Papers

We show that $C^*$-algebras with complexity rank one have real rank zero, as an application, the uniform Roe algebra $C_u^*|\mathbb{Z}|$ has real rank zero.

This paper investigates a parallel machine scheduling problem featuring a single common server responsible for both loading and unloading operations. Each job consists of a unit-time loading operation, non-preemptive processing on one of \(m\) identical machines, and a unit-time unloading operation executed by the same server. The objective is to minimize the makespan. Unlike classical loading-only common-server models, our setting requires the server to handle an unloading operation immediately after a job's processing phase concludes. We prove that the corresponding decision problem is strongly NP-complete when the number of machines is given as part of the input. Furthermore, we analyse the worst-case performance of the List Scheduling (LS) and Longest Processing Time (LPT) heuristics. For three machines, we establish that Algorithm LS achieves an approximation ratio of at most \(5/2\). For an arbitrary fixed \(m \ge 3\), we show that the general LS bound approaches \(4-3/m\) as the number of jobs grows, whereas for Algorithm LPT, we prove the finite-instance ratio $3-\frac{2}{m}+\frac{(m-1)(m-2)}{mn}.$ Thus, for each fixed \(m\), the LPT bound approaches \(3-2/m\) as the number of jobs grows.

Given a countable discrete group equipped with a proper length function, one can construct a natural spectral triple on its reduced group C$^{\ast}$-algebra. A well-studied question in non-commutative metric geometry is whether the Connes pseudo-metric associated with such a triple recovers the weak$^{\ast}$-topology on the state space, thereby yielding a compact quantum metric space in the sense of Rieffel. While this metric property is known to hold for several classes of groups - including those of polynomial growth and word-hyperbolic groups - it was widely expected that not every word-length function induces a compact quantum metric space. Despite this, no explicit counterexample has been identified to date. In this note, we provide the first family of counterexamples by proving that for every integer $d \geq 2$ the canonical spectral triple of the Lamplighter group $(\mathbb{Z}/2\mathbb{Z})\wr\mathbb{F}_{d}$, equipped with the word-length function associated with a finite symmetric generating set, fails to be a spectral metric space.

Given a partition $(n_1,\ldots,n_r)$ of a positive integer $n$, one has the associated $n$-dimensional multiprojective space $\mathbb{P}^{n_1}\times \cdots \times \mathbb{P}^{n_r}$. We show that distinct partitions of $n$ yield non-isomorphic multiprojective spaces, giving a new proof via the extremal contractions of their closed cone of curves. In contrast to the earlier approaches, the argument here is uniform across all partitions, and extends beyond multiprojective spaces. In fact, we further extend it to a more general setting, namely to products of Fano varieties of Picard number one: we prove that a fixed such factor in each dimension makes the products attached to distinct partitions pairwise non-isomorphic. As a consequence, a complete classification of products of smooth quadrics, of dimension $\geq 3$, has been obtained.

For an almost contact pseudo-metric manifold we study a variation of the metric in the spirit of Meli-Ngakeu-Olea. Such variation is defined using a vector field on the manifold and the procedure shifts the signature of the metric by increasing the positive index by two. First we derive conditions on the vector field for the obtained manifold to remain almost contact pseudo-metric. Then we study the cases of almost pseudo co-Kähler manifolds. In particular we obtain geometric characterizations of variations for which the fundamental form remains closed and establish a local decomposition into a product. We further investigate the variation in the setting of statistical structures.

We study a continuous-time $K$-unit online resource allocation problem with nonhomogeneous Poisson arrivals and time-varying valuation distributions. While the optimal dynamic policy generally depends on both the remaining inventory and the time left in the horizon, we focus on a simpler and practically appealing class of stock-based threshold policies, whose limited number of thresholds depend only on the number of units remaining. We evaluate these policies against the multi-unit prophet benchmark, which selects the best $K$ realized values in hindsight. Our main contribution is a new competitive-analysis framework for stock-based thresholds in continuous time. We first reformulate the problem through a type-covering dual. The central challenge for analyzing the dual is that the dual is both infinite-dimensional and non-convex: the adversary can choose time-varying arrival and valuation processes, while the policy performance depends nonlinearly on the stochastic inventory trajectory. We overcome these challenges by reducing the continuous-time adversarial problem to a Poisson optimization $PoisOPTRe_K$, and then proving sharp structural properties of its worst-case solutions. In particular, adversarial arrivals admit cutoff and late-filling structures, which yield an exact four-parameter formulation for two thresholds and a finite nested-interval representation for general thresholds. These reductions make the guarantees directly computable. For example, for two thresholds, we obtain a competitive ratio $0.6269$ for $K=2$; with three thresholds, we obtain the ratio $0.6816$ for $K=3$. In this way, we show that simple stock-based thresholds achieve strong prophet-inequality guarantees despite ignoring calendar time.

In this paper, we study a subcritical Galton-Watson process with state-dependent immigration, where immigration occurs only if the preceding generation is empty. Under heavy-tailed assumptions on the offspring and immigration distributions, we establish central limit theorems for the total progeny up to time $n$, thereby extending existing results for the finite-variance model in the literature.

In a proper vertex coloring $c$ of a graph $G$, a vertex $u$ is called a b-vertex if $u$ is adjacent to a vertex in every other color class. A ${\rm b}^{\ast}$-coloring is a proper coloring in which a b-vertex is adjacent to a b-vertex in every other color class. A Grundy coloring is a proper coloring obtained by the First-Fit (greedy) coloring procedure. A $z$-coloring of $G$ is a ${\rm b}^{\ast}$-coloring that is also a Grundy coloring. The ${\rm b}^{\ast}$-chromatic number (resp., $z$-chromatic number), denoted by ${\rm b}^{\ast}(G)$ (resp., $z(G)$), is the maximum number of colors used in a ${\rm b}^{\ast}$-coloring (resp., $z$-coloring) of $G$. Every graph admits a ${\rm b}^{\ast}$-coloring and a $z$-coloring that can be found using a polynomial-time coloring heuristic. Let ${\rm m}^{\ast}(G)$ be the largest integer $k$ such that a vertex of degree at least $k$ in $G$ has $k$ neighbors of degree at least $k$. We employ list-coloring techniques to prove that if $G$ has a girth of at least $7$, then ${\rm b}^{\ast}(G) = {\rm m}^{\ast}(G)+ 1$. A similar result is obtained for graphs of girth at least $6$ when ${\rm m}^{\ast}=3$. Finally, we obtain some results for the $z$-chromatic number. We prove that if the girth is at least $2m^{\ast}(G)+4$ and $G$ contains a specific tree as an ordinary subgraph, then $z(G)= m^{\ast}(G)+1$.

Large-scale optimal transport (OT) problems involve a vast number of transport variables, leading to prohibitive memory and computational costs. To address these challenges, we propose a multiscale primal-dual interior-point relaxation method (MSIPRM). The multiscale outer framework constructs a hierarchy of standard OT problems at progressively finer levels. At each level, the OT problem is solved over a sequence of adaptively refined active sets initialized based on the solution support at the previous level. This yields a sequence of closely related sparse subproblems, thereby substantially reducing memory requirements. The primal-dual interior-point relaxation method (IPRM) serves as the inner solver for each sparse subproblem. Since IPRM does not require strictly interior iterates, it can readily use the solution of the previous subproblem as a warm start. To efficiently obtain the Newton direction, we solve a reduced Schur complement system derived from the normal equations. Furthermore, we develop an effective support-identification strategy based on the approximate solutions obtained by IPRM. We establish condition number estimates for the Schur complement matrices and analyze the global and local convergence properties of the algorithm. Numerical experiments on large-scale test problems demonstrate the computational efficiency and scalability of MSIPRM and show that it compares favorably with existing solvers. In particular, MSIPRM can handle instances whose full formulations contain trillions of transport variables.

It is known that Chebyshev based polynomial approximation gives a near-optimal rate of convergence for calculating a function of a Hermitian matrix. However, previous work has not discussed the option of true minimax approximation, nor the specifics of writing a high performance implementation with a rigorous analysis of errors. This work provides such an analysis and an open-source implementation of three approximation methods in C++.

We prove new deviation inequalities for infinite-order U-processes, that is, U-processes whose order may increase with the sample size. Our goal is to obtain the sharpest possible dependence of the tail bounds on the order of the process, as well as to characterize the largest range in which the process exhibits sub-Gaussian deviations. We derive closed-form inequalities for processes indexed by Euclidean classes, including VC-subgraph classes of functions. We then illustrate the statistical applications of our results to multivariate depth. In particular, we introduce a multivariate analogue of the Hodges-Lehmann estimator based on Tukey's median, establish sub-Gaussian concentration inequalities with explicit constants for its deviations from the mean, and identify an interesting phenomenon concerning its asymptotic distribution.

The unit square $I=[0,1]^2$ is divided into two rectangles by the vertical line $x=p$, where $p\in(0,1)$. Consider $N$ independent uniformly distributed points on $I$, which we interpret as a population of individuals, with the line $x=p$ representing a community boundary that separates the population into two communities. Whether a pair of individuals share a connection depends on their locations in $I$ and on whether they belong to the same community. Specifically, two individuals in the same community are connected if they are within distance $R_N$ of each other, while two individuals in different communities are connected if they are within distance $R_N'$ of each other, giving rise to a geometric variant of the so-called stochastic block model. A statistician observes the adjacency matrix of the resulting graph together with the geometric locations of the individuals and is tasked with estimating the boundary location $p$. Depending on how $R_N$ and $R_N'$ scale as $N\to\infty$, we establish necessary and sufficient conditions for consistent estimation of $p$. Whenever consistent estimation is possible, we devise an estimator that converges to $p$ as $N\to\infty$ and provide explicit bounds on its estimation error.

We determine the Morita equivalence classes of $2$-blocks with quaternion defect groups of arbitrary $2$-power order, thereby completing the proof of Donovan's conjecture for blocks of tame representation type.

High-speed trains (HSTs) have become a prominent means of transportation, requiring high data rates and reliable communication services for HST passengers. However, the wireless channels in HST communication systems are susceptible to various security threats, including eavesdropping. Addressing these security concerns is therefore of critical importance. One promising technology for enhancing security is the integration of a reconfigurable intelligent surface (RIS) on an unmanned aerial vehicle, referred to as an aerial reconfigurable intelligent surface (ARIS). This technology offers significant potential for improving wireless network performance, though it also introduces unique challenges in terms of physical layer security (PLS). This paper investigates the PLS of ARIS-aided HST communication systems. A problem of maximizing the weighted sum secrecy rate is formulated by jointly optimizing the active beamforming at the base station (BS) and the phase shift at the ARIS, subject to constrains on the BS transmit power and the unit modulus of the ARIS reflecting coefficient. To address this problem, a joint optimization algorithm is proposed using the block coordinate descent method. Specifically, the problem is decomposed into two subproblems: active beamforming design and ARIS phase shift optimization. The active beamforming is optimally designed via the successive convex approximation technique, while the ARIS phase shift is efficiently updated using the alternating direction method of multipliers technique. Simulation results demonstrate the rapid convergence of the proposed algorithm, which achieves a higher secrecy rate compared to existing methods in the literature.

Reconstruction of 3D electromagnetic currents from sparse, multi-frequency, far-field radiation data is severely ill-posed because sparse sampling masks specific current Fourier modes and the missing spectrum enlarges the null space of the forward operator. Although structured-Hankel completion can recover missing spectrum by exploiting the finite rate of innovations structure of a compactly supported geometrically sparse source, it treats current components independently without guaranteeing compatibility with the model. This often renders non-physical solutions that violate model consistency. To resolve this, we propose a training-free Maxwell-model-consistent 3D structured-Hankel framework (MC-Hankel) that alternates joint low-rank Hankel completion with an exact, closed-form projection onto the true Maxwell synthesis subspace while enforcing conjugate symmetry and noise-aware data fidelity. Across tests on two source models, three sparse sampling rates (30%, 40%, and 50% of a Nyquist-sampled grid), and clean versus noisy (10 dB SNR additive white Gaussian noise) conditions, MC-Hankel consistently outperforms sub-sampled Fourier inversion, scale-normalized $\ell_1$-compressed sensing baseline, and the standard joint 3D annihilating filter-based low-rank Hankel matrix completion approach (ALOHA). Across all considered source-noise-sampling configurations, MC-Hankel improves mean peak SNR (PSNR) by 0.65-4.22 dB and tri-planer structural similarity (3D SSIM) by $0.0266$-$0.0703$, reduces relative full-volume $\ell_2$ reconstruction error by $7.4\%$-$38.9\%$, and brings non-physical model residuals down to machine precision over standard joint 3D ALOHA with merely 17% computational overhead. Exact signed-rank tests are underpowered for five paired trials, so statistical conclusions are reported together with paired bootstrap intervals, effect sizes, and small-sample limitations.

Acceleration for deterministic root-finding problems has been extensively studied in recent years; specifically, the anchor-based, or Halpern-type methods achieve optimal convergence rates with respect to the operator norm. However, acceleration via these methods does not directly carry over to stochastic setting due to accumulation of errors, unless one enforces diminishing variance via increasing batch sizes or variance reduction techniques. In this work, we show that another class of acceleration, namely the dual-anchor mechanism, extends to the stochastic setting without such error accumulation, in contrast to anchor-based algorithms. Consequently, we cleanly achieve $O(\epsilon^{-3})$ complexity with iteration-independent batch size, without any variance reduction or double-loop recursive regularization, for stochastic root-finding (resp. fixed-point) problems with cocoercivity (resp. square-nonexpansivity) in expectation. For strongly monotone operators, the same algorithm attains a sharper $\widetilde{O} (\epsilon^{-2})$ complexity, nearly matching the lower bound in terms of $\epsilon$-dependence.

In this paper, we introduce two duality formulas for $\widehat{\mathcal{Q}}$-multiple zeta values ($\widehat{\mathcal{Q}}$-MZVs for short) which are defined by Takeyama and Tasaka. Since taking two different limits of $\widehat{\mathcal{Q}}$-MZV recovers the $\boldsymbol{p}$-adic multiple zeta value and the $t$-adic multiple zeta value respectively, finding the relation of $\widehat{\mathcal{Q}}$-MZVs partially supports the Kaneko–Zagier conjecture and its refined version. Currently, three types of duality formulas for the $\boldsymbol{p}$-adic multiple zeta values are known, and for one of them, the $q$-analogue was studied by Takeyama and Tasaka. We present $q$-analogues of the remaining two duality formulas for the $\boldsymbol{p}$-adic multiple zeta values.

Recently Batanin and Markl developed a theory of Koszul duality for operads over a general operadic category in the spirit of Ginzburg-Kapranov. However, their approach doesn't apply directly to the operadic category of n-trees, which are used to describe categorical structures. In this paper we present a category of (locally constant) 2-nets which addresses the main difficulty of extending their theory to the case of 2-trees. Specifically, our category of 2-nets is an operadic category containing 2-trees as a full subcategory, and every map between 2-nets admits a factorization into a chain of elementary maps.

We establish a sharp lower bound for the total Gauss-Kronecker curvature of convex hypersurfaces in Cartan-Hadamard manifolds with pinched negative curvature. The bound holds in all dimensions when the diameter is small relative to the curvature scale, and in dimensions 4 and 5 without any restriction on the diameter, provided that the pinching is sufficiently tight. The proofs are based on the Chern-Gauss-Bonnet theorem and weighted Hsiung-Minkowski inequalities. As an application, we obtain the isoperimetric inequality of the Cartan-Hadamard conjecture in dimension 5 under sufficiently pinched curvature.

In \cite{feit}, Feit used the Generalized Laguerre Polynomials (GLP) to prove that the groups $\widetilde{A}_{5}$ and $\widetilde{A}_{7}$ occur as Galois groups over $\Q$. Hajir, in \cite{hajir} extended these results to prove that $\widetilde{A}_{n}$ is Galois over $\Q$ whenever $n \equiv 1 \pmod{8}$. A key ingredient of both proofs is the explicit determination of the Hasse-Witt invariant of (the diagonalization of) the trace form of the root fields of the GLP, which relies on the calculation of a certain determinant, $\Delta_t$. The explicit formula for $\Delta_t$ used in \cite{feit} and \cite{hajir} was derived using properties specific to the GLP which do not generalize to other polynomials. In this paper we revisit Feit's original calculation of $\Delta_t$ and situate it in the context of Hankel determinants. We give an alternate derivation of $\Delta_t$ using standard combinatorial arguments and then apply these results to the Jacobi polynomials, a two-parameter family of orthogonal polynomials encompassing the GLP as a special case. We compute an explicit formula for the $\Delta_t$ of the Jacobi polynomials as well as the associated Hasse-Witt invariant. The techniques used in this paper are not specific to the Jacobi polynomials and are widely applicable.

We construct a family of layer barriers for colour-biased tight Hamilton cycles in uniform hypergraphs. For every $k\ge 3$ and every $a\in\{0,\ldots,k-1\}$, we give a red–blue coloured $k$-graph that contains a tight Hamilton cycle, while every tight Hamilton cycle in the construction is perfectly colour-balanced. The construction underlying the higher-uniformity threshold conjectured by Behague, Clemen, Hyde and Morrison corresponds to the boundary case $a=0$ of this family. We show that interior choices of $a$ can yield strictly denser barriers. In particular, for $k=17$ and $a=8$, the asymptotic relative minimum vertex degree of our construction is \[ \frac{5761}{8192}\approx 0.703247, \] which exceeds the conjectured value $d_{17}\approx 0.699277$. This provides a counterexample to the proposed higher-uniformity threshold in Conjecture~6.1 of Behague, Clemen, Hyde and Morrison. Moreover, by choosing the layer appropriately as $k\to\infty$, the family contains barriers whose asymptotic relative minimum vertex degree is \[ 1-O\bigl(k^{-1/2}\bigr). \] Thus the interior members of the layer-barrier family exhibit substantially different behaviour from the previously considered boundary construction in large uniformity.

We introduce a natural generalization of the Bell numbers: the $m$-Bell numbers $B^{(m)}_{n}$, characterized by the property that $m$ applications of the binomial transform reproduce the original sequence shifted $m$ places to the left. Their exponential generating functions satisfy $m$-th order ordinary differential equations whose solutions are hypergeometric (hyper-Bessel) functions, specializing to the exponential function when $m=1$ (classical Bell numbers) and to modified Bessel functions when $m=2$ (yielding "Bessel-Bell" numbers). Mirroring the Bell-Stirling correspondence, we construct $m$-Stirling triangular arrays from the two-term recurrence $S_m (n+1,k) = m \left\lfloor k/m \right\rfloor S_m(n,k)+S_m(n,k-1)$ and prove an elementary shift identity from which the central structure theorem follows: the row sums of the $m$-Stirling triangle reproduce $B^{(m)}_{n}$, and, more finely, the residue-class row sums are precisely the $m$ primitive $m$-Bell sequences. The $m$-Stirling numbers come in dual pairs (with first-kind partners, generalized falling factorials, and Lah-type companions), serve as conversion operators between polynomial bases, admit Dobiński-like formulas, and count congruence-constrained partitions in an urn model as well as restricted permutation insertion histories. Finally, we show that the $m$-Bell numbers govern the moments of the Conway-Maxwell-Poisson distribution with integer dispersion parameter $\nu=m$: the scaled moments are combinations of fixed hyper-Bessel carrier ratios whose integer coefficients are precisely the primitive $m$-Bell sequences, recovering for $m=1$ the classical fact that the moments of the Poisson distribution are the Bell numbers.

Bregman proximal stochastic gradient (BPSG) methods bring variance-reduced composite optimization to objectives whose geometry is poorly captured by Euclidean smoothness. Their performance, however, remains sensitive to the step size: raw stochastic curvature estimates can fluctuate sharply, whereas line searches add repeated proximal evaluations. We introduce Ada-BPSG, a line-search-free BPSG method that couples the SAGA gradient table with a stabilized Barzilai–Borwein (BB) candidate. A mediant aggregates incremental secant information so that nearly singular local ratios receive little weight, and an explicit safeguard translates the resulting curvature estimate into the bounded step-size sequence required for convergence. This design yields a direct analytical chain from relative smoothness and component-wise variance control to convergence in finite-dimensional normed spaces. We prove an $O(n/K)$ ergodic rate for convex objectives, a restarted linear rate under relative quadratic growth, and an $O(1/K)$ bound for a Bregman proximal residual in the nonconvex setting. On logistic regression and sparse nonnegative matrix factorization, Ada-BPSG combines low objective values with substantially less sensitivity to the initial step size than standard variance-reduced baselines, while avoiding line search.

In this paper, we investigate the integrability of Lotka-Volterra (replicator) systems arising from interaction matrices generated from corresponding graph structures, continuing work started by Visomirski and Griffin [J. Phys. A., 58:015701, 2025] and Evripidou et al. [J. Phys. A., 55:325201, 2022] (among others). In particular, we define a new family of graphs, the spoked graphs, and show that all dynamics generated from this family are integrable. In reference to Evripidou et al. (2022), we define a new anti-cloning operator and show that its action on balanced tournament graphs (with odd vertex count) generates new graphs whose dynamics are integrable. Interestingly, we provide numerical evidence that this anti-cloning operation leads to chaotic behaviour when applied to other graph families (e.g., the directed cycles that generate the classically integrable Volterra lattice). This work completes a taxonomy of all integrable dynamics generated by directed graphs with up to six vertices started by Visomirski and Griffin (2025), and suggests several future directions of study on this topic.

Let $X$ be a smooth projective surface with Picard number two, where either $X$ is a geometrically ruled surface or the closed Mori cone is rational polyhedral. In this paper, we characterize $X$ with the bounded cohomology property, i.e., there exists a constant $c_X>0$ such that $h^1(\mathcal O_X(C))\le c_Xh^0(\mathcal O_X(C))$ for every curve $C$ on $X$.

Affine frequency division multiplexing (AFDM) is a promising chirp-based multicarrier waveform for high-mobility integrated sensing and communication (ISAC). Accurate angle, delay, and Doppler estimation is essential for AFDM sensing. Since target delays and Doppler shifts are generally continuous-valued, representing them on a discrete delay–Doppler grid causes energy leakage and peak displacement in the discrete affine Fourier transform (DAFT) domain. The AFDM chirp also induces delay–Doppler coupling in the DAFT-domain response. The resulting DAFT-domain matching-score surface exhibits a local ridge that is not aligned with the normalized-delay and normalized-Doppler axes. To address these issues, this paper investigates joint estimation of angle and continuous-valued delay–Doppler parameters for a colocated AFDM-ISAC sensing architecture. A transform-domain sparse sensing model is formulated from the fractional DAFT-domain response. Based on this model, a coupled-coordinate Newtonized orthogonal matching pursuit (CC-NOMP) estimator is developed. CC-NOMP uses the AFDM-induced coupling coordinate to parameterize the dominant local ridge. It combines coupled-coordinate Newton refinement with safeguarded updates, coupling-aligned delay refinement, and cyclic multi-target refinement to estimate angle, continuous normalized delay, and normalized Doppler. A deterministic Cramér–Rao bound and a dominant-order complexity analysis are also derived. Simulation results with continuous-valued off-grid target parameters show that CC-NOMP achieves lower delay and Doppler error floors than the considered baselines while maintaining comparable angle-estimation accuracy.

We prove the simple-object prediction of Shan–Yan–Zhao and a basis-preserving dual-cell realization for the distinguished vacuum block of the simple affine vertex algebras $L_{-1}(D_\ell)$, $\ell\ge5$. The block has exactly $\ell+1$ simple objects, indexed by the subregular affine left cell containing $s_0$. The proof combines a primitive-ideal inclusion, an independent exhaustion argument, and a finite-length step. A noncritical Sugawara lift supplies finite-dimensional weight-space detectors in the original Shan–Yan–Zhao category-$\mathcal O$ block, so dévissage applies to its ordinary Grothendieck group. We then identify this group, basis by basis, with the $q=1$ specialization of the corresponding dual affine left-cell module. An injective signed normalized-character realization identifies the resulting image with the canonical dual-cell image in the completed singular-orbit module and hence supplies the corresponding abstract $\widehat W$-module structure. We do not identify this action with a functorial action arising from affine twisting functors or Kashiwara–Tanisaki localization. The subregular inverse Kazhdan–Lusztig calculation of Bezrukavnikov–Kac–Krylov also yields uniform character formulas.

The $n$-qubit Pauli group is an essential ingredient to most quantum applications, from computing and error correction to benchmarking and simulation. Despite comprising merely a discrete set of operators, it exhibits many quintessential features of quantum theory, including contextuality, which has been identified as a key resource to quantum advantage in a variety of different flavours. Here, we extend this analysis, introducing the notion of a `noncontextual property' whose nonexistence proves the Kochen-Specker theorem, similarly to and generalising common arguments based on the nonexistence of valuations. We relate this notion formally to the existence of Boolean-valued frame functions, and characterise all such frame functions in the case of the $n$-qubit Pauli group. For two qubits, we show that the Pauli group admits noncontextual properties, despite admitting no valuations. We then establish this as the only nontrivial such case with $n\geq 2$, by proving that any Boolean-valued frame function on stabiliser states is constant for more than two qubits. We also perform a similar analysis for the symplectic theory underlying the $n$-qubit Pauli group, for which nonconstant Boolean-valued frame functions exist for all $n$, yet only in restricted form. By comparison, this shows that contextuality in the $n$-qubit Pauli group is not only a consequence of the projective nature of the Pauli group as a representation of its underlying symplectic vector space, but of the geometry of symplectic polar spaces itself. In geometric terms, our result determines all Cameron-Liebler sets of maximal totally isotropic flats in the binary affine-symplectic space.

The continuous-time quantum walk on an oriented circulant graph is determined by the Fourier eigenvalues of its Hermitian adjacency matrix. We classify perfect state transfer (PST) between distinct vertices in every nonempty oriented circulant graph. We show that each such graph is described by an odd primitive quadratic Dirichlet character of conductor $\Delta$, a set of gcd-classes, and a choice between the two orientations of each selected class. For a graph of order $n$, we derive an explicit formula for every Fourier eigenvalue without assuming that $n/\Delta$ is coprime to $\Delta$. We prove that PST occurs only for $\Delta\in\{3,4,8\}$ and give necessary and sufficient conditions on the connection set for each conductor. Equivalently, the square-free radicands of oriented circulant graphs with PST are exactly $1$, $2$, and $3$. More generally, when $\lambda_j=\sqrt{D}\eta_j$ with $\eta_j\in\mathbb{Z}$, congruences satisfied by the integers $\eta_j$ determine all PST pairs and times, the minimum period, and the largest vertex sets supporting multiple state transfer (MST). In this class, pretty good state transfer is equivalent to PST. We also determine the connected orders and enumerate the resulting graphs.

An $\ell$-congruent spanning tree of a nontrivial connected graph is a spanning tree in which every vertex has degree congruent to one modulo $\ell$. This notion provides a common generalization of classical spanning trees and odd spanning trees. We show, via a constructive greedy algorithm, that every $n$-vertex graph $G$ satisfying $n\equiv2\pmod{\ell}$ and $\delta(G)>\frac{(\ell-1)n}{\ell}$ has an $\ell$-congruent spanning tree. For the special case of odd spanning trees ($\ell=2$), our algorithmic approach simplifies the original proof by Zheng and Wu. We also derive formulas for the numbers of $\ell$-congruent spanning trees in complete graphs and complete bipartite graphs. These formulas specialize to the classical spanning-tree formulas when $\ell=1$ and to the corresponding odd-spanning-tree formulas when $\ell=2$.