Papers

Oscillator Potts machines (OPMs) provide a physics-inspired, energy-minimization framework for solving combinatorial optimization problems described by the $q$-state Potts Hamiltonian. Although OPMs may be viewed as multistate extensions of oscillator Ising machines (OIMs), here, we show that they exhibit dynamical properties absent in the binary case. Specifically, we derive a configuration-dependent local-stability condition for a recently proposed multiharmonic OPM formulation and show that configurations with the same Potts energy need not be dynamically equivalent. In particular, for $q\geq4$, permutations of the Potts labels can alter the Jacobian spectrum and, consequently, the regularization strength required to locally stabilize a given Potts configuration. Thus, different phase encodings of the same Potts solution can exhibit different local stability properties despite having identical Potts energies.

DNA labeling has attracted increasing attention in biomedical applications, including molecular imaging, diagnostics, and genomic analysis. In a DNA labeling process, a set of DNA sequence patterns, referred to as labels, is designed according to the requirements of a specific application. For each DNA sequence, the labeling process generates an output sequence that records the positions of the labels. DNA sequences with different labeling outputs can therefore be distinguished through the labeling process. To quantify this capability, the labeling capacity is defined as the exponential growth rate of the maximum number of DNA sequences that can be distinguished through the labeling process as the sequence length tends to infinity [2]. To date, the labeling capacities of several cases in the single-label setting have been determined. In this paper, we formulate the labeling process as a deterministic channel and show that its zero-error capacity is equal to the labeling capacity. For a single label, the corresponding channel can be represented by a star graph. Thus, characterizing the labeling capacity of a single label is equivalent to determining the zero-error capacity of the corresponding star graph. We derive the zero-error capacities of all star graphs, thereby providing a complete characterization of the labeling capacities for all single-label cases. Furthermore, we develop a general method for constructing capacity-achieving codes. These results apply to labeling problems over arbitrary finite alphabets and are not restricted to the DNA alphabet. Finally, for a fixed label length, we exactly characterize the range of achievable labeling capacities and identify all single-label structures that attain the minimum and maximum capacities.

Let $k$ be a field of characteristic zero and let $n\geq 2$ be an integer. We prove that, with $\Z/n\Z$-coefficients, the category of effective motives with modulus is equivalent to the category of effective Voevodsky motives.

Out-of-distribution (OOD) generalization is a central challenge in scientific machine learning. We study regression problems in which the test distribution differs from the training distribution and ask: under what assumptions on the target function or operator is stable extrapolation possible, and how far beyond the training domain can one extrapolate? Existing theory controls the test error through additive penalties measuring the discrepancy between the training and test distributions. Such guarantees show robustness to small distribution shifts, but can be very pessimistic in comparison to OOD performance observed empirically. We identify classes of holomorphic functions and operators for which the OOD generalization error converges at algebraic rates even in the presence of large distribution shifts. This phenomenon stems from the increasing smoothness of higher-index coordinates, leading to what we term a `blessing of high dimensionality'. For learning with either polynomials, deep neural networks or deep neural operators, we derive explicit rates for arbitrary test measures supported on suitable domains and quantify how the admissible domain depends on the underlying regularity of the function or operator. Our extrapolation guarantees are independent of the test distribution, depending only on its support. We also present a series of numerical experiments across a range of functions and operators that support the main theoretical findings.

Let $G$ be a finite simple graph and let $A_c(G)$ be the Artinian algebra associated with its cover ideal. We prove that $A_c(G)$ has the WLP when $\tau(G)>|V(G)|/2$, where $\tau(G)$ denotes the size of a minimum vertex cover of $G$. As a consequence, $A_c(G)$ has the WLP with high probability when the Erdős-Rényi random graph model is considered. Moreover, we study the borderline case $\tau(G)=|V(G)|/2$ and as a result, classify the WLP for paths, cycles, Ferrers graphs, and well-covered trees.

Polyak step size (PS) and Armijo line search (ALS) have received increasing attention in stochastic optimization, with encouraging empirical performance and theoretical guarantees. However, their convergence theory for stochastic heavy ball (SHB) methods remains limited. In this work, we develop a unified convergence analysis for SHB equipped with PS and ALS. To this end, we introduce a modified Armijo rule that closely parallels the Polyak step size, together with a decoupling analysis that isolates the historical dependence induced by momentum. For SHB with standard PS and ALS, we establish expected convergence for strongly convex, convex, and non-convex objectives without interpolation or restrictive conditions on the momentum parameter. Under interpolation or strong growth, we further strengthen the results to almost sure rates and last-iterate convergence. Moreover, for general settings beyond interpolation, we prove almost sure convergence to the exact optimum or to stationarity for SHB with diminishing variants of PS and ALS. These results provide a more comprehensive theoretical view of Polyak step size and Armijo line search for stochastic heavy ball methods.

Let $A=(a_{ij})$ be an $n\times n$ real matrix and let $\lambda$ be a partition of $n$. Let $\phi^\lambda$ be the class function dual to the Young permutation character, and let $$ \phi^\lambda[A] = \sum_{\sigma\in\mathfrak{S}_n}\phi^\lambda(\sigma) \prod_{i=1}^{n}a_{i\sigma(i)} $$ be the corresponding monomial immanant. Stembridge [Canad. J. Math. 44 (1992), pp. 1079-1099] posed the following open problem: If all minors of $A$ of order at most $r$ are nonnegative, and the partition $\lambda$ has length at most $r$, is it true that $\phi^\lambda[A]\ge 0$? This paper solves the problem for all hook partitions.

We study federated learning (FL) with adversarial clients, where the goal is to minimize the average loss of the honest (non-adversarial) clients without knowing their identity. Under heterogeneity, a single shared model parameter is statistically inappropriate: it cannot capture the distinct data-generating processes across clients, incurring an irreducible model-heterogeneity bias and severely limiting robustness to adversarial clients (a.k.a. Byzantine-robustness). We address this problem through representation learning, where each client learns a personalized linear head, while collaboratively estimating a shared nonlinear representation through Byzantine-robust aggregation. We demonstrate that the heterogeneity among honest representation gradients is controlled by the representation error and statistical errors that decay either with the number of data samples per client ($\tau$) or the number of iterations ($T$). In particular, our non-asymptotic parameter recovery error bound reveals three terms: (i) an initialization-dependent error that goes away with $T$, (ii) finite-sample noise terms that decreases with $\tau$ and the number of honest clients, and (iii) a stochastic gradient variance term that also reduces with $T$. Importantly, with no irreducible model-heterogeneity bias in our bounds. We extend the regression analysis to multiclass classification, and empirically validate it on CIFAR-10, FEMNIST, and School Exam Score datasets.

We study how first-order differential observables affect the zero-frequency behavior of black-hole wave equations. Quasinormal modes and late-time tails do not always change in the same way. We give a simple condition for when an observable removes the leading threshold term of a Green function. For Schwarzschild Regge-Wheeler modes, the relevant operator is determined by the regular static solution. The transformation is nondegenerate at nonzero frequency but becomes globally degenerate at zero frequency. As a result, the nonzero quasinormal-mode problem is unchanged, while the leading fixed-radius late-time tail gains one extra inverse power of time.

We develop Picard Farey symbols for finite-index subgroups of \(G=\PSL_2(\ZZ[i])\) from the Gaussian Farey tessellation of \(\HH^3\) by ideal octahedra. The ambient geometry naturally involves two different index-two extensions of \(G\): the orientation-preserving group \(\PGL_2(\ZZ[i])\) and a reflection extension. This distinction explains the two \(G\)-orbits of Farey faces, which merge into one \(\PGL_2\)-orbit, and the classical extended modular tessellation appears as a totally geodesic slice. A Picard Farey symbol records finitely many Gaussian-rational octahedral occurrences, their local stabilizers, and cooriented face-pairing transformations. We prove that such a symbol reconstructs the finite Picard-cell action and hence the subgroup; moreover, when subgroup membership is decidable, the symbol can be found by a terminating finite double-coset procedure. For torsion-free subgroups we obtain whole-octahedron fundamental domains. The marked symbol also supplies finite orbit relations for the established Gaussian Steinberg edge–face presentation, with its corresponding Bianchi modular-symbol relations. When the natural modular-to-Picard coset map is bijective, a side-connected Kulkarni special polygon lifts, using the same modular representatives, to a Picard fundamental polyhedron whose intersection with the invariant real plane is exactly that polygon; the level-$(2+i)$/level-$5$ family is worked out explicitly. Comparison examples show that octahedral boundary minimization, absolute geometric face minimization, and preservation of such a modular section are distinct problems. Canonical reduction and effective Hecke reduction remain open.

We introduce endpoint k-path-transitive graphs, in which the automorphism group fixing two prescribed vertices pointwise acts transitively on the paths of length k joining them. We study the geodesic case, where k equals the distance between these vertices. We identify the union of these geodesics with the Hasse graph of a finite bounded graded poset and reduce it to proper blocks by complete cuts andmatching compression. Assume that the induced group K acts faithfully and 2-homogeneously on the first distance layer. We first prove that every nontrivial normal subgroup of K is transitive on every internal layer if and only if soc(K) is. Under this normal-basic condition, we classify the proper blocks in the affine case and the equal-width proper blocks with a 2-transitive internal layer in the almost-simple case. The nontrivial design interfaces are Paley or affine symplectic designs in the affine case, and projective designs, the 2-(11, 5, 2) design, the HigmanSims design, or their complements in the almost-simple case. The proper blocks have reduced rank at most four, whereas almost-simple proper blocks can have arbitrarily large reduced rank without the equal-width condition. We also give a stabilizer factorization criterion for assembling blocks while preserving endpoint-geodesic transitivity.

Physics-informed neural networks (PINNs) solve partial differential equations (PDEs) by incorporating governing physical laws into the training loss. For evolution equations, however, their conventional pointwise space–time representation does not explicitly encode temporal dependence, which can hinder accurate prediction. To mitigate this limitation, this work proposes a novel neural architecture termed a causal integral neural network (CinNet). The core module of CinNet is a Volterra-type causal integral term, which aggregates historical features to encode temporal dependence, thereby incorporating temporal causality at the architectural level rather than through training-level modifications as in many existing methods. Building on CinNet, we further develop a causal integral physics-informed neural network (CI-PINN) for solving evolution equations. Extensive numerical experiments on benchmark evolution equations demonstrate that the presented method outperforms various baseline PINN variants in terms of solution accuracy, with pronounced superiority under sparse-collocation scenarios. Additional empirical analyses show that CI-PINN exhibits low sensitivity to hyperparameter choices, while ablation studies confirm the effectiveness of the proposed network components.

The classification of block-transitive 5-designs remains an open problem. The additional parameter condition that $k$ divides $v$ is called the Camina-Gagen condition. In this paper, we investigate block-transitive simple $5$-$(v,k,2)$ designs satisfying the Camina-Gagen condition. Using the classification of finite $2$-homogeneous permutation groups, we consider the affine and almost simple cases separately. We prove that no such design admits a block-transitive automorphism group of affine type. For the almost simple case, up to isomorphism, there are exactly two possibilities: a $5$-$(12,6,2)$ design admitting PGL(2,11) as a block-transitive automorphism group and a $5$-$(24,8,2)$ design admitting PGL(2,23) as a block-transitive automorphism group.

Classical stochastic approximation methods rely on estimators of the first moment (mean) of a random regression function. We study methods that employ estimators of both the first and the second moments, which include modern deep-learning optimizers such as Adam and Muon as special cases. We derive second-moment stochastic approximation methods through the lens of optimal preconditioning for solving matrix equations, and develop a two-stage framework for their convergence analysis. The first stage focuses on the analysis of conceptual (impractical) methods that rely on the exact first and second moments. In the second stage, we replace the exact moments with their respective estimators, and invoke Dvoretzky's theorem to show that the resulting practical methods converge almost surely to a neighborhood of the target solution. The size of the neighborhood depends on the biases and variances of the first- and second-moment estimators. We derive concrete bounds for Muon and a spectral variant of Adam that determine the radius of their neighborhood of convergence.

We study multiple change-point detection in multivariate time series whose distributions change in a piecewise constant manner. Distributional changes can manifest across different moment orders, from shifts in the mean and covariance to changes in higher-order moments. Higher-order moments capture increasingly rich distributional features but become difficult to estimate in high dimensions. Our tensor representation unifies moments of different orders within a common linear algebraic framework, enabling a new method to detect changes in moments of all orders up to a prescribed fixed order $p$. The resulting procedure accommodates temporal dependence and allows the dimension of the time series to grow with the sample size. Under suitable regularity conditions, the proposed procedure achieves a localization error rate that matches a newly developed minimax lower bound. We further derive limiting distributions under both nonvanishing and vanishing moment jumps and construct asymptotically valid confidence intervals in the vanishing-jump regime. Numerical experiments and real-data analyses demonstrate the method's effectiveness in detecting moment changes and a range of distributional shifts.

Interference management in multi-stream multi-input multi-output (MIMO) device-to-device (D2D) networks often leads to weighted sum-rate maximization with sum-log-determinant involving matrix-valued signal-to-interference-plus-noise ratios (SINRs). The state-of-the-art paradigms, including weighted minimum mean-square error (WMMSE) and fractional programming (FP), have achieved tremendous success in link scheduling, power control, and beamforming problems. Recently, an upgraded FP approach, named surrogate-enhanced FP (SEFP) in a scalar form, has attained improved performance in joint uplink scheduling and power control in coordinated multicell SISO networks. The proposed SEFP improves the surrogate construction of the classical Lagrangian dual transform plus quadratic transform for logarithmic fractional objectives with a novel reciprocal-inverse transform (RIT), yet its extension to the matrix form for MIMO settings does not seem straightforward because the matrix ratio and the auxiliary matrix generally do not commute. In this paper, by leveraging mathematical tools from Hermitian functional calculus, we extend RIT to matrix ratios by lifting the scalar RIT along the eigendirections of an auxiliary matrix and recasting the resulting directional construction in an operator form. Following, we develop a matrix SEFP framework for weighted sum-log-determinant maximization. Further, we establish a unified view of SEFP and the recently proposed XMMSE method, specify when the two algorithms attribute to identical minorization-maximization (MM) surrogates and variable update trajectories, and prove XMMSE can be considered as a special case of SEFP under the algorithmic family perspective. Inspired by the unified view, we develop SEFPLinQ for the joint scheduling and beamforming optimization in flexible-association MIMO D2D networks. Numerical results demonstrate consistent performance gains.

The one-dimensional compressible Navier–Stokes–Fourier–Poisson system under the Boltzmann relation is studied for the existence of small-amplitude viscous shock profiles connecting the Rankine–Hugoniot states of the quasineutral Euler system with effective pressure $P_{\mathrm{eff}}=\rho(\theta+1)$. This profile is unique up to translation, of Lax type, and orbitally stable in time without a zero-mass assumption on the perturbation. The proof is based on a center-manifold reduction of the traveling-wave ODE and on the method of $a$-contraction with shifts, together with a relative-entropy functional augmented by electric energy and a sharp estimate of the viscous flux associated with $P_{\mathrm{eff}}$.

In a referential that rotates with Earth, the dynamics of the configurations of satellites in a Delta Walker constellation can be analyzed as a dynamical system as a function of a translation on the torus. This paper extends this type of analysis to routing over a Walker constellation, leveraging fixed ground relays. It considers any deterministic, time-invariant routing rule on such a collection of satellites and relays. The main object of interest is the routing table associated with this rule, which specifies, for all source-destination pairs, a route made of satellites and relays between them, for instance the shortest, together with angular information on next hop at each step of the route, which is essential for beamforming in this context. The main result is that, for each such routing rule, the routing table inherits the dichotomy of the torus flow: when the ratio of the Earth-spin and satellite angular speeds is rational, the routing table process is periodic with the same period as the constellation. When it is irrational, the table is non periodic but admits ergodic long-run averages which can be computed as ensemble averages w.r.t. an explicit invariant measure. Simulations with mixed satellite–gateway greedy routing illustrate these findings in terms of both temporal and spectral properties of the Walker routing table time series. The paper also discusses how to define metrics that cope with such non periodic fluctuations when present. This is illustrated by a comparison of the round-trip times between two far away cities as offered by a Walker constellation using greedy routing on one side, and by the currently available terrestrial fiber on the other side. It is shown how ergodicity can be used to make this comparison between the time varying instantaneous round trip times of the constellation and the constant round trip time of the fiber network meaningful.

We consider the isoperimetric problem with planar density equal to 1 on a horizontal strip and to a constant $\lambda>1$ outside the strip. We resolve a conjecture made by Cañete-Miranda Jr-Vittone by showing that every four-arc region admits a three-arc competitor with the same weighted area and strictly smaller weighted perimeter. The argument applies to every $\lambda>1$ and gives an explicit positive lower bound for the perimeter difference.

Jackson and Thomassen conjectured that every $2k$-strong digraph contains a spanning $k$-strong oriented subdigraph. We prove sharp degree conditions for the existence of such a subdigraph. For every fixed positive integer $k$ and all sufficiently large $n$, every $n$-vertex digraph $D$ with $\delta^0(D)\ge\lfloor(n+k-1)/2\rfloor$ admits a $k$-strong orientation. This threshold is best possible even for the weaker conclusion that $D$ itself is $k$-strong. We also prove a sharp Woodall-type analogue for every fixed positive integer $k$ and all sufficiently large $n$: if $d_D^+(x)+d_D^-(y)\ge n+2k-2$ for every missing arc $xy$, then $D$ admits a $k$-strong orientation, and this bound is again best possible. As a further consequence, we determine the sharp minimum total degree threshold. Finally, the semi-degree result also remains valid when $k\le\alpha n$ for every fixed $0<\alpha<0.094882\ldots$.

We formulate a precise sense in which a BRST anomaly may possess a classical phase space shadow. It is the non-equivariance class of a Hamiltonian symmetry action on covariant phase space. To organize this statement, we introduce the Weil covariant phase space bicomplex, whose basic subcomplex controls equivariant Hamiltonian lifts. We then compare this purely classical class with local BRST descent. For a Hamiltonian admissible mixed transgression, the ghost-number-two descendant $a^{2}_{d-1}$ defines a deghostification map to the charge algebra, which is generated by the failure of the Cartan representative to be equivariant. The induced cohomology class is independent of Bardeen type changes of descent representative. Thus the classical anomaly is a shadow of the BRST anomaly in the precise sense that both are realizations of the same Weil transgression class, while remaining objects of different theories. As an explicit example, we analyze a five dimensional inflow transgression and show that the classical charge cocycle and the BRST descendant arise as two realizations of the same Weil transgression class.

We investigate the problem of which multisets of integer entries yield integer eigenvalues in every square matrix arrangement? Combining module reduction with a dyadic criterion for complete splitting of cubic polynomials, we first show that if the multiset has at least one zero entry, the only possibilities have either at most one nonzero entry or exactly two whose product is a perfect square. We then apply block embedding to extend it to higher dimensions, by obtaining a linear threshold on the number of zeros. For any multiset close to a nonzero constant, we use rank reduction to derive an exact criterion for one exceptional entry, thus yielding nonconstant examples in infinitely many dimensions, and to also derive an exact reduction for two distinct exceptional entries in dimension three. The general fully nonzero three-dimensional case remains open.

In this note, we provide an estimate for the Willmore energy of Klein bottles in $\mathbb R^N$, building on the ideas of Li-Yau and Montiel-Ros for $2$-tori. In particular, we prove that the Willmore energy $\mathcal{W}(\phi)> 6\pi$ for any conformal branched immersion $\phi:\mathbb{K}^2_b\rightarrow \mathbb R^N$, where $\mathbb{K}^2_b$ is a flat Klein bottle with $0.350\lesssim b \lesssim 0.755$. This confirms partially a conjecture of Kusner. Moreover, for a flat immersion $\psi:\mathbb{K}^2_b\rightarrow \mathbb R^N$, we derive a lower bound of $\mathcal{W}(\psi)$, which confirms a conjecture of Hirsch-Mäder-Baumdicker for flat Klein bottles in $\mathbb R^N$. We also construct a smooth family of flat Klein bottles in $\mathbb R^5$, with Willmore energy between $6.912\pi$ and $8\pi$.

Let $\Omega\subset\mathbb{R}^2$ be a bounded planar set that is star-shaped with respect to the origin, and let $A,B,C,D$ be independent random points uniformly distributed on $\Omega$. We consider the random segments $S_{AB}$ and $S_{CD}$ and study the distribution of the smaller angle $\Theta\in[0,\pi/2]$ formed by them, conditional on the event that they intersect. Using the radial function of $\Omega$, together with a parametrization of each segment in terms of its supporting line and the positions of its endpoints along that line, we derive an integral representation for the conditional distribution \[ \Pr\{\Theta\leq\theta | S_{AB}\cap S_{CD}\neq\varnothing\}. \] The resulting expression makes explicit how the geometry of the boundary of $\Omega$ determines the angular distribution. The probability of intersection appears naturally as the normalizing constant and is related to the probabilistic version of Sylvester's four-point problem.

Let \[ M_{k,\ell}=[z^k-w^\ell]\subset H^2(\mathbb D^2), \qquad k,\ell\in\mathbb N,\quad k\ne\ell . \] We study the large-index asymptotics of Yang's numerical invariants and the boundary structure of their generating function. Starting from the exact staircase formula obtained in our preceding work, we prove that \[ \Sigma_j(M_{k,\ell}) = \frac{C_{k,\ell}}{j} + O(j^{-2}), \] where \[ C_{k,\ell} = \int_0^\infty \frac{x^2} {(x+1/k)^2(x+1/\ell)^2}\,dx . \] The first strict descent together with $C_{k,\ell}$ recovers the unordered pair $\{k,\ell\}$, yielding an asymptotic rigidity principle. At the next order we obtain a periodic correction \[ \Sigma_j(M_{k,\ell}) = \frac{C_{k,\ell}}{j} + \frac{\Psi_{k,\ell}(j)}{j^2} + O(j^{-3}), \] whose least period is \(\operatorname{lcm}(k,\ell)\), and we determine the leading amplitudes of the strict drops. For Yang's generating function \[ \mathcal P_{k,\ell}(t) = \sum_{j=0}^{\infty}\Sigma_j(M_{k,\ell})t^j, \] we prove that it has radius of convergence one and a logarithmic singularity at $t=1$; hence it is never a polynomial for $k\ne\ell$, giving a negative answer to Yang's polynomiality question within this quasi-homogeneous family. The periodic higher-order corrections generate root-of-unity polylogarithmic boundary modes. The leading logarithmic coefficient together with the second-order boundary support determines the unordered pair $\{k,\ell\}$, while successive renormalized boundary limits recover the Fourier coefficients of every finite-order periodic asymptotic term.

We prove that the negative $r$-spin tautological relations of Chidambaram–Garcia-Failde–Giacchetto are a consequence of Pixton's $3$-spin relations, for every $r\ge2$, and deduce that they hold in the tautological Chow ring of $\overline{\mathcal{M}}_{g,n}$. In particular, our result for $r=2$ implies that Kazarian–Norbury $K$-relations hold in Chow. The proof factors through two essential parts. The first part shows that the tautological relations arising from Chiodo's class imply the negative $r$-spin relations. This requires an intricate study of limiting behavior of the Givental–Teleman graph contributions. In the second part, we show that the Chiodo relations are a consequence of Pixton's $3$-spin relations as an application of a generalization of Janda's work on tautological relations from CohFTs.

The storage and recovery of a vector of length \(2^d\) can become prohibitively expensive as \(d\) grows, a manifestation of the curse of dimensionality. For certain structured functions and multiscale quantities, the resulting discretized vectors admit compact quantized tensor train (QTT) representations after binary tensorization: a length-\(2^d\) vector is reshaped into a \(d\)-way binary tensor with low tensor-train (TT) rank. Motivated by this construction, we study compressed sensing of tensorized signals that combine bounded TT rank with sparse low-order interaction structure, using randomly sampled Walsh–Hadamard measurements. For this structured model class, we establish a uniform restricted isometry property and derive a measurement condition guaranteeing uniqueness of noiseless recovery. We also propose a structure-aware initializer for the alternating linear scheme (ALS), obtained by projecting the adjoint backprojection onto the low-order interaction space before applying TT-SVD. We prove a uniform initialization error bound whose measurement requirement, for fixed structural parameters, grows polynomially with the tensor order \(d\) rather than with the ambient dimension \(2^d\). Numerical experiments support the predicted RIP and initialization scaling and demonstrate more reliable ALS recovery.

Agentic reinforcement learning (ARL) with verifiable rewards improves the ability of large language models (LLMs) to tackle knowledge-intensive tasks by learning to interleave search and reasoning. However, most existing ARL methods optimize only LLM-generated tokens and treat retrieved evidence as environment observations. This creates an information-credit gap: failures caused by missing or misleading evidence are attributed to the LLM policy rather than to the retriever, which motivates training the LLM and the retriever jointly. In this paper, we show that retrieval and LLM policy learning are order-sensitive: adapting the retriever before optimizing the policy yields a larger reward gain than the reverse order. To preserve this hierarchy while allowing both components to co-adapt, we formulate retrieval-augmented agentic RL as a bilevel optimization problem. To solve it efficiently, we introduce BRIDGE, a memory-efficient first-order bilevel method motivated by a loss-landscape analysis of the RL and retrieval objectives. Across seven open-domain QA benchmarks, BRIDGE achieves the highest average accuracy with both 3B and 7B backbones, improving the multi-hop average over the strongest baseline by 9.6 and 3.4 EM points, respectively. It also achieves the best averaged answer accuracy and reasoning quality across medical QA benchmarks.

We show that for a horizontally stretched Bunimovich billiard, for almost every stretching parameter, there exists a sequence of eigenfunctions concentrating in the rectangular part. Thanks to the work of Markarian–Oliffson Kamphorst–Pinto de Carvalho, we know that for an open set of parameters such billiards are ergodic. The proof is a result of interaction with ChatGPT 6: we were curious if it could establish the existence of bouncing ball modes for Bunimovich or Sinai billiards. That was not successful until we suggested varying the wings. That produced an essentially complete argument for the existence of quasimodes. We then suggested using Hassell's parameter-variation argument which led to the result about eigenfunctions. The argument was then significantly simplified and clarified by the authors.

We explicitly investigate Gorenstein projective, injective and flat modules over the $n$-trivial extension $R\ltimes_{n}M$ of a ring $R$ by an $R$-bimodule $M$. Assume that $fd(M^{\otimes_{R}i}_{R})<\infty$ and $pd(_{R}M^{\otimes_{R}i})<\infty$ for any $1\leq i\leq n$, $fd(\textbf{Z}(R)_{R\ltimes_{n}M})<\infty$ and $pd(_{R\ltimes_{n}M}\textbf{Z}(R))<\infty$. It is proven that a left $R\ltimes_{n}M$-module $(X,f)$ is Gorenstein projective if and only if the sequence $M^{\otimes_{R}n+1}\otimes_{R} X\stackrel{(M\otimes f) \cdots(M^{\otimes_{R}n}\otimes f)}\longrightarrow M\otimes_{R} X\stackrel{f}\longrightarrow X$ is exact and coker$(f)$ is a Gorenstein projective left $R$-module. As a consequence, we characterize Gorenstein projective, injective and flat modules over tensor rings.