algebras with the dual Chevalley property. We introduce a modified generalized path coalgebra $\Bbbk(\mathrm{Q},\mathcal{S})$ associated with a given quiver $\mathrm{Q}$ and a collection of simple coalgebras $\mathcal{S}=\{C_i\mid i\in \mathrm{Q}_0\}$ indexed by the vertices of $\mathrm{Q}$, such that its link quiver coincides with $\mathrm{Q}$. We prove that such a coalgebra admits a graded coquasi-Hopf algebra structure with the dual Chevalley property if and only if $\mathrm{Q}$ is a generalized Hopf quiver and $\bigoplus_{i\in \mathrm{Q}_0}C_i$ forms a cosemisimple coquasi-Hopf algebra. Moreover, we provide a classification of these coquasi-Hopf algebra structures. We then study the link-indecomposable components of a coquasi-Hopf algebra with the dual Chevalley property, and give the generalized dual Gabriel's theorem for such coquasi-Hopf algebras. As an application, we apply the quiver method to classify finite integral tensor categories with the Chevalley property of finite representation type. We also give structural characterizations of coradically graded coquasi-Hopf algebras of tame corepresentation type. Furthermore, we investigate finite braided integral tensor categories with the Chevalley property via the quiver approach.

Let $G$ be a finite group and $p$ a prime. It is interesting to determine when two ordinary irreducible representations of $G$ have the same $p$-modular reduction; this is the same as saying that the corresponding rows of the decomposition matrix are equal, or that the characters of the two representations agree on $p$-regular conjugacy classes. In fact we consider the more general problem of asking when two rows of the decomposition matrix are proportional. In the case where $G$ is a double cover of the alternating or symmetric group, this problem has been solved except when $p=3$. Here we resolve the missing case for spin characters (i.e. characters which are not lifted from the covered group), which completely solves the problem for the double cover of the symmetric group. There are surprising parallels to our solution to the corresponding problem for $p=2$.

Building on preliminary results about lower sections of real-valued functions on a metric space, we provide a constructive counterpart of Ekeland's theorem on the approximate optimisation of real-valued functions.

Covariance estimation yields a fundamental second-order statistic underlying representation learning, dimension reduction, and dependence modeling. While covariance has been well understood in Euclidean spaces, it is ill-defined for random objects residing on nonlinear Riemannian manifolds, which increasingly arise in modern machine learning applications involving shapes, symmetric positive definite (SPD) matrices, etc. This paper introduces an intrinsic Riemannian cross-covariance for manifold-valued random objects. Our approach defines covariance and correlation by transporting local variations to a common tangent space via parallel transport, yielding a second-order descriptor that is independent of arbitrary coordinate choices. We establish that the proposed covariance inherits desirable properties of its Euclidean counterparts and characterize its asymptotic behavior. Numerical studies on spheres and SPD manifolds, together with real-data experiments on heart valve shapes in Kendall's shape space, demonstrate the effectiveness of our estimators and verify the stated properties. Our results position the Riemannian covariance as a fundamental tool for second-order learning and analysis in non-Euclidean representation spaces.

This paper studies the complexity of a dual dynamic programming (DDP) method for solving a class of convex optimization problems with linear coupling constraints. Existing complexity results based on DDP depend on the dimensions of the state vectors and, in particular, grow exponentially with dimension. The goal of this paper is to establish a complexity bound that is independent of the dimension. Our approach first studies an unconstrained strongly convex problem and develops a flexible framework for DDP, called FDDP, for solving the associated dynamic programming equations, and establishes an iteration-complexity bound for it that is independent of the dimension. A dimension-independent complexity bound for the original linearly constrained problem is then obtained by applying FDDP to a corresponding unconstrained strongly convex problem obtained via smoothing and penalization. Inspired by the literature on bundle methods, FDDP updates lower approximations of cost-to-go functions through a generic procedure that includes both the classical multi-cut DDP and a new two-cut DDP variant as special cases. The two-cut variant maintains only two affine cuts per stage at each iteration, making it more memory-efficient while retaining the same theoretical guarantees. Finally, numerical experiments illustrate the practical behavior of multi-cut and two-cut DDP, including their dependence on problem parameters and their performance relative to a direct quadratic constraint reformulation.

The category of noncommutative quadratic quadric hypersurfaces, ${\tt Quad}\text{-}{\tt QHS}$, consists of pairs $(A, f)$, where $A$ is a quadratic algebra and $f \in A$ is a nonzero degree $2$ element. We associate to such $(A, f)$ a pair $(\bar{A}^!, f^!)$, and show that this association makes ${\tt Quad}\text{-}{\tt QHS}$ into a category with duality. We construct a faithful functor from the category of graded modules over $\bar{A}^!$ to the homotopy category of curved DG modules over a canonical curved DG algebra $(A \otimes \bar{A}^!, d, f \otimes f^!)$. If $A$ satisfies the left strong rank condition and $f \in A$ is not a right zero divisor, we show that the restriction of our functor to a natural full subcategory of the category of graded modules over $\bar{A}^!$ is valued in a stable category of noncommutative matrix factorizations of $f$. When $A$ is Koszul of finite global dimension and $f \in A$ is normal and regular, we prove that the even Clifford algebra, $\bar{A}^![(f^!)^{-1}]_0$, is isomorphic to a canonical PBW-deformation of a Zhang twist of the $2$-Veronese subalgebra of the Koszul dual $A^!$. Finally, we study several classes of Artin-Schelter regular algebras to illustrate our results.

Thurston gave a simple way to construct all triangulations of the sphere for which 5 or 6 triangles meet at each vertex, using the Eisenstein integers $\mathbb{E}$. While such triangulations can be defined purely combinatorially, Thurston noticed that given such a triangulation, one can make all the triangles into flat equilateral triangles with the same edge length, and this gives the 2-sphere a flat Riemannian metric except at 12 cone points with angle deficit $π/3$. He showed that up to rescaling, all such Riemannian metrics arise from his procedure. He studied the moduli space $\mathcal{M}$ of all such metrics modulo rescaling, and showed that $\mathcal{M}$ is open and dense in an orbifold $\overline{\mathcal{M}} = \mathbb{PC}^{10}_+/Γ$. Here $\mathbb{C}^{10}_+ = \{ v \in \mathbb{C}^{10} \vert \; Q(v) > 0\}$ for some quadratic form $Q$ of signature $(1,9)$ on $\mathbb{C}^{10}$, $\mathbb{PC}^{10}_+$ is its projectivization, and $Γ$ is a certain discrete group of linear transformations of $\mathbb{C}^{10}$ preserving both $Q$ and the lattice $\mathbb{E}^{10} \subset \mathbb{C}^{10}$. He also showed that $\overline{\mathcal{M}}$ is the moduli space of flat Riemannian metrics on the sphere with at most $12$ cone points and angle deficits that are positive integer multiples of $π/3$. Here we briefly outline the basic ideas behind this work, and illustrate them with examples.

Data-driven modeling shows significant promise for faster-than-real-time forecasting of fluid flows. For real-world engineering applications (e.g., flow control), models must contend with limited, imperfect, and incomplete experimental measurements. In this work, we present an analysis of data-driven surrogate models trained to forecast the time-evolution of experimentally measured cylinder wakes in the subcritical vortex shedding regime. Using a dataset of two-dimensional, two-component particle image velocimetry measurements, we train fully convolutional neural networks, U-Nets, Fourier neural operators, and dynamic mode decomposition-based models to forecast the development of experimentally measured velocity fields. To characterize data-driven approaches contending with transient flow features and limited, imperfect observations, the development of predictions over extended forecast horizons is examined at a fixed Reynolds number (Re = 590). Next, models are trained at a range of Reynolds numbers (Re = 230 to Re = 2920) to investigate the impact of increasingly turbulent and three-dimensional flow phenomena, and the challenges associated with measuring them, on forecast quality. We find that experimentally trained surrogate models can provide meaningful predictions over short time horizons, propagate low-frequency dynamics over longer forecast periods, and achieve faster-than-real-time evaluation. However, the data-driven models struggle to preserve transient flow features and high-frequency energy content when faced with noisy measurements and incomplete state observations. This emphasizes the underlying challenges that remain for data-driven modeling approaches to effectively contend with fluid dynamics in real-world engineering applications, where observations are often imperfect and limited.

A systematic comparison between VMEC and HINT equilibrium calculations has been carried out for Large Helical Device plasmas to clarify the influence of the assumption of the nested flux surfaces at finite beta. Three vacuum magnetic-axis configurations, $R_{\rm axV} = \SI{3.53}{\, m}$, $\SI{3.60}{\, m}$, $\SI{3.85}{\, m}$, are examined for the beta values on the axis in the range $β_0 \in [0.0\%, 5.0\%]$. The magnetic-axis position, the rotational transform on the axis, and the plasma volume enclosed by the last closed flux surface are compared between the two codes. At low-$β_0$, VMEC and HINT give consistent equilibria, indicating that the nested flux surfaces are largely preserved. Above a configuration-dependent critical $β_0$, however, the two solutions begin to diverge, indicating that the nested flux surfaces assumption is compromised. In HINT, the enclosed plasma volume decreases at higher beta because the stochastic magnetic field evolves near the plasma edge, whereas VMEC cannot represent this flux surface breaking due to its assumption of nested flux surfaces. These results show that the 3D equilibrium responses in LHD equilibria become increasingly important from inward- to outward-shifted configurations, mainly through Pfirsch-Schlüter current-driven perturbations of the magnetic field and the resulting edge stochasticity.