Papers

Natural Policy Gradient (NPG) is a well-established Reinforcement Learning algorithm that underlies widely used methods such as Trust Region Policy Optimization and Proximal Policy Optimization, both of which have demonstrated strong empirical success. In this paper, we study exact NPG in finite-horizon Markov Decision Processes with known dynamics and horizon-dependent transition kernels. We provide the first finite-time convergence guarantees for this algorithm in this setting, for which we consider both constant and increasing step size regimes. With a constant step size $\eta_t=\eta$, we prove that NPG converges sublinearly with a rate of $\mathcal{O}(H^{2}/t)$ after $t$ iterations, where $H$ is the horizon length. We also extend this constant step size analysis to linear MDPs in an exact population-projection oracle under a full support projection distribution, recovering the same sublinear rate as in the tabular setting. Furthermore, with increasing step sizes, we prove that this algorithm achieves a linear convergence rate of $\mathcal{O}\left(\left(1-\frac{1}{\vartheta_\rho}\right)^t\right)$ for a problem-dependent constant $\vartheta_\rho > 1$, and the horizon-only robust schedule of the form $\eta_t=\eta_0(H/(H-1))^t$ where $\eta_0>0$ and $H \geq 2$, attains this same geometric rate.

A vertex-centroid move slides one vertex of a polygon along the line through it and the centroid of all vertices; Problem 60 of The Open Problems Project asks whether such moves can make every polygon regular. We prove that two simple labeled $n$-gons, with straight angles allowed, are joined through simple polygons by finitely many positive moves exactly when they have the same orientation; a nonconstant move is positive if the vertex remains on the same side of the centroid of the other vertices. The same holds for flows toward or away from the instantaneous area centroid. Inward-only or outward-only moves reach every target of the same orientation up to translation and positive scaling. The common mechanism is a first-order criterion: a symmetric definite center response makes the one-vertex fields of a smooth affine-equivariant center bracket generating. The response is a weighted covariance for point barycenters and $(d+2)$ times the volume covariance for polyhedral centroids. Once translation and scale are factored out, the fields sum to zero, giving exact one-sided control. We resolve Problem 60 under both conventions for a vertex at the centroid. For the vertex-centroid system on $n$ affinely spanning points in $\mathbb{R}^d$, we classify positive-move reachability and prove a local $3dn$-move bound.

Multi-rendezvous missions for active debris removal and in-orbit servicing incur prohibitive propellant costs when their targets differ in the orientation of their orbital planes, and classical mission design treats the nodal precession induced by Earth's oblateness as a perturbation to be cancelled. This paper presents a drift-aware trajectory optimisation framework that instead exploits that precession as a mission-design resource, benchmarked against the published winning solution to the European Space Agency's Kessler Run trajectory-optimisation competition as a trusted reference. Three methodological contributions are made. First, an enhanced drift orbit shapes the size, shape, and inclination of an intermediate orbit to tune the differential precession rate, substantially cheaper than the altitude-only designs of prior work in the Sun-synchronous regime, where inclination is the dominant lever. Second, a global per-leg time-budget optimiser makes explicit and resolves the coupling between drift-time allocation and downstream nodal geometry that defeats greedy allocation. Third, a sequence-recovery test independently re-derives a removal order consistent with the published reference from the debris catalogue alone. Applied to the full debris campaign on the published mission partition under the competition's transfer-time rules, the framework closely matches the published benchmark cost using only analytical transfer models and minutes of single-core computation. A cross-validation anchored to a widely used flight-dynamics tool quantifies the fidelity limits of the analytic model and bounds the cost of literal execution, and the constructive nature of the algorithm supports rapid in-orbit replanning after a missed manoeuvre.

Let $X = (X_1, \ldots, X_n)$ be a random vector from any Borel probability law on $\mathbb{R}_+^n$. We revisit the problem of deriving a lower confidence bound (LCB) on a scalar parameter of that law. We recast classical work, beginning with Buehler, in purely probabilistic terms to form a more accessible and extensible framework. We then specialize the framework to the case where the components of $X$ are independent. In this context, we prove that Gaffke's bound is Buehler optimal for the order that it induces with respect to the maximum marginal mean parameter: $max_{i \in [n]} E_Q[X_i]$, which reduces to the common mean when the $X_i$ are independent and identically distributed. That is to say, no other valid LCB that orders samples in the same way as Gaffke's bound can improve on it with respect to this parameter.

We develop a mesoscopic framework for analyzing perturbations of finite point sets. Given a reference node set $Y$ with known cubature and approximation properties, we consider a disordered node set $Q$ that is observed only through its populations in cubes at scale $r>0$. By imposing Hall-type (HT) combinatorial constraints on these populations, we prove the existence of a perfect matching between $Y$ and $Q$ with $O(r)$ range. This allows integral approximation estimates on coarser cubes at scale $h>r$ to be transferred from $Y$ to $Q$ with explicit error control and anchors $Q$ to a periodic grid. We then use translation-invariant quasi-interpolation methods to obtain high-order estimates of order $h^s$ as in the quasi-uniform setting, but for a different class of geometries. The key restrictions are the HT conditions and the bound $r\le Ch$, where $C<1$ is scale independent.

We use a gluing procedure introduced by Ginzburg to describe the relative Langlands dual of the hyperspherical Hamiltonian $(\mathrm{GL}_n \times \mathrm{GL}_m)$-variety $T^*(\mathrm{Hom}(\mathbb{C}^m,\mathbb{C}^n))$, and in particular the Rankin-Selberg case $m=n$. We show that the dual is isomorphic to the triangle part of Cherkis-Nakajima-Takayama bow varieties, recovering a result of Nakajima. Using this result, we give a new proof of a theorem of Braverman-Finkelberg-Nakajima describing the regular perverse sheaf in type $A$. Following a suggestion of Ginzburg, we explain how to modify the gluing so that the dual Hamiltonian variety of $T^*\mathbf{N}$, for any gluable finite-dimensional representation $\mathbf{N}$ of a complex reductive group $G$, is naturally equipped with an anti-symplectic involution, and give an explicit formula for this involution in the Rankin-Selberg case.

External difference families (EDFs) are closely related to algebraic manipulation detection (AMD) codes, a cryptographic primitive that protects messages against additive tampering by an adversary who cannot observe the transmitted codeword. We first study a cyclotomic construction of external partial difference families over finite fields and derive a necessary and sufficient condition under which the resulting families are EDFs. For even block sizes up to $14$, we obtain explicit criteria in terms of quadratic and biquadratic residues, yielding new $R$-optimal weak AMD codes. We then construct bounded generalized strong external difference families using cyclotomic classes over finite fields, direct products of finite fields, and generalized cyclotomic classes over integer residue rings. These constructions give infinite families of systematic $G$-optimal strong AMD codes with flexible parameters.

The Karpelevič theorem describes the complex numbers that occur as eigenvalues of stochastic matrices of a fixed order. We give a self-contained proof based on invariant polygons. Their geometry yields a product equation for an eigenvalue of maximal modulus at a fixed argument. Convexity determines this modulus, explicit stochastic matrices attain it, and a comparison between Farey intervals completes the boundary description. As the matrix order increases, the resulting equation gives a uniform relative asymptotic for the gap $1-|z|$ between a boundary point $z$ and the unit circle, even arbitrarily close to arc endpoints. We also determine the optimal contraction factor for a planar rotation and dilation when vector size is measured by scaling a polygon with at most $N$ vertices.

There are eleven finite rings of order $p^{2}$ denoted by $A_p$ to $K_p$ in alphabetical order. In particular, we consider $I_{2}$ and $I_{3}$ which are commutative non-unital rings of orders 4 and 9 defined by generators and relations as \[I_{p}=\left\langle a,b\mid pa=pb=0,\:a^{2}=b,\:ab=0\right\rangle\] for $p=2, 3,$ respectively. Alahmadi et al. studied codes over these rings. In this paper, we study additive complementary dual (ACD) codes over the rings $I_{2}$ and $I_{3}$. We show relations between ACD codes over $I_{2}$ and binary linear complementary dual (LCD) codes using a reduction map from $I_{2}$ to $\mathbb{F}_{2}$, and between ACD codes over $I_{3}$ and ternary LCD codes using a reduction map from $I_{3}$ to $\mathbb{F}_{3}$. Using the first relation, we classify ACD codes over $I_{2}$ with the highest minimum distances for $n=1, 2, 3$ and partially for $n=4, 5$. It turns out that they are maximum distance separable (MDS) codes. Using the second relation, we classify ACD codes over $I_{3}$ with the highest minimum Lee distances for $n=1, 2$ and partially for $n=3$. We generalize the two relations into a relation between ACD codes over $I_{p}$ and $p$-ary LCD codes using a reduction map from $I_{p}$ to $\mathbb{F}_{p}$. This is a correction of the paper published in Advances in Mathematics of Communications, Volume 24, pages 61-76, 2026. In particular, we corrected the statements of Theorems 3.5, 4.7, 4.8, and their proofs.

We prove that a domain $D$ is an RB-domain if and only if the function space $[D\to E]$ is continuous for every domain $E$, thereby resolving a conjecture of Luan and Li. More precisely, for each domain $D$ we construct an explicit algebraic test domain $A_D$ such that continuity of the single function space $[D\to A_D]$ already forces $D$ to be an RB-domain. For countably based domains $D$, the test domain can be chosen as a fixed domain that is independent of $D$. The analogous continuity characterization of FS-domains is false: the closed-disk domain is an FS-domain, but its function space into a suitable pointed algebraic domain is not continuous.

This paper further studies the model companion of an endomorphism acting on a vector space, possibly with extra structure. Let $T$ be a model-complete theory that $\varnothing$-defines an infinite $K$-vector space $\mathbb{V}$. In previous work, we introduced a family $\{T^C_\theta : C \in \mathcal{C}\}$ of extensions of the theory $T_\theta := T \cup \{\text{``$\theta$ is an endomorphism of $\mathbb{V}$''}\}$ that parameterizes all consistent extensions of the form $$ T_\theta \cup \left\{\sum\nolimits_{k}\bigcap\nolimits_{l}\operatorname{Ker}(\rho_{j, k, l}[\theta]) = \sum\nolimits_{k}\bigcap\nolimits_{l} \operatorname{Ker}(\eta_{j, k, l}[\theta]) : j \in \mathcal{J}\right\}, $$ where all sums and intersections are finite, all the $\rho[\theta]$'s and $\eta[\theta]$'s are polynomials over $K$ with $\theta$ plugged in, and $\mathcal{J}$ is some possibly infinite index set. We also presented a sufficient condition that implies that every $T^C_\theta$ has a model companion $T\theta^C$. In this paper, we study the case where $T$ is an o-minimal expansion of the theory of ordered groups. Doing so, we obtain a new family of theories that have $\operatorname{TP}_2$ and $\operatorname{SOP}$, and are $\operatorname{NATP}$, as well as distal non-o-minimal theories of various $\operatorname{dp}$-ranks with and without the exchange property.

We investigate a joint optimization problem of dividend payments and capital injections for a surplus process driven by a general diffusion. Dividend payments are assumed to be absolutely continuous in time, with the dividend rate bounded by a nonnegative concave function of the current surplus; while capital injections are modelled by a general nondecreasing process. We first analyze an auxiliary bail-out problem in which capital injections are required to keep the surplus nonnegative at all times. Under a concavity assumption on the drift, we prove that the associated value function is concave and is a classical solution of the corresponding Hamilton-Jacobi-Bellman (HJB) equation. We further characterize an optimal policy as a refraction-reflection strategy: the surplus is reflected at zero by capital injections, while dividends are paid at the maximal admissible rate whenever the surplus exceeds an optimal threshold. Our main contribution establishes that the general optimization problem exhibits a Lokka-Zervos type dichotomy. More precisely, an optimal policy is either a dividend refraction strategy without injections, in which ruin occurs, or a refraction-reflection dividend-injection strategy. This optimal injection decision is characterized through a simple comparison of the auxiliary value functions at the origin.

Let $k\geq 3$ be a fixed integer and $x$ be a large real number. Let $0<c_0<\theta\leq 1$ be a fixed real number for some $c_0>0.$ In this paper we show that there exists a constant $1/2\leq \delta_k(\theta)<1$ such that the interval $[x, x + x^{\delta_k(\theta) +\varepsilon} ]$ contains an integer of the form $n_1n_2 \cdots n_k$ such that $|n_j-n^{1/k}|\ll n^{\theta/k} \ (j=1,2, \cdots, k)$. Especially we have $\delta_3(1) = \delta_4(1) = 1/2,$ which improves previous results of Chan.

We study online bipartite matching with reusable server capacity and non-stationary rewards. Jobs arrive sequentially, reveal compatible servers, reward rates, and processing durations, and must be accepted or rejected irrevocably. An accepted job occupies one unit of server capacity only during its processing interval, so an assignment may displace an unknown sequence of future jobs. Existing guarantees are typically calibrated by a global reward range, which can become arbitrarily large when rewards drift over a long horizon. We instead impose a locally bounded reward condition: reward rates of jobs that can compete for the same server within a relevant time window differ by at most a factor $\delta$. Under this condition, we develop two BALANCE-type algorithms with time-aware opportunity-cost losses. TS-BAL maximizes cumulative blocking losses over feasible reuse schedules and achieves a competitive ratio of $2\ln(\delta D)+\mathcal O(\ln\ln(\delta\vee D))$. GR-BAL uses a greedy relaxation of this loss and achieves $\ln(\delta D)+\mathcal O(\ln\ln(\delta\vee D))$, matching a lower bound of $\ln(\delta D)$ in the leading term. Numerical experiments demonstrate robust performance under substantial global reward drift and favorable finite-capacity performance.

We study simulation-optimization of systems of optimizers (SOSO): agent-based simulations in which every agent solves a structured optimization - a linear program (LP), mixed-integer program, or dynamic program - at every decision epoch. Such systems arise in supply chains, electricity markets, and logistics, yet standard simulation optimization treats the simulation as a black box, discarding the inner optimization's geometry. We formalize SOSO and develop a framework that converts this geometry into computational advantage. We prove that the inner LP's optimal basis and dual variables propagate through the dynamics to yield an exact, unbiased, single-replication infinitesimal perturbation analysis (IPA) gradient of the outer objective, with measure-zero basis changes as the only obstructions. We derive a common-random-numbers covariance bound governed by a computable basis-disagreement count from the forward simulation. We further prove that IPA variance grows exponentially with feedback depth, formalizing the bullwhip effect, and introduce surrogate-as-decision with an error budget unifying LP-horizon and reinforcement-learning surrogates under one bound. We compose these into PRIME, a stochastic-approximation solver integrating the IPA gradient, adaptive step sizing, and multi-start spatial diversification. On six testbeds spanning the SOSO taxonomy, PRIME achieves the best or tied-best optimality gap at equal budget with near-zero oscillation and narrow seed-to-seed variance; IPA yields a ~2000x per-replication variance reduction over independent finite differences, and common random numbers cut paired-difference variance by over 10,000x in a 1,000-SKU, six-distribution-center supply chain. These results establish exploiting embedded-optimization geometry as a practically significant direction for simulation optimization.

We study the problem of constructing the nearest graph Laplacian matrix to a given Laplacian while enforcing a prescribed connected-component structure. Let the vertex set be partitioned into nonempty disjoint blocks $C_1,\ldots,C_k$, and let $U=[u_1,\ldots,u_k]$ be the matrix of the corresponding block-indicator vectors. The constraint $MU=0$ ensures that these prescribed indicators lie in the nullspace of the optimized Laplacian $M^\star$, and hence the associated graph has at least $k$ connected components. To guarantee exactly the prescribed components, we impose additional block-connectivity constraints on the principal blocks $M_j=M[C_j,C_j]$. These constraints ensure that each prescribed block induces a connected weighted subgraph. The resulting problem is a convex semidefinite optimization problem with a strictly convex Frobenius-norm objective. We prove existence and uniqueness of the minimizer and show that the optimized Laplacian has exactly the prescribed connected components, with nullspace $\operatorname{span}\{u_1,\ldots,u_k\}$. The framework proposed in this work provides a principled tool for quantifying the minimum structural intervention required to transform a graph-based network into one having a prescribed group-separated structure. Numerical examples, including the Sampson monastery positive-affection network, illustrate the nearest faction-consistent weighted reconstruction and the minimum Laplacian perturbation required to realize the prescribed faction structure.

CAR T-cell circuit design enables targeting tumour cells while keeping healthy cells unaffected to a cancer treatment. Designing CAR T-cell circuits that maximize the number of tumour cells eliminated (called efficacy) while meeting the safety constraint of keeping a large number of healthy cells unaffected is a challenging task. We provide a new multistart local search approach for this problem and show that it outperforms the current SOTA approach on a single-cell cancer atlas. Furthermore, we introduce a multi-objective local search approach which provides insights into the trade-offs of efficacy and safety of the treatment. Our experiments also demonstrate that the multi-objective approach yields improved solutions even for the constrained single-objective optimisation problem.

The Constantin-Iyer formula represents a smooth solution of the incompressible Navier-Stokes equations on $\mathbb{T}^3$ as $u=\mathbb{P}\,\mathbb{E}[(\nabla A_t)^\top(u_0\circ A_t)]$, the projected expectation of a stochastic Weber field. We separate this expectation into the product of the means and a centred covariance, and show that the covariance – together with the second moment of the inverse deformation gradient and the two-point covariance – satisfies a closed deterministic advection-diffusion equation: because every Lagrangian label is driven by the same Brownian path, the quadratic covariation collapses into a Laplacian and no moment hierarchy appears. From these equations we obtain an energy/production balance, pointwise and Holder covariance bounds, and an exact degeneracy of the two-point operator in the separation variable. A second group of results concerns the mean displacement, which solves a forced heat equation along the flow; its $L^2$ norm is controlled unconditionally by the initial energy, while an exact family of Navier-Stokes shear flows shows that no analogous Holder bound can hold uniformly down to $t=0$. We further isolate several obstructions, each backed by an explicit counterexample – the norm of an expected deformation gradient does not control the expected norm, even for genuine stochastic flows of smooth divergence-free drifts; a spatial Holder bound on the mean does not imply parabolic Campanato decay – and we prove a gauge-invariant quotient criterion equivalent to the Serrin norm of the velocity, together with a Liouville-type rigidity theorem for ancient solutions whose mean coordinate is affine of rank at least one. No claim is made regarding global regularity; the missing steps are stated explicitly as open problems.

The permanent rank of a matrix $A$ is the size of the maximal square submatrix in $A$ which has a nonzero permanent. In this paper we characterize additive transformations $\Phi$ which preserve matrices of per-rank-one. Under an additional assumption that $\Phi$ is surjective or that it preserves per-rank-one in both directions we prove that $\Phi$ is a composition of a multiplication with diagonal matrices and permutation matrices from both sides, transposition, and injective endomorphism of the base field.

We review the renormalization group framework for signal detection in high-dimensional data, tailored to the regime where the signal may be of extensive rank and does not separate from the noise bulk as isolated spikes. The framework provides a conceptually simple criterion for distinguishing signal from noise within a quasi-continuous spectral region near a random-matrix universality class. This scenario lies beyond the reach of standard methods such as the Baik-Ben Arous-Péché threshold, which requires eigenvalues to be cleanly separated from the bulk. The renormalization group approach, by contrast, directly tracks spectral deformations and consistently yields a lower limit of detection without relying on spike separation. We review results that identify the presence of a signal by testing the stability of the Gaussian fixed point of an effective field theory for the collective behaviour of the degrees of freedom in the spectral tail, where the signal resides. We also discuss how the scale dependence of the canonical dimension, induced by the signal, manifests as a dimensional phase transition.

We develop a systematic framework for quantum mechanics on a finite graph-resolved class of second-countable, T1, non-Hausdorff one-manifolds. The central problem is that local differential expressions do not by themselves determine the quantum theory: non-Hausdorff incidence, bundle transport, smooth extension, and analytic completion can impose additional global constraints on the operator domain. We formulate these constraints using transported jets, Whitney realisability, Sobolev traces, and formal return holonomy, obtaining exact closure results for finite smooth graph resolutions. At first Sobolev order, scalar dynamics reduce to a weighted quantum graph whose edge weights arise from a marked resolving presentation, leading naturally to weighted Kirchhoff Hamiltonians. Multiple-origin lines and circles, branching junctions, finite trees, and split-and-rejoin geometries then yield explicit deficiency indices, reflection and transmission laws, interference conditions, compact dark states, and embedded cavity modes. For finite-rank Hermitian bundles, unitary gluing selects a transmitting fixed subspace, so scattering becomes projection onto the invariant sector of the subgroup generated by the gluing matrices. Compact-group representations therefore turn non-Hausdorff incidence into an invariant-sector selection mechanism; connected compact semisimple groups require at most two relative gluing matrices for full invariant completion, with explicit SU(3) examples. The resulting theory separates topological non-Hausdorff data from the analytic and representation-theoretic structures that remain detectable by quantum dynamics.

We study the combinatorial structure of parity sequences associated with the accelerated Collatz map with the goal of identifying extremal configurations and relating them to the existence of periodic orbits. To each finite sequence of an orbit, we associate a binary word whose ones encode the odd iterates, and we introduce a functional $C(d)$ on such words which provides an explicit expression for the iterates and characterizes possible periodic cycles. We define a natural rotation action on binary words, compatible with the cyclic structure of periodic orbits, and consider the functional $C_{\min}(d)$ as a canonical representative of each rotation class. In this setting, we formulate and solve a discrete optimization problem on the set of binary words of fixed length and prescribed density. We prove that Christoffel words are, up to rotation, the unique maximizers of $C_{\min}(d)$ on $D_{N,r}$, the set of binary words of length $N$ with exactly $r$ ones, thereby establishing a direct connection between the dynamics of the Collatz problem and the classical theory of balanced words. As a consequence, we obtain restrictions on the possible existence of nontrivial cycles and derive explicit bounds for the minimum element of an orbit in terms of its length and the proportion of odd iterates. These results show that the combinatorial structure of parity sequences imposes strong constraints on Collatz dynamics and suggest that extremal configurations are governed by classical objects from the combinatorics on words, exhibiting a pronounced structural rigidity.

A formula is established that counts the number of different balls that can be made by decorating the pentagons and hexagons of a classic football with Truchet-like patterns.

Let $\Gamma$ be a Coxeter graph, and let $S$ be its vertex set. An elemental complex over $\Gamma$ is an abstract simplicial complex $\mathcal D$ on $S$ that contains every pair $\{s, t\}$ with $s \neq t$ and $m_{s,t} \neq \infty$. To each elemental complex $\mathcal D$, we associate a cube complex $\Xi_{\mathcal D}$ on which the virtual Artin group ${\rm VA} [\Gamma]$ acts. We prove that $\Xi_{\mathcal D}$ is connected, simply connected, and locally regular; that the action of ${\rm VA}[\Gamma]$ on $\Xi_{\mathcal D}$ is regular, cocompact, and by isometries; and that the stabilizers of cells are parabolic subgroups. We finally prove that $\Xi_{\mathcal D}$ is CAT(0) if and only if ${\mathcal D}$ is a flag complex.

Understanding the transport of semiclassical measures along generalized bicharacteristics is a key ingredient in the proof of observability for the wave equation. Here, the coefficients are only assumed to be of class $C^1$, which is a limit case for the existence of bicharacteristics. We consider boundary conditions of the form $\partial_\nu u + i \partial_t u = 0$, as a first step towards more general Lopatinskii-type conditions. We derive the transport equation satisfied by the measure arising from the concentration of high-frequency waves that obstruct observability. Observability of solutions associated with positive time-frequencies is then obtained by contradiction under the geometrical control condition.

We introduce a discrete difference operator D_k to study the one-dimensional quantum random walk (QRW) with the Hadamard coin. Explicit combinatorial expressions are obtained for the probability amplitudes a(n,k) and b(n,k), which encode the final step direction and carry alternating signs that reflect the merging of leftward steps. Removing these signs and the coin-state distinction recovers the classical binomial distribution. The symmetric and antisymmetric combinations $a\pm b$ are shown to coincide with diagonal and sub-diagonal entries of the Krawtchouk matrix. Using cross identities among Krawtchouk matrix elements, we prove by induction that the amplitudes satisfy a Parseval identity sum (a^2+b^2)=2^n-1, establishing probability conservation in the Krawtchouk formulation. The operator D_k acts as a discrete Hermite polynomial generator: the ratios h_m = (n choose k)^{-1} D_k^m (n choose k) admit explicit closed forms and converge to Hermite polynomials in the continuous limit. At the discrete level, D_k connects successive Krawtchouk matrices and acts as a coherence generator and a raising operator. The h_m quantify the position-dependent degree of quantum interference on both halves of the distribution, either individually (for x >= 0) or through an inverse-Pascal combination of several h_m (for x < 0), and the ballistic O(n^2) scaling of the variance emerges from the superposition of all excited states. Numerical illustrations for n=6 and n=10 corroborate the analytical results.

We develop a framework for quickest detection of distributional change in signals modeled as rough paths. By representing the pre-change and post-change dynamics as two independent rough paths and modeling the observed signal as their concatenation at an unknown random change-point, we formulate quickest detection as an optimal stopping problem using rough paths. We show that the optimal stopping rule takes the form of the first hitting time to a half-space by a linear functional of the rough path signature, and establish that the same structural form arises independently from the intrinsic geometry of the observed path. Statistical guarantees on detection delay and false alarm probability are derived, and the framework is extended to a distributionally robust formulation in which both the pre-change and post-change models are uncertain. The proposed rules are implemented via zeroth-order stochastic approximation over truncated-signature coefficients and evaluated numerically under Brownian and fractional Brownian dynamics, achieving performance comparable to optimal methods in the Brownian setting while outperforming them and remaining robust to adversarial path perturbations in the fractional Brownian setting.

We study the FiboLaplace and LucasLaplace sequences obtained by applying the Laplace transform to generalized Fibonacci-type and Lucas-type polynomials. We relate these families and show that, for each fixed transform parameter, the FiboLaplace sequence is holonomic. We give a combinatorial interpretation in terms of weighted colored tilings, leading to recurrence relations, identities, and a triangular refinement by the number of dominoes. We also derive two continued-fraction expansions and obtain a second combinatorial interpretation in terms of generalized Motzkin paths. These results extend earlier work of Givens and Moll.

These notes aim to exemplify the concept of action and partial action of a finite monoid $M$, identifying among the so-called indecomposable actions those that are also simple. We present all the simplified examples of monoids (non-groups) of order two and three, explicitly visualizing which are simple and which are not. In addition, we prove three general lemmas relating simplicity to indecomposability in the context of partial actions, and establish a result on the relation between sub-actions of total actions and partial actions.

A semidomain is called bi-UFS if both its additive monoid and its nonzero multiplicative monoid are unique factorization monoids. The Bi-UFS Positive Conjecture predicts that the only positive semidomain with this property is the nonnegative integers. We prove this conjecture for finitely generated algebraic positive semidomains. In the cyclic case, we show that for every positive algebraic number $\alpha$, the semidomain $\mathbb{N}_0[\alpha]$ is bi-UFS if and only if $\alpha \in \mathbb{N}$, equivalently $\mathbb{N}_0[\alpha]=\mathbb{N}_0$. The proof separates the quadratic case, where an analysis of the least additive atom larger than $1$ leaves only the examples $\mathbb{N}_0[\sqrt 2]$ and $\mathbb{N}_0[(1+\sqrt 5)/2]$ to exclude, from the higher-degree case, where explicit multiplicative identities force the minimal polynomial into impossible forms. We then give a Perron-Frobenius argument showing that if $\alpha_1,\ldots,\alpha_n$ are positive algebraic numbers and $\mathbb{N}_0[\alpha_1,\ldots,\alpha_n]$ is bi-UFS then this semidomain is $\mathbb{N}_0$. Finally, we prove a reduction theorem for complex semidomains: every bi-UFS subsemidomain of $\mathbb{C}$ with finitely many additive atoms admits an isomorphic realization as a positive semidomain. Consequently, every finitely generated algebraic bi-UFS semidomain over $\mathbb{C}$ is isomorphic to $\mathbb{N}_0$.