Papers

We derive a new beta-type basic hypergeometric integral identity from the equality of supersymmetric partition functions on $\mathbb{RP}^{2}\times\mathbb{S}^{1}$. Unlike previously known identities obtained from lens-space partition functions, this integral does not appear to arise as a degeneration of the lens elliptic beta integral. Our result enriches the collection of basic hypergeometric beta integrals arising from supersymmetric dualities and has applications to supersymmetric gauge theories, integrable models, and the theory of special functions.

Endo-Pajitnov manifolds are compact non-Kähler manifolds which generalize the Inoue surfaces $S_M$ to higher dimensions. In this paper, we compute their Dolbeault cohomology, show that they satisfy the Hodge decomposition at the level of dimensions and prove that they are rigid with respect to small deformations.

Organizations increasingly use large language models and agentic AI systems to generate probabilistic assessments and candidate actions in high-consequence settings. This creates a managerial problem distinct from prediction: how should organizations allocate decision authority to AI-generated recommendations as evidence quality, uncertainty, and organizational objectives evolve over time? Existing AI governance frameworks emphasize transparency, documentation, oversight, and regulatory compliance, but provide limited quantitative guidance for dynamically allocating decision authority under uncertainty. To address this challenge, we formulate adaptive AI delegation as a Governance-Aware Partially Observable Markov Decision Process (POMDP) in which Bayesian inference estimates the informational state and sequential optimization determines delegated AI authority. The paper also develops a quantitative validation and benchmarking framework for governance policies. Synthetic stress tests, reported LLM-confidence robustness, forecast-accuracy validation, governance-appetite sensitivity, and fragile-AI early-warning experiments evaluate whether the proposed policy exhibits graceful degradation, robustness to confidence-only perturbations, adaptive delegation under improving evidence quality, and interpretable calibration of institutional conservatism. The Governance-Aware POMDP is further benchmarked against five representative governance strategies operating under identical Bayesian beliefs, information, and governance objectives. The results show that while specialized heuristics perform well in stationary settings, sequential Bayesian governance provides the strongest general-purpose governance policy across heterogeneous AI-quality regimes by adaptively allocating organizational decision authority under uncertainty.

We prove a sufficient condition for the absolute convergence of Mayer cluster expansions of log-partition and correlation functions applicable to lattice gases with possibly complex-valued multi-body interactions. Not only are several classical results subsumed but a partition scheme for spanning hypergraphs also makes our methods well-suited for treating stronger multi-body interactions, including higher-order hard-core repulsion in the context of hypergraph independence polynomials. Furthermore, our approach is easily combined with the Gruber–Kunz condition to produce extended convergence results for the polymer expansion of lattice gases, rivalling those obtained not too long ago by Nguyen and Fernández (2024).

We give an explicit, cochain-level algebraic model for the pronilpotent completion of a group with finitely generated first cohomology. To each binomial $\cup_1$-dga $(A,d_A)$ over $R=\mathbb{Z}$ or $\mathbb{F}_p$ ($p$ prime) – a differential graded algebra endowed with a Steenrod $\cup_1$-product and a compatible binomial operation – we associate a pronilpotent group $G(A)$ that depends only on the 1-quasi-isomorphism type of $A$, provided $H^0(A)=R$ and $H^1(A)$ is a finitely generated free $R$-module. This group arises functorially from the 1-minimal model of $A$, which is unique up to isomorphism. When $A=C^*(X;R)$ is the cochain algebra of a connected CW-complex $X$ with $H^1(X;R)$ finitely generated, the group $G(A)$ recovers the Bousfield–Kan $R$-completion of $\pi_1(X)$ when $R=\mathbb{F}_p$, and its pro-torsion-free-nilpotent completion when $R=\mathbb{Z}$. Moreover, the group $G(A)$ comes equipped with a natural inverse system $\{G_n(A)\}_{n\ge 1}$ whose structure maps $G_{n+1}(A)\to G_n(A)$ are surjective. If $A=C^*(X;R)$, then $G_n(A)$ is the quotient of $\pi_1(X)$ by the $(n+1)$th term of the fastest descending central series whose successive quotients are free $R$-modules. We give a purely algebraic necessary and sufficient criterion that, given an isomorphism $G_n(A)\cong G_n(B)$, determines whether $G_{n+1}(A)\cong G_{n+1}(B)$, and we illustrate the use of this criterion with examples distinguishing spaces with isomorphic cohomology rings.

We study standard spherical growth rates of right-angled Coxeter groups through the clique polynomial of the defining graph. We prove that every even degree at least four occurs as the degree of a strongly primitive Salem growth rate: for each $d\geq2$, there are infinitely many connected $K_{2d+1}$-free defining graphs whose full reciprocal-radius polynomial is an irreducible Salem polynomial of degree $2d$. We also establish independence-polynomial obstructions for prescribed Salem polynomials, including a sharp first-coefficient bound $a_1\leq-5$. Finally, an elementary golden-ratio growth gap excludes Lehmer's number from the standard spherical growth rates of right-angled Coxeter groups.

Let $(X,d,\mu)$ be a metric measure space whose measure $\mu$ is uniformly locally doubling and which supports a local weak $(1,p)$-Poincaré inequality for some $p\in[1,\infty)$. Given $\theta\in(0,p)$ and an Ahlfors–David codimension-$\theta$ regular subset $E\subset X$, we provide a complete intrinsic description of the trace-space of the Besov space $B^{\theta/p}_{p,1}(X)$ to $E$. More precisely, we show that the trace operator is well defined and bounded from $B^{\theta/p}_{p,1}(X)$ to $L_p(E,\mathcal H_\theta\lfloor_E)$. We also show that the upper estimate in the Ahlfors–David codimension-$\theta$ regularity condition is necessary for such boundedness under the local weak Poincaré inequality. Conversely, assuming that $E$ is Ahlfors–David codimension-$\theta$ regular, we construct a bounded nonlinear extension operator from $L_p(E,\mathcal H_\theta\lfloor_E)$ to $B^{\theta/p}_{p,1}(X)$. Thus the trace-space is identified intrinsically with $L_p(E,\mathcal H_\theta\lfloor_E)$. This extends the classical limiting case of the trace theorem obtained by Burenkov and Gol'dman. Finally, we apply the general theory to $K$-regular trees, $K\ge 1$, for which we additionally derive a necessary and sufficient criterion for the existence of traces.

Let $N_{\alpha}(d)$ denote the maximum number of equiangular lines in $\mathbb{R}^d$ with common angle $\arccos(\alpha)$. Balla conjectured that, if the spectral radius order $\kappa_{\frac{1-\alpha}{2\alpha}}$ of $\frac{1-\alpha}{2\alpha}$ is finite, then $$N_{\alpha}(d)\leq \max\left\{\frac{(1-\alpha^2)(1-2\alpha^2)}{2\alpha^4},\left\lfloor\frac{\kappa_{\frac{1-\alpha}{2\alpha}}(d-1)}{\kappa_{\frac{1-\alpha}{2\alpha}}-1}\right\rfloor\right\},$$ for any $d\geq 1$. The conjecture has previously been verified only for $\alpha\in\left\{\frac{1}{3},\frac{1}{5},\frac{1}{1+2\sqrt{2}}\right\}$. In this paper, we prove that this conjecture holds for $\alpha=\frac{1}{1+2\sqrt{3}}$ and $\alpha=\sqrt{5}-2$. On the other hand, we show that Balla's conjecture fails for infinitely many $\alpha$.

This dissertation develops a framework for embedding arbitrary connected graphs isometrically into Cayley graphs of abelian groups, with applications to harmonic analysis on networks. It addresses representing irregular graph-structured data within highly symmetric algebraic hosts, on which classical Fourier theory applies verbatim rather than by analogy. The theoretical core is twofold. First, we introduce edge relations phi, Phi, and Psi that detect metric parallelism, a strict generalization of the Djokovic-Winkler relation beyond bipartite and partial-cube structures, with a transitive prune operation converting them into candidate same-generator edge partitions. Second, we prove the Cocycle/Quotient Labeling Theorem: any edge partition induces a most-generic consistent vertex labeling as a GF(2) quotient of dimension k = t - rank(A), where A is the cycle-class parity matrix; the labeling can fail only by shortcuts, never by stretching. With a shortcut-repair loop terminating in the isometric spanning-tree embedding, this gives a universal algorithm: every connected graph G embeds isometrically into a Cayley graph of (Z_2)^k with k <= n-1, verified exhaustively on all 995 connected graphs of at most seven vertices. A bounds theory follows: k >= max(diam(G), ceil(log2 n)); stars satisfy k_min(K_{1,q}) = ceil(log2 q) + 1, exponentially below the naive dimension; odd cycles require k = n-1. We then generalize the quotient machinery from GF(2) to Z via the Smith Normal Form, giving embeddings into products of cyclic groups. The primary application is harmonic analysis: these embeddings ground Fourier analysis, convolution, and wavelet transforms on graph signals, preserving translation-modulation duality, convolution theorems, and Plancherel identities that matrix-based graph signal processing lacks. We name this framework Group-Embedding-based Graph Signal Processing (GE-GSP).

The dipole and its descendants model subsurface scattering with a radial reflectance profile fitted to a flat, semi-infinite slab. This assumption introduces a systematic geometry error on thin and curved objects. We isolate the effect by comparing the dipole with the finite-slab multipole under the same diffusion model and boundary condition. In slab geometry the diffuse-albedo error has a material-independent leading rate, $C e^{-2\tau}$ with $\tau=T/\ell_d$, while the prefactor remains material dependent; the same image series gives the transmitted flux, whose leading decay is $e^{-\tau}$. We give the closed-form albedo and transmittance, relate the exponents to killed random walks, and extend the interpretation to spatially varying media through optical distance. A brute-force volumetric path tracer fits a reflectance-deficit rate of 1.99 and a transmittance rate of 0.99, matching the round-trip and single-pass predictions. The resulting thickness predictor is a useful thin-feature heuristic, but stress tests show that curvature and illumination can dominate away from the slab setting. For the remaining geometry-dependent term we solve the screened-Poisson diffusion problem directly inside the signed-distance domain with Walk on Spheres, without an interior mesh or a tangent half-space approximation; the estimator matches closed-form tests to 0.75%. Against a four-case path-traced benchmark it improves the back-lit, thickness-governed case but not every front-lit or curved case, showing that the method reduces geometry error within diffusion and does not replace radiative transport.

For integers $1\le r\le n+1$, let $N(n,r)$ denote the least number of chains in the Boolean lattice $B_n=2^{[n]}$ that cover every strict $r$-term chain. The case $r=1$ is the classical chain-decomposition problem and is generalizing Dilworth's theorem and Sperner's theorem. We study two complementary regimes. First, when $r>1$ is fixed and $n\to\infty$. Let $$M(n,r):= \max_{\substack{ a_0+\cdots+a_r=n a_0,a_r\ge 0,\ a_i\ge 1\ (1\le i\le r-1) }} \binom{n}{a_0,\ldots,a_r}.$$ We prove that lower and upper bounds which differ only by a logarithmic factor: \[ M(n,r)\le N(n,r)\le \left(\frac r2+o(1)\right)\log n\cdot M(n,r). \] Second, we consider the near-maximal regime $N(n,n-t)$, where $t>0$ is fixed. We prove that \[ N(n,n-t)= (\gamma_{t-1}+o(1))n!, \] where $\gamma_d$ is the limit of the density of a minimum size subset of the hypercube $Q_n$ that meets all $d$-dimensional subcube.

We consider self-maps of a sphere in the critical Sobolev space with a given Brouwer degree. Our main result is that the (directed) distance between maps of different degrees is equal to an explicit constant times the difference in degrees. In the case of the 2-sphere this resolves an open problem by Brezis.

We formulate a notion of oriented polytope, including Street's oriented simplices and Gray's oriented cubes, and use this to prove an oriented version of the Street–Roberts conjecture, presenting $(\infty,\infty)$-categories as sheaves on suitable families of oriented polytopes, generalizing work of Campion. This allows us to understand $(\infty, \infty)$-categories from a geometric perspective, as directed analogues of homotopy types. These familes of oriented polytopes induce basic operations in higher category theory: for instance, the join, Gray tensor, and bicone arise from the geometry of the orientals, cubes, and orthoplexes, respectively. We study the interaction of these operations and derive some geometric formulae, generalizing work of Ara–Maltsiniotis, Verity, and others.

Let $V \subset \mathbb{A}^2(\mathbb{C})$ be an algebraic curve such that $\mathrm{deg} X \neq \mathrm{deg} Y$, where $X, Y$ denote the coordinate functions on $\mathbb{A}^2(\mathbb{C})$ restricted to $V$. We prove there exists an effectively computable constant $c$, that depends linearly on the height of $V$, such that $\max \{h(x), h(y)\} \leq c$ for every $(x, y) \in V$ with $x$ and $y$ both CM $j$-invariants. This establishes, for such curves, an effective version of the André–Oort conjecture that has a better dependence on the height of $V$ than previous effective results.

In this work, we propose a multi-level machine learning framework for solving inverse scattering problems with multi-frequency data. The multi-level neural network is built along the frequency axis of the scattering problem, wherein at each fixed frequency, a new level of network is added to the existing architecture to update the reconstruction. By marching through the frequency levels, the proposed multi-level computational framework is able to obtain higher-order Fourier modes of the imaging target as the depth of the neural network grows and higher-frequency data are used. Furthermore, the overall learning problem is decomposed into a sequence of simpler local tasks, each associated with a single frequency. This decomposition significantly reduces the complexity of the optimization problem and mitigates the risk of convergence to undesirable local minima, resulting in a robust and reliable training procedure for solving inverse scattering problems. We conduct various numerical experiments for the inverse source scattering problem and the inverse medium scattering problem to illustrate the effectiveness and robustness of the proposed machine learning framework. In addition, theoretical analysis in the neural tangent kernel regime shows that the proposed multi-level architecture progressively recovers the higher-order Fourier components of the imaging target.

Balance-oriented multi-warehouse inventory allocation is a recurring decision problem in large-scale e-commerce supply chains, in which a fixed replenishment quantity is distributed across warehouses to balance post-allocation inventory coverage while accounting for demand forecasts and heterogeneous allocation constraints. In practice, allocation requirements are often scenario-dependent and expressed in semi-structured or natural-language form rather than as ready-to-solve operations research (OR) formulations. We propose an OR-guided Large Language Model (LLM) for Allocation (ORLA) that uses solver feedback to generate, verify, and select OR formulations. ORLA integrates automatic "Problem-Model-Code (PMC)" generation, learning-based formulation selection, and feasibility restoration. We develop three complementary mixed-integer programming formulation families based on deviation minimization, soft band compliance, and knapsack-inspired allocation, together with solver-ready mixed-integer linear programming reformulations, modular constraint extensions, and a penalty-based relaxation mechanism for infeasible cases. The LLM component generates candidate formulations and executable solver code from textual or semi-structured specifications, while the solver provides verification signals for executability, feasibility, and solution quality. To address instance heterogeneity, ORLA estimates the expected quality of candidate formulations, selects promising candidates, and combines their outputs through score-aware aggregation. Experimental results on 29 production evaluation batches from JD.com show that the best single OR formulation improves allocation accuracy by 3.4 percentage points over the incumbent approach, while the full ORLA framework achieves a 4.5 percentage-point overall improvement and improves allocation accuracy in 26 of the 29 evaluation batches.

We construct a Hermitian covariance form on hyperbolic components in parameter spaces of complex Hénon maps, associated to the full complex unstable derivative cocycle. The form measures infinitesimal variations in the marked complex unstable multiplier spectrum. Using a recent multiplier rigidity theorem by Cantat–Dujardin, we prove that it induces a distance on every hyperbolic component. Motivated by Sullivan's dictionary and by the thermodynamic interpretation of the Weil–Petersson metric, our result gives a first higher-dimensional holomorphic-dynamical counterpart of pressure-type metric structures. On the other hand, the construction differs from the one-dimensional theory in an essential way: it replaces the real geometric potential measuring unstable expansion by the full complex unstable derivative cocycle. This also suggests a complex derivative cocycle counterpart to pressure-type metric structures in Teichmüller theory and Anosov representation theory.

Let $f_3(N)$ be the least integer such that every set $A\subseteq\{1,\ldots,N\}$ of size at least $f_3(N)$ contains distinct elements $a,b,c\in A$ such that $a+b\in A$, $a+c\in A$, and $b+c\in A$. We prove that $f_3(N)\le 5N/8+O(1)$. Together with the standard construction $[N/8,N/4]\cup[N/2,N]$, this gives $f_3(N)=5N/8+O(1)$, resolving Erdős Problem 865. The proof is self-contained. An earlier conditional version of the reduction has also been formalized in Lean 4/Mathlib with no sorries and no added axioms.

Let $X$ be a $PD_4$-complex and let $\pi=\pi_1(X)$. If $\pi$ is torsion-free and $\pi_2(X)$ is a finitely generated projective $\mathbb{Z}[\pi]$-module then either $\pi$ is free or $\pi$ is $FP$ and $c.d.\pi=4$. If, moreover, $H^3(\pi;\mathbb{Z}[\pi])=0$ then $\pi$ is a free product of $PD_4$-groups and a free group.

In this paper, we study the initial-boundary value problem of the 1D compressible Navier–Stokes/Cahn–Hilliard system with vacuum. We establish the global existence and uniqueness of strong solutions to this initial-boundary value problem. No any initial compatibility conditions are required via time weighted techniques, which leads to a loss of regularity near the initial time. Therefore, the uniqueness of solutions obtained in this paper is even more challenging. To address this issue, we establish refined growth estimates and singular-in-time weighted energy estimates that induce a Gronwall-type structure, which ultimately allows us to close the uniqueness proof in Eulerian coordinates without passing to Lagrangian coordinates.

In this paper, we study the long-time behaviour of the two-dimensional stochastic damped Euler equation on the torus driven by bounded random forcing. Unlike stochastic Navier-Stokes or fractionally dissipative Euler equations, the model possesses no viscous regularisation, so the classical parabolic smoothing is unavailable. We prove that when the damping coefficient is sufficiently large, the associated Markov semigroup admits a unique invariant measure and converges exponentially fast to equilibrium. The key ingredient is the establishment of a global-in-time $W^{1,\infty}$ estimate for the vorticity. This estimate yields a compact absorbing mechanism in $C(\mathbb{T}^2)$, which enables us to establish the uniqueness of the invariant measure and exponential mixing. To the best of our knowledge, this is the first exponential mixing result for a genuinely inviscid stochastic Euler-type equation. Our approach demonstrates that sufficiently strong linear damping can effectively replace the compactness mechanism usually provided by viscosity and is expected to be applicable to other inviscid or weakly dissipative stochastic partial differential equations driven by bounded random forcing.

Virtual sensing – recovering full-field structural response from sparse sensor measurements – is a fundamental challenge in structural health monitoring (SHM). Fracture assessment, fatigue evaluation and remaining useful life (RUL) prognosis all take the applied load as their input, and that load frequently acts on surfaces that cannot be instrumented. An improved load estimate is therefore of direct practical value for SHM and for prognostics and health management (PHM). The mode-based approach reconstructs the displacement field accurately from a small number of sensors and has been applied successfully for that purpose. The load inferred from this reconstructed displacement is far less reliable, because the residual displacement error is amplified when it is mapped back through the dynamic stiffness. This study therefore formulates virtual sensing as a PDE-constrained optimization (PDE-CO) problem in which the elastodynamic equation is an equality constraint and the applied load is the optimization variable, so that displacement and load are estimated together rather than in sequence. To manage the computational cost, the framework separates offline finite element model preparation from online reconstruction. The assembled matrices are reused during online optimization, and the forward and gradient evaluations can use GPU-based parallel computation. The numerical comparisons support improved noise robustness of PDE-CO displacement reconstruction relative to the mode-based approach within the tested settings. These results provide a theoretical and computational foundation for the development of virtual sensing for SHM and PHM.

This paper addresses the existence of nontrivial solutions to a class of mixed local-nonlocal problems involving a mixed interpolated Hardy potential. We first establish a concentration-compactness principle for mixed local and nonlocal operators. This result is combined with Ricceri's variational principle to obtain an existence result for quasilinear elliptic problems under different growth assumptions on the nonlinearity. Furthermore, we apply the classical mountain pass theorem to obtain a second existence result in the superlinear case.

The growing demands for higher throughput and cost-efficient wireless communications drive the need for receivers that are both simple to deploy and robust to hardware impairments and nonlinear environments. While classical model-based receivers and recently proposed deep neural network ( DNN) architectures provide complementary benefits, they either rely on simplified linear Gaussian assumptions, require considerable computational resources, or are tailored for a given setting and modulation. In this work, we propose a compact and modular DNN augmentation that universally refines the soft outputs of existing receivers (model-based or data-driven), addressing two distinct operating regimes: structurally incomplete soft information arising from reduced-complexity detectors, and degraded soft outputs caused by hardware impairments and synchronization errors. A key property of the proposed framework is its task-agnostic nature: operating without any knowledge of the specific source of unreliability, it produces well-calibrated log-likelihood ratios (LLRs) suitable for channel decoding. Our design leverages an element-wise scaled convolutional neural network tailored to perform learned interference cancellation across users and neighboring subcarriers, combined with a training algorithm that encourages accurate LLR s for soft channel decoding. Numerical results demonstrate that the proposed augmentation consistently improves diverse receiver algorithms in challenging channel conditions while incurring minimal overhead.

We study a class of mixed-integer bilevel stochastic programs in which the leader commits to a first-stage decision before uncertainty is realized, and the follower solves a mixed-integer optimization problem for each revealed scenario. Due to the hierarchical structure and discrete variables at both levels, these problems are inherently $\Sigma_2^p$-hard, rendering standard single-level reformulations computationally intractable. To address this challenge, we develop an exact algorithm that combines deterministic value-function reformulations with scenario-wise decomposition. Specifically, we propose an extended single-level reformulation and a corresponding relaxation that enable scenario decomposition. We then introduce a stochastic subgradient cutting-plane scheme that dynamically generates follower optimality cuts and updates the Lagrange multipliers. We prove that, under boundedness assumptions, our algorithm converges in finite time to a global optimum and provides valid upper and lower bounds throughout its execution.

We study long-time optimal control of control-affine semiautonomous neural ordinary differential equations (SA-NODEs) with $\ell^1$-regularized controls. Three results are established. First, optimal state-control pairs satisfy an exponential turnpike property: they remain exponentially close to a stationary optimal pair for most of the time horizon, with decay rate and prefactor independent of the horizon length $T$. Second, $\ell^1$ penalisation induces one-sided temporal sparsity: optimal controls are active at full amplitude on an initial arc $[0,T^*]$ and vanish identically on $(T^*,T)$, where $T^*$ is independent of $T$ for $T$ large. Third, an integral turnpike estimate shows the time-averaged deviation from the stationary pair is bounded uniformly in $T$. The proofs combine dissipativity inequalities, uniform adjoint bounds via the Pontryagin optimality system, and a time-rescaling argument adapted to the semiautonomous architecture. Numerical experiments on a Duffing oscillator and a damped pendulum confirm the three-phase turnpike profile and the one-sided sparsity structure, and demonstrate a $30\times$ parameter reduction over vanilla NODEs with no loss of stabilization performance.

In this paper, we introduce controlled semiautonomous neural ordinary differential equations (controlled SA-NODEs) for the approximation and learning of nonlinear controlled dynamical systems. The proposed framework extends semiautonomous neural ODEs to control-affine systems while preserving reduced parameter complexity through time-independent trainable coefficients. We establish a universal approximation theorem showing that controlled SA-NODEs approximate trajectories of nonlinear controlled systems uniformly on compact sets of initial conditions and admissible controls. Under additional Sobolev and Barron regularity assumptions, we derive quantitative approximation estimates of order $\mathcal{O}(P^{-1/2}+Q^{-1/2})$. We further prove that approximate controllability properties of the original nonlinear system are preserved under the controlled SA-NODE approximation. Numerical experiments on controlled pendulum and Duffing oscillator systems demonstrate that the proposed framework achieves accurate trajectory reconstruction and controllability performance with significantly fewer trainable parameters than classical neural ODE architectures.

Cycle rank is a depth parameter for digraphs introduced by Eggan in 1963. Gruber (DMTCS 2012) and Giannopoulou, Hunter, and Thilikos (DAM 2012) asked whether the problem of determining if a given digraph has cycle rank at most $w$ is fixed-parameter tractable parameterized by $w$. We provide such algorithms for semi-complete digraphs, and for digraphs of bounded directed clique-width. Specifically, we show that given an $n$-vertex semi-complete digraph~$G$ and an integer $w$, one can in time $2^{\mathcal{O}((w+1)4^{w})} n^2$ determine whether $G$ has cycle rank at most~$w$. The proof is reduced to the case of bounded directed clique-width, and we then show that given an $n$-vertex digraph $G$ with a directed clique-width $k$-expression and an integer $w$, one can in time $2^{\mathcal{O}((w+1)4^k)}n$ determine whether $G$ has cycle rank at most $w$. Additionally, we consider the Minimum Feedback Arc Set problem on semi-complete digraphs, and show that it can be solved in time $n^{\mathcal{O}(w)}$, when a cycle rank decomposition of depth $w\ge 1$ is given.

This paper investigates the capacity of multipleinput multiple-output (MIMO) optical wireless communication (OWC) channels under a total average-power constraint. Since different nonnegative input vectors can be mapped to the same image vector and thus induce the same output distribution, we formulate a nonnegative basis pursuit (NN-BP) problem to identify the minimum-l1-norm input vector for each image vector. Based on the NN-BP characterization, we derive an equivalent expression for the channel capacity in terms of the image-vector distribution. We then establish computable lower and upper capacity bounds for both nT >= nR and nT < nR cases, and prove that the proposed bounds are asymptotically tight in the high signal-to-noise ratio (SNR) regime. Numerical results for indoor and outdoor OWC scenarios demonstrate that the proposed bounds improve upon existing ones and close the constant gap in the high-SNR regime.

We study the double affine Hecke algebra (DAHA) of type $(C_n^\vee,C_n)$ from the perspective of deformation theory. First, we provide a zeros-and-residues realization of this algebra, extending the construction of Ginzburg, Kapranov, and Vasserot to the non-reduced affine root system setting. Specializing the parameters of the DAHA to the base point gives the crossed product of a quantum torus algebra with the finite Weyl group of type $C_n$. We then show that for all $n$, the completed DAHA is the formal universal deformation of this crossed product algebra, extending Oblomkov's result for $n=1$. Our proof explicitly identifies the completed DAHA with the undeformed crossed product algebra equipped with a formal star product.