Papers

We consider the reflected stochastic reaction-diffusion equation on $[0,1]$: \begin{align*} \left\{ \begin{aligned} d u(t,x) &=\frac{1}{2}\partial_{xx} u(t,x)dt +b(u(t,x))dt + \sigma(u(t,x)) W(dt,dx)+L(dt,dx),\\ u(t,x)&\geq 0, \quad t\geq 0, \ x\in [0,1],\\ u(0,x)&=u_0(x)\geq 0, \quad x\in [0,1],\\ u(t,0) &= u(t,1) = 0, \quad \forall\ t\geq 0, \end{aligned} \right. \end{align*} where the initial value $u_0$ is non-negative on $[0,1]$ satisfying $u_0(0)=u_0(1)=0$, and $ W(dt,dx)$ is a space-time white noise. The $L$ in the equation is a random measure on $[0,\infty)\times(0,1)$, which is a part of the solution pair $(u, L)$. In this paper, we establish the existence and uniqueness of invariant measures, as well as exponential mixing for the reflected stochastic reaction diffusion equation under the dissipative condition $$(b(x)-b(y))(x-y)\leq -\alpha (x-y)^2,$$ which include the coefficients having polynomial, even exponential growth. The big obstacle of utilizing the dissipative condition is the lack of the Itô formula/energy equality for such equations. To circumvent the problem, we use the newly found method in our paper (arXiv:2606.26619, 2026) to fully exploit comparison principles of reflected stochastic reaction-diffusion equation.

We prove a version of Heisenberg's uncertainty principle for a rather general class of locally compact abelian groups. We compare the lower bound provided by our approach with the optimal lower bound in the Euclidean case, and formulate the Heisenberg uncertainty principle for local fields.

Settling a problem raised by Eccles in 2015, Narayanan in 2026 considers a two-player game in which two captains alternately select players while the opponent decides to which team each selected player is assigned. Moreover, the two teams are required to have equal cardinalities, and Narayanan proved that the second player has a non-losing strategy. In this paper, we study a natural variant in which the teams are allowed to have different cardinalities, and the winner is determined by comparing the average strengths of the two teams. We show that, in this setting, the parity of the total number of players completely determines which player has a non-losing strategy: the first player has a non-losing strategy when the number of players is even, while the second player has a non-losing strategy when it is odd.

We first rewrite the Chmutov and Vignes-Tourneret's three-permutation formula as an explicit hyperedge-partial-duality formula in the two-permutation model, and show that in this model partial duality acts exactly by preserving the support and length of every hyperedge while reversing the $\alpha$-cycles corresponding to the selected hyperedges. Next, using the Cori and Hetyei's construction of the medial map, we define for each hyperedge subset $E'\subseteq E(H)$ a black/white smoothing state $S_{E'}$, and prove rigorously that the state circles of $S_{E'}$ are in bijection with the vertices of the partial dual $H^{E'}$. Consequently, $H^{E'}$ is Eulerian if and only if every state circle has even length. On this basis we prove the following two main theorems: \[ \begin{aligned} H^{E'}\text{ is Eulerian} &\Longleftrightarrow \exists\text{ a crossing-total direction $\Omega$ of }M(H) \\ &\hspace{3.35em}\text{such that } E'=D(\Omega)\cup T',\quad T'\subseteq T(\Omega), \end{aligned} \] \[ \begin{aligned} H^{E'}\text{ is bipartite} &\Longleftrightarrow \exists\text{ an all-crossing direction $\Phi$ of }M(H) \\ &\hspace{3.35em}\text{such that } E'=C(\Phi). \end{aligned} \] Here $D(\Omega)$, $T(\Omega)$ and $C(\Phi)$ denote, respectively, the sets of all $d$-type, $t$-type and $c$-type hyperedges. Unlike the ribbon-graph case, the hypermap setting exhibits a genuine new obstruction: if some hyperedge-partial dual is bipartite, then every hyperedge of the original hypermap must have even length.

Financial resilience concerns the rate at which a position recovers, or further deteriorates, in response to adverse conditions. As a first step, Laeven, Ferrari, Rosazza Gianin, and Zullino (arXiv:2505.07502) introduced the resilience rate, defined as the expected instantaneous rate of (favorable) change of a price or risk-assessment process. Since this quantity captures only the conditional mean of future increments, it cannot distinguish between positions having the same expected recovery but different conditional risk profiles. We obtain a richer characterization by evaluating such increments through a genuine, possibly nonlinear, dynamic risk measure. More precisely, for an Itô process $\pi$ and a normalized, cash-additive dynamic risk measure $\rho$, we define the resilience evaluation by \[\mathcal D_s^\rho\pi_t := L^1\text{-}\lim_{\varepsilon\to0^+} \frac{1}{\varepsilon}\rho_s(\pi_{t+\varepsilon}-\pi_t), \qquad 0\leq s\leq t<T,\] whenever the limit exists. When $\rho$ is a convex dynamic risk measure induced by a BSDE with a Lipschitz or quadratic driver, we prove that this limit is well-posed and admits an explicit dual representation. It is given by the worst-case conditional expectation, over a zero-penalty class of measure changes, of an effective drift combining the drift of $\pi$ with the risk adjustment assigned by $\rho$ to its volatility. We further establish attainment of the optimal scenario and illustrate the scope of the construction, as well as the role of the assumptions, through examples and counterexamples.

We introduce a data-driven probabilistic framework for learning systems based on Gibbs measures on hierarchical structures. Unlike standard empirical risk minimization, where a dataset is used to identify a single optimal parameter, our approach transforms the empirical loss function into an interaction potential defining an energy-based model. The resulting Gibbs distribution describes a family of equilibrium learning states generated by the data. We formulate the consistency conditions of the associated finite-volume distributions and derive nonlinear integral fixed-point equations whose solutions characterize the admissible learning states. These equations provide a rigorous connection between empirical loss landscapes and probabilistic inference on trees. For translation-invariant solutions, the problem reduces to the analysis of positive compact operators induced by data-dependent kernels, allowing us to establish existence and uniqueness conditions in the one-dimensional setting. Furthermore, we show that hierarchical learning systems may exhibit phase-transition phenomena: for certain empirical kernels on Cayley trees, multiple Gibbs measures emerge beyond a critical inverse temperature, corresponding to distinct equilibrium prediction regimes. Numerical experiments with non-separable kernels illustrate the appearance of multiple solution branches and demonstrate the coexistence of several data-induced learning states. Our results provide a new perspective on energy-based learning, where data do not merely determine an optimal model through minimization but define an entire probabilistic landscape of possible inference states.

Let $k\ge4$ and $j\ge2$ be integers with $j$ even. Harder's conjecture predicts a congruence between a primitive elliptic cusp form of weight $2k+j-2$ and a degree-two vector-valued Siegel Hecke cusp eigenform of weight ${\det}^{k}\mathrm{Sym}^{j}$. We prove the corresponding Harder-type congruence for the spinor $L$-polynomials under explicit arithmetic hypotheses on a congruence prime and an auxiliary imaginary quadratic field. Our approach uses Hermitian automorphic forms on the quasi-split unitary group $\mathrm{U}_{2,2}$. We construct Hermitian spin lifts of Siegel cusp forms and show that the Hermitian cusp form produced by a Hermitian Klingen-Eisenstein congruence lies in the image of this lift. The key step is to prove that the Galois representation attached to this Hermitian cusp form is conjugate invariant. For this, we combine the endoscopic classification for quasi-split unitary groups with Selmer-group vanishing results. This allows us to identify the Hermitian cusp form as the spin lift of a Siegel cusp eigenform and thereby obtain the Harder-type congruence. The argument applies uniformly to both even and odd $k$.

Scale-resolving simulations of wall-bounded turbulent flows remain prohibitively expensive at high Reynolds numbers, owing to the stringent near-wall resolution requirements. High-order compact gas-kinetic schemes (CGKS) are accurate, robust, and efficient for compressible flows, making them an attractive foundation for reducing this cost. Building on the fifth-order scheme CGKS-5th, we develop a wall-modeled CGKS framework that alleviates the near-wall resolution burden through a pressure-gradient-based non-equilibrium wall model while preserving the resolving power of the outer solver. CGKS-5th resolves the outer flow and supplies the wall model with data at the exchange location. On coarse near-wall meshes, the wall model reconstructs the under-resolved viscous wall stress, while CGKS-5th provides the inviscid wall flux directly; the two combine to form the wall momentum flux. To capture non-equilibrium effects in adverse-pressure-gradient and separated regions, the wall model retains a pressure-gradient source term together with a pressure-gradient-corrected near-wall damping function. We assess the framework on two distinct flows: bluff-body separation past a circular cylinder, and a shock-induced separation bubble on the transonic RAE 2822 airfoil, using near-wall meshes far coarser than wall-resolved simulations require. For the RAE 2822 case, this corresponds to a twentyfold coarsening in the wallnormal direction, with comparable coarsening in other directions. In both cases, the wall-modeled CGKS-5th reproduces the separated flow structures and markedly improves near-wall predictions over its wall-model-free counterpart, most notably the skin-friction coefficient. The framework thus delivers accurate predictions of these separated flows at substantially reduced near-wall cost, while its lightweight coupling adds less than 1% runtime overhead in a multi-GPU implementation.

We prove Springer's Odd Degree Theorem for quadratic forms over LG rings, and Scharlau's and Knebusch's norm principles for quadratic forms over semilocal rings. We present applications to the flat cohomology of spin groups and étale norm groups.

Let $G$ be a compact Lie group, let $M$ be a smooth manifold with a $G$-action. Then all the data of this model is contained in the action groupoid $G\ltimes M$. If $U_y$ is a small enough neighbourhood of $y\in M/G$, the slice theorem says that \begin{equation*} \pi^{-1}(U_y)=S_{x}\times_{G_{x}} G \end{equation*} where $x$ is a point in the $y$ orbit, $S_x$ is the slice of $x$ and $G_x$ is the isotropy group of $x$. An alternative approach to describe group actions on spaces is through the language of groupoids. Local properties of Lie groupoids are often studied via linearization theorems. One can compute the equivariant cohomology $H_G^*(\pi^{-1}(U_y))$ of $\pi^{-1}(U_y)$ using the Weil model or the Cartan model. Also by the homotopy theory, the equivariant cohomologies $H_G^*(\pi^{-1}(U_y))$ and $H_{G_x}^*(S_x)$ are isomorphic. In this paper, we explicitly construct a natural chain map between the Weil (and respectively between the Cartan) models of $(\pi^{-1}(U_y), G)$ and $(S_x, G_x)$, and prove that it induces an isomorphism in equivariant cohomology. We then introduce the notion of slice (or locally linearizable) groupoids, which are locally modeled on Lie group actions on manifolds with gluing data, several examples and applications are discussed. In the last section, we generalize the equivariant theory to these groupoids using sheaf-theoretic methods. We further show that when a slice groupoid is equivalent to an action Lie groupoid, the equivariant cohomology of the slice groupoid coincide with the equivariant cohomology of the Lie group action on a manifold.

We prove~\cite[Conjecture 2.12]{BGS20} on the signature tail asymptotics of pure rough paths and extend it to arbitrary reasonable tensor norms. In more details, let \[ \mathbf X_t=\exp(tl) \,\text{ with }\, l=l_1+\cdots+l_m\,\text{ and }\, l_r\in\mathcal L_r(V), \] be a pure $m$-rough path over a finite dimensional real or complex Banach space, and equip the tensor powers of $V$ with arbitrary reasonable tensor algebra norms. We prove that \[ \limsup_{n\to\infty}\left(\left(\frac{n}{m}\right)!\left\|\pi_n(\exp l)\right\|_n\right)^{m/n}=\|l_m\|_m . \] In particular, this identifies the signature tail with the local $m$-variation of the pure rough path. The upper bound was obtained in~\cite{BGS20}; the main contribution of the paper is the matching lower bound. Its proof is based on finite dimensional developments and a norming cyclic construction. For every top-level tensor $l_m$, we also build a contractive development in which $\|l_m\|_m$ appears as an eigenvalue at degree $m$.

We propose two deterministic momentum-based first-order methods for minimizing lower-bounded nonconvex functions with Lipschitz-continuous gradients and Hessians. Both methods attain an $\varepsilon$-approximate first-order stationary point in $O(\varepsilon^{-7/4})$ function and gradient evaluations, and their update parameters do not depend on the target accuracy or a prescribed terminal iteration. The convergence guarantees concern the best point among weighted averages of the iterates. Instead of relying on momentum-resetting restarts, both methods employ a velocity-control mechanism that adaptively reduces the velocity. The first method uses a differentiable velocity-control rule, making its last iterate differentiable with respect to the initial point. To the best of our knowledge, this is the first deterministic first-order method to combine this differentiability property with the stated complexity. The second method adaptively estimates the gradient and Hessian Lipschitz constants and requires no prior knowledge of their values. Numerical experiments show that it outperforms existing parameter-free methods on some test problems. Both methods are derived by discretizing a new continuous-time model described by an ordinary differential equation. We develop parallel continuous- and discrete-time convergence analyses within the performance estimation problem framework.

Axial algebras are non-associative algebras generated by idempotents, called axes, whose adjoint action satisfies a fusion law. When this fusion law is graded, axes naturally lead to automorphisms of the algebra, and so such axial algebras are inextricably linked with groups. This article is meant to complement the recent survey \cite{ms} by significantly expanding the list of interesting open problems suggested by the specialists in the field, and providing a further discussion of the related concepts and available results.

In this note, we prove that the transcendental basepoint-free conjecture for Calabi-Yau manifolds holds if it holds for its hyperkähler factors in its Beauville-Bogomolov decomposition. Based on a contraction theorem due to Bakker and Lehn, we show that the conjecture holds for a big and nef class $\alpha$ on a hyperkähler manifold under a mild condition on the dimension of the space generated by classes of rational curves on which $\alpha$ vanishes.

We determine the exact region of Chatterjee's rank correlation $\xi$ and Blomqvist's $\beta$, that is, the set of all pairs $(\xi(C),\beta(C))$ where $C$ ranges over all bivariate copulas. The region equals $\{(x,y)\in[0,1]\times[-1,1]:|y|^3\le2x\}$, so every copula satisfies the sharp inequality $|\beta(C)|^3\le2\xi(C)$. Its left boundary $\xi=|\beta|^3/2$ is attained uniquely by an explicit family of copulas that perturb independence by a signed tent function centered at the median, and its right boundary $\xi=1$ by deterministic interval-exchange copulas. The proof reduces the problem to a one-dimensional minimization of an $L^2$-norm over $1$-Lipschitz functions. We further show that the region does not shrink when restricted to radially symmetric copulas and that positively quadrant dependent copulas attain exactly its upper half. For copulas under which the second variable is stochastically increasing in the first, the exact region is $\{(x,y)\in[0,1]^2:y^3\le2x,\ 3(1-y)^2\le4(1-x)\}$, and its right boundary is attained uniquely by explicit singular two-branch copulas. In particular, under stochastic increasingness $\beta=0$ is compatible with every value $\xi\in[0,1/4]$, whereas $\xi>1/4$ forces $\beta>0$.

The recently introduced formalism of chiral cluster seeds replaces quantum cluster variables with deformed vertex operators. In this framework, a decorated quiver associated with a seed encodes the operator product expansions of the corresponding vertex operators. This formalism is applied to several $(q,t)$-deformed W-algebras, including $\mathcal{W}_{\mathfrak{q},\mathfrak{t}}(\mathfrak{gl}(N|M))$, $U_q(\widehat{\mathfrak{sl}}_2)$, and the deformed Bershadsky–Polyakov algebra. In particular, it is shown that different free field realizations of the currents are related by mutations of the associated chiral cluster seed. The second part of the paper introduces a $(q,t)$-deformation of the subregular W-algebras, denoted by $\mathcal{W}_{\mathfrak{q},\mathfrak{t}}^{\text{sub}}(\mathfrak{sl}(N))$. All free field realizations obtainable through seed mutations are described. An embedding of $\mathcal{W}_{\mathfrak{q},\mathfrak{t}}^{\text{sub}}(\mathfrak{sl}(N))$ into the free field realization of $\mathcal{W}_{\mathfrak{q},\mathfrak{t}}(\mathfrak{sl}(N))$ tensored with a rank-two Heisenberg algebra is constructed. This embedding may be viewed as a deformed analogue of inverse quantum Hamiltonian reduction. The relation between the subregular algebras and $\mathcal{W}_{\mathfrak{q},\mathfrak{t}}(\mathfrak{gl}(1|N))$ is also discussed.

This paper investigates learning-assisted transmit beam selection for indoor millimeter-wave (mmWave) systems operating with hybrid beamforming and joint transmission. A synchronized SDR-based testbed at 26.5 GHz band is deployed to collect wideband channel measurements in a realistic office corridor environment. Using the measurement dataset, beam selection is formulated as a supervised learning problem aiming to approximate the SNR-optimal beam obtained through exhaustive sweeping. Two complementary approaches are examined: a geometry-driven Deep Neural Network (DNN) that predicts the optimal beam from spatial features, and a pilots-only method that infers suitable beams using a limited number of sounded pilot beams without positional information. Experimental results demonstrate high prediction accuracy and significant reduction in beam search overhead compared to exhaustive sweeping, highlighting the effectiveness of measurement-driven learning for practical indoor mmWave beam management.

In this paper, we study a particular family of solutions of the Vlasov-Navier-Stokes system posed on $\mathbb{R}^d$ (with $d\geq 2$), and show their convergence to the unique solution of the pressureless Euler-Navier-Stokes system. A global existence result for the latter system, in the small data regime, was established in \cite{MonENS}. Here we place ourselves in a multiphase framework, introduced and studied by Zakharov in \cite{Zakharov1,Zakharov2}, in order to carry out an analogous analysis for a system that we will call multiphase pressureless Euler-Navier-Stokes. We then study the single-phase limit and obtain a rigorous link between the Vlasov-Navier-Stokes system and the pressureless Euler-Navier-Stokes system.

We investigate adaptive increasingly rare Markov chain Monte Carlo algorithms and the associated time-average estimator for approximating expectations. Under a simultaneous Wasserstein contraction assumption on the underlying family of Markov kernels we derive explicit bounds for the mean squared error. We illustrate the applicability of our estimate through adaptive stereographic algorithms and Metropolis-Hastings schemes that employ normalizing flows for adaptation. We also consider a generic adaptive algorithm for doubly intractable problems and provide a corresponding cost analysis to achieve a desired precision.

The Kohayakawa–Nagle–Rödl–Schacht conjecture predicts that locally dense graphs contain, asymptotically, at least as many homomorphic copies of any fixed graph as the random graph of the same edge density. We prove that every graph with at least one edge satisfies a natural $L^p$ relaxation of this conjecture in the graphon setting. More precisely, let $F$ be a graph with $m>0$ edges, and let $n$ be the number of non-isolated vertices of $F$. If $$ p\ge \binom {n}{2}/m, $$ then for every $\rho$-locally dense graphon $W$, $$ t(F,W^{\circ p})\ge \rho^{pm}. $$ Equivalently, if $$ W_F(\mathbf x)=\prod_{ij\in E(F)}W(x_i,x_j), $$ then $$ \|W_F\|_{L^p}\ge \rho^{e(F)}. $$ The proof is based on a Hölder uniformization over vertex relabellings, in the spirit of Conlon–Lee. We also prove a more general comparison principle with edge-transitive KNRS supergraphs, yielding sharper exponents whenever $F$ embeds into an edge-transitive KNRS graph. Finally, positive-semidefinite methods give theta-subdivision results: Sidorenko-good graphs are closed under arbitrary uniform theta-subdivisions; the non-uniform theta theorem of Im–Li–Liu admits a Sidorenko-good lift, under the same divisibility assumptions, after removing the parity restriction; and uniform theta-subdivisions of KNRS graphs are regular-KNRS.

We say that a hypersurface $\Sigma \subset\mathbb{R}^{n+1}$ is $\alpha$-stationary if it is a critical point of the Euler-Dierkes-Huisken functional $\mathcal{E}_\alpha(\Sigma)=\int_\Sigma|X|^\alpha\, d\mathcal{H}^n$, introduced by Dierkes and Huisken in \cite{[DH-24]}. In this paper, we prove that every smooth, complete, connected, embedded $\alpha$-stationary hypersurface in $\mathbb{R}^{n+1}$ passing through the origin with $\alpha>0$ is a linear hyperplane.

In joint work with Lacini and Sheridan, we proved that the first and second secant varieties of a smooth projective complex variety embedded by the complete linear system of a sufficiently positive line bundle are projectively normal. The purpose of this paper is to establish effective results on how positive the embedding line bundle must be for this result to hold. We also provide effective conditions under which the defining ideal of the first secant variety is generated by cubics, and furthermore, generated by $3 \times 3$-minors of a matrix of linear forms. The latter result gives an effective version of a theorem of Agostini and the second author.

We consider nonparametric estimation of the regression function in a model where individuals share a common noise component and repeated measurements are available for each individual. We propose a projection estimator which minimizes a least-squares contrast that accounts for the covariance structure resulting from the common noise. We analyze its risk measured either as the expectation of the empirical norm or as the expectation of the theoretical norm associated with the contrast. We discuss how the number of repeated measurements affects the estimation rates in the common noise model, and precisely characterize the dependence on the number of repetitions. In addition, we propose a data-driven projection estimator and establish risk bounds in terms of the expected empirical norm. The results are illustrated with some simulation experiments.

We are interested in conducting political polls sequentially, so that one can stop acquiring data as soon as possible while safely yielding statistically significant results. Building off e-values, which have recently become a useful tool to create sequential testing methods, we develop a theory of posterior optimal e-values. We use voting as a convenient example on which to illustrate our method. First, we design statistical tests for Condorcet and Borda voting system, and also for Schulze voting system which we are the first to tackle statistically. Then, we study the construction of optimal sequential e-values in the deceptively simple setting of multivariate Bernoulli data, with general composite null and alternative hypothesis sets $\mathcal{H}_0$ and $\mathcal{H}_1$. We give a way to compute these e-values using an efficient Frank-Wolfe algorithm, giving a pretty general way to compute Reverse Information Projections, even when $\mathcal{H}_0$ corresponds to a non-convex parameter set. Finally, we illustrate the efficiency, both in terms of power and sample size of our method. We compare with state of the art in both simulated and real data experiments, with application to French 2022 presidential election data.

We prove Strichartz estimates for the space-time density $\rho$ of solutions to the free Liouville equation on flat tori. In dimension one, we obtain the optimal range of estimates for the density $\rho \in L^p_{t,x}$ in terms of $f_0 \in L^{a}_vL^{b}_x$. In higher dimensions, we prove that such estimates cannot hold and that a weight has to be added: $\rho$ can be bounded in terms of the norm of $|v|^\gamma f_0$. We conjecture a range of optimal estimates, and partially prove them. Finally, these results have natural applications to the $X$-ray transform and Kakeya problems on Euclidean cylinders.

We consider two coupled linear heat equations on different spatial domains that interact through a lower dimensional interface. This models conjugate heat transfer. The problem is solved using Dirichlet–Neumann waveform relaxation. This allows us to couple separate codes for the subproblems, a so-called partitioned approach. Our overall goal is to develop more efficient partitioned methods, and to this end, we want reliable error estimates. We use an exponentially weighted Fourier technique to derive new error estimates in $L^2$ for finite time $T$ in both continuous and time-discrete settings. We identify an optimized relaxation parameter that guarantees superlinear convergence. Our new continuous estimate predicts linear convergence when $k^2/T$ is small, and superlinear when $k^2/T$ is large, with $k$ the iteration index. Our new time-discrete estimate closely mirrors its continuous counterpart for small $\Delta t$. Otherwise, we obtain linear convergence for large $k$. We also show that convergence is fast when the contrast is large, provided that the small physical parameter domain (e.g. air) is using the Dirichlet transmission condition, and the large physical parameter domain (e.g. steel) is using the Neumann transmission condition in the Dirichlet-Neumann waveform relaxation method. Numerical experiments confirm these findings, and we demonstrate the relevance on the analysis for a nonlinear conjugate heat transfer test case.

Unsourced integrated sensing and communication (UNISAC) has emerged as a promising paradigm for supporting massive connectivity in 6G networks. However, existing approaches predominantly rely on fixed-position antennas at the base station (BS) and user equipment (UE). In uplink transmission with huge access density and limited resource budgets (i.e., finite blocklength, FBL), the fixed arrays are constrained by their physical aperture and static spatial sampling, which lead to severe multi-user interference and an unavoidable pilot collision error floor. To conquer the bottleneck derived from fixed-position physical constraint and utilize the abundant spatial diversity within compact space, this paper proposes a novel unsourced ISAC framework incorporating a fluid antenna system (FAS) at the user side. The proposed scheme exploits the positional flexibility of FAS to reconfigure the channel environment by continuously adjusting antenna ports in the spatial domain. Numerical results demonstrate that the proposed FAS-aided approach significantly reduces the per-user probability of error (PUPE) and enhances angle-of-arrival (AOA) sensing accuracy. Specifically, the proposed scheme provides a 40 dB capacity gain over traditional TDMA at 1000 active users. It should be noted that the FAS considered in this paper is only deployed at the transmitter. In our future work, we will try deploying FAS at both the transmitter and receiver.

The Schatten-$q$ quasi-norm is a widely used nonconvex rank surrogate and matrix factorization is an effective approach to reduce computational cost. In this paper, we consider the equivalent group-sparse factorized reformulation of Schatten-$q$ norm regularized low-rank matrix recovery problem. Though this factorized model exhibits favorable performance, two issues remain: (i) the error bound of critical points is unexplored; (ii) the proximal operator of $\|\cdot\|_2^q$ lacks a closed-form solution for general $q$, limiting algorithms to adopt fixed $q$ like $1/2$ or $2/3$. This paper addresses both issues. We investigate the properties of critical points for the factorized problem and show that, compared to nuclear norm, the Schatten-$q$ norm implicitly endows critical points with column orthogonality. From this insight, we introduce the notion of S-critical points under mild conditions that ensure column orthogonality with easily operable criterion for identifying. We show that global optimal points must be S-critical points and we derive an error bound between S-critical points and the true matrix. We further present an inexact proximal alternating linearized minimization method for the factorized problem, along with practically computable inexact proximal operator for $\|\cdot\|_2^q$ and criteria to find solutions satisfying inexactness conditions, and we establish the whole sequence convergence and a convergence rate guarantee under Kurdyka–Łojasiewicz condition. Moreover, we prove that the factorized model with least-squares loss has KL exponent $1/2$ at S-critical points, then the iteration converges linearly under suitable condition. Extensive numerical experiments validate the effectiveness of our algorithm and confirm the theoretical properties of the factorized model.

We study composition operators induced by a smooth symbol between weighted Bergman spaces of the polydisc. We first prove a compactness criterion that only requires knowing what happens on the distinguished boundary. Then we prove simple geometric characterizations of boundedness and compactness on some $A^2_\beta(\mathbb{D}^d)$, particularly for $\beta > d-3$.

Inspired by nonequilibrium phenomena in game dynamics and behavioral evidence on the impact of extreme events on decision making, we investigate the nonlinear dynamics of a discrete-time multiagent learning rule in population congestion games under extreme events affecting one of the actions. The population state, following a risk-sensitive variant of the Multiplicative Weights Update (MWU), is coupled with a belief variable capturing the agents perceived risk and updated through an adaptive expectation rule. We perform a two-parameter bifurcation analysis with respect to the agents controlled parameters, identifying regions of qualitatively distinct behavior. Equilibria are studied first from both game-theoretic and dynamical perspectives. The resulting two-dimensional system exhibits complex behavior, including multi-stability among fixed points, invariant curves, periodic and chaotic attractors. Despite this complexity, the attractors can be grouped into distinct families, while the Cesàro averages of the trajectories are shown to converge to the stationary equilibrium. The incorporation of risk associated with the extreme event leads to new dynamical phenomena: attracting invariant curves arise and give rise to phase-locking Arnold tongues, within which the dynamics is qualitatively similar. In this setting, codimension-two resonances are identified as organizing centers, both within individual tongues and along the bifurcation curves associated with the fixed-point family. Chaotic attractors emerge and are destroyed through Feigenbaum cascades and forward or reverse boundary crises, with interior and merging crises also observed, along with transient chaos and narrow periodic windows. For each qualitatively distinct region, representative phase portraits and the associated basins of attraction are examined.