Papers

Classical Floquet-Bloch theory decomposes abelian periodic problems over the character torus of the lattice. For nonabelian nilpotent lattices, the non-type I obstruction rules out a comparable parametrization of the full unitary dual. We do not attempt to remove this obstruction. Instead, we construct an exact Bloch-type replacement on the representation-theoretic part of the theory which is visible from rational Kirillov data and from finite-dimensional rational fibers. Let $\Gamma$ be a torsion-free finitely generated nilpotent group and let $G$ be its Malcev completion. For an irreducible unitary representation $\pi_l$ of $G$ attached to a rational Kirillov parameter $l\in\mathfrak{g}_{\mathbb Q}^{*}$, we prove an exact restriction theorem for $\pi_l|_\Gamma$. The branching is first described by induced representations attached to rational polarizations. On the rational odd locus relevant to finite-dimensional representations, it further decomposes into finite-dimensional irreducible representations of $\Gamma$. On these finite-dimensional rational fibers we construct a positive finitely additive Plancherel measure. It gives Fourier inversion and normalized trace identities for nilpotent lattices, recovering Pytlik's formula in the discrete Heisenberg case.

We consider three discrete-time models for species evolution. In all three models, at each time step $n$, with probability $p$, a species is born with an independent $\text{Uniform}[0,1]$ fitness value and, with probability $1-p$, a species is killed. The mechanism for selecting which species to kill when a death occurs distinguishes the three models: in the first model, the least fit species is always killed; in the second model, with probability $r$, the least fit species is killed and, with probability $1-r$, a species chosen uniformly from the population is killed; in the third model, with probability $r$, the least fit species is killed and, with probability $1-r$, the species with the largest fitness less than an independent $\text{Uniform}[0,1]$ outcome is killed. We establish asymptotic results as $n \to \infty$ for the three models. These results demonstrate that small changes to the death mechanism of the model can lead to vastly different asymptotic behaviour. To prove our results we develop a novel approach that relies on coupling arguments and mean-field limits.

We study obstructions to the existence of separated good moduli spaces for open substacks of the moduli stack $\mathcal{M}_{g,n}^r$ of $A_r$-stable curves. Our approach is based on an analysis of families of curves over $\Theta_R$ and $\overline{\text{ST}}_R$, building on prior work on the local geometry of $\mathcal{M}_{g,n}^r$. We prove that $\mathcal{M}_{g,n}^r$ is neither $\Theta$- nor $\textsf{S}$-complete. We then construct an open substack $\mathcal{U}_{g,n}^r \subset \mathcal{M}_{g,n}^r$ and show that the counterexamples identified in $\mathcal{M}_{g,n}^r$ do not occur within this substack. Moreover, we prove that $\mathcal{U}_{g,n}^r$ cannot be strictly contained in any other substack of $\mathcal{M}_{g,n}^r$ that admit a separated good moduli space. Furthermore, we show that the inclusion $\mathcal{U}_{g,n}^r \subset \mathcal{M}_{g,n}^r$ is both $\Theta$- and $\textsf{S}$-complete. These results will be used in a forthcoming paper to prove that $\mathcal{U}_{g,n}^r$ admits a separated, and indeed proper, good moduli space for $r \leq 5$.

Finite groups are rigid algebraic objects, whose Cayley graphs expose a rich network geometry through which group-theoretic structure can be measured, compared, and learned. In this paper, a dataset of $131{,}406$ Cayley graphs is constructed, covering all groups of order at most $767$ except order $512$, recording exact algebraic labels for group properties together with a broad collection of graph, cycle, distance, and spectral statistics. This census aims to provide novel benchmarks for studying how finite-group properties are reflected in Cayley graph observables. It also yields new enumerative contributions: alongside recovering known OEIS sequences for standard group classes, new sequences for monolithic groups and for groups generated by at most three, four, and five elements are contributed to the OEIS. The accompanying network analysis identifies several empirical regularities and formulates testable conjectures, including relationships involving square clustering, Cayley graph diameter, average graph disorder, and spectral eigengaps of nilpotent groups. Finally, a comparison between classical models, an MLP, and graph neural network architectures is performed for predicting algebraic group properties directly from Cayley graph data. The results show that engineered graph statistics are highly informative, while GNNs, especially GIN and in some fixed-order settings GCN, can recover substantial structural signal directly from the graph. Such that graph-aware architectures show phases of optimality on these group-theoretic graph representations.

We revisit the classical expansion of the planetary Hamiltonian for small eccentricities and mutual inclinations of the orbits, presenting a derivation based entirely on vector formalism. We demonstrate that the secular part of the disturbing function, as well as any other Fourier harmonic, can be expressed in terms of scalar products of vectors lying in the system's invariant plane. Furthermore, we show how to express any such term using the angular momentum and eccentricity vectors of the orbits.

Streaming contraction certificates, which determine in real time whether observed data is sufficient to certify a safe control action, face two structural challenges: the disturbance distribution shifts during operation, and the system consists of coupled subsystems whose joint model is unavailable. This paper addresses both. First, we develop a Wasserstein-robust certificate beta_cert(t,epsilon) = beta_hat(t) - rho(t)(1+2epsilon(t)), where epsilon(t) is estimated online from the empirical excess kurtosis of recent residuals, so the certificate degrades gracefully under distributional shift rather than failing catastrophically. Second, we prove that local certificates beta_A and beta_B, estimated independently from each subsystem's data, compose into a network-level guarantee via beta_net = (beta_A+beta_B)/2 - sqrt[(beta_A-beta_B)^2/4 + gamma^2] > 0 whenever gamma < sqrt(beta_Abeta_B), with no joint model required. On a five-node G5 benchmark under three noise regimes - Gaussian, heavy-tailed Laplace, and spike events - the Wasserstein certificate remains valid in 73% of spike-regime timesteps versus 33% for the Gaussian baseline (2.2x improvement), while the Gaussian certificate never authorizes deployment during the spike window. The compositional framework correctly identifies all three coupling regimes from local data alone, with gamma_warn = sqrt(beta_A*beta_B) is approximately 0.98, precisely predicting network-level contraction loss.

As wireless visual intelligence evolves from isolated task inference to ordered skill workflows, the communication bottleneck shifts from transmitting a single semantic representation to coordinating reusable skill states under channel constraints. Existing DeepJSCC and prompt-guided visual transmitters usually treat each task as an independent full-token transmission, with limited reuse of execution memory across semantic workflows. This is inefficient for workflows such as Detect, Segment, and Keypoint, where later stages often require only state-relevant semantic updates. To this end, we propose SkillComm, a skill-driven semantic communication framework that uses reusable skill states as shared context for workflow-aware token prioritization and memory-assisted token-grid reconstruction. A shared Skill-Book maps a high-level visual intent into a synchronized executable skill sequence at the transmitter and receiver. Conditioned on this workflow, adaptive token selection exploits cross-step memory to transmit only state-active tokens through joint source-channel coding, while the receiver reconstructs a task-ready token grid by combining decoded tokens with local historical memory. Experiments on the MS COCO 2017 validation set for the Detect-Segment-Keypoint workflow show that SkillComm reduces token transmission cost by 51.2% while retaining 99.4% upper-bound-normalized average precision at high SNR. These results demonstrate that reusable skill states enable selective semantic update delivery for future agentic and embodied visual intelligence.

The accurate prediction of brittle crack initiation, propagation, and complex topological evolution remains computationally prohibitive when utilizing traditional high-fidelity phase-field finite element methods. To overcome these computational bottlenecks, a physics-informed conditional Denoising Diffusion Probabilistic Model (DDPM) is proposed for the full-field spatiotemporal forecasting of fracture evolution across diverse loading regimes and energy decomposition methods. The generative architecture is conditioned on rolling historical damage states and explicitly derived kinematic proxies – phase-field velocity and gradient magnitude – ensuring temporal coherence without non-physical artifacts. The principal contribution is spatially-localized uncertainty quantification without modification to the trained model. Ensemble variance concentrates at crack branching junctions ($\sigma_\mathrm{max} = 0.222$ at Y-junction bifurcations; zero high-uncertainty pixels in four deterministic propagation cases), while the high-$\sigma$ tail identifies high-error predictions with 90% precision – an 18-fold improvement over random selection. One-step crack tip localization achieves sub-pixel accuracy (0.12 px mean error) across both held-out validation subsets (shear-star and tension-spect), confirming cross-regime generalization. In closed-loop autoregressive rollout over 50 steps, the DDPM maintains Dice = 0.929 $\pm$ 0.010 while a deterministic U-Net collapses to Dice = 0.423 under error accumulation, a 2.2$\times$ gap that establishes the value of stochastic re-sampling for long-horizon stability. Per-step inference requires approximately 3.6 s on an H100 GPU, approximately 28$\times$ faster than the FEM reference and 1{,}000$\times$ slower than a deterministic U-Net a cost that buys the stochastic diversity enabling uncertainty quantification.

We develop an enhanced in-context learning (ICL) framework to improve the performance of pilot-based beamforming in multi-user multiple-input single-output (MU-MISO) systems. The proposed scheme integrates the ICL-Transformer backbone with the pilot encoder-decoder network (EDN) and the beamformer EDN. A crucial feature of our ICL network is that it can handle multiple channel models without retraining, enabled by the construction of model-specific context datasets. To improve convergence and robustness, we introduce three key innovations: (a) a curriculum learning (CL) strategy that smoothly transitions from supervised LMMSE-labeled imitation to unsupervised sum-rate maximization, (b) a self-evolving mechanism that dynamically expands and refines the context datasets for all channel models during CL-based training, and (c) a mismatch-aware extension that incorporates several mismatches into the general ICL framework and bypasses explicit channel calibrations. Ablation studies validate the effectiveness of the in-context architecture and enhanced training strategies. Simulation results over diverse communication environments show that the proposed scheme is able to rapidly adapt to both seen and unseen channel models without gradient-based parameter updates, and can mitigate the mismatch issues via intelligent context constructions. Furthermore, our scheme consistently outperforms the existing beamforming schemes under pilot-based settings, including the WMMSE benchmark and the recent Transformer-based methods.

We study deterministic first-order minimization of a convex function without prior knowledge of the objective's growth, smoothness regime, or associated parameters. We develop anytime, parameter-free bundle-level methods that adapt simultaneously to these unknown properties and attain best-known oracle complexities. For nonsmooth Lipschitz objectives satisfying quadratic growth, the proposed bundle-level W-certificate method (BLW) achieves the optimal complexity without requiring the growth modulus or target accuracy as input. We then introduce an accelerated variant, A-BLW. Without knowing the Hölder smoothness parameters, the quadratic-growth modulus, or the target accuracy, A-BLW attains the optimal rates in the nonsmooth, weakly smooth, and smooth regimes. Central to both methods is an affine W-certificate, a condition based on the descent-slowness of an affine minorant that converts the geometry of a bundle model into an optimality-gap guarantee under quadratic growth. A stopping-time analysis further shows that the same A-BLW algorithm, without modification, achieves the corresponding best-known rates for general convex objectives and for objectives satisfying Hölder growth of order at least two. Numerical experiments illustrate the practical performance of the proposed methods.

A $4$-uniform $2$-cycle in a $4$-uniform hypergraph of length $t$ is a cyclic ordering of $2t$ vertices $v_1v_2\cdots v_{2t}v_1$ such that $v_{2i+1}v_{2i+2}v_{2i+3}v_{2i+4}$ are edges for $0\le i\le t-1$ while the addition is modulo $2t$. For every $\gamma>0$ and large $n$, we characterize the $n$-vertex $4$-uniform hypergraphs such that every triple of vertices is contained in at least $(1/3+\gamma)n$ edges and admits a Hamilton $2$-cycle. Up to the error term $\gamma n$, the assumption on the minimum codegree is best possible and verifies a conjecture of Garbe and Mycroft. As a consequence, this gives a polynomial-time algorithm that decides whether an $n$-vertex $4$-uniform hypergraph with minimum codegree $(1/3+\gamma)n$ contains a Hamilton $2$-cycle. This stands as a steep contrast to the graph case where such a hardness gap has size $o(n)$.

We consider a two-species reaction-diffusion system in one space dimension that is derived from an epidemiological model in a spatially periodic environment with two types of pathogens: the wild type and the mutant. The system is of a hybrid nature, partly cooperative and partly competitive, but neither of these entirely. As a result, the comparison principle does not hold. In the previous work, we studied the propagation properties of the solutions to the Cauchy problem for this system and showed, among other things, that the spreading speeds of the fronts to the right and to the left directions, denoted by $ c^*_R$ and $ c^*_L$, can be characterized by using certain principal eigenvalues, and studied the homogenization limit as the spatial period $L$ tends to $0$, and also discussed the long-time behavior of solutions behind the fronts. In the present paper we prove the existence of pulsating traveling waves in the right direction (resp. left direction) with speed $c$ for any $c\geq c^*_R$ (resp. $c\geq c^*_L$), where $c^*_R$ and $c^*_L$ denote the aforementioned spreading speeds in the right and left directions. We also prove that the leading edge of any traveling wave has the exponential decay rate that is anticipated from formal linear analysis, thus extending part of the results of Hamel 2008 to systems of equations. Finally, we present an example in which the two speeds $c^*_R$ and $c^*_L$ are different. This is done by considering a multi-scale singular limit problem. This result highlights a marked difference between our system and scalar KPP type equations.

The memory kernel of an open quantum system obeys Kramers–Kronig (KK) relations if and only if its Laplace transform is analytic in the upper half-plane – a property known as Hardy-space analyticity. Here we show that non-unitary exchange statistics, the defining property of paraparticles, intrinsically breaks Hardy-space analyticity. The metric $\eta$ that guarantees a real closed-system spectrum for these particles necessarily differs from the physical Born inner product ($\|\eta - I\|_F / \|I\|_F = 0.51$) – a mathematical consequence of the R-matrix's non-unitarity, not a parameter choice. This metric is a "shadow metric": Schur's lemma forces it to commute with every bilinear observable, making the distortion physically invisible in the closed system. But when the paraparticle is coupled to a bath, any coupling operator that lies outside the symmetry algebra – that is, any interaction that sees the internal flavour structure – exposes the distortion. The memory kernel then develops upper-half-plane poles at coupling $g_c \approx 0.1$, breaking standard dispersion relations before the closed-system spectrum complexifies. Fermions and bosons, whose exchange is unitary ($\eta = I$ as an analytic fact of the canonical anticommutation algebra), are immune at any coupling, because there is no distortion to expose. The violation is intrinsic: it distinguishes non-unitary exchange statistics from ordinary particle statistics at the level of the memory kernel's analytic structure.

We completely determine the magnitude homology of tope graphs of real hyperplane arrangements. Their ranks can be described as the Hilbert functions of the Stanley–Reisner rings of certain simplicial complexes naturally associated with the arrangements. For Coxeter arrangements, this gives a computation of the magnitude homology of the Cayley graph of the corresponding Coxeter group. We also prove the homological reciprocity for central arrangements conjectured by Koizumi–Liu. The proof combines poset combinatorics, the Edelman–Walker theorem, and Alexander duality.

We investigate almost Ricci–Bourguignon solitons on three-dimensional contact metric manifolds. Under natural curvature assumptions, we show that the additional freedom introduced by allowing the soliton function to vary is rigidly constrained by the contact geometry. Using a local orthonormal \(\varphi\)-basis on the non-Sasakian region, we derive the full component form of the almost Ricci–Bourguignon soliton equation. As applications, we consider the cases where the potential vector field is pointwise collinear with, or orthogonal to, the Reeb vector field. For contact metric three-manifolds satisfying \(Q\xi=\sigma\xi\), we prove that a collinear potential field must vanish on the non-Sasakian region whenever \(\xi(\sigma)=0\). In the orthogonal case, when \(\sigma\) is constant and the manifold is non-Sasakian, the almost soliton function is forced to be constant; hence the soliton reduces to a Ricci–Bourguignon soliton. In fact, the metric is Einstein, and if the orthogonal potential field is not identically zero, then the metric is flat.

Let $V$ be a six-dimensional complex vector space and let $G=\operatorname{Gr}(2,\Lambda^2V^\vee)$ parametrize pencils of skew-symmetric $6\times6$ matrices. The group $\operatorname{PGL}(V)$ has $11$ orbits on $G$, classified by the Jordan-Kronecker canonical form. We compute the Chow class of every orbit closure in the Schubert basis of $\operatorname{CH}^*(G)$.

$\hat\delta$-rings are a globalization of $\delta$-rings. Motivated by their application to obstructing spectral lifts, we develop the theory of animated $\hat\delta$-rings. They extend the discrete notion in the same way that animated $\delta$-rings extend that of $\delta$-rings, but are not given by the process of animation. We also generalize the definition to schemes and extend the obstruction to this setting. We apply it to concrete examples such as number rings, closed subschemes of $\mathbb P^n$, and various group schemes.

Some results in ergodic theory on joinings and multiple mixing and the proof of Steve Kalikow's theorem on 3-fold mixing of 2-fold mixing rank one automorphisms are discussed.

The sample complexity is the minimum number of copies required to learn an accurate classical description of a quantum state. Bosonic and fermionic Gaussian quantum states are families of quantum states that play a key role in quantum science and technology, from quantum optics and many-body physics to quantum chemistry, quantum computing, and quantum information theory. Despite their importance, their sample complexity had not been fully determined. We settle this open problem and show that both bosonic and fermionic Gaussian states can be learned using a number of copies that scales quadratically in the number of modes, regardless of whether the state is pure or mixed, and independently of any energy bound on the state. We derive these results by using the representation theory of Gaussian unitaries and by putting forth a generalization of the random purification channel to this setting and beyond.

Many numerical algorithms for optimal control leverage an elimination of the state via the control-to-state map such as condensed approaches or preconditioned conjugate gradient methods for the optimality system. As such, the norm of the control-to-state map directly enters the convergence estimates for these methods, e.g., via the condition number of the associated linear system. In this work we show that using feedback transformations one may reformulate the optimal control problem to decrease the norm of the (feedbacked) control-to-state map, leading to a drastic improvement of the involved condition numbers. We illustrate the abstract approach for ordinary and partial differential equations such as parabolic, hyperbolic or elliptic equations. For each of these problem classes we provide a constructive method to improve solution operator norms via feedbacks. Further, we showcase the efficacy of the method by means of various numerical examples with elliptic, parabolic and hyperbolic partial differential equations.

We study twisted Weyl group multiple Dirichlet series attached to symmetrizable Kac-Moody root systems, using the Chinta-Gunnells method to construct their $p$-parts. Our main result is a decomposition theorem for functions invariant under the twisted Chinta-Gunnells action: under natural analytic hypotheses, such a function has a unique expansion in terms of shifted Chinta-Gunnells averages, indexed by the dominant weights in the highest weight module determined by the twisting parameter. In particular, we show that this decomposition holds for twisted multiple Dirichlet series over rational function fields. For finite root systems, these results were proved by Friedlander. We also show that the relevant Chinta-Gunnells averages admit analytic continuation to the interior of the complexified Tits cone. In the affine $\widetilde{A}_1$ case, we prove extra functional equations, not arising from the Weyl group, for the untwisted average and for averages twisted by fundamental weights. As a consequence, we obtain an explicit formula for the multiple Dirichlet series with square-free twisting parameters, and show that it also satisfies an extra functional equation.

In this article, we study a new class of collective motion for identical coupled Stuart-Landau oscillators on graphs. This model was previously known to converge to the synchronized state for a certain class of initial data. Here, we show that when the interaction matrix is circulant, there exists another class of attractors, in which the particles are uniformly distributed on a circle, reminiscent of the twisted states known for the Kuramoto model, but which rotate around the origin. However, contrary to the classical Kuramoto case, here the rotation is caused by the asymmetry of the graph structure, and not by the natural frequencies. We identify conditions for the existence and local stability of the rotational twisted states, both in the Stuart-Landau model and in the Kuramoto model. We also provide sufficient conditions for the existence and stability of the synchronized state, which can co-exist with the rotational twisted state in a certain parameter region. We provide a generalization of the class of interaction matrices able to generate rotational twisted states via leader-follower interaction matrices. We show that heterogeneous rotational twisted states can also exist for a system of heterogeneous oscillators, attracted to different individual target amplitudes. This study is accompanied by numerical simulations that illustrate the possible behaviors of the system, which also include metastable dynamics and a chimera state.

This article presents a novel, numerically viable algorithm for solving sparse robust optimal control problems in continuous time. We consider a constrained linear noisy system governed by an ordinary differential equation (ODE), with an $L^1$-type objective function in line with the sparse optimal control literature. The resulting optimal control problem is shown to admit a semi-infinite programming (SIP) formulation. Building upon this insight, we develop a new framework that enables the computation of exact solutions – to our knowledge, the first such achievement in the context of sparse optimal control. We demonstrate that a finite and computationally viable convex optimization problem can be solved to recover, in a lossless manner, both the optimal value and the corresponding optimizers of the original SIP, while also guaranteeing satisfaction of uncountably many constraints. We also show that the parameter-dependent noisy systems and the minimum attention problem fall into our framework and can be solved efficiently via our algorithm. The efficacy of our algorithm is illustrated through a benchmark numerical example.

In multi-objective combinatorial optimization, unsupported non-dominated points typically outnumber supported points and are often significantly more challenging to compute. Recent studies show that extreme supported non-dominated points provide high-quality representations of the non-dominated set for certain binary problems. We demonstrate that this observation does not generalize to capacitated network optimization problems: representation quality decreases with increasing arc capacities, whereas supported non-dominated points consistently provide high-quality representations with respect to several quality indicators. However, supported point sets may still be too large in practical applications, where only a small, fixed number of alternatives is typically desired. Selecting fixed-size representations from the non-dominated set requires its computationally expensive generation and thus diminishes the computational advantages that representations are intended to provide. We therefore suggest the (extreme) supported points as alternative candidate sets in subset selection problems. Our numerical results show that restricting the candidate set to supported non-dominated points yields fixed-size representations of nearly the same quality as those selected from the complete non-dominated set. Overall, supported non-dominated points serve both as high-quality representations and as reasonable candidate sets for subset selection.

In earlier work [21], we posed a stability question for Serrin's overdetermined problem under Dirichlet perturbations and proved that the answer is negative in dimensions $n\ge3$. Here we resolve the question in the planar convex class and obtain a sharp quantitative theory without any a priori geometric nondegeneracy. Let $u_\Omega$ solve \[ -\Delta u_\Omega=1\ \text{in }\Omega,\qquad \partial_\nu u_\Omega=-\frac{|\Omega|}{P(\Omega)}\ \text{on }\partial\Omega, \qquad \int_{\partial\Omega}u_\Omega\,d\sigma=0, \] and set $O(\Omega):=\text{osc}_{\partial \Omega}u_\Omega$. We construct fixed-area annuli with $O(\Omega_k)\to0$ that remain far from every disk, showing that convexity is essential in dimension two. By contrast, if $\Omega_k\subset\mathbb R^2$ are convex, $|\Omega_k|=\pi$, and $O(\Omega_k)\to0$, then, up to translations, $\Omega_k$ converges in Hausdorff distance to the unit disk. Moreover, \[ R_\Omega-r_\Omega+\inf_{z\in\mathbb R^2}d_H(\Omega,B_1(z)) \le C\,O(\Omega) \] for all planar convex $\Omega$ with $|\Omega|=\pi$ and sufficiently small $O(\Omega)$, and the linear order is optimal. The proof combines a new mechanism excluding long-thin degeneration, the rough-domain Serrin rigidity theorem of Figalli–Zhang, new tangential-gradient and linear boundary-growth estimates, a boundary $P$-function estimate, and the reverse-Serrin identity of Magnanini–Molinarolo–Poggesi. We also study the weaker deficit \[ A(\Omega):=\frac1{P(\Omega)}\int_{\partial\Omega}u_\Omega,d\sigma-\min_{\partial\Omega}u_\Omega. \] In the planar convex class, $A(\Omega_k)\to0$ still forces convergence to a disk, and \[ R_\Omega-r_\Omega+\inf_z d_H(\Omega,B_1(z)) \le C A(\Omega)^{2/3} \] for $|\Omega|=\pi$ and sufficiently small $A(\Omega)$.

In this thesis we define $n$-duals of VB $n$-groupoids over Lie $n$-groupoids and study their properties. For $n = 0$ this returns the dual vector bundle construction, while for $n = 1$ this returns Pradines's construction of the dual of a VB groupoid over a Lie groupoid, which includes the cotangent symplectic groupoid of Coste, Dazord and Weinstein. For $n = 2$, we propose a new construction that shows that VB 2-duals exist for VB 2-groupoids and they are VB 2-groupoids themselves. Their canonical dual pairings are nondegenerate up to homotopy in the same sense as shifted symplectic structures. In particular, we can apply this construction to the tangent of a Lie 2-groupoid and obtain a cotangent VB 2-groupoid (the 2-cotangent) which is canonically 2-shifted symplectic. We apply this in two ways: First, to characterize 2-shifted symplectic structures on a Lie 2-groupoid as Morita equivalences between its tangent and 2-cotangent groupoid. Second, to compute the 2-cotangent of a Lie 1-groupoid and show it is symplectic Morita equivalent to the bar construction of the 1-cotangent. Along the way, we develop the theory of $n$-duals for simplicial vector spaces, which covers the case where the base is a point. In this case, $n$-duals always exist, as they are defined by a mapping space construction. By a reformulation of the Eilenberg-Zilber theorem in terms of mapping spaces, we obtain that the canonical $n$-dual pairing is nondegenerate up to homotopy for all $n$-types.

Let $(M,\omega,\mathcal{F}_1,\mathcal{F}_2)$ be a bi-Lagrangian manifold. For a distribution $\mathcal{D}\subseteq TM$, we characterize when the subbundle $N^{*t}\mathcal{D}:=\mathcal{D}\oplus N^*\mathcal{D}$ of the Whitney sum $W=TM\oplus T^*M$ is a Dirac structure for both the untwisted and $(H,\theta)$-twisted Courant brackets. Consequently, every bi-Lagrangian structure canonically determines a pair of transverse Dirac structures. the associated bi-Lagrangian connection, also called the Hess connection $\nabla$ induces a linear connection on $W$ with respect to which both Dirac subbundles are parallel, thereby establishing a natural link between bi-Lagrangian geometry and Courant Dirac geometry. We also construct induced bi-Lagrangian structures on $TM$, $T^*M$, and $W$, and study the relation between their affineness and that of the original structure. We also lift to these bundles the canonical action of the symplectomorphism group on bi-Lagrangian structures.

We give explicit quantum Latin squares of order $6$ with cardinalities $19$, $21$, $23$, $25$, $27$, $32$, and $35$, where vectors differing only by a global phase are identified. Cardinalities $19$ and $21$ arise from symmetric Schur products of columns of dephased Butson matrices. A parameterized direct-sum construction in $\C^6=\C^4\oplus\C^2$ yields cardinalities $23$ and $25$ by a controlled splitting of two pairs of rays. Cardinality $27$ is obtained from mixed Schur products of a $BH(6,6)$ matrix and a row-permuted copy. Two further mixed constructions give cardinalities $32$ and $35$: the first has exactly two three-element phase classes, while the second has exactly one two-element phase class. In every Butson case, Hadamard orthogonality and the ray count are certified by finite arithmetic with exponent matrices. Combined with previously known attainable values and the general impossibility of cardinality $7$, these constructions realize every order-six cardinality except $7$ and the single currently unresolved value $29$.

The aim of this paper is twofold. First, we establish a Riesz conjugate theorem for weighted harmonic Bergman spaces. More precisely, we prove that if $f=u+iv$ is a harmonic $K$-quasiregular mapping in $\mathbb{D}$ and the real part $u$ belongs to the weighted harmonic Bergman space $a_\alpha^p$, $0<p<\infty$, then the imaginary part $v$ also belongs to the same space, together with a quantitative norm estimate. Moreover, for $1<p<\infty$, the corresponding constant is shown to be independent of the weight parameter $\alpha$. Second, we establish Riesz–Fejér inequalities for weighted harmonic Bergman spaces for $1<p<\infty$. In the special case $p=2$, we further improve the corresponding constant by using the Hilbert space structure and orthogonality techniques. As applications of our main results, we establish Riesz conjugate theorems and Riesz–Fejér inequalities for the Möbius invariant spaces $Q(n,p,\alpha)$ introduced by Zhu [Illinois J. Math. 51 (2007), pp. 977–1002] and their harmonic counterparts $Q_h(n,p,\alpha)$ introduced by Sun, Liu, and Wang [Potential Anal. 65 (2026), Article no. 12].

We prove that all nondegenerate simplices have the canonical Ramsey property, thereby resolving a central open problem in canonical Euclidean Ramsey theory and providing a canonical counterpart to the celebrated simplex Ramsey theorem of Frankl and Rödl~[JAMS, 1990].