Papers

Rational Heun operators on the $q-$linear grid are presented. They are second-order $q-$difference operators $W_q$ constructively defined from the requirement that they have a raising action on rational functions of type $[n/n]$, namely $W_q: [n/n] \rightarrow [n+1/n+1]$, with poles on $q-$linear grids. It will be observed that these operators are related to one family of the Ruijsenaars-van Diejen-Takemura Hamiltonians. A distinguished subclass of $W_q$ called classical which shifts the pole structure while preserving the rational function type and a prescribed basis is also characterized.

We define and study the rectangular finite free heat flow, a dynamical system on polynomials that plays the role of the heat equation in the setting of rectangular finite free probability. We show several equivalent characterizations of the evolution (including PDE and gradient flow formulations), establish basic properties of the dynamics, and determine the asymptotic distributions of the polynomial roots in the long-time and high-degree limits. We also discuss connections with Calogero-Moser systems and Dunkl processes, and we show that the rectangular finite free heat flow describes the mean curvature expansion of a family of compact Lie group orbits.

While algorithmic stability is a central tool for understanding generalization of learning algorithms, existing high-probability guarantees typically rely on uniform boundedness or sub-Gaussian/sub-Weibull tail assumptions, which can be overly restrictive for modern settings with heavy-tailed or unbounded losses. We develop a stability-based framework that requires only a finite $L_p$ moment condition. Our first contribution is sharp concentration inequalities for functions of independent random variables under $L_p$ constraints, extending McDiarmid's bounded-differences techniques beyond the classical regime. Leveraging these results, we derive sharp high-probability generalization bounds across a range of learning paradigms, including empirical risk minimization, transductive regression, and meta-learning. These guarantees show that $L_p$ stability suffices for robust generalization even when boundedness fails, substantially weakening the standard assumptions in the stability literature.

Let $M$ be a simple connected binary matroid, and let $e$ and $f$ be distinct elements of $M$. It is well known that, when the only circuits containing $e$ are spanning, $M$ is a circuit with at least three elements. This paper proves that if every circuit containing $\{e,f\}$ is spanning, then the canonical tree decomposition of $M$ is a path in which each vertex is labeled by a circuit, a copy of $U_{1,3}$, or a binary spike having one non-tip element deleted.

We study parabolic second order tangent sets in semialgebraic optimization when the supplied polynomial representation is singular. Our focus is the representation problem: formal differentiation of singular defining equations may lose the intrinsic second order geometry. The main result is a real radical branch recovery theorem. If the local real minimal prime branches of the equality ideal are smooth, then the branches selected by a direction and their exact parabolic second order tangent sets can be recovered from a local real radical decomposition and first- and second order polynomial data, without a preassigned stratification. The formula is invariant under changes of regular branch generators, while counterexamples delimit the roles of real radicalization, branch regularity, and constraint qualification. In optimization, the recovered models reduce each directional second order necessary test to finitely many linear programs in the parabolic correction and hence to branchwise multiplier certificates. A parameterized example shows that the formal second order system of the original singular representation may falsely reject a strict local minimizer, whereas branch recovery gives the correct test. Intrinsically singular branches are separated from representation singularities and can be incorporated when an exact higher order model is available.

Flexible intelligent metasurface (FIM) technology has emerged as a promising technology for enhancing wireless communication performance by dynamically reshaping the propagation environment. Compared with conventional rigid reconfigurable intelligent surfaces (RIS), an FIM is composed of multiple electromagnetic (EM) scattering units, each of which can flexibly modify its displacement in the direction normal to the surface, thereby cooperatively morphing the overall surface shape. This additional degree of freedom (DoF) enables improved beamforming and interference mitigation, particularly in complex multicell scenarios. In this paper, an optimization problem for maximizing the weighted sum-rate (WSR) in a multicell multi-user multiple-input single-output (MU-MISO) system assisted by an FIM deployed at the cell boundary is investigated. We jointly optimize the transmit beamforming at the base station (BS), the phase shift matrix, and the FIM surface shape, subject to constraints on the transmit power budget, unit-modulus reflection coefficients, and surface shape morphing range. Due to the non-convex objective function with highly coupled variables, solving the formulated optimization problem is challenging. To tackle this challenge, we propose an efficient alternating optimization framework that leverages the weighted minimum mean square error (WMMSE) method to reformulate the problem and the block coordinate descent (BCD) algorithm to iteratively update the variables. Specifically, the Riemannian conjugate gradient (RCG) algorithm is leveraged to optimize the phase shift matrix, while the projected gradient descent (PGD) method is adopted to optimize the surface shape of the FIM. Additionally, the optimal beamforming vectors are obtained in closed form.

In this paper, we develop a Constraint Energy Minimizing Generalized Multiscale Finite Element Method (CEM-GMsFEM) for solving linear elasticity problems in heterogeneous perforated domains. The presence of numerous perforations introduces multiple scales into the computational domain, making direct fine-grid simulations computationally expensive. The proposed method follows the standard offline–online decomposition of CEM-GMsFEM. In the offline stage, local spectral problems are solved on coarse elements to construct auxiliary spaces, and localized energy-minimizing basis functions are then computed on oversampled regions to capture fine-scale geometric information induced by the perforations. In the online stage, residual-driven basis functions are constructed in enlarged coarse neighborhoods to incorporate source-term information and improve the accuracy of the multiscale approximation adaptively. We establish convergence results for both the offline and online stages. In particular, we derive error estimates for the localized multiscale approximation and prove the convergence of the adaptive online enrichment algorithm. Moreover, we show that the oversampling regions used in the online stage can be determined locally, leading to a reduction in computational cost while maintaining convergence properties. Numerical experiments on perforated media with different geometric configurations demonstrate the accuracy and efficiency of the proposed method.

Motivated by the B-coloring defined by Gyárfás and Sárközy, we introduce a new edge coloring called D-coloring. For a graph $G$, a D-coloring of $G$ is a proper edge coloring such that every diamond subgraph is rainbow. The D-chromatic index of $G$, denoted by $\chi'_D(G)$, is the minimum number of colors needed for a D-coloring of $G$. Denote by $\Delta$ the maximum degree of $G$. We prove that $\chi'_D(G)\le \frac{9}{16}\Delta^2+\frac{1}{2}\Delta$, conjecture that $\chi'_D(G)\le \frac{1}{2}\Delta^2+\frac{1}{2}\Delta$, and verify this conjecture for $\Delta\le 5$.

We present a partial classification of two-generator discrete and faithful subgroups of the trivalent tree automorphism group, specifically for cases where the generators satisfy a restriction on a small geometric quantity. When the restrictions on the geometric quantity or tree valency are relaxed, we discuss the possible reduced quotient graphs for these subgroups and construct infinite families of graphs of groups on each. Additionally, we include a generalized Poincaré algorithm that determines whether a given set of tree automorphisms generates a discrete subgroup.

For a graph G = (V, E) embedded in the Klein bottle, let F(G) denote the set of faces of G. A C_4-face-magic Klein bottle labeling on G is a bijection f: V(G) to {1, 2,..., |V(G)|} such that for any F in F(G) with F isomorphic C_4, the sum of all the vertex labelings along C_4 is a constant. We say that a C_4-face-magic labeling X={x_{i,j} : 0< i,j< 5} on the 4x4 Klein bottle grid graph is horizontally (or vertically) pairwise balanced if x_{2i-1,j} + x_{2i,j}=17 for 0< i <3 and 0< j \le <5 (or x_{i,2j-1} + x_{i2,j}=17 for 0< i <5 and 0< j <3). We show that the 4x4 Klein bottle grid graph has 192 C_4-face-magic labelings up to symmetries on a Klein bottle. We classify these labelings into two categories depending on whether a C_4-face-magic label preserving permutation of the labeling is either horizontally pairwise balanced or vertically pairwise balanced. These results extend known results on C_4-face-magic labelings on an mxn Klein bottle grid graph.

In this paper, we study the Seshadri constants and Gromov widths of polarized toric varieties whose moment polytopes are generalized permutohedra. We show that both invariants coincide with the lattice width of the moment polytope, and provide an explicit formula for them in terms of the defining submodular function.

The transformer's emergent ability to perform in-context learning (ICL) has sparked a wide range of studies designed to understand its underlying mechanisms. Existing works often study how training task diversity, defined either as the number of ICL training task vectors or as the number of function classes from which the task vectors are drawn, shapes both the learning dynamics and generalization capabilities of ICL. While both definitions have uncovered many interesting phenomena, many observations under the latter definition remain theoretically unexplained. This paper presents a minimal analytical model under which these phenomena provably emerge from the properties of the training data. By modeling the training task vectors as a mixture of low-rank Gaussians, we show how training task diversity, defined by the number of non-overlapping columns between subspaces that parameterize the covariance matrices, improves both the generalization and optimization trajectory of ICL with linear attention. In particular, we show that our model can explain (i) why training with task diversity shortens the ICL plateau and (ii) why ICL appears to achieve out-of-distribution generalization. We conclude by empirically demonstrating how our results extend to nonlinear transformers and nonlinear function classes. Overall, our work presents a tractable framework to unify existing observations.

We introduce a measure-based nudging framework for assimilating macroscopic observations into microscopic mean-field particle dynamics. The central difficulty is a representation mismatch: the forecast is a labeled particle system, while the observations specify only a smoothed, permutation-invariant density. To address this mismatch, we define the forecast-observation discrepancy as a quadratic functional on probability measures after applying the same smoothing operator used by the observation process. The Wasserstein gradient of this functional induces a transport velocity on state space, which yields a particle-level correction without constructing particle-to-particle matching, linearizing the dynamics, or estimating ensemble covariances. For a fixed observation scale, we prove well-posedness of the assimilated McKean-Vlasov dynamics and propagation of chaos for the interacting particle approximation. Under exact smoothed observations and an observability condition at the kernel scale, we establish an $L^2$-stability estimate showing exponential decay up to a bias floor controlled by model misspecification. Numerical experiments on linear, bimodal, chaotic, kinetic, and collective-motion systems demonstrate that the method can recover macroscopic structure from incomplete density-level observations.

This paper studies information rates of noisy duplication channels with memory, motivated by nanopore DNA sequencing. In nanopore sequencing, the measured signal is affected by both inter-symbol interference (ISI), caused by multiple DNA bases residing in the pore, and random sample duplications, where variable translocation speed causes each base to generate a random number of samples. These two effects make direct theoretical analysis difficult. To address this, we derive a new decomposition of the information rate into two interpretable terms: one capturing the channel memory through an auxiliary ISI channel, and another capturing the uncertainty in the segment boundaries caused by random duplications. This decomposition separates the dominant channel distortions and replaces the direct analysis of the full channel with two more readily tractable components. We then study the second term through a soft alignment functional closely related to Soft-DTW, which yields a strong asymptotic equipartition property result and an alternative proof of the Markov-constrained coding theorem. Finally, we develop a lower bound on the information rate that depends on the distribution of jump distances between adjacent nanopore levels. This bound gives a simple geometric explanation of channel synchronisability and provides a tractable framework for computing achievable rates of Oxford nanopore sequencers.

Let $K_{KR}$ be the least constant governing uniform approximation of approximately additive real-valued set functions by finitely additive signed measures. We prove that $$K_{KR}\le\frac{374167}{20000}=18.70835.$$ Starting from a norming functional for best uniform approximation, we form two families of almost extremal sets with complementary point frequencies. Intersections and biregular expanders then give two recombination schemes. The sharper estimate comes from using the schemes on different ranges and from cancelling the positive and negative errors with unequal weights. This improves the previously published bound $38.8$.

This paper focuses on parallel-plate electrostatic actuators, which are fundamental structures found in Micro-Electro-Mechanical Systems (MEMS) for numerous modern devices. It considers the equations used to model these actuators. In particular, we investigate the behavior of the solutions to a MEMS model that is represented by a second-order ordinary differential equation, where the spring is characterized as soft, linear, or hard. We present the mathematical structure underlying the pull-in and touchdown phenomena that characterize the behavior of systems incorporating soft and hard spring, based on the behavior of linear springs. The pull-in phenomenon corresponds to the degeneration of the solutions to the model equation. The touchdown phenomenon corresponds to the finite-time singularity of the solutions. These global dynamics and structures are systematically revealed through geometric approaches based on Poincaré-type compactification, the center manifold theorem, and blow-up technique. These methods successfully induce and resolve the dynamics at infinity.

We study finite-sample change detection for one-dimensional noisy dynamical systems using partition-based empirical approximations of stationary behaviour. Given observations from an interval-valued process, we partition the state space, estimate a finite transition matrix from observed transitions between partition elements, and apply a small Doeblin-type regularisation to ensure a unique stationary distribution. From an initial reference segment, we compute a baseline empirical stationary distribution \(\widehat{\pi}_{0,\rho}\). For each later sliding window, we compute \(\widehat{\pi}_{t,\rho}\) and define the score \[ S_t=\|\widehat{\pi}_{t,\rho}-\widehat{\pi}_{0,\rho}\|_1. \] Large values of \(S_t\) indicate a change in stationary behaviour relative to the baseline. The statistic detects changes in invariant density or stationary law, but not all possible changes in transition dynamics. Under explicit assumptions on empirical transition concentration, finite-state stationary distribution stability, partition approximation, regularisation bias, and noise stability, we derive a finite-sample bound for the empirical stationary density. The bound separates sampling error, regularisation bias, partition approximation error, and noise bias. We then obtain a single-window false-alarm guarantee and a sufficient detection condition when the invariant density changes by more than the estimation error. We illustrate the method on synthetic noisy beta-map change-point experiments.

We study the spectral properties of a class of unbounded linear operators on a non-Archimedean Hilbert space $E_{\omega}$. More precisely, we consider operators of the form \[ T=D+F,\qquad F=\sum_{j=1}^{m} u_j\otimes v_j, \] where $D$ is an unbounded diagonal operator and $F$ is a finite-rank perturbation. This work extends the spectral analysis of Diagana and McNeal for rank-one perturbations of diagonal operators to the case of arbitrary finite-rank perturbations. The main objective is to describe the spectrum, point spectrum, and essential spectrum of such operators in terms of the diagonal sequence associated with $D$ and the Fredholm properties of $\lambda I-T$. The theory of Fredholm operators plays a central role, particularly in the computation of the essential spectrum and in the study of stability under finite-rank perturbations.

AI systems increasingly delegate decisions to specialized models, evaluators, tools, and supervisory controllers. The central AI problem is no longer only model accuracy, but uncertainty-aware governance: how much autonomy to grant, which evidence should calibrate trust, what performance ceiling a delegated AI system can sustain, and when human intervention becomes necessary. We propose the Minimum Sufficient Oversight Principle (MSO), a variational principle for principled autonomy delegation: minimize governance burden on the Fisher information manifold subject to a delivery constraint. The resulting Euler-Lagrange solution yields a water-filling allocation of governed delegation across the task space. Building on a revealed-action governed delegation channel model, we prove a capacity theorem for stationary symbolwise review policies, derive a local first-order approximation relating workflow complexity to quality degradation, and give a drift-dominated autonomy-time scaling law linking intervention timing to effective capacity, complexity, and drift. Within this framework, masking appears as a structural AI-governance pathology: corrected performance can hide the competence signal needed to calibrate trust. Synthetic simulations and a semi-real reconstructed workflow support design prescriptions including upstream-first correction, sensitivity-based intervention, and explicit feasibility checks before autonomy is expanded. The result is a computable framework for uncertainty, planning, and oversight in delegated AI systems. A companion Python package is available at https://github.com/crbazevedo/delegation-lab.

On the centennial of Erdal İnönü's birth, this article reflects on his scientific legacy and his role in shaping modern theoretical physics in Türkiye. We briefly discuss his life, scientific vision, and contributions to academic institutions, and then turn to his most celebrated scientific achievement: the İnönü-Wigner contraction. Through the simple geometric example of a sphere becoming a plane, we present an accessible introduction to this important idea and its significance for modern physics.

We identify a previously unrecognised structure in the finite-temperature geometry of Blahut–Arimoto (BA) rate-distortion optimisation. The starting point is an exact partition identity. For every source density (p) and every inverse temperature $\beta>0$, the BA partition function $Z(x)=\int q^*(y)e^{-\beta|x-y|^2}dy$ satisfies $$ Z(x)=\left(\frac{\pi}{\beta}\right)^{d/2}p(x). $$ This identity, obtained from the BA fixed-point equation, implies that the BA effective score $g_\beta=-\nabla\log Z$ coincides exactly with the classical Fisher score $s=-\nabla\log p$ for all temperatures. Moreover, if $v=-\nabla\log q^*$ denotes the translation mode generated by the quadratic-distortion symmetry, then its BA projection satisfies $\mathcal P v=-s$. These observations lead to the central identity $$ J(p)=\mathcal R(v):=\langle v,\mathcal G v\rangle_{L^2(q^*)}, $$ where $\mathcal G$ is the BA relaxation kernel. Thus Fisher information is exactly the Rayleigh quotient of the translation mode and is therefore a temperature-invariant spectral quantity in the BA framework. This yields a geometric interpretation of the Fisher information inequality: the inequality $$ J(X+Y)^{-1}\ge J(X)^{-1}+J(Y)^{-1} $$ becomes the parallel-combination law of a Rayleigh quotient under convolution. The entropy power inequality then follows through the standard heat-flow argument. The contribution is not a new proof of the entropy power inequality, but the identification of a hidden geometric structure: Fisher information as the spectral charge of the translation mode in BA rate-distortion geometry, with the entropy power inequality emerging as a consequence of this temperature-invariant fact.

We study whether a labeled tree can be recovered from the numbers of larger labeled trees that contract to it. We first characterize containment using the edge splits induced on a proposed survivor set and derive an exact formula for hosts with one marked display. We then study several displays in the same host. After contracting every edge invisible to all \(k\) displays of an \(n\)-vertex target, the remaining collision core has at most \(k(n-1)+1\) vertices. Once the coincidences and relative positions of the surviving labels are recorded, these bounded cores give a finite exact expansion of each collision count. We also prove that almost every sufficiently large labeled tree contains any fixed target, with an exponentially small exceptional proportion. We obtain an exact avoidance formula and generating function for star displays in which the center survivor is the largest label of the host. We call such displays top-centered. We derive fixed-star asymptotics and a coarse threshold for this restricted containment problem.

Robust Subspace Recovery (RSR) aims to identify an underlying d-dimensional subspace from a dataset heavily corrupted by outliers. Complexity-theoretic results establish a threshold for the problem's computational hardness based on the dimension-scaled signal-to-noise ratio (DS-SNR): the problem is SSE-hard when the DS-SNR is strictly less than 1, and solvable via practical algorithms when it is greater than 1 under general position assumptions. However, the exact behavior of practical algorithms at the critical boundary DS-SNR = 1 has remained unknown. This work resolves the behavior of Tyler's M-estimator (TME) at this critical boundary, consequently establishing a sharp phase transition. Specifically, we prove that TME converges exactly to the true subspace for DS-SNR \geq 1 under a new stability condition, which is less restrictive than the general position assumptions used in prior literature. Our analysis utilizes a decomposition of the TME iterates within a majorization-minimization framework.

Let \(X\) be a smooth complex projective variety of dimension \(n\geq 2\), and let \(D=D^p+D^c\), \(D^c=\sum_i D_i^c\), be a simple normal crossing divisor. We regard \(D^p\) as the cusp divisor and the components \(D_i^c\) as compact orbifold divisors with standard weights \(\alpha_i=1-1/p_i\). Let \(\mathcal X=X[\sqrt[p_i]{D_i^c}]_i\) be the root stack along the compact components. We study the canonical parabolic Higgs bundle \(E_*=(\Omega_X^1(\log D^p)\oplus\mathcal O_X)_*\), whose compact weights are the \(\alpha_i\) on the conormal lines of \(D_i^c\), while the parabolic structure along \(D^p\) is trivial. Assume that \((E_*,\theta)\) is polystable with respect to some ample line bundle and that equality holds in the parabolic Bogomolov–Gieseker inequality. We prove that the trace-free adjoint Higgs bundle is flat. The associated principal \(\mathrm{PU}(n,1)\)-variation gives a faithful monodromy representation \(\rho:\pi_1^{\mathrm{orb}}(\mathcal X^o)\to\mathrm{PU}(n,1)\) and a period map to the complex ball \(\mathbb B^n\). The period map is unramified in the orbifold sense and identifies \((\mathcal X,D^p)\) with the canonical orbifold toroidal compactification \((\mathcal T_\Gamma,D_\Gamma^{\mathrm{tor}})\) of a finite-volume ball quotient, where \(\Gamma=\rho(\pi_1^{\mathrm{orb}}(\mathcal X^o))\) is a finite-volume lattice satisfying the regular log-root condition. We also formulate this standard-weight uniformization as an equivalence of categories: the compactified quotient construction and the monodromy construction define quasi-inverse functors between the regular log-root lattice category \(\mathsf{Lat}_n^{\mathrm{reg}}\) and the standard Bogomolov–Gieseker category \(\mathsf{BG}_n^{\mathrm{std}}\).

We establish optimal inequalities relating the systole and the inradius to the volume of finite-volume hyperbolic 3-manifolds. In the cusped orientable case, we refine a theorem of Gendulphe by proving a sharp systole-volume inequality whose unique extremal manifold is the figure-eight knot complement. Excluding the figure-eight knot complement, we obtain a stronger inequality whose extremal manifold is the sister of the figure-eight knot complement. We also establish analogous optimal systole-volume inequalities for closed orientable hyperbolic 3-manifolds, where the extremal manifolds are the Weeks-Matveev-Fomenko manifold, the manifold Vol3, and the Meyerhoff manifold. In the second part of the article, we study the inradius. We prove optimal inradius-volume inequalities for orientable and nonorientable cusped hyperbolic 3-manifolds, identifying respectively the sister of the figure-eight knot complement and the Gieseking manifold as the extremal cases. We also prove that the Gieseking manifold is the unique cusped hyperbolic 3-manifold of minimal inradius, thereby completing a result of Gendulphe, who had previously established the corresponding lower bound.

Given $R$ a finite commutative ring with identity and $k \in \mathbb{N}$, we consider the $k$-th unitary Cayley graph $G_R(k)=Cay(R,U_{R,k})$ with $U_{R,k} = \{ x^k: x \in R^*\}$, and its symmetrized version $\mathcal{G}_R(k) = Cay(R,T_{R,k})$, with $T_{R,k}=U_{R,k} \cup (-U_{R,k})$. If $R$ is a local ring with maximal ideal $\frak m$, we give the blow-up decompositions for the graphs: namely, we have $G_R(k)= (G_{R/\frak m}(k))^{(|\frak m|)}$ and $\mathcal{G}_R(k)= (\mathcal{G}_{R/\frak m}(k))^{(|\frak m|)}$ for any $k$ such that $(k,|R|)=1$. If the ring $R$ has Artin decomposition $R=R_1 \times \cdots \times R_s$ in local rings $R_i$, we give the Kronecker product decompositions $G_R(k) = G_{R_1}(k) \otimes \cdots \otimes G_{R_s}(k)$ and $\mathcal{G}_R(k) = \mathcal{G}_{R_1}(k) \otimes \cdots \otimes \mathcal{G}_{R_s}(k)$. In further $(k,|R|)=1$, these decompositions can be given in terms of generalized Paley (GP) graphs over finite fields, that is $G_{R_i}(k) = \Gamma(k_i,q_i)$ and similarly for $\mathcal{G}_{R_i}(k)$, for $i=1,\ldots,s$. Also, the reduced graphs correspond to the graphs of the reduced rings, i.e.\@ $\big(G_{R}(k)\big)_{red}\simeq G_{R_{red}}(k)$ and $\big(\mathcal{G}_{R}(k)\big)_{red} \simeq \mathcal{G}_{R_{red}}(k)$. By using these decompositions in terms of GP-graphs, we study some basic structural properties of the graphs such as directedness, bipartiteness and connectedness.

The fundamental limits of over-the-air computation (AirComp) are explored in a two-transmitter, two-receiver MIMO Gaussian network, where both receivers demand the same aggregation of source symbols originating at the two transmitters. An AirComp degrees of freedom (ACDoF) metric is defined, constrained by an asymptotic mean-squared error threshold. For a generic MIMO setting where the two transmitters are equipped with $M_1, M_2$ antennas, and the two receivers with $N_1, N_2$ antennas, the AirComp DoF value is shown to be almost surely equal to $\min\{M_1,M_2,N_1,N_2,(1/3)\max\{M_1+M_2,N_1+N_2\}\}$. For SISO settings results are extended beyond generic channels to arbitrary channel realizations. For finite signal-to-noise ratio(SNR) settings, an iterative alternating optimization algorithm is explored.

We characterise e-processes for testing non-parametric hypotheses that require the conditional mean of a finite-dimensional statistic to lie in a prescribed convex set. Our main result is a complete-class theorem: every such e-process is pointwise dominated by a predictable product of affine one-step e-variables. Consequently, restricting to test supermartingales entails no loss: at any rejection threshold, the dominating process rejects no later on any observation path. We further show that a minimal complete class exists and consists of predictable products of maximal affine one-step e-variables. These results extend previous complete-class results from single-step constrained testing and bounded one-dimensional conditional mean testing to a broader conditional sequential setting.

In this paper, the first of a two-part series, we explore 2-cell embeddings of cubic graphs, particularly those with small genus. Using local rotations, we introduce a new way of describing the space of 2-cell embeddings and their mutual relationship for any fixed (cubic) graph. We introduce the unstable dual of an embedding of a cubic graph, a subgraph of the dual graph, and describe how the genus of the corresponding embedding can be recovered from properties of the unstable dual. Finally, we characterize the unstable duals of embeddings with genus at most 2 of cubic cyclically 5-edge connected planar graphs and use these to generate those of genus 3 that have connectivity at most 2.

We investigate a new class of infinite-horizon backward stochastic differential equations for ergodic optimal control where the cost and state dynamics are time and path-dependent. The state process is defined on an unbounded underlying domain and satisfies an extended dissipativity condition. In contrast with the time-homogeneous Markovian setting, the optimal ergodic cost in our framework is characterized by the asymptotic behavior of a deterministic function, rather than by a single real constant. We obtain well-posedness, verification and stability properties, which extend the previous results in the literature on the Markov case.