Papers

We study the problem of maximum likelihood estimation for moving average (MA) time series models from the perspective of algebraic statistics, with a focus on the structure and number of solutions of their likelihood equations. Of particular interest is to classify the critical points that lead to non-invertible models. We consider the composite likelihood as an alternative estimation method and analyze its critical points. We extend our algebraic analysis to autoregressive processes (AR). We provide algebraic closed form formulas for the parameters when possible. We also explore in simulations how methods from numerical algebraic geometry perform against traditional optimization for these models.

Long forward error correction codes are typically constructed by concatenating shorter component codes that are then decoded through iterative Soft-Input Soft-Output (SISO) of their components. The recently introduced Soft Output Guessing Random Additive Noise Decoding (SOGRAND) has been shown to enable accurate SISO component decoding for a broad range of component codes. Here we establish that by specializing its SISO computation to Single Parity Check codes, SOGRAND offers an alternative existing Check Node (CN) update for decoding Low Density Parity Check codes. Simulation results demonstrate similar or better decoding performance than Gallager's sum-product algorithm and norm-min-sum, while offering two distinct low complexity, hardware friendly CN update algorithms.

We study an extension of the algebra of the infrared to curve–valued potentials, focusing on the elliptic curve case. Given a finite configuration of points on an elliptic curve, we construct associated \(L_\infty\)-algebra. Adding an extra point produces a directed \(A_\infty\)-algebra and an \(L_\infty\)-morphism to their directed Hochschild deformation complexes. Maurer–Cartan elements of the \(L_\infty\)-algebra determine \(A_\infty\)-deformations of the directed algebra. We formulate a conjectural comparison between Maurer–Cartan deformations of the directed algebra and the directed Fukaya–Seidel category associated to compatible finite collections of thimbles.

We prove that for every $n\ge 2$ the Higman–Thompson group $F_n$ has a maximal subgroup of infinite index isomorphic to itself. In fact, we construct a chain of subgroups $F_n=H_0>H_1>H_2>\cdots$, all isomorphic to $F_n$ and with trivial intersection, such that for every $i$ the only subgroups of $F_n$ containing $H_i$ are $H_i,H_{i-1},\ldots,H_0=F_n$; in particular, each $H_{i+1}$ is maximal in $H_i$. We prove that for all $n\ge m\ge 2$, every closed maximal subgroup of $F_m$ isomorphic to $F_n$ arises from a homeomorphism between the $n$-ary and $m$-ary Cantor spaces given by a finite semi-synchronizing transducer–a variation of the synchronizing transducers of Bleak, Cameron, Maissel, Navas and Olukoya. We characterize the homeomorphisms of Cantor spaces conjugating $F_n$ into $F_m$ as the order-preserving or order-reversing rational homeomorphisms whose minimal transducer is semi-synchronizing. At the heart of the paper is a machinery bridging transducers and Stallings $2$-cores of subgroups, which reduces the conjugation of finitely generated closed subgroups by such homeomorphisms to an algorithmic procedure. As applications, we prove that Jones' ternary oriented subgroup $\vec F_3\le F_3$ is isomorphic to $F_4$, answering questions of Aiello, and that all known maximal subgroups of infinite index of Thompson's group $F$ which act minimally on $(0,1)$ are isomorphic to Higman–Thompson groups. That raises the problem of whether all maximal subgroups of infinite index of $F$ which act minimally on $(0,1)$ are isomorphic to Higman–Thompson groups. We briefly discuss related results regarding fast groups of homeomorphisms and maximal subgroups of Thompson groups.

We study sequential posted pricing for selling a single item under an exogenous deadline: a seller makes take-it-or-leave-it price offers to buyers one at a time, stopping when some buyer accepts or the selling opportunity expires. We consider both deterministic deadlines and random deadlines drawn from a known distribution, and allow buyer valuations to be either independent or arbitrarily correlated. Despite its practical relevance, this collection of deadline-constrained sequential pricing models has received limited algorithmic study beyond special cases. We develop the first approximation algorithms for revenue maximization across these settings.

Eternal domination is a process in which a set of guards occupying a dominating set on a graph protects against an infinite sequence of attacks. After a vertex is attacked, one guard must move along an edge to the attacked vertex and each of the remaining guards may move along an edge so that the guards again occupy a dominating set on the graph and can defend the next attack. The minimum number of guards needed in a graph $\Gamma$ is the eternal domination number, denoted by $\gamma_{\mathrm{all}}^\infty(\Gamma)$. In this paper, we show that the eternal domination number of a vertex-transitive graph with an efficient dominating set is equal to its domination number. We show that a Cayley graph on a generalized dihedral group whose connection set contains few or many reflections is efficiently dominated. Then, we provide an infinite family of connected Cayley graphs for which $\gamma_{\mathrm{all}}^{\infty}(\Gamma) = \gamma(\Gamma)+1$, generalizing a result of [Braga et al., J. Combin. Math. Combin. Comput. 96 (2016), 13–22]. Finally, we build an infinite family of connected Cayley graphs with $\gamma_{\mathrm{all}}^\infty(\Gamma) \geq \gamma(\Gamma)+2$.

In the binary trust-gossip dynamics, $n$ agents each hold a directed binary opinion, trust or distrust, of every other agent, and a gossip step lets one agent copy another agent's opinion of a third whenever the copier trusts the source. We prove that under any fair schedule of such steps, every trajectory reaches an absorbing state in finitely many steps, one in which no gossip step changes any opinion; in particular, there are no limit cycles. The proof is constructive and rests on a single descent measure: the number of ordered pairs of distinct agents linked by a chain of trust. Trust-adding gossip leaves this count unchanged and trust-removing gossip can only lower it, so it never rises, and it is zero only at the all-distrust state. From any non-absorbing state we exhibit a finite run of steps that reaches an absorbing state or strictly lowers the count; since the count is a non-negative integer at most $n(n-1)$, the process halts at an absorbing state.

Physical sensing and actuation noise floors should inform how much belief resolution a decision-making system can reliably use. We introduce Finite Reliability Representations (FRR), a framework for covering belief spaces by reliability cells: regions within which the optimal action-value function Q*(b,u) varies by at most a tolerance epsilon, uniformly over actions. The framework is formulated on beliefs rather than states and uses a cover rather than an equivalence quotient, because approximate decision-closeness is not transitive in general. A central technical point is that noisy Bayesian updates should not be treated as globally contractive on arbitrary beliefs. We therefore separate three objects: the fixed-observation filter map, the predictive observation law, and the controlled belief-transition kernel. For nonlinear continuous-state systems, FRR is obtained under a reachable-set Lipschitz modulus for the belief-transition kernel. For finite-state POMDPs, the same construction becomes exact on the belief simplex: prediction is linear, Bayesian correction is a normalized positive linear map, sensor noise enters through observation-distribution distinguishability, and actuation uncertainty enters through an action-execution channel. Under the corresponding action-value Lipschitz condition, an FRR cover supports a cell-constant policy whose suboptimality is bounded by 2 epsilon/(1 - gamma). We also introduce reliability entropy, the logarithm of the minimal number of reliability cells, as a measure of certified decision-relevant belief complexity. The framework distinguishes representation sufficiency from fundamental performance floors imposed by sensing, process, and actuation noise. It applies to finite POMDPs, linear-Gaussian filters, locally linearized nonlinear filters, and particle-filter implementations through analytic or empirical certification of reliability cells.

We prove that the reduced odd Khovanov homology of a link $L$ is naturally a module over the exterior algebra of the first homology of the link's branched double-cover. We then describe this module structure more geometrically and related it to the odd Khovanov maps induced by link cobordisms. As an application, we will give a combinatorial proof of a recent result of Spyropoulos-Vidyarthi-Zhang about the odd invariant for $2$-knots in the special case where the $2$-knot is a ribbon $2$-knot. Additionally, we will show that Levine-Zemke's main result from their 2019 paper on Khovanov homology and ribbon concordance remains true for odd Khovanov homology with rational coefficients and with coefficients in $\mathbb{Z}_{2^k}$.

Many stabilizer quantum error-correcting codes are built from a finite pattern repeated across space or time, such as lattice codes, translation-invariant graph states, and quantum convolutional codes. Ordinary stabilizer ZX-diagrams capture only finite truncations of such systems, obscuring the repeated structure that defines them. We introduce the delayed stabilizer ZX-calculus, a finite graphical language for these infinite, translation-invariant processes. It extends the odd-prime-dimensional stabilizer ZX-calculus with a single new generator, the delay, which feeds data from one time step to the next. We equip the calculus with two semantics. In the first semantics, we interpret the behaviour of a delayed ZX-diagram as an equivalence class of sequences of quantum channels; where two sequences are identified if they have the same information content. We show that the behaviour of a delayed ZX-diagram uniquely determines an infinite stabilizer group. In the second semantics, we interpret the delay as a formal variable, encoding the translation-invariant families of Pauli operators as generating functions. This allows us to represent a delayed ZX-diagram in terms of a tableau of generating functions, from which the infinite stabilizer group can be recovered. Finally, we give a complete axiomatization of the delayed stabilizer ZX-calculus, featuring generalised Euler decomposition and colour change rules. Using generalised forms of local complementation and pivoting, we reduce every diagram to a unique normal form. This establishes soundness, universality, and completeness for the generating tableau semantics.

Ample sets of hypercubes, introduced by A. Dress in 1995, constitute a combinatorial structure with rich properties and important examples. Ample sets can be characterized in a multitude of combinatorial, graph-theoretical, recursive, and geometrical ways, and they are equivalent to lopsided sets introduced by J. Lawrence in 1983. In this paper, we define and investigate ample sets of Cartesian products $U=U_1\times\cdots\times U_m$. This is done using minor-subproducts of $U$, which correspond to products of partitions of factors: each minor-subproduct is obtained by partitioning each $U_i$ into blocks and contracting blocks into singletons. For a minor-subproduct $M$ and a set $S$, we define the notions of shattering of $M$ by $S$, of copy of $M$ in $S$, of projection $S_M$ of $S$ on $M$, and of strong-projection $S^M$ of $S$ on $M$. We call a set $S$ ample if for any minor-subproduct $M$ that is shattered by $S$, there exists a copy of $M$ included in $S$. We prove that several characterizations of ample sets can be extended to ample sets of Cartesian products. In particular, we show that ampleness of $S$ is equivalent to the ampleness of the complement $S^*$, to superisometricity (isometricity of $S^M$ for any minor-subproduct $M$), and commutativity $(S^M)_{M'}=(S_{M'})^M$ for all minor-subproducts $M,M'$ with disjoint supports. We also provide more efficient characterizations of ampleness, in particular, by showing that $S$ is ample iff S is isometric and both $S_e$ and $S^e$ are ample for some elementary minor-subproduct, iff the intersection of S with any interval [u,v] with u,v in S is ample in the classical sense. We characterize ampleness by push downs and provide a decomposition theorem, allowing us to prove that their prism complexes are contractible. We provide new examples of ample sets arising from payoff games, prism-like polyhedra, and quasi-median graphs.

We prove a Hölder-type regularity estimate for the martingale integrand of a backward stochastic Volterra integral equation (BSVIE). The estimate is formulated in $L^p(\Omega)$ after averaging in $L^2$ over the first time variable, and gives an averaged Hölder estimate of order $1/2$ in the second time variable. Our approach is based on the approximation of the BSVIE by a system of BSDEs. For this system, we establish a uniform regularity estimate for the martingale component using Malliavin calculus, and then pass to the limit to obtain the result for the BSVIE. We allow for a general Malliavin differentiable free term and generator. In particular, neither is assumed to come from a forward stochastic differential equation or to have a specific functional form. We also propose an explicit Euler scheme for the approximating BSDE system and show that the regularity estimate yields a convergence rate for the resulting discrete approximation of the BSVIE.

We establish finite-sample closed-loop stability guarantees for Model Predictive Path Integral (MPPI) control applied to discrete-time Linear Time-Invariant (LTI) systems with additive Gaussian process disturbances. The key observation is that, for unconstrained LTI/quadratic systems with the DARE terminal cost, the exact finite-horizon MPC law has the same first control action as the infinite-horizon LQR law for every planning horizon. Thus, finite-sample MPPI can be analyzed as a stochastic perturbation of LQR. First, we show that the MPPI control law approximates the LQR feedback with high probability. The approximation error decomposes into a Monte Carlo term that decreases with the sample count and an infinite-sample temperature bias that persists at finite temperature but vanishes as the temperature is reduced. The resulting constants are written in terms of the horizon-dependent stacked cost matrices, making explicit that the finite-sample certificate is parametrized by the selected planning horizon. Second, we use a Lyapunov perturbation argument to prove practical exponential stability in expectation. On sample paths that remain in a compact Lyapunov sublevel set over a finite operating horizon, the expected state norm decays exponentially up to three residual floors: a process-noise floor, an MPPI approximation floor, and a confidence floor from the per-step sampling failure probability. The sufficient sample threshold is explicit and computable from the DARE solution, LQR stability margin, MPPI sampling parameters, temperature, and planning horizon. In the joint limit of infinite samples and vanishing temperature bias, the result recovers the stochastic LQR stability bound.

For every $n\geq 2$, we exhibit an $n$-dimensional K-polystable toric $\mathbb{Q}$-Fano variety $X_n$, defined by the face fan of an explicit lattice polytope, and whose alpha invariant is exactly $\tfrac{2}{2n+1}$. This answers a question of Liu and Zhuang whether there exists an $n$-dimensional K-semistable $\mathbb{Q}$-Fano variety whose alpha invariant is between $\tfrac{1}{n+1}$ and $\tfrac1n$. The main result of this paper was obtained by Chatgpt 5.5 pro, and the Danus system based on the Rethlas system.

We study collision properties of simple random walks in unimodular random rooted graphs. This work continues the study initiated in~\cite{HutchcroftPeres2015}: under recurrence and an integrability condition on the root, two independent random walks collide infinitely often a.s. We prove that, under transience and an integrability assumption involving the Green function on the root, two independent random walks collide only finitely often a.s. We apply these results to several random graphs with unbounded degree: the Gilbert graph, the Delaunay graph, the Gabriel graph; and the long-range percolation model. We use these collision properties to characterize stationary measures of the voter model on these graphs.

This article addresses the following general question: Given a one-dimensional manifold $M$ and $1 \le r < s \le \infty$, does there exist a $C^s$ orientation preserving compactly supported diffeomorphism of $M$ that is undistorted in the group $\mathrm{Diff}_{c,+}^s(M)$ of such diffeomorphisms while distorted in the bigger group of $C^r$ diffeomorphisms? Interestingly, the answer is known to be positive in the case $(r,s)=(1,2)$ and negative in the case $(r,s)=(2,\infty)$, according to arXiv:2004.07055 and arXiv:2507.13770, respectively. The first part of this note originates from a failed attempt to extend the ideas of arXiv:2004.07055 to the case $(r,s)=(2,3)$. More precisely, in regularities $C^1$ and $C^2$, obstructions to distortion are provided by drifts of cocycles for isometric actions of $\mathrm{Diff}_{c,+}^r(M)$ on Banach spaces for $r=1$ and $r=2$ (namely, the logarithmic and projective derivatives $f\mapsto \log Df$ and $f\mapsto D\log Df$, respectively). On $\mathrm{Diff}_{c,+}^3(M)$, the so-called Liouville cocycle is a natural candidate when looking for new obstructions, but we show that its drift vanishes for $C^2$-distorted diffeomorphisms (and this holds more generally for any "similar" cocycle). This does not rule out the existence of $C^2$-distorted diffeomorphisms that are $C^3$-undistorted. However, at least in the case of the real line, such a diffeomorphism should have very low regularity. Indeed, extending the methods and results of arXiv:2507.13770, in the second part of this article, we show that every compactly supported $C^2$-distorted diffeomorphism of the real line is $C^r$-distorted provided its differentiability class is larger than $C^{2r+2}$.

In an article, S. Adlaj showed the construction of the $F_4 = 65\,537$-gon ($F_4$ a Fermat prime). By extending the author's method to cases where the multiplicative group is not of order $2^n$ but has an additional prime factor, I here construct the $641$-gon. $641$ is the smaller prime factor of the composite Fermat number $F_5 = 2^{2^5} + 1$.

Time-dependent processes are often described by machines with an internal state which is updated as time evolves. An external observer cannot see this state and learns about a process only through finite observations of its inputs and outputs, each of which imposes a constraint on the trajectories the process can exhibit. We introduce a semantic construction in which two stateful processes have the same behaviour when they have the same constraints, as determined by finite observations, independent of their internal state. The construction is defined over any preorder-enriched monoidal category with a compatible notion of discarding, which we call a discard bicategory, capturing partial, non-deterministic, probabilistic, and quantum processes. The resulting category of behaviours provides a functorial semantics for free feedback categories in the sense of Katis, Sabadini, and Walters. For non-deterministic systems, we prove a categorified compactness theorem: every compatible family of finite observations between compact Hausdorff spaces extends uniquely and functorially to an infinite closed relation. Restricted to affine relations over finite fields, the compactness theorem recovers Willems' notion of behaviour for linear time-invariant systems.

We develop methods for forcing $\lim^1 \mathbf{A} \ne 0$, where $\mathbf{A}$ is a particular inverse system of abelian groups introduced by Mardešić and Prasolov in their computation of certain strong homology groups. These methods allow us to extend previous nonvanishing results of Casarosa and Lambie-Hanson for $\lim^k \mathbf{A}$ for $k \geq 2$. Specifically we show that, for a given $n$, it is relatively consistent with ZFC that $\mathfrak{b} = \mathfrak{d} = \omega_n$ and $\lim^k \mathbf{A} \ne 0$ whenever $1 \leq k \leq n$ (previously established with $2 \leq k \leq n$). We also show it is relatively consistent with ZFC that $\mathfrak{b} = \mathfrak{d} = \omega_{\omega+2}$ and $\lim^k \mathbf{A} \ne 0$ for all $k \geq 1$ (previously established with $k \geq 2$). We also adapt proofs of Kamo to show that $\lim^1 \mathbf{A} = 0$ holds in many finite support iterated forcing extensions.

Informally, the capacity of list-decoding in a given adversarial error model is the largest rate at which we can list-decode with list size polynomial in the block length. The capacity of list-decoding from insertions and deletions is a basic, yet poorly understood, aspect of coding against synchronization errors. For example, when dealing with a $\delta>1/2$ fraction of insertions, the best known lower bounds give little more than the fact that the capacity is positive. Beyond that regime we also only have loose bounds, with the lower bounds stemming from the analysis of uniformly random codes. We make progress in our understanding of the limits of list-decoding binary codes from insertions and deletions. We show that the capacity of list-decoding from a $\delta$-fraction of insertions is exactly \begin{equation*} (1+\delta)\left(1-h\left(\frac{\delta}{1+\delta}\right)\right) \end{equation*} for all $\delta\in[0,1]$, achieved with high probability by a code sampled according to a symmetric $2$-state Markov chain. Curiously, we complement this by showing that such an approach does not beat uniformly random coding in list-decoding from deletions. We also give an improved upper bound on the capacity of list-decoding from a $\delta$-fraction of deletions, showing in particular that it behaves like $1-h(\delta)$ when $\delta\to 0$. This matches the asymptotic behavior of the capacity of the binary deletion channel for vanishing deletion probability.

We solve in semi-explicit form a class of non-Markovian stochastic optimal control problems with path-dependent rewards, using path signatures. We reformulate the control problem as the computation of Laplace transforms of signature functionals thanks to the Boué-Dupuis representation. Exploiting recent signature representations of such transforms on tensor algebras, we determine the value process and the optimal control through an infinite-dimensional system of Riccati equations on the extended tensor algebra. We establish an explicit feedback representation of the optimal control and the value process as an infinite linear combination of the time-extended signature of the controlled process, with time-dependent coefficients. The expansions being intrinsically local, we propose a dynamic recentering algorithm to ensure a global representation over the entire time horizon. We illustrate the approach on genuinely path-dependent, non-linear examples that go beyond the tractable linear-quadratic setting, including the tracking of linear functionals of the signature and signature lifts of Volterra control problems.

In a 1934 Annals of Mathematics paper, Samuel S. Wilks posed the problem of computing joint moments of disjoint principal minors of Wishart random matrices, describing the general case as “extremely complicated.” These moments arise in classical multivariate statistics, including multivariate regression, analysis of variance, generalized correlation coefficients, and likelihood ratio testing. We solve Wilks' problem for nonnegative integer exponents and any finite number of disjoint principal minors using Stein's method, specifically the Wishart Stein characterization recently developed by Bailly et al. (2026). The Wishart Stein identity yields a degree-lowering recursion that terminates after finitely many steps and evaluates the desired moment exactly. Numerical experiments validate our recursive formulas against direct Monte Carlo simulations from the Wishart distribution.

The basic reproduction number is the spectral radius of a matrix, $R_0=\rho(K)$. Taking that definition literally, we ask what $K\mapsto\rho(K)$ discards. A matrix carries three kinds of information: its dominant eigenvalue, its dominant eigenvectors, and its departure from normality. $R_0$ keeps only the first; the other two are where the epidemic's social structure lives. The right eigenvector is the burden distribution, the left the source distribution; they coincide when the system is normal and diverge under heterogeneity. Across the $177$ national contact matrices of Prem et al., the operator is never normal, and once age-specific susceptibility is included, its source and burden eigenvectors are misaligned by a median of $26^{\circ}$, exceeding $40^{\circ}$ in some countries: the groups that drive transmission are systematically not those that bear it. We prove that under reciprocal contact this misalignment obeys a Kantorovich bound set by the susceptibility contrast $q_{\max}/q_{\min}$ alone, and zero when susceptibility is uniform, with the excess in real, non-reciprocal matrices contributed by contact asymmetry. Transient amplification, by contrast, stays small, so the operative social content is the misalignment, not transient blow-up. The omission also has teeth: because minimizing $R_0$ protects those who spread infection, while minimizing deaths protects those who die from it, the two target different age groups; the former sometimes raises average infection fatality even as it lowers the scalar. When contact is strongly structured and susceptibility is heterogeneous, we suggest reporting $R_0$ along with its eigenvectors rather than reporting it alone.

Russell recently introduced a family of three-colored partitions in analogy with earlier work by Alladi and Gordon. He showed that their enumerations are surprisingly connected to the classic Rogers-Ramanujan identities. Using the method of linked partition ideals, we elaborate on Russell's results by further counting each part color. Unlike Russell's proofs, our arguments do not require any use of computer algebra systems. With these multivariate generating function relations, we are led to confirm a conjecture of Russell. Furthermore, our results offer a combinatorial understanding of a new triple Rogers-Ramanujan type identity.

We develop a static mean field game formulation for optimal wind energy capacity siting in Alberta, Canada. The revenue model for agents is based on the expected wind resource at a location and the covariance of this wind resource with other locations. Agents choose locations to maximize long-run revenue while accounting for spatial correlations in wind resource availability. Spatial wind dependence is incorporated through an empirically calibrated covariance structure, combining a PCA-based component estimated from historical weather station data with a parametric residual kernel, to capture variability across locations. The equilibrium investment problem can be formulated as a quadratic program, which we solve for four policy scenarios that progressively restrict the feasible siting area, incorporating viewscape and transmission constraints beyond a baseline of minimal siting restrictions. These policy and infrastructure-driven land use restrictions are represented as constraints on the agents' action space, allowing us to examine their economic implications through a comparison of the MFG equilibria across policy scenarios.Our findings highlight the trade-off between regulatory land use objectives and economic efficiency in the use of renewable energy resources in the short and long term, and provide quantitative insights into how policy design shapes the spatial distribution and financial performance of wind investments. The static MFG formulation could be used to inform electricity system planning including, transmission planning and renewable energy development planning.

This paper presents a new nonlinear two-level acceleration method for solving the particle transport equation with anisotropic scattering. The method is formulated with the projection operator approach. The low-order equations are defined for the angular moments using projection operators and closures of the second-moment method. A nonlinear prolongation operator is applied to the scattering term to derive the high-order transport equation. The nonlinear system of high-order and low-order equations is equivalent to the original transport equation. The equations are approximated in space by the lumped linear-discontinuous Galerkin method. Numerical results are presented to demonstrate the performance of the proposed numerical method.

We derive the equality between one-dimensional sums associated with tensor products of Kirillov-Reshetikhin column crystals of classical affine types and Lusztig q-analogues of weight multiplicities. The matching of the corresponding root systems is suggested by Howe duality. Our main tool is the dual Cauchy formula for Koornwinder polynomials due to Mimachi, which we combine with specializations in these polynomials. The mentioned dualities are proved for one-dimensional sums of all (twisted and untwisted) classical affine types except types B_n^(1) and D_n^(1). On another hand, all the Lusztig q-analogues of classical type are covered by our dualities, but they may have different parameters depending on the length of the roots in the underlying root system.

We present an L1-optimal control problem class with linear nonnegative costs subject to multiplicative Itô diffusion processes with elementwise linear input constraints. Forward invariance of the positive orthant is established for the considered stochastic dynamics, and a simulation method consistent with this invariance property is proposed. Both finite-horizon and discounted infinite-horizon stochastic L1-optimal control problems are considered. These problems admit explicit solutions characterized by a vector-valued ordinary differential equation in the finite-horizon case and by an algebraic equation in the infinite-horizon case. Notably, the optimal value function and feedback policy coincide with those of the corresponding deterministic problem, demonstrating robustness to multiplicative stochastic uncertainty. A portfolio example illustrates our results.

We study the online routing problem in large language model serving, where requests arrive sequentially and must be dispatched to parallel decode workers under tight batch-size and KV-cache constraints. Unlike widely used routing heuristics that are not tied to explicit service-level objectives (SLOs) and offer limited control over latency-throughput trade-offs, we introduce a multi-objective optimization framework that formulates routing as an online linear programming with interpretable decision rewards. We apply an efficient bid-price control policy based on the online linear programming that admits requests when their SLO-weighted benefit exceeds their shadow prices. To meet millisecond decision requirements, we develop a warm-started, projected first-order updates that track the evolving dual shadow prices online with predictable runtime. We integrate our router into the Vidur simulator and demonstrate substantial improvements over standard baselines across multiple SLO regimes, including end-to-end latency, time-to-first-token, throughput, and tail performance. A big picture from our result: a science-based approach outperforms others based on heuristics.

We give a lower bound on the Ulrich complexity of hypersurfaces in terms of their dimension.