Papers

We prove that every group $G$ definable in the pure $p$-adic field $\mathbb{Q}_p$ is weakly regular. We show that every one-parameter subgroup is definable. We also show that its one-parameter core $G_u$, together with a suitable regular open subgroup $G_\Omega$, is definable. Finally, we show that if $H$ is a dfg component of $G$, then $G_u=\langle(H^g)_u:g\in G\rangle$. In fact, $G_u$ is a finite product of the one-parameter cores of finitely many conjugates of $H$. In particular, when $G$ is definably amenable, $G_u=H_u$.

We study finite local-unit-aligned totally ordered monoids, that is, finite totally ordered monoids in which each element has coinciding greatest right and left local units. We prove that every such monoid admits a canonical rigid chain-indexed direct-system representation, and conversely that every rigid system of the corresponding kind reconstructs a finite local-unit-aligned totally ordered monoid. The representation is induced intrinsically by the local-unit map $\tau$, through the canonical stratification of the positive idempotent skeleton into $\tau$-multiplication-coherent blocks. More precisely, from $\tau$ we construct component monoids and transition maps forming a strictly compatible finite chain-indexed direct system from which both the ambient order and the ambient multiplication are recovered. In the finite case, strict compatibility forces every proper transition map to be unit-constant; this rigidity makes the canonical components $\tau$-multiplication-cohesive and yields a Clifford-type ordinal-sum-like reconstruction theorem for finite local-unit-aligned totally ordered monoids.

Given a dual uniform algebra $A=X^*$ with maximal ideal space $M_A$, we provide the first sufficient condition in terms of the Gleason distance of $A$ for a sequence in $M_A\cap X$ to be interpolating for $A$. We prove that a sequence in $\mathbb{D}^N$ is uniformly separated if and only if it is interpolating for $H^\infty(\mathbb{D}^N)$ and its sequence of norms satisfies the Blaschke condition, and then use this characterization to classify the interpolating sequences for $H^\infty(\mathbb{D}^N)$ in terms of its Gleason distance. We also study interpolating sequences for $\mathscr{H}^\infty$, the algebra of bounded Dirichlet series, obtaining necessary and sufficient conditions for a sequence in $\mathbb{C}+$ to be interpolating for this space, and relating the geometry of such sequences to that of the interpolating sequences for $H^\infty(\mathbb{C}+)$. Finally, we show that a sequence in the Shilov boundary of the second dual of a uniform algebra $A$ is interpolating for $A^{**}$ if and only if it is discrete for the $w^*$-topology.

Let $d,h$ be positive integers with $h$ even. A set $\mathcal{A} \subseteq\mathbb{Z}^d$ is a $B_h$-set if the sums of $h$ elements of $\mathcal{A}$ are distinct up to the order of the summands. A $B_2$-set is also called a Sidon set, a Golomb ruler, and a Babcock set. Let $K\subseteq \mathbb{R}^d$ be a convex body containing the origin, and let $A_K(n)$ be the number of elements of $\mathcal{A}$ in $\bigcup_{t<n}t\ast K$. We prove that if $h$ is even then \[ \liminf_{n\to\infty} \frac{A_K(n)}{\left( vol(n\ast K)/\log n \right)^{1/h}} \le \left(\frac{4}{d\log 2} \cdot \frac{\Gamma(1+h/2)^2\Gamma(1+d/2)^2}{\Gamma(1+d/h)^{h}}\right)^{1/h}. \] The constant does not depend on $K$. As a special case, this explicitly bounds $\liminf_n A(n) \left(\frac{\log n}{n^d}\right)^{1/h}$, where $A(n)= | \mathcal{A}\cap[0,n)^d | $, which Rackham-Šarka (2010) had proved finite.

Traffic shockwaves are stop-and-go waves that propagate upstream through the streams of vehicles and are one of the major causes of traffic congestion, fuel inefficiency, and increased accident rates in modern transportation systems. Although Connected and Autonomous Vehicles (CAVs) offer a promising opportunity to mitigate such shockwaves, most existing control strategies rely on global traffic state information, making them impractical for early-stage deployment of Vehicular Ad-hoc Networks (VANETs). In this paper, we propose a decentralized Multi-Agent Reinforcement Learning (MARL) framework that integrates a Graph Neural Network (GNN) to enhance the control architecture of connected and autonomous vehicles. The proposed approach enables vehicles to learn cooperative control policies using locally available information and interaction with neighboring vehicles. The effectiveness of the proposed scheme is evaluated using a scalable simulation environment under realistic highway traffic conditions. Simulation results show that the proposed GNN-based MARL framework can reduce the propagation of traffic shockwaves by up to 80%, even when only 10% of the vehicles are connected.

We prove tight asymptotics of the cover time $\tau_{\mathrm{cov}}(d)$ of a continuous-time branching random walk on the Hamming graph $\{0,1,\dots,b-1\}^d$, as $d\to\infty$. We focus on the slow-branching regime, where particles move at rate one and branch at rate $\lambda\in(0,1)$. For $b>2$, we show that $\tau_{\mathrm{cov}}(d)=x_\star d+\lambda^{-1}\log d+O_{\mathbb P}(1)$. For $b=2$, we show that $\tau_{\mathrm{cov}}(d)=x_\star d+\chi^{-1}\log\log d+O_{\mathbb P}(1)$. Here, $x_\star$ and $\chi$ are explicit positive constants depending only on $b$ and $\lambda$. Our results sharpen previously known linear-order estimates. The dichotomy reflects the geometry of the last uncovered region: for $b>2$, there are exponentially many antipodes, whereas the binary hypercube has a unique antipode and its neighbors govern the final coverage. Our proofs combine classic spine change of measure techniques and many-to-few estimates with a multiscale decomposition of the genealogy and a weighted martingale analysis of the early population.

For $m=5$ and for every $m \geq 7$ we construct families of pairwise non-isomorphic free-by-cyclic groups isomorphic to $F_{2m} \rtimes \mathbb{Z}$ having isomorphic profinite completions. This answers a question of Bridson & Reid from 2015.

Stochastic matching on hypergraphs is an important topic for its versatility in capturing real-life systems, from living donor transplant to ride-hailing. Nevertheless, finding necessary and sufficient criteria for stability is a long-standing problem. One of the key difficulties is the fact that greedy policies, whilst maximally stable for stochastic matching on graphs, no longer achieve maximal stability region on hypergraphs. So far, no alternative families of policies with similar properties have been known. In this work, we introduce online assignment policies, in which each item is assigned to a matching hyperedge type upon arrival. We show that this is a good generalisation to greedy policies, by proving that they are maximally stable. Their natural amenability to analysis allow us to derive several necessary and sufficient criteria for stability, which generalise the known criteria for graphs. Furthermore, the constructive proof gives a maximally stable arrival-rate agnostic policy.

The discovery of scaling laws has motivated training neural networks on ever increasing quantities of data. This is typically done with a constant decoupled weight decay which causes the network weights to shrink steadily over the course of training. Taking inspiration from the Robbins–Monro conditions, we propose to scale weight decay by the fraction of the peak learning rate $\eta/\eta_{\max}$. We prove that this scaled weight decay preserves the asymptotic stationarity guarantees of the corresponding unregularized methods for both stochastic gradient descent and the non-Euclidean spectral optimizer Muon, thereby avoiding the additional asymptotic bias introduced by constant decoupled weight decay. This retains the stability benefits of weight decay without changing the asymptotic optimization target. Using a steady-state analysis, we explain why under standard weight decay the weight norm shrinks steadily as training proceeds, whereas under scaled weight decay it settles to a roughly constant value. When applied to the training of mixture-of-experts models, Muon with scaled weight decay (Muon-SW) consistently outpaces Muon with identical hyperparameters, reaching the same validation loss $\mathbf{30\%}$ faster at our largest scale across models from $72 - 930$ million parameters trained at $\sim 600$ tokens per active parameter. If this trend continues to hold, the method promises to substantially accelerate the pre-training of frontier models while requiring only a few lines of code to implement.

A $t$-$(v,k,\lambda)$ covering is a collection of $k$-subsets (blocks) of a $v$-set such that every $t$-subset of points lies in at least $\lambda$ blocks; the covering number $C_\lambda(v,k,t)$ is the least number of blocks in such a collection, and one writes $C(v,k,t)$ when $\lambda=1$. The recorded bounds for $C(12,6,4)$ have been $40 \le C(12,6,4) \le 41$. We show that no $4$-$(12,6,1)$ covering with $40$ blocks exists, and hence that $C(12,6,4)=41$. A counting argument shows that in a hypothetical $40$-block covering every point lies in exactly $20$ blocks, the link of every point is an optimal $3$-$(11,5,1)$ covering with a forced degree sequence, and the six pairs of points of degree $10$ form a perfect matching; an exhaustive case analysis over the orbits of a group of order $3840$, carried out by satisfiability solving, then shows that no optimal $3$-$(11,5,1)$ covering occurs as such a link. Each of the $81$ formulas in the primary proof has an unsatisfiability certificate checked by drat-trim and by the formally verified checker cake_lpr; two additional cross-encoding certificates are checked by the same pipeline. The lower-bound argument uses no tabulated covering number: its only numerical input, $C(10,4,2) \ge 9$, is itself certified. As a by-product the certificates yield a self-contained certified proof that the optimal $3$-$(11,5,1)$ covering is unique up to isomorphism. Equivalently, the Turán number $T(12,8,6)$ is $41$; the new value propagates to improved lower bounds for $C(13,7,5)$, $C(14,8,6)$, $C(15,9,7)$ and $C(16,10,8)$.

We explore the prospect of basis-like results for the class of Aronszajn lines which are consistent with the Continuum Hypothesis (CH). In particular, we prove that each of the following statements is consistent with CH: Any two Countryman lines contain isomorphic or anti-isomorphic uncountable suborders; for any coherent Aronszajan tree $T \subseteq {}^{< \omega_1} \omega$, the filter $\mathcal{U}(T)$ is an ultrafilter. The first result confirms a conjecture of Shelah in the context of CH which was previously shown to follow from the Proper Forcing Axiom. On the other hand, the weak diamond principle $2^\omega < 2^{\omega_1}$ implies that there does not exist a two element basis for the Countryman lines.

We consider a three-dimensional fluid-structure interaction problem coupling the incompressible Navier-Stokes equations in a time-dependent domain with a square-root damped plate equation, which is posed on the moving upper boundary of the fluid. We prove, a priori, the exponential decay of strong solutions for initial data that are sufficiently small in a suitable Sobolev space. The proof combines higher-order energy estimates, Stokes-type regularity bounds, and a nonlinear bootstrap scheme that closes under the smallness assumption.

We study set-defined graph classes: hereditary classes whose vertices are assigned fixed-length numerical tuples, with adjacency determined solely by equality patterns among coordinates. These classes arise in structural graph theory, communication complexity, logic, and adjacency labeling schemes. We ask when they are $\chi$-bounded, that is, when chromatic number is bounded in terms of clique number throughout the class. First, we prove a decomposition theorem: every graph in a set-defined class can be partitioned into a number of parts polynomially bounded in its clique number, each inducing a union of a bounded number of shift-colorable graphs, that is, graphs admitting a homomorphism to a shift graph. Thus bounded unions of shift-colorable graphs form the fundamental obstruction to $\chi$-boundedness in set-defined classes. For full set-defined classes, consisting of all graphs realizable by a fixed Boolean rule on equality patterns, we prove a stronger dichotomy: every such class is either polynomially $\chi$-bounded or contains shift graphs of arbitrarily large chromatic number. Moreover, we provide an algorithm that, given a Boolean-function description of a full set-defined class, decides $\chi$-boundedness of the class. It reduces the problem to feasibility of tropical linear programs, and its correctness follows from a duality with winning strategies in mean-payoff games. Conversely, every integer system of tropical inequalities, and hence every mean-payoff game, can be encoded in strongly polynomial time as a set-defined class whose non-$\chi$-boundedness is equivalent to feasibility. This provides a graph-theoretic counterpart of tropical feasibility and mean-payoff-game solvability, linking structural graph theory, tropical algebra, and game-theoretic algorithms.

A $3$-uniform hypergraph (or $3$-graph) $H$ on $n$ vertices is \emph{$(n,d,\mu)$-dense} if $e_H(X,Y,Z)\ge d|X||Y||Z|-\mu n^3$ for all $X,Y,Z\subseteq V(H)$. This is one of the weakest standard notions of quasirandomness for $3$-graphs and is also known as linear quasirandomness. In this paper, we determine the sharp diagonal thresholds for tight Hamilton cycles in $(n,d,\mu)$-dense $3$-graphs $H$ under conditions on the minimum vertex degree $\delta_1(H)$ and the minimum codegree $\delta_2(H)$. We actually prove a general result: define \[ f(d):=\frac{1-\sqrt{(4d-1)/3}}2. \] We prove that $(n,d,\mu)$-density together with $\delta_1(H)\ge\alpha\binom{n-1}{2}$ forces a tight Hamilton cycle whenever $d > 1/3$ and $\alpha>f(d)$. In particular, $f(1/3)=1/3$, which answers Problem~8.3(i) of Araújo, Piga and Schacht and confirms Conjecture~8.1 of Han, Shu and Wang. For the minimum codegree condition, the sharp diagonal threshold is $(\kappa,\kappa)$, where $\kappa$ is the unique real solution of $\kappa=(1-\kappa)^3$. Since $\kappa\approx0.3177>1/4$, this gives a negative answer to Problem~8.3(ii) of Araújo, Piga and Schacht and disproves Conjecture~8.2 of Han, Shu and Wang. The two proofs use a common Hamilton-framework reduction, but the two degree conditions lead to distinct dominant-component lemmas for $(n, d, \mu)$-dense $3$-graphs, which are of independent interest and whose proofs do not rely on the absorption method.

In this paper, we investigate the subtractive closure operator in commutative hemiring theory. While this operator satisfies standard closure properties, it does not distribute over ideal addition, intersection, or multiplication in general. We introduce Kuratowski hemirings (where closure preserves joins) and nucleus hemirings (where closure preserves meets), providing explicit counterexamples and positive results for each. Key findings include: the standard semiring $\mathbb{N}_0$ and the max-plus algebra are nucleus hemirings with multiplicative distributivity; the interval semiring $T = [0,1]$ is Kuratowski and multiplicatively distributive but not nucleus; and polynomial hemirings $S_0[X,Y,Z]$ over zerosumfree hemirings form a broad class of non-nucleus counterexamples. We also correct a recent claim regarding distributivity over arbitrary intersections.

In Question~3.1 of his 1995 paper on depth and transfer, Carlson asked whether the depth of a finite-group cohomology ring is always realized by the dimension of one of its associated primes. We give a negative answer. Let \[ G=\SG{128}{859},\qquad k=\kbar. \] An exact presentation certificate proves that $\depth H^*(G;k)=2$. Okuyama's associated-prime theorem would convert an associated prime of dimension two into a rank-two elementary abelian subgroup $E\leq G$ satisfying $\depth H^*(C_G(E);k)=2$. We enumerate all $75$ rank-two elementary abelian subgroups of $G$ and obtain six centralizer types. Duflot's theorem gives depth at least three for four types, while exact ideal-quotient certificates exhibit regular sequences of length three for the remaining two. Hence every rank-two centralizer has cohomological depth at least three, so $H^*(G;k)$ has no associated prime of dimension two. The finite group presentation, the three cohomology-ring presentations, the enumeration summary, and the exact algebraic certificates are included for independent verification.

Let \(G=\Spin(10)\) be the split spin group over a field of characteristic different from \(2\), and let \(T\subset G\) be a split maximal torus. We determine the image of the integral Chow restriction map \(\CH(BG)\to \CH(BT)^W\), equivalently \(\CH(BG)\) modulo torsion. The main new geometric ingredient in the proof is a construction of the class \(c_2c_3c_5\), where the \(c_i\) are the elementary Chern classes after restriction to \(T\). This class is obtained from the proper equivariant push-forward associated with the affine cone over the spinor variety in its half-spin embedding for the special Clifford group \(\Gamma^+(10)\). Modulo two, the image is the subring generated, over the smallest Steenrod-stable subring of \(\F[c_2,c_3,c_4,c_5]\) containing \(c_2^2,c_3^2,c_4^2,c_5\) and \(c_2c_3c_5\), by the torus restriction of the top Chern class of a half-spin representation. The integral characteristic image is the full inverse image of this mod-two subring under reduction modulo \(2\).

We develop a pluripotential approach to complex Hessian equations on compact Hermitian manifolds. In this setting, the lack of closedness of the background metric introduces torsion terms that prevent a direct extension of the Kähler theory. Our main result is a uniform $L^\infty$ estimate for bounded $\omega$-$m$-subharmonic solutions of the equation \[ (\omega + dd^c u)^m \wedge \omega^{n-m} = cf\,\omega^n, \] under the assumption that $f \in L^p$, $f \ge 0$ for some $p>1$. The proof combines a weak comparison principle with torsion error, a capacity theory adapted to the Hermitian setting, and a nonlinear iteration scheme controlling the decay of sublevel sets. As applications, we obtain existence, stability and compactness results for weak solutions with $L^p$ densities. These results extend several aspects of the pluripotential theory. of complex Hessian equations beyond the Kähler framework.

We study the problem of recovering latent inner products from a random geometric graph with anisotropic Gaussian latent points. More precisely, for an i.i.d. sample $x_1, \dots, x_n \sim N(0,\Sigma)$ where $\Sigma \in \mathbb{R}^{d \times d}$, an edge $(i,j)$ is present in the graph if and only if $\langle x_i, x_j \rangle \ge \zeta$ for a threshold $\zeta$. We assume the threshold $\zeta$ to be chosen such that the average edge density of the graph is of constant order. To address the undesired degree fluctuations amplified by the anisotropy of the latent points, we consider the doubly centered adjacency matrix of the graph, and estimate the latent inner products using a rank-$d$ spectral approximation of the doubly centered matrix. The estimator obtains a mean squared error with a rate involving the stable rank of the covariance matrix $\Sigma$. Notably, the rate of estimation matches the state of the art for the isotropic case $\Sigma = I_d$, and permits an ill-conditioned covariance matrix with a diverging condition number. The analysis of the spectral method proceeds via the entrywise Hermite expansion of the doubly centered adjacency matrix with respect to the latent inner products. Instead of the standard trace method, it uses a decoupling argument recently introduced by Kaushik, Romberg, and Muthukumar (2025) to control nonlinear error terms.

Huang and Shen developed a novel class of high-order BDF-IMEX consistent-splitting schemes for the incompressible Navier-Stokes equations, giving the first rigorous stability and error analysis for a fully decoupled splitting scheme of temporal order higher than two. Extending their analysis from unit viscosity to arbitrary viscosity, this work reveals that the error upper bound coefficient contains inverse powers of the viscosity. Our numerical experiments show that the scheme can break down at high Reynolds number. To save the scheme from this failure, we stabilize it by adding to the velocity update a symmetric positive-semidefinite spectral vanishing viscosity operator, built from the directionally applied Maday-Kaber-Tadmor kernel, which selectively damps the high, under-resolved modes at no additional asymptotic cost and leaves the structure of the error analysis intact. We establish stability and error estimates for the stabilized scheme in which the spectral vanishing viscosity provides viscosity-independent coercive control of the high modes. Three two-dimensional tests demonstrate the robustness and accuracy of the stabilized scheme. For a manufactured solution, the stabilized scheme retains its design order for k=2,3,4, whereas the unstabilized scheme diverges. For a perturbed Kovasznay flow, it accurately resolves the boundary layer at Re=10^4 and drives the perturbation back to the steady state, while the unstabilized scheme blows up. For the Kelvin-Helmholtz instability problem, it reproduces the reference integral diagnostics throughout the reliable regime, whereas the unstabilized scheme produces spurious solutions or blows up.

For the canonical problem of estimating a multivariate normal mean under squared error loss, we demonstrate, for the first time, the existence of proper Bayes minimax multiple shrinkage estimators by introducing a general approach for their explicit construction. As opposed to minimax shrinkage estimators that shrink towards a single prespecified target, minimax multiple shrinkage estimators adaptively shrink towards the more promising of a set of prespecified targets, substantially increasing the region of potential risk reduction while maintaining the protection of always being at least as good as the maximum likelihood estimator. These estimators are particularly useful in practice as they address the challenge of selecting a minimax shrinkage estimator when prior information suggests more than one viable shrinkage target to choose from. In contrast to previous formal Bayes minimax multiple shrinkage estimators, which were built on mixtures of superharmonic marginals, these proper Bayes minimax multiple shrinkage estimators are obtained via mixtures of square-root superharmonic marginals. Examples of such proper Bayes minimax multiple shrinkage estimators include an adaptive convex combination of the rescaled Strawderman shrinkage estimators.

We introduce sequence distortion spectrum, a quasi-isometry invariant recording the large-scale distance profiles of sequences indexed by $\mathbb N$ or $\mathbb Z$ in a metric space. For a rate function $f:\mathbb N\to(0,\infty)$, extended by $f(t)=0$ for integers $t\le 0$, a sequence $(p_n)$ in a metric space $X$ is \emph{$f$-distorted} if there exists an integer $C\ge 1$ such that for all $m,n$ we have $$\frac{1}{C}f(\lfloor \frac{1}{C}|n-m|-C\rfloor)\le d(p_n,p_m)\le C f(C|n-m|+C)+C.$$ This definition implies that $f(N)=O(N)$. For rate functions, realizability depends only on the ambient quasi-isometry type and the growth type of $f$. We classify the possible power rates $f(N)=N^\alpha$ (where $0<\alpha\le 1)$ for Euclidean spaces: in $\mathbb R$ only the linear rate $\alpha=1$ occurs, while in $\mathbb R^k$, $k\ge2$, the realizable exponents are exactly $1/k<\alpha\le 1$. For a geodesic $\delta$-hyperbolic space $X$, no power rate $N^\alpha$ with $0<\alpha<1$ occurs. The hyperbolic plane also realizes the logarithmic rate. An exponential packing bound for $X$ rules out every $o(\log N)$ rate, but a proper CAT$(-1)$ surface of unbounded geometry realizes a log–log rate. In an arbitrary simplicial tree, every realizable rate is linear up to constants. Finally, we construct two pairs of proper geodesic spaces: the first has equivalent basepoint packing functions and the second equivalent uniform packing functions; both pairs have equal asymptotic dimensions and filling-function growth classes, and isometric asymptotic cones at the chosen wedge points for every common scaling sequence and ultrafilter. Yet sequence distortion distinguishes each pair, and the second pair has bounded geometry.

We study continuous data assimilation for the two-dimensional Navier–Stokes equations on a smooth, bounded, connected domain with Navier-slip boundary conditions, using no interior observations. The available data consist only of finite-dimensional measurements of the tangential velocity on a non-empty relatively open subset $\Gamma\subset\partial\Omega$. We prove that sufficiently strong boundary feedback, constructed from sufficiently fine observations, generates a coercive spectral gap for the assimilation error. The limiting gap is identified with that of a mixed-boundary problem obtained by imposing a homogeneous Dirichlet condition on $\Gamma$, and is shown to be of order $\nu$. Combining this feedback-induced coercivity with an estimate of the non-linear error production in terms of the long-time averaged symmetric-gradient energy of the reference solution, we obtain a sufficient criterion for exponential synchronisation. We verify this criterion in the unforced case, for sufficiently small forcing when the unnudged Navier-slip form has an $\mathrm{L}^2$ spectral gap, for sufficiently large viscosity on domains without tangential rigid motions, and for sufficiently large viscosity in the presence of positive boundary friction. We also treat perfect slip on domains admitting tangential rigid motions, where synchronisation follows under a smallness condition on the non-rigid solenoidal component of the forcing.

In this article, we investigate conformal Killing vector fields on closed Riemannian manifolds under a curvature pinching condition. By establishing a new Bochner-type identity for the $1$-form dual to a conformal Killing vector field, we derive a sharp gradient estimate via a Moser iteration procedure. Based on this estimate, we prove that, under a suitable upper bound on the Ricci curvature, every nontrivial conformal Killing vector field must be nowhere vanishing. Consequently, on even-dimensional manifolds with non-zero Euler characteristic, every conformal Killing vector field vanishes identically, which in turn implies that the conformal transformation group of such a manifold is finite. Our results extend the classical rigidity theorems of Yano and Bochner from the setting of non-positive Ricci curvature to that of small positive Ricci curvature, and generalize the recent results of Chen and Han from Killing vector fields to conformal Killing vector fields.

Societies often struggle to prioritize important challenges in a timely manner, with substantial costs from delayed action on issues like climate change and pandemic mitigation. A persistent puzzle is that broad concern on issues often fails to translate into collective priority. We argue that a key driver lies in how concern is formed across competing issue domains. Some issues depend heavily on social learning, where individuals infer importance from others, often because direct experience is limited. Others depend more on individual learning from firsthand experience. We develop a dynamic model in which two subgroups form issue-specific concerns through individual and social learning, and these concerns are aggregated into collective priority. The model yields three insights. First, with two issues of equal objective severity, the one that depends more on social learning tends to be underprioritized when both issues are severe. Second, gradual increases in severity delay reprioritization of the issue, with the delay growing as reliance on social learning increases. Third, this bias can be reduced by reducing social learning or by increasing intergroup learning beyond a critical threshold. These results offer a general mechanism for why severe problems can remain neglected in collective action despite widespread concern, and why intergroup interaction or experiential simulations may help align collective priorities with objective risks.

In 1995, Brakke proposed a formulation of the Plateau problem for a given boundary $\Gamma$ that allows for triple junctions and tetrahedral singularities. Let $\Gamma$ be a smooth closed curve and let $G=\pi_1(\mathbb{R}^3\setminus \Gamma)$. For each proper finite index subgroup $N$ of $G$, Brakke constructs a $(\mathrm{\mathbf{M}}, 0, \infty)$-minimal surface $\Sigma_N$, which is obtained as the projection of the boundary of a perimeter-minimising fundamental domain in the covering space associated to $N$. We advance the theory in two ways. Firstly, we extend Brakke's construction to include all normal subgroups $N\triangleleft G$, i.e. possibly with infinite index. Secondly, we prove a compactness result which implies that there exists a proper normal subgroup $N_0\triangleleft G$ such that \[ \mathrm{Area}(\Sigma_{N_0})=\inf_{\{N\triangleleft G, N \neq G\}}\ \mathrm{Area}(\Sigma_N). \] A similar result holds when $\Gamma$ has many connected components. Furthermore, we study the spanning and minimising properties of the $\Sigma_N$s and $\Sigma_{N_0}$.

Originating from the complex approximation of holomorphic functions, the Runge approximation for elliptic equations has evolved into a fundamental tool for inverse problems and even learning-based numerical methods since its proposition by Lax, Malgrange, with quantitative characterizations further established by Rüland and Salo. It should be remarked here that Runge approximation is ill-posed. In numerical analysis, explicit quantitative estimates are required to characterize the dependence of the approximant's growth on the outward continuation distance of the original solution. This paper investigates the spatial dependent quantitative Runge approximation for the Helmholtz equation using cylindrical harmonics, considering both interior and exterior boundary value problems. We explicitly derive the relevant indices for the three-circle configuration and obtain asymptotic indices for general geometric settings. The derived results provide norm estimates for the expansion coefficients, which are crucial for the implementation of regularization methods. Furthermore, the established bounds enable the construction of spectrally accurate numerical approximations for solutions to the Helmholtz equation.

In this paper, we study the Cartan decomposition and Iwasawa decomposition of the Chevalley group from the root category. Then we investigate the Iwasawa decomposition for totally positive submonoid of the Chevalley group.

De Finetti theorems convert permutation symmetry into approximate mixtures of product states and thereby justify a wide range of reductions in classical and quantum statistics. In this work we study de Finetti hierarchies in which the feasible states are additionally constrained to be fixed points of quantum channels, a condition that subsumes invariance under arbitrary compact symmetry groups. Combining the mean-ergodic theorem with the structure theory of conditional expectations, we prove a tight bound on the entanglement-assisted classical capacity of the dual of a conditional expectation, block-wise distortion bounds for informationally complete measurements adapted to fixed point algebras, and an exact type-based refinement of the chain rule for permutation-invariant states. From these tools we derive several de Finetti theorems: a double-sided extension theorem with $O\left(\sqrt{\log n}/n\right)$ convergence, an interpolation theorem whose dimension dependence is governed solely by the block structure of the fixed point algebras and which recovers the dimension-independent classical behavior for maximal tori, and a Bose-symmetric variant. Exploiting Schur-Weyl duality and Gelfand-Tsetlin bases, we further show that the rounding scheme producing certifiably good separable inner approximations for (constrained) separability problems can be implemented in time polynomial in $1/\epsilon$ for fixed local dimensions, complementing the known efficient outer hierarchies. Applications to bilinear optimization under symmetries and to approximate quantum error correction are discussed.

This paper shows that surpassing the Chu limit on $Q$-factors via time-modulated matching requires non-smooth switching strategies. We derive a nonlinear differential condition to show that differentiable modulation functions are sub-optimal, then show that the optimal switching trajectory for maximizing the violation of the Chu limit is a piecewise-constant (Bang-Bang) profile. Finally, we establish an upper-bound connecting the antenna size, switching speed, and bit error rate, which reveals that switching time becomes longer as antennas become electrically smaller.