Papers

In this paper, we develop a physics-based post-processing technique for data-driven reduced-order models (ROMs) of transport-dominated problems. Besides the slow decay of the Kolmogorov n-width, ROMs based on globally supported bases often produce unphysical oscillations when approximating solutions with shocks or sharp gradients, a phenomenon analogous to Gibbs oscillations in spectral approximations. To address this issue, we introduce a post-processing framework based on Gegenbauer polynomial reconstruction. The key idea is to re-project the ROM solution onto a Gegenbauer polynomial basis over each interval of analyticity. Originally developed for spectral approximations, Gegenbauer reconstruction achieves spectral accuracy while effectively suppressing Gibbs oscillations. We extend this technique to data-driven ROMs and consider three representative approaches: Proper Orthogonal Decomposition (POD)-Galerkin ROM, Operator Inference (OpInf), and nonlinear manifold ROMs based on convolutional autoencoders (CAE). Numerical results show that the proposed post-processing consistently removes spurious oscillations and substantially improves solution quality for all three ROMs. For one-dimensional problems, the method is straightforward to implement once discontinuities are detected. We further develop a practical extension to two-dimensional problems using line-by-line reconstruction in each coordinate direction. Extensive numerical experiments demonstrate that the proposed method reduces errors by up to one or two orders of magnitude for inviscid transport problems and significantly outperforms total variation regularization in both numerical accuracy and the sharp resolution of discontinuities.

For a quasinilpotent operator $T$ on a Banach space $X$, Douglas and Yang defined $k_{x}=\limsup\limits_{\lambda\rightarrow 0}\frac{\ln\|(\lambda-T)^{-1}x\|}{\ln\|(\lambda-T)^{-1}\|}$ for each non-zero vector $x$, and called $\Lambda(T)=\{k_x:x\neq 0\}$ the power set of $T$. In this paper, we prove that $\Lambda(T)$ always contains $1$ for every quasinilpotent operator $T$ on $X$. Moreover, we introduce the concept of a Banach space $X$ having uniform multiplicity infinity and prove that some classical Banach spaces possess this property. As an application, we show that if $\sigma\subset [0,1]$ is right closed and contains $1$, then there exists a quasinilpotent operator $T$ on a class of Banach spaces with uniform multiplicity infinity such that $\Lambda(T)=\sigma$.

This paper presents an actionable human resource (HR) decision-support framework for small firms and startups based on a multi-domain matrix (MDM). The framework addresses three key challenges faced by small organizations: complex interdependencies among organizational components; the lack of systematic analytical tools for HR decision-making; and the need for rapid responses in fast-changing organizational environments. The proposed framework formulates startup human resource management as a multi-domain structural modeling problem, where members, skills, and projects are interconnected domains within an integrated MDM. Based on this representation, the framework provides qualitative analysis guidelines and quantitative metrics for diagnosing an organization's HR state and supporting personnel decisions on workload redistribution, hiring, and capability development. A case study of MDM-based HR decisions for an early-stage technology startup is conducted to demonstrate the framework's practical applicability. The application shows that the framework can identify workload imbalances, reveal a key member with an unsustainable workload, and inform a subsequent hiring decision. The framework can be further applied after the hiring of a new member to track changes in the organization's multi-domain structure and support continuous HR diagnosis.

In this paper, we prove that the solution map of Camassa-Holm equation with linear multiplicative noise $$ \left\{ \begin{array}{l} {\rm d}u+(u\partial_xu+\partial_xP[u])\,{\rm d}t=\beta u\,{\rm d}W, u(0,x)=u_0(x), P[u]=(1-\partial_x^2)^{-1}\left(u^2+\frac 1 2(\partial_x u)^2\right) \end{array} \right. $$ depends almost surely continuously on the deterministic initial data in $H^s$ for $s>3/2$. Furthermore, we prove the existence and non-uniqueness of an invariant measure for the Camassa-Holm equation with linear multiplicative noise.

Since the complexity of quantum state tomography (QST) scales exponentially with system size, exploiting priors such as low-rankness, tensor-network structures, and neural-network representations is essential for scalable QST in terms of sample complexity and parameter complexity. In this paper, we introduce a unified framework, termed structured factorization, that builds on BurerMonteiro-type factorization by parametrizing the density matrix as $FF^\dagger$, where the factor $F$ is constrained to belong to a structured model class. This factorization guarantees physical validity by construction while allowing a broad range of structural priors to be incorporated directly through the choice of the factor space, ranging from the generic Cholesky decomposition to low-rank matrices, matrix product operators, and neural density operators based on multilayer perceptron and transformer architectures. Building on this structured factorization framework, we formulate QST as an optimization problem over the factor space from measurement data. We first develop a unified statistical analysis of the sample complexity of least-squares estimation for a broad class of structured quantum states. We then propose a projected gradient descent method that operates directly on the factor space and accommodates a wide range of structural parametrizations and reconstruction objectives. To further exploit the geometry of the maximum-likelihood estimation formulation and the constraints on the factors, we derive a power method that yields a step-size-free algorithm with fast convergence, recovering Covers method as a special case when the factor is unconstrained.

Let $F$ be a non-Archimedean local field. Let $N$ be the unipotent radical of the standard parabolic subgroup of $\mathrm GL_{2n}(F)$ of type $(n,n)$ with fixed nondegenerate additive character $\psi$. For an irreducible admissible representation $\pi$ of $\mathrm GL_{2n}(F)$, a theorem due to Gomez–Gourevitch–Sahi on generalized Whittaker models gives a criterion for the vanishing of the twisted Jacquet module $\pi_{N,\psi}$ in terms of the wave-front set. We translate this orbit-theoretic answer into Langlands–Zelevinsky data: if $\pi=L(\mathfrak m)$, then $\pi_{N,\psi}=0$ if and only if the Zelevinsky dual $\mathfrak m^{\mathrm t}$ contains a segment of length at least $n+1$. We do this in response to a conjecture proposed by D$.$Prasad about the vanishing of $\pi_{N,\psi}$ in terms of the adjoint $L$-function $L(s,\pi\times\pi^\vee)$. We prove that, for every irreducible representation $\pi$, vanishing of $\pi_{N,\psi}$ implies the pole inequalities predicted by D$.$Prasad. However, we show that the converse implication is false by an explicit counterexample for $\mathrm GL_4(F)$. For the generalized Steinberg constituents $v_{P_\beta}^G$ of the principal series containing the trivial representation, we make an explicit calculation of when $\pi_{N,\psi}$ is zero. In particular, for $\mathrm GL_6(F)$, exactly three of the $32$ constituents of such a principal series violate the converse direction of the conjecture proposed by D$.$Prasad.

We consider the conditional regularity of the mild solution $v$ of the $3-D$ incompressible Navier-Stokes equations with initial data $v_0\in \dot{H}^{\frac 1 2}$ and vorticity $\Omega_0\in L^{r_0}$ for some $r_0\in (1,2)$. We prove that if the solution associated with initial data $v_0$ blows up at a finite time $T^\ast$, then for any $2<p<\infty$, and any unit vectors $e$ in $\mathbb{R}^3$, the integral $$\int_0^{T^\ast}\left\Vert (v(t)|e)_{\mathbb{R}^3}\right\Vert_{\dot{B}^{\frac 1 2+\frac 2 p}_{2,\infty}}^p{\rm d}t$$ blows up at $T^\ast$. The conclusion improves the recent results in Chemin et al. (Arch Ration Mech Anal 224(3):871-905, 2017) and Han et al. (Arch. Rational Mech. Anal. 231:939-970, 2019).

The $d$-Hoggatt triangle is a lower triangular matrix whose entries are given by specific minors of Pascal's triangle formed by consecutive $d$ rows and $d$ columns. The cases $d=1,2,3$ correspond to Pascal's triangle, the Narayana triangle, and the Baxter triangle, respectively. In this paper, we present the infinite log-concavity of the row and column sequences, the log-concavity of the sequences along transversals, and the eventual log-convexity of the sequences along rays of the $d$-Hoggatt triangle. In addition, we prove the asymptotic normality of the row sequences and total positivity of the $d$-Hoggatt triangle.

The transformation of the $h$-vector of a finite simplicial complex under an $\mathcal{F}$-uniform subdivision is encoded by a transformation matrix. Mu and Welker conjectured that the transformation matrix of the barycentric subdivision is totally positive. In this paper, we give a new combinatorial proof of this conjecture. We also prove the total positivity of the transformation matrix of the interval subdivision. In addition, we establish a sufficient condition for the transformation matrix of a uniform subdivision to be totally positive of order $2$ (TP$_2$), thereby partially answering a question of Mu and Welker. As an application, we show that the transformation matrix of the $r$-colored barycentric subdivision is TP$_2$.

This paper presents a novel layered planning and control framework based on multi-rate nonlinear model predictive control (MR-NMPC) that enables quadrupedal robots to perform hybrid bipedal locomotion with wall-assisted support in constrained environments. Real-time trajectory optimization for this locomotion presents significant challenges, as the controller must simultaneously plan for both the contact points and the continuous trajectories of the robot's center of mass (CoM) and orientation within the robot's nonlinear dynamics while accounting for unilateral contact constraints, underactuation, and the switching nature of the robot's dynamics. At the high level of the control framework, an MR-NMPC is proposed, which dynamically plans both the discrete-time trajectories of the contact points and the continuous-time trajectories of the CoM and orientation, using a single rigid body (SRB) dynamics model. By incorporating contact-point planning within the multi-rate optimal control framework, this approach enhances dynamic stability compared to heuristic foot placement strategies. At the low level of the control framework, a nonlinear whole-body controller (WBC) based on virtual constraints and a quadratic program enforces full-order dynamics and tracks the MR-NMPC references. The proposed approach is validated through extensive numerical simulations demonstrating the robust wall-assisted bipedal locomotion of a Unitree A1 quadrupedal robot on rough terrains and under external disturbances in a constrained environment. Comparative analysis shows that the proposed MR-NMPC achieves a 2.9 times higher success rate compared to conventional MPC with heuristic-based foot placement strategies in negotiating irregular terrain at high speeds.

For $m\in\mathbb{Z}_{\geq 0}$, let \[ N_{n,m}(x)={}_2F_1(-n,-n-m;m+1;x), \] which specializes to the Narayana polynomials of types $B$ and $A$ for $m=0$ and $m=1$, respectively. We prove that the associated basis transformation \[ T_{N_m}\left(\sum_{k=0}^n a_kx^k\right)=\sum_{k=0}^n a_kN_{k,m}(x) \] maps every real-rooted polynomial with nonnegative coefficients to a real-rooted polynomial. The proof is based on the rectangular additive convolution of polynomials. We then apply this result to products of lower triangular matrices and obtain a general criterion ensuring that their row generating functions remain real-rooted. As consequences, we recover this property for powers and products of several classical triangular matrices, including Pascal's triangle, the Stirling triangles, and the Narayana triangles of types $A$ and $B$. We conclude with conjectures concerning the squares of the Eulerian and Delannoy triangles.

Two generalizations of the Frobenius integrability theorem are proved concerning distributions which are piecewise-C1 but may fail to be continuous. The criteria presented are sufficient, but not necessary, for complete integrability of such distributions with bi-Lipschitz coordinates.

Optimal encoding of classical data for quantum-assisted statistical inference is investigated from an information-theoretic perspective. We prove that the accuracy of any quantum-computing inference procedure is upper bounded by the maximal quantum leakage from the classical data through its quantum encoding, establishing leakage as a universal, task-agnostic quality measure for encoders. The optimal encoding strategy, i.e., an encoding strategy that maximizes the maximal quantum leakage, is proved to be attained by pure states. When there are enough qubits, basis encoding is proved to be universally optimal. However, when the dimension of the system is small, phase encoding is optimal. For the latter, the optimal encoding is not unique. That is, any tight frame, any ensemble whose average state is the maximally mixed state, is in fact optimal. Within tight frames, equiangular tight frames (ETFs) are distinguished as the uniquely symmetric optimal encodings, i.e., they saturate the Welch lower bound on pairwise overlaps. Prominent special cases are the qubit trine, the regular simplex, and symmetric informationally complete positive operator-valued measures (SIC-POVMs), for which the ETF structure and explicit codeword constructions are provided. Numerical examples are presented to validate the theoretical predictions.

We introduce a new copula class, called Lancaster copulas, built from orthogonal expansions of continuous Lancaster probabilities. We derive infinite-series representations for the copula and its density, study truncation effects, and show in numerical experiments that low-order truncations already provide accurate approximation.

In this paper, we consider the automated discovery of iterative optimization algorithms. We formulate the algorithm discovery task as a discrete optimization problem and search for new update functions using latent space Bayesian Optimization. The proposed framework first learns a continuous representation of the discrete space of update functions using variational autoencoders, transforming the algorithm discovery task from a discrete to a continuous search problem. The continuous representation is subsequently used to search for new algorithms using Bayesian optimization. Application to two case studies shows that the proposed approach can discover new update functions in symbolic form without any assumptions on the functional form of the update function. Moreover, the computational time required to discover the new update functions is lower than existing mathematical programming-based approaches.

The concept of O-metrics was recently introduced as a generalization of several metric-type structures by replacing the addition operation of the standard triangle inequality with a binary operation that may fail to be associative. This non-associativity naturally to generalized polygon inequalities and patterned compositions. In this work we develop an analytic framework for inequalities arising in such settings. By introducing generalized $\om$-series governed by admissible control functions $\varphi$, we establish a separation principle that resolves inequalities of the form $u \leq v \, \om \, \varphi(k,u)$. This result provides convergence criteria for patterned compositions and determines intervals for the admissible contraction parameter $k$. As application, some fixed point theorems of Ćirić type are obtained for mappings on O-metric spaces, as well as corresponding results for b-metric spaces. The approach provides a unified analytic framework for studying contractive conditions and iterative processes in generalized metric spaces.

Preferential attachment (PA) network is a widely used model for capturing the growth dynamics of real-world networks, in which newly arriving vertices are more likely to connect to existing vertices with higher degrees. In this paper, we consider a setting in which an anomalous vertex appears at some time point and receives edges with an additional attachment advantage governed by a parameter $\beta$, while ordinary vertices continue to follow the PA mechanism with parameter $\delta$. Detecting such anomalies is challenging due to the high variability in degree growth in PA networks and the limited information available when the anomaly arrives late in the network evolution. We propose an iterative parameter estimation procedure together with a likelihood-based detection framework. Simulation results show that the proposed procedure provides accurate estimation of the parameters $\beta$ and $\delta$. Detection performance depends on the time of anomaly occurrence: anomalies arising at midway stages of the network evolution are detected most reliably, whereas very early and late anomalies remain challenging, particularly when the parameter $\beta$ is small.

Physics-informed neural networks (PINNs) provide a mesh-free framework for solving partial differential equations, but their training is often affected by loss imbalance, optimization stiffness, and difficulty in capturing localized or multi-mode solution structures. Hard-soft PINNs (HSPINN) alleviate part of this difficulty by embedding Dirichlet or periodic constraints directly into the trial space, but the resulting fixed admissible representation can still be poorly conditioned for sharp or heterogeneous residual fields. This paper proposes a reliability-aware hard-soft PINN (RA-HSPINN) that preserves exact embedded constraints while introducing a bounded learnable reliability field to modulate the interior representation. The method combines this reliability-aware ansatz with inverse-EMA global loss balancing and lightweight regularization, while retaining the standard mean-square residual form. The reliability field is a numerical modulation variable, not a physical parameter or calibrated probability. RA-HSPINN is evaluated on nonlinear Burgers equations, periodic convection, a mixed-boundary Poisson problem, and a mixed first-order Poisson system. Compared with HSPINN, it reduces the relative error by $98.65%$ for sharp-gradient Burgers, $72.42%$ for Burgers data with noisy and incompatible initial conditions, $61.18%$ for smooth periodic convection, $60.02%$ for localized periodic convection, $29.36%$ for mixed-boundary Poisson, and $82.17%$ for a multi-mode mixed first-order Poisson system. The results show that reliability-aware modulation is most beneficial when hard-soft trial spaces are admissible but difficult to optimize, especially in localized, unreliable-data, and multi-mode PDE regimes.

Let $M_n$ be an $n\times n$ random matrix whose entries are independent Rademacher random variables, and put $N=\binom n2$. We prove \[ Pr(M_nM_n^T=M_n^TM_n)=2^{-N+O(n)}. \] This gives the sharp exponential order for the probability that a random sign matrix is normal. The lower bound is supplied by symmetric sign matrices. We also record the immediate consequence that random $0$-$1$ matrices have the same sharp exponential normality probability. The proof of the matching upper bound is combinatorial: after conditioning on the symmetric/skew-symmetric type pattern of the off-diagonal entries, a mod-$4$ reduction gives a system of linear equations over $F_2$; a rank-duality argument converts the sum over type patterns into a count of commuting symmetric pairs over $F_2$; and this count is bounded by summing over rational canonical types, using balanced symmetric bilinear forms and the standard finite-field centralizer formula.

We prove a complete asymptotic expansion for quantum neural network operators when they approximate arbitrary quantum channels. This is the non-commutative analogue of the classical Voronovskaya theorem. The expansion reveals that the approximation error splits into three fundamentally different parts: integer powers of \(1/n\) involving ordinary Fréchet derivatives; fractional powers governed by Marchaud fractional derivatives, which capture the Hölder smoothness of the channel; and purely quantum commutator terms that have no classical counterpart. The remainder is bounded sharply by an explicit constant: \[ \norm{R_{m,n}(\Phi,\bullet)}_\diamond \le C_{m,\gamma,d} \|\Phi\|_{\cC^{m,\gamma}} \, n^{-(m+\gamma)} (\log n)^{3m/2}. \] We present a numerical test for a classical analogue that confirms the predicted convergence rate and the logarithmic correction, directly validating the asymptotic theory. Based on this expansion, we obtain three major advances: a quantum central limit theorem for the fluctuations of quantum neural network operators, a method to construct optimal interpolation geodesics between quantum channels via Kubo-Ando means, and a systematic understanding of how fractional smoothness limits the acceleration of quantum neural network approximations. The numerical test further demonstrates that the theoretical rates are sharp and that logarithmic enhancements are unavoidable. Altogether, our work builds a rigorous bridge between classical approximation theory, fractional calculus, and quantum machine learning, offering both theoretical insight and practical tools for designing and analyzing quantum neural networks in finite dimensions.

In this paper, we study the properties of linear second-order recurrences whose coefficients are non-integer rational numbers. We examine the set of indices at which the terms in the sequence are integers. Specifically, we prove two results. First, after fixing coefficients to satisfy certain coprimality assumptions, we show that there exist coprime integer initial conditions so that the set of integer terms of that sequence is as large as desired. Second, using the same coprimality assumptions, for each fixed pair of initial conditions, we give an explicit cutoff index after which no further terms are integral.

In our previous work [arXiv:2504.06248], we adopted Koopman theory to link the existence of different constants of motion to the presence of specific network motifs of Kuramoto oscillators. Yet, it remains to be shown how the partial integration can be carried out using the Koopman generator and its eigenfunctions. In this paper, we construct a random graph from network motifs that admit Koopman eigenfunctions and conserved quantities, and use it to define a partially integrable Kuramoto model. We perform the partial integration of the introduced model when there are monomial eigenfunctions and conserved cross-ratios, while providing an operator-theoretic derivation of the Watanabe-Strogatz transformation based on Magnus expansion and a recent result on closed forms of the Baker-Campbell-Hausdorff formula [arXiv:1502.06589].

We compute the number of unlabeled chordal graphs on $n$ vertices, both the total count (OEIS A048193) and the connected count (OEIS A048192), extending two sequences whose published values had remained at $n=15$. The method is a Polya-Burnside enumeration: the number of unlabeled graphs in a class closed under relabeling is the average over $S_n$ of the number of labeled graphs fixed by each permutation. The technical core is the evaluation, for an arbitrary permutation $\pi$, of the number of $\pi$-invariant labeled chordal graphs. We give a dynamic program for this quantity that lifts the evaporation-based labeled chordal counting of Hebert-Johnson, Lokshtanov and Vigoda to the equivariant setting. Its central structural ingredient is a divisor-bundle decomposition: when a connected piece spans a cyclic orbit of size $c$, it forms, for each divisor $d \mid c$, a $d$-fold bundle whose constituent is an object of the same kind in the cyclic world of order $c/d$, computed by the same program recursively. We prove the decomposition and the correctness of the resulting recurrences, and we prove that the full Burnside computation runs in sub-exponential time $n^{O(\sqrt{n})}$. We report the new terms through $n=20$ and describe four independent validations, including exact agreement with all previously known values of both sequences and an Euler-transform consistency check.

Let $\mathcal{F}\subset\binom{[n]}{k}$ be an intersecting family, $\Delta(\mathcal{F})=\max_{x\in[n]}|\{F\in\mathcal{F}:x\in F\}|$, and $\varrho(\mathcal{F})=\Delta(\mathcal{F})/|\mathcal{F}|$. Frankl and Wang conjectured that if $n>100k$ and $|\mathcal{F}|>\binom{n-3}{k-3}$, then $\varrho(\mathcal{F})\ge 3/7$; the constant $3/7$ is sharp because of the Fano-plane construction. In this note we obtain three results. First, we show that no linear threshold $n>Ck$ can be sufficient: using a truncated Fano-plane construction we exhibit, for every constant $C$ and all large $k$, an intersecting family with $n>Ck$, $|\mathcal{F}|>\binom{n-3}{k-3}$, yet $\varrho(\mathcal{F})<3/7$. In particular, the original condition $n>100k$ does not guarantee the conclusion. Second, for $k=3$ we prove that $\varrho(\mathcal{F})\ge 3/7$ holds for every nonempty intersecting $3$-uniform family; the proof is nontrivial and does not rely on any assumption on $n$ or $|\mathcal{F}|$. Third, using the classical pseudo-sunflower bound $|\mathcal{F}|\le t^k$ (for families containing no pseudo-sunflower of size $t+1$), we obtain a completely explicit polynomial threshold for all $k\ge4$: if $n>(k-3)(7k^4+k)+3$ and $|\mathcal{F}|>\binom{n-3}{k-3}$, then $\varrho(\mathcal{F})\ge 3/7$. In particular, the simplified bound $n>7k^5$ is sufficient for every $k\ge4$.

Width-balance conditions at a junction are often associated with reflectionless transmission, or transparency, in some one-dimensional wave models on networks. We show that, on cyclic networks, this identification is incomplete: local balance is a vertex-level condition, while transparency also requires synchronization of the path travel times. To make this precise, we consider a shallow-water wave model and derive the channel-width-weighted scattering law for a vertex of arbitrary degree and introduce a source-relative definition of balance, reflecting the fact that the same junction may be reflective or reflectionless depending on which edges carry incoming waves. Under this definition, a balanced vertex is transparent only to the synchronized incoming-amplitude direction. For an \(N\)-path island, the resulting first-generation upstream reflection is governed in frequency space by the width-weighted distribution of path delays. Exact broadband cancellation requires all path travel times to agree; when they do not, commensurability of the path-length differences produces a periodic comb of frequency-selective zeros, independent of channel widths. At $N\ge 4$ we further exhibit a hybrid regime in which the reflection factor vanishes without commensurability among the path lengths.

The theory of $q$-series and basic hypergeometric series plays a crucial role at the intersection of combinatorics, number theory, and representation theory. From the classical partition identities of Euler and Jacobi to modern developments in class field theory, vertex operator algebras, and the Monstrous Moonshine conjecture, $q$-series provide the analytic framework for a wide range of profound applications. In this paper, we discuss the formalization of this theory in the Lean proof assistant, a process that requires careful design of scalable and versatile structures to reconcile formal algebraic identities with analytic convergence properties. We address these foundational challenges by focusing on the construction of $q$-Pochhammer symbols, $q$-binomial coefficients, Bailey's Lemma and similar primitives. To demonstrate the utility of this work, we provide fully verified proofs of the Jacobi Triple Product formula and the celebrated Rogers-Ramanujan identities, which serve as both historical and technical benchmarks for the field. This work establishes a rigorous computational foundation for the future formalization of mock theta functions, modular forms, and the diverse algebraic structures that underpin their applications across mathematics and physics. AxiomProver was used to produce the formalizations in this paper.

We extend some methods of our previous work on special elements in Milnor K-theory of algebraic tori, exhibiting in particular a $\mathrm{GL}_n(\mathbb{Q})$-equivariant isomorphism between a chain complex of simplicial cones, computing the homology of $S^{n-1}$, and the trace-fixed part of the weight-n Gersten complex for the Milnor K- theory of $\mathbb{G}_m^n$ over $\mathbb{Q}$. Via a relationship between graded pieces of algebras of cones and Steinberg modules, this refines a result of Charlton-Radchenko-Rudenko.

We present the necessary and sufficient conditions on the order $q$ of a finite field $\mathbb{F}_q$ such that every irreducible polynomial of the form $x^2+bx+c \in \mathbb{F}_q[x]$, with $b\neq 0$ and $c$ a primitive element of $\mathbb{F}_q$, is a primitive polynomial. As a by-product of this result, we also present a new infinite family of finite fields $\mathbb{F}_q$ for which it is easy, in a different way, to determine when an irreducible polynomial of degree two is primitive.

We check that Levine's simplicial–cubical comparison argument for Bloch's cycle complexes also works over an arbitrary DVR. As a result, the sheaf of cubical Bloch cycle complexes computes motivic cohomology for smooth schemes over Dedekind bases.

Parametrized partial differential equations (PDEs) arise in many-query simulation, optimization, control, and uncertainty quantification, where repeated full-order solves restrict the number of high-fidelity snapshots that can be generated. This limitation is particularly pronounced in computational energy science, where multiscale models of porous-media flow, transport, and energy materials often make large snapshot datasets impractical. Proper orthogonal decomposition (POD) constructs compact reduced bases from solution snapshots, but it may exhibit limited out-of-sample predictive capability when the snapshots insufficiently sample the solution manifold. To address this limitation, we propose a spectral-subspace-augmented POD-Galerkin method (SS-POD) tailored to limited-data regimes. SS-POD starts from a problem-adapted spectral approximation space, partitions it into orthogonal subspaces, and performs POD locally on the projected snapshot matrices. An energy-balancing rule determines the spectral partition so that the resulting local POD problems are assigned comparable amounts of snapshot energy. For nonlinear parametrized PDEs, SS-POD is coupled with the discrete empirical interpolation method (DEIM). Numerical experiments show that SS-POD improves out-of-sample accuracy over standard POD-Galerkin while retaining compact reduced bases in limited-snapshot regimes. In particular, for a Laplace-Beltrami problem on the unit sphere with only 5 snapshots, SS-POD achieves a relative error of $3.9*10^{-8}$ using 91 basis functions, whereas the standard POD error saturates at $7.8*10^{-4}$ and the spectral-Galerkin method requires 256 basis functions for comparable accuracy. These results indicate that SS-POD provides an effective strategy for high-fidelity reduced-order modeling from limited snapshot data.