Papers

In this paper, we derive thresholds and fluctuations for arithmetic progressions with prescribed color patterns in sparse random colorings of $[n]:=\{1, 2, \ldots, n\}$, where each element of $[n]$ is colored independently according to a given probability vector. For any admissible ordered palette of colors, we determine the full multi-parameter threshold region for the appearance of a colorful arithmetic progression. The threshold is governed by two competing mechanisms: a global first-moment condition and a local color-availability condition, resulting in a polyhedral satisfiability region, with a piecewise-polyhedral threshold surface. In the satisfiability region we establish asymptotic normality for the number of colorful arithmetic progressions of a given length, with an explicit rate of convergence in Wasserstein distance. On the threshold surface, we identify three distinct asymptotic regimes: Poisson, compound Poisson with mixed Poisson jumps, and compound Poisson with uniform jumps, after an appropriate normalization. These results provide a complete description of the threshold and fluctuation behavior of general colored arithmetic progressions under sparse random colorings, in a unified framework that interpolates between classical uncolored/monochromatic progressions in binomial random subsets and multicolored, including rainbow, arithmetic progressions.

We present shape calculus techniques and a scalable automatic shape differentiation framework for multi-phase topology optimisation on unfitted discretisations defined by several level-set functions. First we establish general, exact shape calculus expressions in the discrete case for multi-phase systems by leveraging concepts from convex geometry. To complement this theoretical foundation, we introduce an open-source multi-phase automatic shape differentiation framework based on polytopal cutting. This computational framework is validated against both finite differences and our established exact expressions, matching the latter to near machine precision. Furthermore, the proposed automatic shape differentiation is scalable across distributed computing environments, demonstrating near ideal weak scaling up to 1.65 billion finite elements across 13,824 computer cores. We demonstrate our implementation by solving unfitted multi-phase topology optimisation problems for anisotropic diffusion, linear elasticity, and fluid–structure interaction. Together, these theoretical and computational contributions provide a robust and accessible foundation for advancing multi-phase topology optimisation using unfitted finite element methods. In particular, the methods enable the solution of topology optimisation problems involving multi-phase and multi-physics systems with non-trivial boundary conditions. The open-source software is available at https://github.com/zjwegert/GridapTopOpt.jl.

The arithmetic graph $B_n$ joins distinct $a,b\in\N$ when $\max(a,b)/\gcd(a,b)\le n$. We prove $\chi(B_{205})=206$, disproving the conjecture that $\chi(B_n)=n$ for every $n$, equivalently the Rainbow Cascades Conjecture. The proof reduces an arbitrary tiling by the arithmetic exponent tile to a periodic tiling, then to two families of finite quotients, which are excluded using exact computations. We also construct a $208$-coloring using $\Z_{104}\times\Z_2$ and prove $212\le\chi(B_{211})\le213$. The lower bound at $211$ follows from prime-cardinality tiling rigidity and the published nonexistence of a cyclic logarithm of length $211$; we give a direct proof of the required rigidity statement. Finally, we record the equivalence with the List Cascade Coloring Conjecture and the conjecture on ironic decorations, and deduce finite graph counterexamples to both. The least $n$ with $\chi(B_n)>n$ is either $195$ or $205$; determining which remains open.

We give a short construction refuting the Köthe Conjecture. The construction is based on the point module approach to the problem, introduced by the first-named and second-named authors together with Easton and Ford.

The singular value decomposition has been established for matrices, Hilbert–Schmidt operators, trace-class operator, compact operators, and bounded operators, but surprisingly not for unbounded operators. Unfortunately, most interesting operators in applied math are unbounded, as any operators involving some form of derivatives — gradient, exterior derivatives, Laplacians, Fourier and other transforms of derivatives, Hamiltonians, etc. — are likely unbounded. In this article, we fill in this last missing piece by establishing the existence of singular value decompositions for unbounded operators in three natural forms: multiplication-operator, direct-integral, and operator-valued-measure. We show it inherits classical properties of finite-dimensional singular value decomposition including approximation results, relationships with fundamental subspaces, and the Moore–Penrose inverse. This discovery opens the door to the singular value decompositions of a myriad of well-known unbounded operators in mathematics, physics, statistics, and finnance — gradients on Euclidean spaces and manifolds, Petrov–Galerkin method, finite-difference operators, Hilbert–Schmidt operators, Hilbert complexes, supersymmetric quantum mechanics, Sturm–Liouville theory, nonparametric density estimation, and the Black–Scholes equation. The resulting decompositions reveal a number of novel insights, among many others: bosonic and fermionic states in supersymmetric quantum mechanics arise as left and right singular vectors of generalized ladder operators; the Riesz transform appears as the left singular operator of the Euclidean gradient; and the Hodge decomposition follows directly from the singular value decompositions of the exterior derivatives.

In this paper, we show that partial positivity is not preserved by marginalization in both the real and complex settings. We also give a sufficient condition for the preservation of partial positivity under marginalization.

We prove the non-uniqueness of weak solutions to the three-dimensional hyper-viscous compressible Oldroyd-type model with general pressure laws and Newtonian viscous. Using an intermittent convex integration scheme, we construct non-unique weak solutions in $L_t^\gamma W_x^{s,p}$ with $(s,p,\gamma)$ lies in the supercritical regime relative to the Ladyzhenskaya-Prodi-Serrin criteria. In addition, the Newtonian viscosity we considered contains the fractional viscosity $(-\Delta)^\alpha$, where the viscous exponent $\alpha$ can be larger than the Lions exponent $5/4$. In the classical viscous case $\alpha=1$, our construction also yields sharp non-uniqueness in $L_t^pL_x^\infty$ for $1<p<2$. Furthermore, we obtain two singular-limit results. First, we prove the strong vanishing viscosity limit for any weak solutions in $H_{t,x}^{\widetilde{\beta}}$ to the compressible fundamental elastodynamic system. Second, $H_{t,x}^{\widetilde{\beta}}$ weak solutions to the incompressible Oldroyd-type model can be obtain as a low Mach number limit (incompressible limit) of a sequence of weak solutions to the compressible Oldroyd-type model. The main ingredient of our result is a new cancellation mechanism that overcomes the difficulty caused by the relative rigidity of the pressure. To the best of our knowledge, this is the first non-uniqueness results of weak solutions to the hyper-viscous compressible hydrodynamics equations.

The Komlós conjecture is a classic problem in discrepancy theory; it asks whether an absolute constant $K$ exists such that given any $n$ vectors $a_1,\ldots,a_n$ inside the $m$-dimensional Euclidean ball, regardless of how large $m,n$ are, there is always a selection of signs $\varepsilon_1,\ldots,\varepsilon_n$ guaranteeing $$\|\varepsilon_1a_1+\ldots+\varepsilon_na_n\|_\infty \leq K.$$ We show that if the $\varepsilon_i$'s are allowed to take not just the values of $\pm 1$ but any unit modulus complex number, which we refer to as complex discrepancy, then the above inequality holds for a finite, explicit constant $K_{\mathbb{C}}$. Here, the $\ell^\infty$ norm of the resulting vector in $\mathbb{C}^m$ is the largest modulus of its entries, and thus the complex discrepancy of real vectors is equivalent to their rank-$2$ vector discrepancy. This quantity provides an upper bound (modulo uniform constant prefactor) on Gaussian discrepancy – a discrepancy measure introduced by Chewi, Gerber, Rigollet and Turner. Thus, we also resolve the Komlós conjecture for Gaussian discrepancy. Our paper builds upon the recent work of Bansal and Jiang on the Beck-Fiala and Komlós conjectures, which we approach from the formalism of Burkholder and the Bellman function method from probability and harmonic analysis. Our work was in part motivated by the realization that the complex discrepancy of the columns of any unitary matrix is equal to 1, a fact that follows from a straightforward calculation based on Idel and Wolf's generalization of the Sinkhorn normal form for unitary matrices.

This paper studies the existence of pullback $\mathcal{D}$-attractors for non-autonomous Kirchhoff wave equations with strong damping and delay effects in the phase space $\mathcal{E} = C_{H^1_0(\Omega)} \times C_{L^2(\Omega)}$. The model contains a strong damping term $-\Delta \partial_t u$, a nonlocal Kirchhoff term $\Psi(\|\nabla u\|^2)$, a state-dependent delay term $\phi(t, u_t)$, and a time-dependent external force $h(x, t)$. There appear to be no results on the pullback attractors of Kirchhoff wave equations involving both strong damping and delay effects in the literature. To this end, we establish delicate uniform estimates and employ the contraction function method to prove the pullback asymptotic compactness of the associated process, which yields the existence of pullback $\mathcal{D}$-attractors.

Let $E$ be a slope-semistable vector bundle on smooth projective curve $C$. Assume, $ \pi : G= Gr_{C}(r, E) \to C$ denote the relative Grassmannian of rank $r$ quotients of $E$. We prove that for every very ample polarization $L= O_{G}(1) \otimes \pi^{*} A$ the polarized variety $(G,L)$ possess an Ulrich bundle. The construction globalizes homogeneous ulrich bundle on Grassmannian fibres. The remaining cohomology vanishing problem reduced to an acyclicity criterion on the base curve. We finally use Falting's cohomological charaterization of semistability.

We determine the exact time exponent of space-optimal reversible pebbling on chains as $1.331742379256310\ldots$. The growth rate of space-optimal reach exists as a limit and admits a variational formula. The same exponent governs complete computations at minimal space, uniformly in the chain length.

This paper applies the Projection Characterization and the Symmetry Bootstrap for two operators $A, B$ (with $A$ unitary or self-adjoint) nearby commuting operators $A', B'$ (with $A'$ unitary or self-adjoint) from arXiv:2412.20795 to problems related to structured almost commuting unitaries, self-adjoint operators that almost anti-commute, and unitaries that almost commute up to a rational phase. We prove several new stability theorems for these types of relations for matrices. We also provide estimates for how close the constructed operators are for several known stability theorems that did not previously have asymptotic estimates.

One of the remarkable contributions of Don Zagier was the Kronecker limit formula for a real quadratic field, where he connects the double series $\mathcal{Z}(s,w,w^\prime)$ to the Dedekind zeta function associated to a real quadratic field. Later, Ishibashi determined all the Laurent coefficients of $\mathcal{Z}(s,w,w^\prime)$ at $s=1$. Recently, Choie and kumar have studied the analytic behaviour of the analogous double series $\tilde{\mathcal{Z}}(s,w,w^\prime)$. In this article, we derive all the Laurent coefficients of $\tilde{\mathcal{Z}}(s,w,w^\prime)$, akin to Ishibashi. These Laurent coefficients involve an interesting function $\mathfrak{F}_k^0(x)$, which was earlier studied by Dixit et. al. (Ramanujan for $k=1$), where they obtained a beautiful symmetric relation for $\mathfrak{F}_k^0(x)$. We establish both the two term and the three term functional equation of $\mathfrak{F}_k^0(x)$, derive the action of the period-like Hecke operator on $\mathfrak{F}_k^0(x)$ and connect an important integral with $\mathfrak{F}_k^0(x)$.

Let $Y_s=\sqrt{s}X+Z$, where $Z$ is standard Gaussian and independent of the real random variable $X$. We prove that, under the square-exponential moment condition $\mathbb{E}e^{\beta X^2}<\infty$ for some $\beta>0$, the scalar minimum mean-square error $\operatorname{mmse}_X(s)$ is analytic at zero signal-to-noise ratio if and only if $X$ is Gaussian, with constant random variables included as degenerate Gaussians. The proof converts estimation in the Gaussian channel into a backward heat flow acting on the moment-generating function $M(z)=\mathbb{E}e^{zX}$. Under the stated tail condition, every non-Gaussian input forces $M$ to have a nonzero complex zero. We show that each zero cluster produces a finite singularity in its localized Borel transform at the action $\xi=z_0^2/2$. After removing the action scale, the Borel coefficients have a nonzero $n^{-1/2}$ prefactor for a simple zero. A zero of multiplicity $m\geq 2$ splits according to the roots of a Hermite polynomial and instead contributes a prefactor $n^{-m/2}e^{r_m\sqrt{2n}}$. A finite-disc localization and relative-cycle continuation argument then show that at least one such singularity survives in the full Borel transform. Thus, for every non-Gaussian input in the stated class, the formal zero-SNR expansion is Gevrey-1 but divergent. Rational-MMSE rigidity and the analogous analyticity criterion for mutual information follow as corollaries.

Let \(\eta=(\eta_n)_{n\ge0}\) be a complex sequence such that \(F_\eta(z)=\sum_{n\ge0}\eta_nz^n\in\Hol(\D)\), and let \(\mathcal R_{(\eta)}\) be the associated Rhaly operator. We give a complete characterization of boundedness and compactness of \(\mathcal R_{(\eta)}:\mathcal D^p_\alpha\to\mathcal D^q_\beta\) for \(1<p,q<\infty\) and \(\alpha,\beta>-1\). The characterization is formulated in terms of the \(H^q\)-norms of the dyadic blocks of \(F_\eta\). For \(-1<\alpha<p-2\), boundedness and compactness coincide and reduce to \(F_\eta\in\mathcal D^q_\beta\). For \(\alpha=p-2\) and for \(\alpha>p-2\), boundedness is characterized by membership in analytic truncated Besov spaces and analytic Besov spaces, respectively; compactness has the same characterization for \(q<p\) and is characterized by the corresponding little spaces for \(p\le q\). We also obtain operator norm and essential norm estimates. As applications, we obtain boundedness and compactness characterizations for Rhaly operators between weighted Bergman spaces and for Cesàro-type operators induced by positive measures. In particular, we characterize \(C_\mu:B^p\to B^p\), answering a question left open by Sun, Ye and Zhou and later noted by Tang. For each \(p>2\), we also construct a symbol \(F_\eta\in H^\infty\cap\lambda^p_{1/p}\) for which \(\mathcal R_{(\eta)}\) is not bounded on \(H^p\).

The Matrix Spencer conjecture asserts that for all symmetric matrices $A_1,\ldots,A_n\in\mathbb{R}^{n\times n}$ with $\|A_i\|\le1$ there are signs $\varepsilon_1,\ldots,\varepsilon_n\in\{-1,1\}$ with $\|\sum_{i=1}^n\varepsilon_iA_i\|=O(\sqrt n)$. We prove it: a signing of discrepancy below $8\sqrt n$ always exists. We also give a randomized algorithm that finds a signing of discrepancy below $12\sqrt n$ with failure probability at most $p$. The algorithm uses $n^{3+o(1)}\operatorname{polylog}(1/p)$ arithmetic operations in the real-arithmetic model. This matches the size $n^3$ of the dense input up to subpolynomial factors. In the other direction, we prove that for every $n$ there are collections of symmetric matrices such that every signing has discrepancy at least $(2-o(1))\sqrt{n}$. We present three different proofs of the matrix Spencer conjecture. The key to every proof is a hereditary small-ball estimate. This is a lower bound on the Gaussian measure of the spectral body $\{x\in \mathbb{R}^n:\|\sum_ix_iA_i\|\le R\}$ that holds for every subfamily of the matrices. The other ingredient turns that Gaussian measure into a partial signing. We give three approaches to obtain such a signing. The first one covers the cube by partially signed faces through Gaussian concentration with a constant $7\cdot10^9$. The second proof replaces the covering by a projection lemma with explicit parameters for a constant $156000$. The third proof turns Gaussian measure into signs by a lossless coding, with no union bound. It proves the estimate at the right radius with smooth spectral barriers and certified coefficients. It gives a constant below $7.88$. For algorithms, the main idea is to project Gaussian points onto a smoothed spectral body. The $n^{3+o(1)}$ time algorithm tracks the Gibbs matrix of that body across coordinate-descent steps with sketched increments and random refreshes.

We give an explicit combinatorial criterion for an Ekedahl-Oort (EO) stratum in the moduli stack of principally polarized abelian varieties of dimension $g$ in characteristic $p > 2$ to meet the supersingular locus. The criterion is obtained by translating a non-emptiness criterion for basic affine Deligne-Lusztig varieties, into an explicit condition on the elementary sequences indexing the EO strata. Furthermore, we give explicit bounds for the number of EO strata of $p$-rank zero which are disjoint from the supersingular locus. As a result, we prove that asymptotically almost every EO stratum of $p$-rank zero meets the supersingular locus. As an application, we also give a lower bound for the number of EO strata meeting the supersingular locus on unitary PEL Shimura varieties of signature $(a,b)$.

We study lazy single-site Metropolis dynamics $P_{\beta}$ at inverse temperature $\beta \geq 0$ on $\{-1,+1\}^d$ for an Ising star with additional signed interactions among the leaves. If the absolute row sums of the leaf-interaction matrix are at most $\kappa\le1/2$, the worst-case total-variation mixing time of $P_{\beta}$ is at least of order $d\exp\{c\beta(d-1)\}$ for $\beta\ge1$, with universal $c>0$. Averaging over global spin reversal reduces the mixing time to $\mathcal O(d^2(1+\beta))$ for both $GP_\beta G$ and $(P_\beta+G)/2$, where $G$ is the Gibbs kernel induced by the partition of the state space into spin-reversal orbits $\{-x,x\}$. Partition-function interpolation gives the lower bound for $P_\beta$. The upper bounds follow from Wu's Dobrushin inequality and a decomposition into projection and restriction chains. This gives an explicit example in which group averaging turns an exponentially slow chain into a polynomially fast one.

We prove that, for every integer $t\geq 1$, the unit two-sphere admits a spherical $t$-design consisting of exactly $54t^2$ points. More generally, such a design exists with exactly $6q^2$ points for every integer $q\geq 3t$. The proof builds on the topological degree method of Bondarenko, Radchenko and Viazovska, using an explicit equal-area partition of the sphere based on the map of Roşca and Plonka. By choosing the cell centers to minimize the average squared geodesic distance and deriving sharper sampling estimates, we obtain the stated quadratic bound on the size of spherical $t$-designs on $\Sph$.

This paper introduces the Higher-Order Quadripolar Argumentation Framework (HQAF), which extends bipolar frameworks by allowing attacks and three types of supports—necessary, deductive, and evidential—to interact at higher order. In HQAF, each interaction may have as its source and target an argument, an attack, or a support. We provide a 3-valued semantics via adjacent complete labellings and equational systems, and define encoded semantics directly in propositional logics, mainly Łukasiewicz three-valued logic for the 3-valued case and fuzzy logics based on continuous t-norms (Gödel, Product, Łukasiewicz) for the fuzzy case. Model equivalence between HQAF and its normal encoding is established. The fuzzy equational semantics is shown to be a specific formalism of the fuzzy encoded semantics, and ternarization connects fuzzy models to 3-valued complete labellings. The study is confined to non-set HQAF; collective interactions are reserved for future work.

We establish two results on the asymptotic equipartition property (AEP) for quantum channels. First, the AEP formulated using smoothing over completely positive trace-preserving maps fails in general. Second, the AEP with smoothing over subchannels (completely positive trace-nonincreasing maps) is equivalent to the strong converse for parallel quantum channel Stein's lemma, to a sharp threshold for the channel hockey-stick divergence, and to an AEP for the channel Lorenz divergence. The counterexample reveals a surprising separation between channel and subchannel smoothing. Although states and classical channels admit a sharp, dimension-independent completion bound, which we establish, the corresponding single-shot gap for quantum channels is unbounded already for qubits. This separation persists asymptotically: we construct qutrit channel pairs with finite max-relative entropy whose CPTP-smoothed rate strictly exceeds the regularized channel relative entropy. The asymptotic gap between channel and subchannel smoothing is unbounded across the family, even when the subchannel approximation error decays exponentially. To establish the equivalences, we develop a uniform filter that converts Lorenz smoothing into a single nearby subchannel. The four conjectured formulations thus share a common asymptotic threshold, whose validity for general channel pairs remains open. Together, these results separate this threshold question from the obstruction imposed by exact trace preservation.

The Borel distinguishing number $D_B(\mathcal{G})$ of a Borel graph $\mathcal{G}$, recently introduced by Bilge and Kaya, is the minimum number of colors required to break the symmetry of $\mathcal{G}$ in a Borel way. In this paper, we investigate the Borel distinguishing number of Schreier graphs induced by the free part of the shift action $\Gamma \curvearrowright n^\Gamma$. We prove that $D_B(\mathcal{G})\le n+1$ for $\Gamma=\mathbb{Z}^d$ equipped with the standard generators. Moreover, we show that $D_B(\mathcal{G})\ge n+1$ if $ \Gamma$ is amenable and $ \{\gamma \in \mathrm{Aut}(\mathrm{Cay}(\Gamma,S)) \mid \gamma(e) = e \}$ is non-trivial. We also show that $D_B(\mathcal{G})$ is finite if $\Gamma$ is finitely generated, and give some applications of our results. These results answer some questions raised by Bilge and Kaya.

Connection formulas for the asymptotic solutions of a system of n> 1 coupled Painleve'-II equations with symmetry-breaking parameters are written explicitly. An asymptotically exact WKB approach to these formulas relies on the quantum-mechanical independent crossing approximation for an explicitly time-dependent Schroedinger equation.

First-arrival traveltime tomography provides a computationally efficient means of imaging subsurface velocity structure, while boundary-conforming deformed grids allow rugged topography to be represented without abandoning a logically structured mesh. In matrix-free Gauss-Newton inversion, however, the linearized full-metric Eikonal transport and its transpose must be applied repeatedly at a fixed background model. Directional sweeping consequently revisits the same state-dependent dependency structure for every Krylov right-hand side. We recast the frozen tangent transport as a directed graph and distinguish loss of the background-traveltime ordering from genuine algebraic cyclicity. Strongly connected components identify the irreducible part of the transport, whereas the remaining dependencies admit exact scalar substitution after reordering. The condensation graph and local block factorizations are constructed once for each frozen source state and then reused for both primal and transpose applications. Numerical experiments on deformed grids and a three-dimensional tomography problem show that the resulting block-triangular formulation preserves the frozen sensitivity action while substantially reducing the repeated transport cost. By eliminating this dominant inner-iteration cost, the proposed method markedly accelerates Gauss-Newton computation and makes full-metric matrix-free inversion on deformed grids practical at much lower cost.

We study the Perron extremal solutions of the stationary one-phase Bernoulli problem. These solutions are important in several applications, but not much is known about the regularity of their free boundaries due to their non-variational construction. We show that, in fact, Perron extremal solutions retain some variational features; specifically, they are solutions in the sense of inner variations. We establish this property by showing that extremal solutions arise as infinite-time limits of monotone solutions of the parabolic Bernoulli problem. As a consequence, we are able to describe the structure of singularities for Perron extremal solutions in two dimensions: largest subsolutions have smooth free boundaries, while smallest supersolutions have free boundary points that are either smooth, or whose blow-up limits are two-plane wedges with equal slope.

Consider the task of online vector balancing for stochastic arrivals $X_1,\ldots,{X_T}$, where the $X_i$ are independent uniformly random $d$–sparse binary vectors in $\{0,1\}^n$. This is a random analogue of the online Beck–Fiala problem. We show that uniformly for $2\le d\le n/2$ and $T = \Theta(n)$, the optimal online prefix discrepancy $\max\limits_{t\leq T}\left\|\sum_{i=1}^t\sigma_i X_i\right\|_\infty$ is of order \[ \Theta\big(\max\{\sqrt d,\log\log n\}\big). \] The upper bound is achieved by an efficient online algorithm. Thus, for $d\le(\log\log n)^2$, the optimal discrepancy is $\Theta(\log\log n)$ and is independent of the sparsity up to constant factors, whereas above this scale it is $\Theta(\sqrt d)$, matching the order of the offline discrepancy. This identifies the threshold at which sparsity begins to govern the online discrepancy of the random Beck–Fiala model.

We study continuous-time and possibly high-dimensional stochastic control problems where drift coefficients and running reward functions are unknown. Due to these missing model primitives, we take the exploratory, reinforcement learning (RL) framework of Wang, Zariphopoulou, and Zhou(2020) with relaxed controls and entropy regularization. The objective is to develop theoretically grounded, efficient and scalable RL algorithms to learn both the optimal value functions (which also solve the exploratory HJB equation) and optimal exploratory feedback control policies. When the diffusion coefficients do not contain control, we employ probabilistic representations of both the optimal value function and its gradient based on an auxiliary state process depending only on the diffusion part of the original dynamics. With a delicate analysis on some properly defined mappings and their fixed points, this leads to the introduction of our policy iteration algorithms and their convergence. We demonstrate the performance of our algorithms through various numerical examples. Finally, we study a special control-dependent diffusion case where probability representation of the Hessian is called for.

We introduce discrete $C^*$-Frobenius algebras as categorical data for constructing infinite-index extensions of Möbius covariant nets. The underlying representation is a countable direct sum of simple dualized modules, while the direct sum itself is not necessarily dualizable. Accordingly, rather than requiring a single bounded multiplication morphism, we work with compatible families of bounded left and right multiplication morphisms. Using Bin Gui's theory of categorical extensions, we associate to every discrete $C^*$-Frobenius algebra two (generally non-local) Möbius covariant extensions generated by left and right charged fields, respectively. If the discrete $C^*$-Frobenius algebra is commutative, the two extensions coincide and yield a local Möbius covariant extension. As applications, for every $C^*$-tensor category generated by an invertible object with trivial self-braiding such that its fusion product rule is given by $\mathbb{Z}$, we construct two classes of examples: the simple current extension by $\mathbb{Z}$ and the discrete Longo-Rehren construction.

We study decentralized learning of Nash equilibria (NE) in infinite-horizon discounted Markov games under bandit feedback, focusing on Markov $\alpha$-potential games. We develop KL-projected natural policy gradient (NPG) algorithms in two settings: an episodic setting with frozen policies during sampling and a fully online setting in which players receive a single realized cost sample per time step and update their policies asynchronously along a continuing trajectory. We establish finite-time high-probability NE regret bounds of order $\widetilde O(T^{-1/4})$ and $\widetilde O(T^{-2/15})$ for the episodic and fully online settings, respectively, up to fixed approximation terms. Crucially, our bounds eliminate the distribution-mismatch coefficient, which can scale prohibitively with the size of the state space, while accommodating potential approximation, estimation-oracle bias, and transition sensitivity. We further identify a state-wise potential structure that yields sharper guarantees with additive dependence on the potential approximation error $\alpha$. We specialize the framework to independent-resource Markov congestion games (IMCGs), establish their approximate-potential and transition-sensitivity properties, and construct decentralized estimation oracles from realized costs. As an application, we introduce strategic online job scheduling on stochastic machines and obtain a scalable decentralized algorithm for learning stable dispatching policies. Overall, our results provide the first finite-time high-probability NE regret guarantees for fully online asynchronous decentralized learning in Markov $\alpha$-potential games, remove distribution-mismatch coefficients from the regret bounds, accommodate fixed estimation-oracle bias, and provide scalable decentralized learning with finite-time guarantees for IMCGs.

Consider a simple undirected graph $G$ on $N$ vertices, with each vertex holding an opinion from one of two options. Starting from the initial opinions on day 1, we run majority dynamics: on each subsequent day, all vertices simultaneously adopt the majority opinion among their neighbors, retaining their current opinion in case of a tie. A well-studied conjecture of Benjamini, Chan, O'Donnell, Tamuz, and Tan states that if $G$ is drawn from the random binomial model $\mathbb{G}(N,p)$, and if the starting opinions are chosen uniformly at random, then as long as $pN \to \infty$ the network will converge to 99% consensus with high probability. We confirm this conjecture as long as $pN \ge N^{\varepsilon}$ for any fixed $\varepsilon > 0$, showing that full unanimity is reached with high probability in $O(1/\varepsilon)$ days. This result improves on a line of results by Fountoulakis, Kang, and Makai, who achieve $\varepsilon = 1/2$; Chakraborti, Kim, Lee, and Tran, who achieve $\varepsilon = 2/5$; and Jaffe, who achieves $\varepsilon = 1/3$. The proof technique involves iteratively revealing the "opinion histories" for each vertex through time, keeping track of the degrees between every vertex and each of $2^k$ opinion history classes on day $k$. Our analysis of this process requires intricate estimates for degree-constrained random graph models using graph enumeration tools from the work of McKay and Wormald as well as Canfield, Greenhill, and McKay, and their extensions by Liebenau and Wormald. In doing so, we connect the discrete dynamics to a deterministic idealized process, whose leading-order behavior is described by conditional Gaussian probabilities and expectations.