Papers

We prove a common generalization of two structure theorems: the dimension-free decomposition theorem for idempotent Schur multipliers and the idempotent theorem in harmonic analysis. Roughly speaking, our result shows that an invariant integer-valued kernel with Hilbert-space factorization norm $\gamma$ admits a signed decomposition into at most $2^{O(\gamma^4)}$ elementary pieces. In the matrix setting these pieces are blocky matrices, while in the group setting they are indicators of cosets. In the locally compact abelian setting this quantitatively strengthens the theorem of Green and Sanders, while in the non-abelian setting it gives a quantitative strengthening of Host's idempotent theorem and, for finite groups, of Sanders's quantitative result. It also improves the exponent in the dimension-free matrix decomposition from $\gamma^6$ to $\gamma^4$.

Spatially restricted immune access can undermine tumour control even when total immune recruitment appears sufficient. We develop a nondimensional reaction–diffusion–chemotaxis model for hepatocellular carcinoma that couples tumour growth and immune-mediated killing with effector diffusion, saturating chemokine-dependent recruitment, and migration along an effective CXCL9/CXCL10/CXCL11–CXCR3 signal. We establish nonnegativity, uniform tumour and mass bounds, and global boundedness of classical solutions in one dimension and, in arbitrary dimensions, when chemokine production is tumour-driven. Analysis of the homogeneous dynamics shows that the threshold $\sigma_0>\delta$ is only local and does not preclude bistable tumour persistence. A mode-wise dispersion relation identifies stationary finite-wavelength and oscillatory instabilities, whose critical sensitivities, dominant modes, growth rates, and frequencies are quantitatively recovered by conservative finite-volume simulations. Using matched initial states and identical mean recruitment, we show that a well-mixed model may predict clearance while margin-limited recruitment preserves a stable interior tumour refuge. Chemotaxis improves interior effector access and reduces, but need not eliminate, this refuge. Spatial-overlap biomarkers further connect model-generated phenotypes to pathology and spatial-omics observables. These results distinguish immune abundance from effective spatial access and identify access-limited recruitment as a mechanism of incomplete tumour control.

For every uncountable regular cardinal $\alpha$, $\mathbb S^{\ast}(\alpha)$ is an explicitly typed four-coordinate forcing motivated by the ladder-system construction of Hayut and Magidor. The forcing is $\sigma$-closed and, after adjoining a formal maximum, $\alpha$-strategically closed. For $\alpha\geq\omega_2$, every nonempty countable family of designated generic branches has a stationary and costationary common trace on the generic ladder-coordinate set $L_\alpha$, while no countable family of cofinal branches generates $L_\alpha$. These conclusions persist under a Kurepa-style level-size bound. In the unrestricted forcing, for every infinite cardinal $\mu<\alpha$ in the ground model, some level of the generic tree contains a copy of $({}^\mu 2)^V$. Consequently, the endpoint-corrected restriction family indexed by $\mathcal P_{\omega_2}\alpha$ is too wide, whereas the scaled restriction system indexed by $\mathcal P_\alpha\alpha$ has all levels of size less than $\alpha$ exactly when $\alpha$ is strongly inaccessible in the ground model. When these equivalent conditions hold, the branch-covering number of $L_\alpha$ relative to the scaled system is at least $\omega_1$. The low-cofinality empty-value convention also ensures that the set of domains of $L_\alpha$ contains no club in $\mathcal P_\alpha\alpha$. The unrestricted tree clause of the motivating presentation is retained, without asserting forcing equivalence.

Developing innovative system architectures increasingly relies on advanced modeling and optimization techniques to frame the architecting process and define the corresponding computational problems. For complex System-of-Systems (SoS), high-fidelity multiphysics and multidisciplinary simulations are essential for capturing detailed behaviors. However, their computational expense and the risk of evaluation failures make direct optimization challenging. To overcome these limitations, surrogate-based approaches, like Bayesian optimization, have emerged as effective tools for managing expensive, black-box simulation tasks. This work introduces a hierarchical Bayesian optimization framework that leverages Gaussian process meta-modeling to handle discrete architectural choices, conditional dependencies, and heterogeneous design variables inherent to SoS problems. Results show that the hierarchical formulation improves search efficiency and robustness compared to conventional surrogate-based methods, enabling the exploration of large and structurally diverse design spaces with limited simulation budgets. We apply the approach to an aircraft-based multi-agent system for wildfire suppression, a use case developed within the EU-funded COLOSSUS project that illustrates how SoS principles can coordinate heterogeneous aerial platforms with complementary roles, supporting both sustainable mobility and emergency response missions. Our framework provides a scalable methodology for SoS architecting and model exploration, offering transferable insights for applications in aviation, sustainable mobility, and resilience-oriented system design. By combining hierarchical representations with surrogate-based optimization, this work is among the first practical demonstrations of hierarchical Bayesian optimization applied to real-world SoS problems, advancing both methodology and practice.

In this paper, a new non-homogeneous linear second-order partial differential equation (PDE) is proposed and investigated, designed to describe the dynamics of correlated pair objects of the microworld (analogues of Cooper pairs or biphotons). The fundamental complex solution of the proposed equation initially contains a spatial background and a doubled phase, which allows for a natural localization of the pair condensate in space. In this work, the continuity and energy balance equations are derived, the discrete spectrum in potential fields is calculated, and specific effects of tunneling and nonlinear interaction of the pair conglomerate with electromagnetic radiation are described.

This note makes available, in facsimile form, a set of handwritten notes from September 1993 devoted to Bochner-type maximum principle computations for a general class of quasilinear parabolic equations which contains, as a special instance, the normalized parabolic $p$-Laplacian \[ u_t = \Delta u + (p-2)\,|\nabla u|^{-2}\,\nabla^2u(\nabla u,\nabla u). \] The notes were acknowledged in the introduction of the paper of Jin and Silvestre \cite{JS}, and the scheme they propose has recently been implemented in \cite{BGM} to obtain the sharp Li-Yau inequality for this equation. They are reproduced here verbatim, with no alterations, preceded by a brief introduction explaining their content and the subsequent history of the equation.

Geometric community detection seeks to recover latent communities in networks where connectivity depends jointly on community structure and continuous spatial geometry. Existing exact-recovery approaches typically discretize the underlying space, which can impose restrictive structural assumptions on the connectivity functions. We develop a polynomial-time, discretization-free algorithm for exact recovery in the Geometric Hidden Community Model (GHCM), operating directly on the continuous geometry. Our method succeeds even when connectivity functions coincide on a nontrivial portion of their visibility range and when two communities can be distinguished only through their connectivity to a third community. We prove exact recovery under these weaker conditions and provide experiments showing that the algorithm succeeds beyond the theoretically guaranteed regime.

The main objective of this paper is to develop Krein-Milman-type theorems and Ulam-type stability results for graphs. To establish these results, we introduce several meaningful definitions of vertex-weighted convex graphs inspired by the concept of sequential convexity. We also present a close relationship between the two discrete structures, namely sequential convexity and perfect binary trees. We show that if a graph satisfies a certain convexity property approximately, then this property can be made exact by minimally perturbing the weights assigned to its vertices. Furthermore, we study several structural characterisations, formulate convex minorants for weighted graphs, and derive sandwich-type results. Special emphasis is placed on trees, and an investigation of extremal value problems is also carried out

For a prime number $p$, we denote by $k_p$ the greatest odd divisor of $p-1$. Tonelli–Shanks algorithm is a classical algorithm to compute a square root of a quadratic residue $n \in \mathbb{Z}$ modulo $p$ as the product of $n^{\frac{k_p+1}{2}}$, which we call {\it Tonelli–Shanks power of $n$}, and a correction term given as a power of a quadratic non-residue modulo $p$. We prove the transcendence over $\mathbb{Q}$ of the images of the following in the ring $\mathscr{A}$ of integers modulo infinitely large primes: the greatest odd divisor $k_p$ of $p-1$, the composite $f(\frac{p - 1}{k_p})$ of any injective map $f \colon \mathbb{Z} \to \mathbb{Z}$ and the greatest $2$-power $\frac{p - 1}{k_p}$ dividing $p-1$, Tonelli–Shanks power $n^{\frac{_p+1}{2}}$ of any $n \in \mathbb{Z} \setminus \{-1,0,1\}$, and the correction term of Tonelli–Shanks algorithm for such $n$.

We prove that for every prime power $q$ and every $p \in (0, 1-1/q)$, a random $\mathbb{F}_q$-linear code of rate $1 - h_q(p) - \epsilon$ is $(p, C_{p,q}/\epsilon)$-average-radius list-decodable with probability at least $1 - q^{-\Omega(n)}$, i.e., for every center $y \in \mathbb{F}_q^n$, the $C_{p,q}/\epsilon$ codewords closest to $y$ have average fractional Hamming distance at least $p$ from $y$. This extends a similar result for (standard) list-decoding due to Guruswami, Håstad, and Kopparty (2010) to the stronger average-radius guarantee, with the same $O(1/\epsilon)$ list size. For average-radius list-decoding, such a result was previously known only for binary linear codes (Guruswami, Li, Mosheiff, Resch, Silas, and Wootters, 2021) and for general (non-linear) random codes over arbitrary alphabets (Elias, 1991).

We organize several natural notions of limit set for discrete subgroups of $\mathrm{PSL}(3,\mathbb{C})$ around a common asymptotic structure encoded by pseudo–projective degeneration and by the full or partial flag data carried by divergent sequences. In the $\theta$-divergent case, the two projections of the full-flag limit set recover the attracting and repelling projective boundary sets, while the Myrberg limit set is the union of the limiting projective lines. Thus the equicontinuity region is the complement of a canonical line configuration; under the usual three-line general-position hypothesis, this same configuration is the Kulkarni limit set and the equicontinuity and Kulkarni ordinary regions coincide.

We study the spatiotemporal dynamics of the expansion of a bacterial colony on a hard substrate. Nutrient from the substrate diffuses into the colony and is taken up by the bacterial cells for them to grow and divide, expanding the initial monolayer and then pancake-shaped colony. The concentration of nutrient determines the local cell growth rate in the colony. Mass conservation relates such local growth rate with velocity which is approximated to be proportional to the pressure gradient by Darcy's law. Altogether, the growing colony is modeled as a moving-boundary problem with the nutrient concentration and pressure solving a reaction-diffusion equation and Laplace's equation, respectively. We analyze the self-consistent moving-boundary model with respect to different geometrical setting. For a general three-dimensional cylindrically symmetric model, we study the steady-state nutrient concentration. We construct and analyze a one-dimensional model for vertical expansion and a two-dimensional disk model for radial expansion of the colony. Our analysis finds that the nutrient depletes into the colony and slows down the vertical expansion of the colony. The vertical level where the nutrient concentration reaches the Monod constant, a threshold below which individual bacteria hardly grow, lowers down exponentially fast. We also estimate the asymptotic radial expansion rate. Moreover, we establish that the region where the nutrient concentration is above the threshold, allowing bacteria to grow and the colony to expand radially, is a ring-shaped peripheral region of fixed thickness. All these are consistent with experiment and agent-based simulations reported in literature.

Q-learning with linear function approximation can be unstable because an arbitrary approximation architecture need not preserve the Bellman contraction. We develop a stable infinite-dimensional linear function approximation framework for Q-learning from a single Markovian behavior-policy trajectory. The learning variable is a coefficient field $\theta\in C(\mathbb L)$ on a compact latent metric space $(\mathbb L,\rho)$. The framework uses a reconstruction operator that maps $\theta$ to a continuous Q-function and a compression operator that maps Bellman updates back to latent coordinates. Nonexpansiveness of both operators induces a contractive latent Bellman map on $C(\mathbb L)$, with a unique fixed point $\theta^*$ whose reconstruction approximates the optimal Q-function up to representation error. We propose two stochastic approximation (SA) algorithms and establish their sup-norm convergence bounds with a leading term of order $\widetilde O(n^{-1/2})$. The infinite-dimensional formulation provides a powerful abstraction for identifying the structures that govern statistical difficulty. Smoothness of the compression map in $\rho$ is inherited by $\theta^*$ and the SA iterates, allowing uniform estimation errors to be controlled through covering numbers of $(\mathbb L,\rho)$ rather than the dimension of $C(\mathbb L)$. Remarkably, the SA algorithms we propose are agnostic to the choice of $\rho$, and thus can automatically adapt to both the smoothness and the geometry. We further illustrate the framework through Q-measure-learning with linear density approximation and output-layer neural weight training under a frozen pretrained network.

A set $\{a,b,c\}$ of distinct positive integers is called a Diophantine triple if $ab+1$, $ac+1$, and $bc+1$ are perfect squares. Dujella formulated the conjecture that the only integral $x$-coordinates on the attached elliptic curve $$y^2=(ax+1)(bx+1)(cx+1)$$ are $0,d_-,d_+$, together with $-1$ when the triple contains $1$, where $d_{\pm}=a+b+c+2abc \pm2\sqrt{(ab+1)(ac+1)(bc+1)}$. The weaker question was stated as Problem 4.8 in Dujella's list of open problems: must every integral point with $x\ne-1$ make $ax+1$, $bx+1$, and $cx+1$ all perfect squares? We construct infinitely many triples $\{5,115,c\}$ admitting an integral point with $x\ne-1$ for which all three factors are nonsquares.

We completely characterize a two-depth version of Journé's covering lemma in the bi-parameter setting. Let $\mathcal U$ be an incomparable family of bi-parameter dyadic rectangles contained in $\Omega$, and write $e_i(I):=e_i(I;\Omega)$ for the two embeddedness depths. For every coordinatewise nonincreasing $w\colon\mathbb N^2\to[0,\infty)$, the estimate \[ \sum_{I\in\mathcal U}w(e_1(I),e_2(I))|I|\lesssim_w|\Omega| \] holds uniformly in $(\mathcal U,\Omega)$ if and only if the following fixed-total condition holds: \[ \|w\|_{\mathrm{ft}}:=\sup_{N\geq0}\sum_{a=0}^Nw(a,N-a)<\infty. \] For instance, $\|(a+1)^{-s}(b+1)^{-(1-s)}\|_{\mathrm{ft}}\sim[s(1-s)]^{-1}$ for $0<s<1$, so this allows weights that are not summable in either variable, while one-depth theory holds precisely for summable weights. Exactly the same characterization holds for dyadic Zygmund rectangles $I=I^1\times I^2\times I^3$ with $\ell(I^3)=\ell(I^1)\ell(I^2)$. We use probabilistic packing methods to prove the sufficiency, and these techniques also yield sparse refinements of the preceding estimates, which self-improve to various forms of exponential integrability.

Stochastic reaction networks are continuous-time Markov chain models for interacting populations, with applications in biochemistry, epidemiology, ecology, and related areas. Coupling multiple stochastic reaction networks effectively is a fundamental challenge in stochastic simulation. Existing variance-reducing coupling methods are typically designed for pairs of processes, though specialized multi-path constructions have been developed for particular applications. To fill this gap, we introduce the multi-path stacked coupling (MSC), a space-time Poisson construction that jointly simulates a finite collection of stochastic reaction networks with common reaction channels and reaction vectors, but possibly different intensity functions and initial conditions. Each pairwise marginal of MSC has the same law as the corresponding split-coupled pair, allowing existing variance bounds to transfer directly; in finite-state settings, we also obtain first-order expansions for the mean and second moment of finite-difference numerators. We demonstrate the framework through three sensitivity estimation tasks requiring several nearby parameterized paths: estimating many first derivatives simultaneously, estimating a single first derivative using a wider finite-difference stencil, and estimating higher-order derivatives. Numerical experiments on a processive phosphorylation network demonstrate strong performance in each of the three application areas considered, consistent with the theoretical advantages of MSC: across all three applications, MSC achieves the smallest root mean square error (RMSE) among the methods considered over the tested computational budgets.

We study one-dimensional Jacobi operators of divergence-gradient type on $\ell^2(\mathbb{Z}_{\geq 0})$, with coefficients generated by the doubling map, $a_n(x)=a(2^n x \mathrm{mod} 1)$. For continuous positive sampling functions, we show that the almost-sure essential spectrum is an interval containing the bottom of the spectrum. Under the assumption $a\in C^1(\mathbb{T})$, we prove square-root asymptotics for the integrated density of states and obtain a small-energy expansion for the Lyapunov exponent as $E\to0^+$. The leading term of the Lyapunov exponent is linear and positive precisely under a nondegeneracy condition on the sampling function. In this case, we prove a large-deviation estimate for the transfer matrices with an explicit rate depending on $E$. As consequences, we obtain local Hölder continuity of the Lyapunov exponent and the integrated density of states near the bottom of the spectrum, as well as the Anderson localization. Some of these results extend the small-coupling results of Chulaevsky–Spencer and Bourgain–Schlag for Schrödinger operators generated by the doubling map to divergence-gradient type operators. The arguments adapt methods from those works, but require uniform control in the small-energy parameter and refined correlation estimates reflecting the divergence-gradient structure.

We develop a structural framework for irregularity in linear dynamics centered on the space $\Upsilon$ of observation schemes. This approach separates the underlying dynamical behavior from the observation mechanism and provides a unified setting in which classical notions such as Li–Yorke chaos, mean Li–Yorke chaos, and distributional chaos arise as particular cases corresponding to Dirac measures and Cesàro averages. We establish two abstract criteria ensuring the existence of large linear structures, leading to dense-lineability and spaceability results for both absolutely $(\mu_m)$-irregular and distributionally $(\mu_m)$-irregular vectors. Furthermore, under a natural density assumption, we obtain a trichotomy describing the global behavior of irregularity across $\Upsilon$, together with rigidity phenomena for the classes of observation schemes generating each type of chaotic behavior.

In a real Hilbert space $\mathcal H$, we study the following inertial differential inclusion $$ \ddot x(t)+\gamma\dot x(t)+\partial\phi(\dot x(t)) +\nabla f(x(t)+\beta\dot x(t))\ni0, $$ where $\gamma>0$ is the viscous damping coefficient, and $\phi$ is a convex potential with a sharp minimum at the origin that models the dry friction damping (typically $\phi=r\Vert\cdot\Vert$ where $r>0$ is the dry-friction parameter). The function $f$ represents the smooth potential to be minimized, and the shifted-gradient evaluation $\nabla f(x(t)+\beta\dot x(t))$ is known as the implicit Hessian-driven model. Here $\beta\geq 0$ represents the corresponding Hessian-driven parameter. Both the explicit and the implicit Hessian-driven dynamics are know to attenuate the oscillations that occurs in inertial systems. Our contribution concerns the quantitative analysis of this continuous dynamic and its temporal discretization counter-part under the action of these three combined dampings: viscous damping, dry friction, and implicit Hessian-driven damping. We establish a global well-posedness, an exact Lyapunov analysis adapted to the implicit Hessian-driven damping, finite length of the trajectory and its strong convergence to an approximate critical point $x_\infty$ of $f$ satisfying: $-\nabla f(x_\infty)\in\partial \phi(0)$. We show finite-time stabilization under a strict interior condition on the terminal force. We also compare the explicit and the implicit Hessian-driven dynamics and show that their trajectories differ by $O(\beta^2)$ on finite horizons. A temporal semi-implicit discretization of the dynamic above leads to a proximal implicit Hessian-driven algorithm based on one shifted-gradient evaluation. We derive its discrete Lyapunov analysis, asymptotic convergence and, under a strict terminal margin, finite convergence of the discrete iterates.

During clinical trials evaluating a drug's effect on a survival endpoint, intermediate events often occur before the primary event. The treatment can affect the primary event through multiple pathways via intermediate events. With multiple intermediate events, each is a treatment-induced confounder of transitions out of the others, so we cannot identify natural or separable effects. We instead consider stochastic interventions that assign each transition the identifiable observed-history hazard of either arm. Contrasts between the resulting counterfactual cumulative incidences define transition-, event-, and path-targeted effects, identifiable without cross-world or isolation assumptions. Based on the efficient influence functions for the counterfactual cumulative incidences under hypothetical interventions, we construct multiply robust and semiparametrically efficient estimators for these targeted treatment effects. Our proposed framework quantifies treatment effects through each transition, on each event, and along each path. By analyzing data from the LEADER trial, we find that liraglutide significantly reduces the risk of cardiovascular and microvascular events. The reduction in all-cause mortality is primarily mediated by its effects on expanded major adverse cardiovascular events.

We present a unified framework for the study of Wishart matrices (W_p(n,\Sigma)), which generalize the chi-squared distribution to matrix-variate settings and model the covariance structure of multivariate Gaussian data. After recalling their defining properties - additivity under independent summation (W_1 + W_2 \sim W_p(n_1+n_2,\Sigma)), equivariance under linear maps (A W A^T \sim W_q(n,A\Sigma A^T)), and their role as sample covariance matrices - we embed the positive-definite cone (S_p^+) within Monge-Ampere geometry. Here (S_p^+) acquires a Hessian manifold structure with affine-invariant metric and volume form (\omega = \det(\Sigma)^{-(p+1)/2},d\Sigma), under which the Wishart density acts as a soliton of natural geometric flows. We then show that the collection of Wishart distributions forms a symmetric monoidal category (\mathcal{W}), whose objects are (W_p(n,\Sigma)) and whose morphisms are linear maps (A:\mathbb{R}^p\to\mathbb{R}^q). The tensor product encodes block-diagonal coupling, with braiding given by block permutation, and the axioms enforce Monge-Ampere functoriality, additivity, and convex duality via the Legendre transform. Applications to quantum error correction are discussed: Wishart laws model correlated noise, Wasserstein geodesics optimize error-mitigation cost, tensor structure captures independent error channels, and Legendre duality underpins entropy-driven decoding.

This sequel to our paper `On the “main conjecture” of equivariant Iwasawa theory' studies the possibility of the vanishing of $SK_1(QG)$, when $QG$ is the total ring of fractions of the Iwasawa algebra $\Lambda G=Z_p[[G]]$, with $G$ the Galois group of the Galois extension $K/k$ of loc. cit. The vanishing is equivalent to $SK_1(D)=0$ for all division algebras $D$ in the Wedderburn components of $QG$, so can be studied via the classification $D_{r,s,F}$ of these $D$'s, which provides a crossed product order $\Delta$ in $D$ and the valuation $v_\bullet$ with valuation ring $\Delta_\bullet$. $\Delta$ contains a special element $\Pi$, which generates a maximal subfield of $D$ over its centre with $V_\bullet(\Pi)=1$ and $\Pi\Delta\Pi^{-1}=\Delta$. The induced $\Pi$-filtration on $\Delta$ enables a study of the reduced norm built on a congruence for $nr(d)$ mod $\Pi\Delta$ for $d\in\Delta$. Given $d\in\Delta^\times$ with $nr(d)=1$, and $n\ge 1$ maximal with $d\in 1+\Pi^n\Delta$ (called the level of $d$), the main problem is to find a suitable commutator product $c\equiv d$ mod $\Pi^{n+1}\Delta$. Then $c^{-1}d\equiv 1$ mod $\Pi^{n+1}\Delta$ hence, setting $d'=c^{-1}d$, has $nr(d')=1$ and level $n'>n$. Repetition ends in $[\Delta^\times,\Delta^\times]$ when the level gets sufficiently large.

We introduce Menger and Rothberger selection principles and games for convergence spaces. Alice plays families that meet every convergent filter, and Bob's selections are required either to retain this property or merely to cover the underlying set. When the convergence is topological, both games recover the classical games. The winning condition requiring an $L$-cover satisfies analogues of the Hurewicz and Pawlikowski characterizations, the condition requiring a cover of $X$ is represented by the weak Menger and Rothberger games. We also show that, for a regular convergence space, a winning strategy for Bob in $\mathsf G_{\fin}(\mathcal C_L,\Cov(X))$ implies an Alster-type covering property. Under hereditary Lindelöfness the space is moreover a countable union of compactoid subsets, which are compact in the pretopological case.

In 1989 Olshanskii introduced a three-parameter family of coordinate-surjective homomorphisms from the genus-two surface group to a direct product of two rank-two free groups. When the common quotient of the two coordinate images is finite of order $n$, restriction to the corresponding regular cover produces an epimorphism \[ \pi_1(S_{n+1})\longrightarrow F_{n+1}\times F_{n+1}. \] We call these maps Olshanskii epimorphisms. They form an explicit high-genus test family for the standardness problem for splitting epimorphisms. We prove that every Olshanskii epimorphism is standard. The genus-two homomorphism determines a Heegaard splitting of a Seifert fibered $3$-manifold over $S^2$ with at most three exceptional fibers. In the finite-quotient cases, classical Seifert theory shows that its universal cover is $S^3$; Waldhausen's theorem then implies that the lifted genus-$(n+1)$ Heegaard splitting is standard. The proof uses neither Perelman's theorem nor the general Poincaré theorem. We also give a constructive treatment of the quaternion case $Q(2,2,2)\cong Q_8$, whose covering surface has genus nine. A maximal tree in the quaternion Schreier graph yields the covering handlebody and explicit Schreier bases. Using geometric longitude–meridian pairs and explicit surface automorphisms supported on the nine one-holed tori, we transform the lifted meridian words into a free basis. A separate fixed-rank Andrews–Curtis certificate reduces the associated balanced presentation. This paper supplies the detailed proof of the result announced in the previous Kharlampovich, Vdovina paper.

We develop a new framework for flexible, nonlinear, and interpretable off-policy evaluation for infinite-horizon reinforcement learning. To handle large state spaces and support transparent decision-making, we model the Q-function using a nonlinear function class with a sparse additive structure. We derive high-probability finite-sample error bounds for estimating the value function of a target policy and show that the bounds depend only logarithmically on the ambient dimension $d$, thereby alleviating the curse of dimensionality. In contrast to most existing theory for off-policy evaluation, which typically assumes access to many trajectories, our analysis guarantees accurate value estimation when either the number of trajectories or the time horizon is sufficiently large. In addition, we propose a group-sparsity-based feature screening procedure that identifies, with high probability, a reduced feature set containing all relevant covariates. Numerical experiments demonstrate the effectiveness of the proposed approach.

Gradient random walk (GRW) methods represent a solution's spatial derivative by weighted particles and recover the solution by cumulative summation. We examine their accuracy for the one-dimensional heat, scalar cubic reaction-diffusion, and Burgers' equations. Thirty-seed ensembles, paired reconstructions, and deterministic controls identify the sources of two apparent accuracy limits. For the heat equation, comparing cumulative bin sums at bin centers introduces a first-order alignment error, which the exact reflected-particle distribution predicts and bin-edge comparison removes. An identity for the expected squared error predicts the remaining floor and the particle count for a target root-mean-square accuracy. With unequal weights, a count chosen in advance gives an error within 1% of its prediction. Reaction-front position, speed, and shape errors decrease under refinement. For Burgers' equation, exact transformed boundary data reduce the deterministic Cole-Hopf error about 490-fold, while fixed-endpoint particle runs approach their deterministic reference.

In 2025, Boaz Klartag proved that there exists a lattice sphere packing in $\mathbb{R}^n$ whose density is at least \[ c n^2 2^{-n}, \] where $c>0$ is an absolute constant. Our aim is to refine the analysis of Klartag's argument and to show that the constant can be taken as \[ c = \frac{1}{2e} - o(1), \qquad n\to\infty. \]

Many studies on finite-horizon stochastic optimal control, including covariance steering, parameterize control policies as state-history-affine. This parameterization enables a convex reformulation, thereby yielding a tractable solution method. However, the necessity of dependence on previous states has not been well established. Is this dependence necessary, or merely an artifact of the convex reformulation? We show that it is an artifact that can be removed losslessly. Given an optimal solution of the state-history-affine formulation, we construct a deterministic Markovian policy which is affine in the current state. We show that, even for the covariance steering problem with a broad class of commonly used state and control safety constraints, the synthesized Markovian policy almost surely produces the same control actions as the history-dependent policy and therefore the same state trajectories, cost, and moments. Thus, every optimum of the history-dependent formulation admits a lossless Markovian transformation. Geometrically, the history-dependent formulation lifts the policy space for convexity, and its optimal solution can be projected back to the Markovian policy space. We extend the analysis to output feedback and a convex upper-bounding surrogate for value-at-risk costs.

We give a short proof that for a given planar set of terminals $P$ there is at most one locally minimal tree with prescribed directions at $P$ (i.e. two locally minimal trees cannot coincide in $B_\varepsilon(P)$). The new ingredient is a combinatorial result which says that a certain embedding of a bipartite cubic graph has the same numbers of balanced and unbalanced vertices.

We characterize all first cosyzygy modules of a given finitely generated module over a local ring. We do this by introducing the notion of a canonical cosyzygy module, and showing how other cosyzygy modules can be obtained from this module. As an application, we characterize Gorenstein rings via properties of canonical first cosyzygy modules.