Papers

We consider the real noncentral sample covariance matrices $\mathcal{W}=YY^\top$ with $Y=A+\Sigma^{1/2}X$. Here $A\in\mathbb{R}^{M\times N}$ is deterministic, $\Sigma\in \mathbb{R}^{M\times M}$ is a deterministic positive definite population covariance matrix and $X\in\mathbb{R}^{M\times N}$ has independent centered entries with variance $N^{-1}$. We prove local laws near regular right edges down to optimal spectral scales without requiring the commutativity of $AA^\top$ and $\Sigma$. As a consequence, we obtain optimal eigenvalue rigidity at the rightmost regular edge and delocalization of the corresponding left and right singular vectors. We also show that, with high probability, there are no eigenvalues in the adjacent spectral gap beyond the optimal $N^{-2/3}$ edge scale, up to an arbitrarily small $N^\varepsilon$ loss. Finally, we establish edge universality at the rightmost regular edge: after centering and scaling, the largest eigenvalue converges to the Tracy–Widom distribution. The main technical ingredient is a stability analysis of the matrix Dyson equation (MDE) associated with the linearization of $Y$, whose self-energy operator does not satisfy the flatness condition of the general MDE theory. Exploiting the special block structure, we reduce the stability analysis exactly to a two-dimensional operator. This reduction yields regularity of the spectral density and square-root behavior at regular right edges, together with sharp stability bounds near such edges.

We prove the spherical Hadwiger theorem in every dimension: for $n\geq1$, every continuous $\mathrm{SO}(n+1)$-invariant valuation on all closed spherical convex sets in the unit sphere $\mathbb{S}^n$ is a unique linear combination of the spherical intrinsic volumes $V_0,\ldots,V_n$. The same classification holds on sets contained in an open hemisphere, with unique continuous invariant extension to the full domain. We first classify simple invariant valuations by an independent induction on dimension. Averaging a signed cone identity gives the spherical volume term; the remaining terms are signed coning transforms that vanish by the lower-dimensional simple classification. A second induction, together with extension to non-proper sets, gives the general classification. The cone–sphere correspondence yields, for $d\geq2$, the corresponding classification on all closed convex cones in $\mathbb{R}^d$, without normalization at the zero cone. In both settings, $\mathrm{SO}$-invariance implies $\mathrm{O}$-invariance.

Let $n\ge3$ and let $h:\A(r,1)\to\A(R,1)\subset\mathbb R^n$ be an onto homeomorphism with harmonic coordinate functions. We prove the sharp Nitsche bound \[ R\le R_{n,+}(r):=\frac{nr}{n-1+r^n}, \] and, when $h$ interchanges the two ends, the strictly stronger sharp bound \[ R\le R_{n,-}(r):=\frac{nr^{n-1}}{1+(n-1)r^n}. \] Both critical cases are rigid: equality forces, up to an orthogonal transformation, the corresponding end-preserving or end-reversing radial harmonic homeomorphism. No continuous extension to the closed annulus, boundary homeomorphism, boundary Jacobian, or sign condition on the Jacobian is assumed. The proof converts the nonzero degree of each interior direction map into a probability coupling and establishes a sharp contraction principle for vector measures under positive zonal kernels, using the strict concavity of spherical-cap barycenters. At either critical value, a second-order endpoint defect forces equality for a limiting transfer kernel, whose equality classification yields an orthogonal coupling graph. The remaining trace is locked by a Dirichlet-to-Neumann spectral gap in the end-preserving case and by endpoint Hölder regularity and uniform convergence of the direction maps in the end-reversing case.

In the study of Nash equilibria of finite-player games, one often seeks equilibria that are compatible with predetermined constraints, either determined by the players or by an external agent. We discuss the algebraic loci, called Nash loci, of games whose Nash equilibrium scheme intersects a fixed algebraic variety in a product of projective spaces. We determine their dimensions and multidegrees in multiprojective space, and their equations for two-player games and for small multiple-player games. The multilinear equations defining the Nash equilibrium scheme allow us to describe Nash loci in the language of Grassmannians and Plücker coordinates. Motivated by this fact, we relate Nash loci to multigraded associated varieties, which are subvarieties in products of Grassmannians that generalize the multigraded Cayley-Chow hypersurfaces of Osserman and Trager.

We show that power semigroups of groups, and more generally reduced finitary power monoids, arise naturally as convolution monoids in Kleisli categories of powerset monads: (1) for the non-empty powerset monad, the Kleisli Hom-space $\mathrm{Hom}_{\mathbf{Kl}(\mathscr P_+)}(1,G)$ is isomorphic to the power monoid $\mathcal P_+(G)$; (2) for the reduced finite powerset monad on pointed sets, the Kleisli Hom-space $\mathrm{Hom}_{\mathbf{Kl}(\mathscr P_{\mathrm{fin}})}$ $(\mathbb Z/2\mathbb Z,H)$ is isomorphic to the reduced finitary power monoid $\mathcal P_{\mathrm{fin},1}(H)$. This unifies several constructions in power semigroup theory: Kleisli convolution representations of semigroups, base change along surjective group homomorphisms, and rigidity of automorphism groups. As an application, we prove that for every proper numerical monoid $S$, the Kleisli Hom-monoid $\mathrm{Hom}_{\mathbf{Kl}(\mathscr P_{\mathrm{fin}})}(\mathbb Z/2\mathbb Z,S)$ is rigid. This gives a proof of the Tringali–Yan conjecture in the language of Kleisli categories. It should be mentioned that this conjecture was already proved by Bhowmik and Tringali in a preprint posted on arXiv on July 25, 2026.

For smooth hypersurfaces over rational function fields of characteristic greater than the degree and with sufficiently large constant field, we improve the number of variables required in Birch's theorem from an exponential function of the degree to a quadratic one. This agrees, up to constants, with the sharp quadratic threshold for the unconditional existence of rational points on smooth hypersurfaces. A circle method argument reduces the required cancellation to lower bounds for the codimensions of certain singular loci associated with complete exponential sums over finite fields. Our main innovation is a new method for proving these bounds: we introduce the notion of multiplication rank for the linear functionals indexing these exponential sums and combine the resulting rank stratification with a weighted degeneration of the Jacobian equations to obtain a codimension estimate that grows linearly with multiplication rank.

In this article, we construct noncommutative cluster varieties, $\mathcal{A}_{R,S}$, for each reduced root system $R$ and marked surface $S$ simultaneously generalizing the cluster varieties of Fock-Goncharov, Li, Goncharov-Shen, Berenstein-Retakh, Goncharov-Kontsevich, and our previously introduced polygonal cluster algebras. Additionally, we define a large class of algebraic groups, we call Jordan split groups. Given a reduced root system $R$ and a family of Jordan algebras, the Lie algebra for $G$ is constructed by unifying the Tits-Kantor-Koecher construction for a single Jordan algebra with the construction of a split Lie algebra. The notion of Jordan split groups is closely related to a grading of its Lie algebra by the root system $R$. We show that these gradings are usually induced by a choice of standard parabolic subalgebra $\mathfrak{p}_\Theta$ and we classify $R$-graded pairs $(G,\Theta)$ via a condition depending only on the subset $\Theta\subset \Delta$ of the set of simple roots. Next, we define Jordan algebra points of $\mathcal{A}_{R,S}$ which parameterize $G$-local systems on $S$ with boundary decoration related to cosets $G/U_\Theta$ when $G$ is Jordan split of type $R$. When $S$ is a disk, points of $\mathcal{A}_{R,S}$ parameterize configurations of decorated flags. We use this to give noncommutative cluster structures on the double $R$-Bruhat cells of $G$, generalizing the cluster algebras of Berenstein-Fomin-Zelevinsky. When each Jordan algebra is formally real, we say that $G$ has a positive structure with respect to $\Theta$. This defines a positive semigroup in $G$. For real algebraic groups, the pairs $(G,\Theta)$ which have positive structures are exactly those which admit a positive structure as defined by Guichard-Wienhard and we give algebraic proofs of many of the properties of positive configurations of flags and of positive representations.

In this paper, we study the ordered solutions of the matrix Pythagorean equation $X^2+Y^2=Z^2$ over $\mathrm{GL}_2(\mathbb{Z})$ . Exploiting the independent sign symmetry of the equation, we first reduce the full solution set to the trace-nonnegative subset $\mathcal M$, consisting of those solutions for which all three traces are nonnegative. We then determine a canonical decomposition of $\mathcal M$ into orbits under simultaneous integral conjugacy.

Let $1<p<n$ and let $v(x)=(1+|x|^{p/(p-1)})^{-(n-p)/p}$ be the standard radial extremal for the sharp Sobolev inequality. We determine all eigenvalues and eigenspaces of the linearized $p$-Laplacian at $v$, defined by its closed quadratic form in $L^2(\mathbb{R}^n,v^{p^*-2} dx)$. After decomposition into spherical harmonics, an explicit gauge transformation and a change of variables identify each radial operator with a shifted Jacobi operator. This yields a complete eigenbasis indexed by $(\ell,k)\in\mathbb{N}_0^2$, where $\ell$ is the angular degree and $k$ is the radial mode number.

We propose a novel method that combines a multilevel decomposition with the framework of Interacting Particle Systems to compute the probability that given quantities of interest take values above or below a certain pre-specified value. Through the application of a sequential sampling scheme, the algorithm is able to achieve convergence rates faster than those achieved by the classical Multilevel Monte Carlo algorithm, and competes with the current state of the art techniques, while also being dimension independent. Furthermore, the sample points generated by the algorithm form a sequence which concentrates near the critical value of the quantity of interest. Our results are rigorously established and later verified through numerical experiments.

We show that if $G$ is a real semisimple Lie group and $\Gamma<G$ is a discrete subgroup with slow growth, in the sense that its growth indicator function is smaller than $\rho$, then $\Gamma$ acts totally dissipatively on the Furstenberg boundary of $G$. Moreover, for $G= SO(n,2)$ and $n\geq 3$, we construct infinite-covolume discrete subgroups that act ergodically on the Furstenberg boundary of $G$, providing a counterexample to a conjecture of Margulis for $G = SO(n,2), n > 2$. The examples arise from lattices $\Gamma<H= SO(n,1)$ and their deformations in $G$. Perhaps more importantly, we describe a new approach to studying deformations of such lattices by relating them to deformations of the smooth right-translation action of $SO(n-1,1)$ on $\Gamma\backslash H$. This correspondence allows us to apply recent results of DeWitt and Dolgopyat on smooth group actions and to give examples where the right-translation action of $SO(n-1,1)$ on $\Gamma\backslash H$ can fail to be $C^0$-locally rigid.

The marine science community increasingly relies on Marine Autonomous Vehicles (MAVs) to collect the critical environmental data required to understand global ocean systems. However, as these operations scale, manually routing and planning large autonomous fleets becomes exponentially complex and time-consuming. To address this, we propose a mixed-integer linear programming (MILP) model designed to automate and optimise MAV deployment schedules. The model accounts for strict operational constraints, including battery capacities and time windows for data collection, while aiming to maximise total data collection and minimise both the number of deployed vehicles and their energy consumption. A key novelty of this framework is integrating conventional ship itineraries, allowing MAVs to support vessels with mid-mission battery swapping or accelerated transit between waypoints. Computational experiments demonstrate that the model is highly scalable, solving routing problems for fleets of dozens of MAVs in seconds, and scaling to hundreds of vehicles in only a few minutes. Beyond operational scheduling, the framework serves as a robust simulation tool for evaluating 'what-if' scenarios and analysing the impact of varying parameters on deployment strategies. Finally, the solution generates a suite of visualisations designed to enhance explainability and support strategic decision-making for stakeholders.

We present results on the approximate computation of stable invariants for filtrations of finite metric spaces in the context of persistent homology. We establish novel approximation algorithms in the setting of $n$-point metric spaces where the growth of the doubling dimension is in $o(\log n)$ and the diameter is bounded. In the $1$-parameter case, by revisiting known techniques (greedy permutations) in a new way, we derive the first linear-time algorithms for the problem of computing additive $\varepsilon$-approximations of any stable barcode. By deriving bounds on the convergence rate and the approximation quality of uniform samples, we extend the approach to selected multiparameter filtrations. We show that for normalized measure bifiltrations, including the multicover and subdivision-Rips bifiltration, any stable invariant can be probabilistically approximated in time constant in $n$. The constants in the running times of our algorithms depend on the doubling dimension, the diameter and the success probability. We further study the problem through the lens of fine-grained complexity and show that computing the rank of a matrix reduces to that of approximating the barcode of the Vietoris–Rips or Čech filtration. We present two variants of the reduction, one for sufficiently good additive approximations and the other for any constant factor multiplicative approximations.

In this paper, we investigate first- and second-order alignment models with non-universal interaction, time delays and possible communication failures, extending the results in [17] to interaction digraphs that are only assumed to be rooted and to a weaker Persistence Excitation Condition. In particular, we allow the amount of interaction over time intervals of fixed length to decay polynomially in time. For the first-order Hegselmann-Krause type model, we prove asymptotic convergence to consensus under a suitable condition relating the decay exponent of the communication weights to the maximal distance from the root. For the second-order Cucker-Smale model, we establish asymptotic flocking under an additional assumption on the decay of the influence function. These results show that collective behavior can still emerge under progressively weakening communication and without requiring strong connectivity of the interaction graph.

We study entire solutions of the Allen–Cahn equation $\Delta_g u = W'(u)$ on Cartan–Hadamard manifolds of the form $u = U\circ b$, where $b$ is a Busemann function and $U\in C^2(\mathbb{R})$; we call these Busemann profiles. Our main result is a rigidity principle requiring no homogeneity of the manifold: if $\inf \Delta_g b>0$ and $U$ has finite limits $\ell_\pm$, then $W(\ell_-)\le W(\ell_+)$, with equality exactly when $U$ is constant. In particular no nonconstant Busemann layer joins the two wells of a balanced double-well potential, in any dimension $n\ge 2$; under $K\le-\kappa_2<0$ the hypothesis is automatic by Hessian comparison, and if $W'$ is locally Lipschitz it holds under the mere nonvanishing of $\Delta_g b$. The mechanism is a dissipation identity for the Hamiltonian $\tfrac12(U')^2-W(U)$ along the flow lines of $\nabla b$, in which the mean curvature of the horospheres acts as a wave speed. Conversely, when $\Delta_g b\equiv c$ and $W$ is bistable and unbalanced, strictly increasing Busemann fronts joining the wells exist precisely when $c$ is the Fife–McLeod speed; their interfaces are then horospheres of constant mean curvature. On warped-line and rotational surfaces we test sharpness: monotone warpings admit no equal-level profile and rotational surfaces no radial two-well profile, while an even convex warping carries a nonconstant monotone layer whose nodal set is the central geodesic leaf.

Operator learning in scientific machine learning is concerned with approximation of maps between infinite-dimensional function spaces; such maps frequently arise as the solution operators of partial differential equations (PDEs). Neural operators have demonstrated broad empirical success at approximating such maps from data. However, most existing neural operator architectures enforce boundary conditions indirectly through training from data even though the boundary condition is often known exactly. Furthermore, existing modifications and approaches that do enforce boundary conditions explicitly suffer from impractical restrictions, including boundary smoothness, uniform grids, and separable, box-like domains. In this work, we propose an architecture which, independently of training, satisfies homogeneous Dirichlet boundary conditions, whilst simultaneously retaining the expressivity of existing kernel-integral neural operator architectures. This is achieved by enforcing the property that the output of each layer is contained in the span of a subset of the homogeneous Dirichlet eigenfunctions of the Laplacian on the output domain. The method requires only that the output domain be bounded with Lipschitz boundary and places no restriction on the choice of discretization, making it applicable to arbitrary mesh data and general geometries. We prove universal approximation for the resulting architecture; furthermore the approach we adopt in the analysis proves universality for a broad class of kernel-integral neural operators thereby uniting existing theory for a variety of operator learning methods. We validate the proposed method on maps defined by the coefficient to solution map in 2D PDEs: Darcy flow on a square domain and the Helmholtz equation on a circular domain. Comparisons are made with alternative methods.

Let $k=\mathbb F_q$. We classify, up to two-sided $k$-Möbius equivalence, all separable indecomposable tame exceptional maps $\mathbf P^1_k\to\mathbf P^1_k$ whose geometric Galois closure has genus one. The resulting arithmetic fixed-field forms are encoded by Frobenius-stable affine elliptic quotient data recovered from the cover; we prove a converse and an exact equivalence criterion. The same data determine branch arithmetic, geometric and arithmetic monodromy, the constant field, and behavior over every finite extension. In particular, over each $\mathbb F_{q^r}$, permutation is equivalent to exceptionality. We obtain explicit rank-two support formulas, necessary and sufficient occurrence criteria, and exact two-sided class counts in every tame signature, including the surviving cases in characteristics $2$ and $3$. In characteristic greater than $3$, tameness is automatic. A sharp $3$-adic obstruction arises only for a stronger cubic self-endomorphism realization problem, not for the fixed-field classification itself.

In this paper, we establish a structural compatibility between the Kronecker product of bipartite graphs that admit polarity and their polarity quotients, and provide a sharp upper bound on the diameter of these graphs. We use these properties to construct new families of graphs, all of which have orders that asymptotically approach the Moore bound. We show that the bipartite compatibility graphs have a natural polarity, and that the corresponding polarity quotients comprise a new family of graphs of any even diameter $2k$ with degree $(q+1)^k$ for any prime power $q$, much smaller than those of other known families of that diameter and field order $q$. These families generalize the Erdős-Rényi-Brown polarity graphs. Kronecker products of these polarity quotients give another new family of graphs of any even diameter for every pair of prime powers $q,r$. Certain generalized polygons are known to admit polarity with polarity quotients approaching the Moore bound asymptotically. Kronecker products of these polarity quotients give another new family of graphs of diameters $2$, $3$ and $5$, including a diameter-$3$ degree-$18$ graph of order larger than any previously known for that degree and diameter.

We show that purely topological torus symmetries impose strong restrictions on the fundamental groups of almost non-negatively curved closed manifolds and on their actions on higher homotopy groups, even when the torus action is unrelated to the metric.

We prove an inverse optimal-transport principle for the ABP contact set of animmersed submanifold. A fixed function on the submanifold determines, for every probability density on a bounded convex target, a source measure and an optimal plan. For a uniform ellipsoidal target, disintegration over the submanifold gives conditional measures whose fiberwise $L^\infty$-norms satisfy a sharp weighted average estimate with constant $\omega_n/\omega_{n+m}$ in codimensions $1\leq m\leq4$. This yields the sharp ellipsoidal Michael–Simon–Sobolev inequality in the same range, including its equality cases and symmetrization consequences. To the best of our knowledge, even in the Euclidean case, the sharp constants in codimensions three and four were previously unknown.

A long-standing question posed by Fukuda (1995) and Ziegler (2000) inquires about the asymptotic behavior of $g(n)$, the maximum number of facets that an $n$-dimensional $0/1$-polytope can have. A remarkable result by Bárány and Pór (2001) via probabilistic methods established that $g(n)$ is at least superexponential in $n$. In this paper, we propose a drastic change of perspective, which leads us to show that for each $n\geq 10$ there exists a $0/1$-polytope having at least $(n-\lceil 2\log_2 (n)\rceil - 1)!$ facets. This provides a significant improvement over the currently known lower bounds for $g(n)$. Furthermore, when combined with known upper bounds, our construction establishes the asymptotic behavior of $\log g(n)$ up to an error of $O((\log n)^2)$. The methods employed throughout this paper are elementary and fully deterministic. The underlying ideas in our proof stem from the combinatorics of hypersimplices and permutohedra.

In this paper we study the problem of realizing finite group schemes as fundamental group schemes of smooth projective varieties. We establish Bertini-type results and prove the triviality of fundamental group schemes of mildly singular complete intersections in projective space. With this in hand, we are able to go through a Godeaux–Serre construction and prove that, over an infinite perfect field $k$, any finite group scheme $G/k$ can be realized as the $S$-, extended Nori and Nori fundamental group schemes of a connected smooth projective variety of any dimension at least $\dim \text{Lie}(G)+2$.

Modern communication systems impose strict demands on data rate, reliability, and power efficiency. In this context, emerging communication paradigms such as identification via channels have become an important topic in post-Shannon information theory, offering the potential for substantially higher identification rates than in conventional channel coding.Deterministic identification is particularly interesting for specialized communication scenarios because it provides a balance between implementation complexity and the communication gains due to higher identification rates. It is therefore a promising communication scheme for future communication systems, including molecular communication systems. Additive Gaussian channels, particularly the additive white Gaussian channel, are among the most important channel models for analyzing the performance of communication systems in information and communication theory. This importance stems from both their mathematical tractability and their ubiquitous appearance in practical applications. To date the deterministic identification capacity for additive Gaussian channels remains unknown even for the simplest case of the additive white Gaussian channel. In this paper, we establish tight bounds on the deterministic identification capacity of additive Gaussian channels by introducing a new perspective on deterministic identification. To this end, we apply results from lattice theory to obtain new capacity results.

We prove the Arnold-Givental conjecture in full generality: given a closed symplectic manifold $(X, \omega)$, an anti-symplectic involution $\tau_X: X \to X$ with fixed point set $L={\rm Fix}(\tau_X)$, and a Hamiltonian diffeomorphism $\phi: X \to X$ such that $\phi(L)$ intersects transversely with $L$, the following inequality holds: \[ \# \big( \phi(L) \cap L \big) \geq {\mathrm dim}_{{\mathbb F}_2} H_*(L; {\mathbb F}_2).\] The proof combines the methods of integral Floer theory of the first and fourth authors, a reduction to Hamiltonian Floer cohomology due to Lu, and a new idea related to localization in a $\mathbb Z/2$-equivariant Floer theory tailored to the problem.

We study the maximum average quantum transition probability between distinct vertices of a triangle-free graph, equivalently the minimum average return probability, at times inversely proportional to the order. For every compact interval of positive scaled times below $\tau_{\mathrm c}\approx3.79099$, the balanced complete bipartite graph is the unique maximizer for all sufficiently large orders. The threshold is sharp: it solves $\tau_{\mathrm c}=4\sin(\tau_{\mathrm c}/2)$, and the balanced complete bipartite graph is eventually suboptimal at every larger scaled time. The corresponding graphon functional combines the edge density with alternating even cycle densities. We prove that the balanced complete bipartite graphon is its unique maximizer through the critical time. On a nonempty interval immediately afterward, the unique maximizer is a balanced bipartite graphon with constant cross-edge weight below one. After normalization by the squared time, we obtain uniform square-root $L^1$ stability through the critical endpoint and at zero time. We also solve the bipartite problem at every positive time and establish sharp cut distance stability. The proofs combine spectral interpolation, an exactly certified six-vertex moment inequality, vertex-measure variation, and a minimum-degree argument. A uniform finite approximation bound and vertex cloning connect the graphon and exact finite results.

Let $\mathsf k$ be a field, $S=\mathsf k[x,y,z]$, and $R=S/I$ be a standard graded Artinian Gorenstein $\mathsf k$-algebra of codimension three. The $h$-vector of such an algebra is known to be symmetric and unimodal. Miró-Roig proved that if $\mathsf k$ is algebraically closed of characteristic zero and the $h$-vector of $R$ has at least three peaks, then $R$ has the weak Lefschetz property. In this article, we extend this result to any infinite field of arbitrary characteristic, using a different, elementary, and more direct argument. In particular, we recover Miró-Roig's theorem without the hypothesis that $\mathsf k$ is algebraically closed. Along the way, we also prove a statement of independent interest that holds over any field: if the $h$-vector of $R$ has at least two peaks, and if $s$ is the largest degree of a peak, then the elements of $I$ of degree at most $s$ have no common factor.

Given a group $G$ and three subsets $S_\ell, S_r, S_m \subset G$, we consider di-Cayley graphs $DX(G;S_\ell,S_r,S_m)$ and di-Cayley sum graphs $DX^+(G;S_\ell,S_r,S_m)$, directed generalizations of the bi-Cayley (sum) graphs $BX(G;S_\ell,S_r,S_m)$ and $BX^+(G;S_\ell,S_r,S_m)$. We refer to these four kinds of graphs collectively as $X^*(G;S_\ell,S_r,S_m)$. First, we give the basic properties of these graphs and compute their adjacency matrices. Then, we obtain the eigenvalues of $X^*(G;S_\ell,S_r,S_m)$ in terms of the spectra of the associated Cayley graphs $X(G,S)$ with $S\in \{S_\ell,S_r,S_m\}$ in two ways, using adjacency matrices and using irreducible characters of $G$.

Building on our previous work on 2-Morita equivalence for $E_2$-algebras using topological pictures, we develop a systematic framework for Morita equivalence of topological orders in different dimensions in terms of $n$-Morita categories $\mathrm{Mrt}_{E_n}(\mathcal{C})$. In this framework, various notions of $n$-Morita equivalence are unified as equivalences of objects in $\mathrm{Mrt}_{E_n}(\mathcal{C})$. We also compare the constructions of higher Morita categories due to Haugseng and to Gwilliam–Scheimbauer. For $n=1,2$, we prove that the functor $\mathrm{Mod}_n:\mathrm{Mrt}_{E_n}(\mathcal{C})\to \mathrm{Mrt}_{E_{n-1}}(\mathrm{LMod}^{\mathrm{rep}}(\mathcal{C}))$ is an equivalence, relating the algebraic description of higher Morita theory to its realization in terms of module categories. In this formulation, the notion of a bi-bimodule arises naturally and provides a common framework for local and confined modules. Its explicit orientation data, together with the corresponding fusion rules, clarifies the relations among defects arising from condensation in topological orders.

For a graph \(G\) admitting a real symmetric realization with exactly two distinct eigenvalues, \(MB(G)\) is the minimum, over all such realizations, of the smaller of the two eigenvalue multiplicities. Adm, Fallat, Meagher, Nasserasr, Plosker, and Yang asked for this parameter for the complement of a path on at least eight vertices. We answer their question completely by proving $$ MB(\overline{P_n})=3 \qquad (n\ge 6). $$ In particular, this resolves the previously unresolved orders \(n\ge 9\) divisible by three. The proof is exact and constructive. We exhibit six vectors in \(\mathbb{R}^3\) whose mutual inner products vanish exactly for consecutive indices, whose rank-one outer products form a basis of \(\mathbb{S}^3\), and which admit a strictly positive Parseval scaling. An elementary absorption lemma then permits any finite faithful orthogonal extension of this vector chain to be added with small positive weights while the six original weights are corrected to retain the Parseval identity. The resulting Gram matrix is a rank-three orthogonal projection in \(\mathcal{S}(\overline{P_n})\). A local two-dimensional orthogonality obstruction gives the matching lower bound. For completeness, we include self-contained proofs of the exceptional small orders: \(MB(\overline{P_3})=1\), whereas \(q(\overline{P_4})=4\) and \(q(\overline{P_5})=3\).

We present an open-source Python package for sampling the Gaussian, Laguerre, Jacobi, and Circular $\beta$-ensembles of random matrix theory. The package implements the Dumitriu–Edelman and Killip–Nenciu constructions, allowing efficient generation of random spectra for general $\beta > 0$. In addition to spectrum generation, it includes tools for the analysis of spectral statistics, from standard nearest-neighbor spacings and spacing ratios to non-adjacent $k$-spacings and the spectral form factor. These tools can be applied to generic spectral data, allowing users to compare them with and fit them to $\beta$-ensemble predictions. In this note, we review the $\beta$-ensembles, describe the package interface, and illustrate its use through several numerical experiments motivated by applications to quantum chaos. Our numerical results include an analysis of $\beta$ as a continuous fitting parameter in spacing ratio statistics, an examination of the numerical evidence for the conjectured $k$-spacing ratio distributions, and a study of the spectral form factor for general values of $\beta$.