Papers

The discrete algebraic Riccati equation (DARE) is used to solve for the optimal feedback gain in linear-quadratic regulator (LQR) control of discrete-time systems. In this letter, we consider the effect of perturbations to the LQR data on the DARE solutions. We present a new frequency domain approach to derive an explicit bound on the difference in DARE solutions under such perturbations. Our bounds are similar to other results that have recently appeared in the literature with two distinctions. First, we use a frequency domain constant that measures the degree of closed-loop stability, in contrast to other results that use a time domain measure. Second, we start from weaker detectability assumptions on the LQR state cost matrix.

Let $ G $ be a non-compact connected reductive real Lie group and $ P $ its parabolic subgroup. We give a list of equivalent conditions for $ P $ to have an abelian unipotent radical. These conditions are related to skew multiplicity free and multiplicity free actions, and using the theory of Clifford algebra and the cubic Dirac operator we deduce formulas for the de Rham cohomology of compact symmetric spaces $ G/P $.

We prove the existence of classical solutions for nonlinearly elastic, single-director surfaces or shells characterized by finite mid-plane strains and curvature. We employ topological methods, yielding solution continua far (in norm) from an unloaded, stress-free reference configuration, i.e., a global implicit function theorem of sorts. We first establish the existence of a local solution path, which is subsequently shown to be part of a global solution branch, viz., a maximal connected, locally compact set, emanating from the unloaded state.

We present a novel algorithm for calculating the preimage of an affine space through a product ${J}={J}_{T-1}\cdots{J}_0$ of matrices ${J}_t$ of special form: finding the largest input space $\mathbf{X}$ such that $\mathbf{x}\in\mathbf{X}$ implies ${J}\mathbf{x}\in \mathbf{Y}$, where $\mathbf{Y}$ is a given output affine space. These special matrices arise in AD, where the Jacobians $J$ describing the linearized computation have precisely this structure: the product of a series of linearized primitive numeric operations. This allows us to use the new algorithm to formulate Null-A mode preimage AD, which finds the affine preimage through the Jacobian or Jacobian transpose of a numeric computation. This is a generalization of the inverse AD problem of solving ${J}\acute{\mathbf{x}}^{\ast}=\acute{\mathbf{y}}^{\ast}$ or ${{J}}^{T}\grave{\mathbf{y}}^{\ast}=\grave{\mathbf{x}}^{\ast}$. The key is to represent affine spaces in a fashion which lends itself to efficient preimage calculation, in a compositional and quasi-local fashion, through a succession of matrices ${J}_t$. Unlike previous methods, Null-A preimage mode AD allows the ${J}_t$ matrices to be non-square, corresponding to a computer program whose number of active variables swells and shrinks during the computation. When ${J}$ is square and the initial affine space is a single point, this finds the conventional inverse. But in the more general case, having the entire affine space provides freedom which can be leveraged in a problem-specific manner. We apply the method to small problems on-CPU where the $J_t$ are linearized scalar unary or binary numeric functions; and to larger problems on-GPU where the $J_t$ are linearized aggregate array operations like convolution and attention.

We provide one full solution, and another nearly complete solution, to two problems from Jim Propp's 1999 article "Enumeration of matchings; problems and progress" (namely, Problems 30 and 31) involving divisibility properties of the number of matchings of two non-bipartite triangular graphs. We extend our method of proof of the second problem to show that the number of matchings of any graph composed of tetrahedral cells, such that the number of cells is suitably few relative to the number of vertices, is divisible by a power of 3; in particular, we then exhibit a collection of non-planar graphs whose number of matchings is divisible by 3.

We determine the maximum population size for which exactly two defective items can always be identified using at most $k$ adaptive quantitative tests, for every $0\le k\le10$. Each test reports the number of defectives in a chosen subset. The exact maxima for eight, nine and ten tests are 65, 112 and 192, respectively. We also classify all pairs of sizes of two disjoint sets, each containing exactly one defective, that can be resolved within each test budget through eight. These exact results are supported by explicit strategies and maximality certificates checked by programs separate from the search implementation. Additional results include partial nine-test boundaries and an eleven-test strategy for 328 items; maximality of the latter remains open.

We formally derive a coupled Enskog system for a gas–particle mixture from the hard-sphere BBGKY hierarchy. The derivation retains the finite separation between colliding centres, includes collisions within and between species and explicitly states the direction of spatial shifts and the mass-dependent prefactors of the four collision operators. The resulting system provides a kinetic starting point for two-phase flows for which the volume effect cannot be ignored. For unit contact factors, we derive the corresponding gas and particle balance laws, and establish an entropy inequality under suitable regularity assumptions.

Hyperdimensional computing and vector symbolic architectures often represent quantized scalar levels by dense binary codebooks whose level-to-level similarity is intended to follow a prescribed function of scalar separation. At finite dimensionality, randomized scalar codebook constructions deviate from this target because of sampling noise, random-start imbalance, update-count fluctuations, component dependence, and finite-capacity effects. We develop a transition-based derandomization framework for dense binary scalar codebooks across two target-similarity families, with similarity decaying exponentially or linearly with level separation. The framework separates the target similarity law, the derandomization variant, and the concrete generator construction, making explicit how initialization, selection, update, and capacity-handling mechanisms shape the induced similarity profile. We formalize derandomization variants that separately constrain initial Hamming weight, update-count variability, and update balance, thereby controlling distinct sources of finite-dimensional error. For each family and variant, we derive the induced mean similarity, identify realization-wise and mean target-matching regimes, and derive exact finite-dimensional expressions for bias, variance, and root-mean-square error. Simulations across dimensions, quantization ranges, reference scalar levels, and generator constructions validate the theory and show how each constraint removes or reduces a specific source of similarity mismatch. The results provide practical guidance for choosing scalar codebook generators that more closely match a desired similarity law under finite-dimensional and hardware-relevant constraints.

We study a family of polytopes interpolating between the hypercube and the associahedron. We give an explicit description of their normal fans and describe their combinatorics in terms of chord diagrams associated to their vertices. This yields interesting number sequences generalizing Pell numbers and asymptotic to Catalan numbers. We describe the transition between the normal fans in terms of star subdivisions and relate them to polytopes associated to Dyck paths as in [Veronica Calvo Cortes and Hadleigh Frost. Dyck paths, Configuration Spaces and Polytopes for Linear Nakayama algebras. https://arxiv.org/abs/2602.04571] and hypergraphic polytopes defined in [Carolina Benedetti, Nantel Bergeron, and John Machacek. Hypergraphic polytopes: combinatorial properties and antipode. Journal of Combinatorics. 2019]. Furthermore, all polytopes in our families yield binary geometries.

Grokking shows that fitting the training data and learning the underlying signal can occur at very different stages. However, existing theories offer limited quantitative insight into how this delayed generalization depends on inductive bias and signal structure. Our work addresses the gap by developing a statistical theory that characterizes how regularization geometry and signal sparsity govern generalization near interpolation. In particular, we focus on the prototypical setting of high-dimensional regression and identify regimes in which sparsity-promoting regularization makes exact interpolation much more accurate than approximate fitting. In strongly overparameterized noiseless problems, we prove a zero–one generalization law and construct a family of convex norms whose interpolators transition from the trivial risk of the all-zero predictor to exact recovery, while keeping the training error equal to $0$. Furthermore, when feature dimension and sample size are proportional, we provide a precise characterization of training and generalization errors along $\ell_r$-regularization paths. This in turn allows us to quantify the generalization gain that remains near interpolation: we show that this gain increases as the norm becomes more sparsity-promoting and as the target becomes sparser, with a sharp drop in generalization reached for noiseless data and $\ell_1$ regularization. Experiments on diagonal linear networks and transformers trained on modular arithmetic demonstrate the generality of our theoretical predictions. Finally, beyond grokking, our work reveals a statistical instability in minimum-norm interpolation: small perturbations in the regularization strength can lead to drastically different generalization, while preserving small training error.

This work provides a comprehensive characterization of the first dynamic transition of a general two-component reaction-diffusion system on a bounded interval in one spatial dimension. By rigorously addressing the difficulties arising from the non-linear interactions between the two components, this study significantly extends the scope of previous studies on dynamic transitions of reaction-diffusion systems, which focused on single-equation cases. The analysis is conducted by employing center manifold reduction within the standard classification of dynamic transition theory. A key contribution of this study is that the nonlinear terms are treated to involve arbitrary powers of the unknowns $u$ and $v$. Furthermore, the theoretical framework is complemented by the analysis of a representative and simplified system. As a result of this generalization, the dynamics of a broad class of fundamental reaction-diffusion systems can be systematically classified.

A recent manuscript produced by Claude under the direction of Levent Alpöge constructs a 1-parameter family of compact complex threefolds diffeomorphic to $S^6$. Our goal is to describe the geometric ideas behind the construction, give a self-contained proof, and remark on possible future research directions.

We determine the decidability of Karp's NP-complete problems on structures which are first-order definable over the theory of equality, also known as orbit-finite sets with atoms or nominal sets.

Can a local solver return the guaranteed globally optimal solution to a nonconvex mixed-integer polynomial power grid optimization problem? By mixing low-rank semidefinite programming (SDP) and moment-based lifting in the Lasserre hierarchy, this paper provides anecdotal evidence in the affirmative. Specifically, we iteratively tighten a moment-based relaxation of the AC Optimal Transmission Switching (AC-OTS) problem, including higher order moment-based constraints, rooted in the Lasserre hierarchy, until it is tight (i.e., primal feasible). We then re-solve this problem using a local, low-rank SDP solver (Knitro), plugging the associated dual variables into a bounding Lagrange dual function. By demonstrating primal and dual objective agreement and primal feasibility, we confirm the solution as the exact, global solution. From these initial results, we hypothesize that mixing (i) low-rank (i.e., highly scalable) SDP methods with (ii) moment-based relaxation tightening could be a scalable alternative to global branch-and-bound-based methods.

Real-world optimization problems are difficult to model accurately because many objectives and constraints reside in domain experts' tacit knowledge, making them hard to formalize. As a result, optimization models often contain miscalibrated objectives, missing constraints, or omitted decision variables, leading to solutions that fail to reflect operational realities. We address this challenge by automatically repairing misspecified formulations using historical data consisting of past solutions and subsequent user overrides. Traditional approaches such as inverse optimization and constraint learning tend to overfit sparse data and produce complex formulations. Our central idea is to combine the reasoning capabilities and prior knowledge of LLMs with the formal grounding provided by optimization. We realize this idea through two complementary paradigms. Top-down, an LLM proposes structural repairs, including new constraints and variables, whose numerical parameters are calibrated and validated through optimization. Bottom-up, optimization infers cuts from observed decisions, which the LLM contextualizes into interpretable, generalizable modeling constraints. We evaluate our approach on 38 misspecification scenarios spanning nine classes of optimization problems, several drawn from real-world applications, and show that TACIT can repair 78.9% of them (vs. 60.5% for the best baseline).

The predictable square function measures the size of a martingale. Applied to parameter differences, it also defines a random pseudometric. We explain how to perform a majorizing-measure argument directly in this realized geometry, without conditioning the terminal field to be Gaussian and without first replacing the metric by a random multiple of a deterministic one. For a finite family of predictable Gaussian sums in a $(2,D)$-smooth Banach space, and a fixed probability measure $\mu$ on the parameter set, we prove \[ \norm{\max_j\osc_T f_j}_{L^p} \le CD\norm{\G_\mu(d)+\sqrt p\,\Delta_d}_{L^p},\qquad p\ge1, \] where $\Delta_d$ is the diameter of the parameter space in the terminal square-function metric and $\G_\mu(d)$ is its ball-mass integral. The proof combines a localized exponential inequality, pathwise averaging over random balls, and a stopping-time argument. Atomic measures recover logarithmically weighted maximal inequalities; Haar measure connects the result with homogeneous entropy and continuity estimates. We also give extensions, examples, and a precise account of the restrictions on choosing the averaging measure.

We study transient solute dispersion in pressure-driven micropolar flow through a concentric annulus representing a catheterized artery. The catheter is impermeable to solute, whereas the arterial wall combines irreversible removal with reversible surface retention. The exact steady micropolar velocity field is coupled to the Gill-Sankarasubramanian generalized-dispersion framework; the resulting bulk-surface hierarchy is advanced with Rannacher-damped Crank-Nicolson time stepping to determine $K_0(t)$, $K_1(t)$ and $K_2(t)$. The exchange coefficient is independent of the micropolar parameters and satisfies $-K_0(0^+)=2(\beta+\theta Da)/(1-\lambda^2)$, whereas $K_2-Pe^{-2}\sim\sigma_v^2t$ initially, so the leading short-time dispersion is hydrodynamic and independent of wall kinetics. In the weak-reaction regime, micropolarity modifies convection approximately linearly and shear dispersion quadratically through the velocity-amplitude factor. A narrow-gap analysis with $\varepsilon=1-\lambda$ gives $\bar v\sim(2-N_c)\varepsilon^2/6$ and $K_2-Pe^{-2}\sim(2-N_c)^2\varepsilon^6/7560$ for a fixed pressure gradient, revealing a sixth-power suppression of shear dispersion as the catheter approaches the arterial wall. Increasing $\lambda$ from $0.01$ to $0.30$ reduces effective convection by about a factor of $2.45$, while chemically passive shear-induced dispersion falls by a factor of $24$. The coupled formulation also yields the exact partition $\Phi_m+\Phi_s+\Phi_a=1$, separating mobile, reversibly retained and irreversibly absorbed solute. The reconstructed field quantifies reaction-modulated transverse non-uniformity, while the results distinguish hydrodynamic effects of confinement and microrotation from kinetic wall effects.

We exhibit a cubic graph with total chromatic number equal to four and list total chromatic number five. This disproves the List Total Colouring Conjecture of Borodin–Kostochka–Woodall, Juvan–Mohar–Škrekovski and Hilton–Johnson from the late 1990s. ChatGPT 6 Astra Ultra discovered the counterexample with little input from the author.

We propose a novel local-polynomial estimator of the ratio $r=f/g$ of two $d$-dimensional densities $f$ and $g$, from which independent samples are available. The estimator is shown to achieve pointwise minimax optimal rates over Hölder classes of arbitrary smoothness index without additional logarithmic factors, and with smoothness being only assumed of $r$ but not of $f$ nor $g$. Our analysis remains valid for points on the boundary of the support of the distribution associated to $g$ under a mild geometric assumption on the (unknown) boundary. We also derive a concentration inequality for the estimator, which can be useful in applications to classification, and give a rate in the supremum norm, where the supremum is also taken over boundary support points. The smoothness class over which the rate is obtained is sufficiently large that the individual densities $f$ and $g$ cannot be consistently estimated uniformly over this class. Direct estimators of the partial derivatives of $r$ together with rates of convergence are also provided. We also obtain asymptotic normality of the estimators under quite general assumptions and again including boundary points, with a consistent variance estimator allowing for data-driven studentization. Moreover, we show how our estimator can be used to estimate the Kullback-Leibler information, in a construction which additionally uses debiasing. We provide the parametric rate together with asymptotic normality under sufficient smoothness of $r$ relative to the dimension $d$.

We construct a separable reflexive Banach lattice $X$ and a closed sublattice $E\subseteq X$ such that every bounded linear operator from $X$ into $\ell_2$ is compact, whereas there exists a positive operator from $E$ into $\ell_2$ that does not attain its norm and admits a positive maximizing sequence converging weakly to a nonzero vector. This answers negatively the question posed by Dantas, Jung and Martínez-Cervantes in [4] concerning the inheritance of the weak maximizing property by closed subspaces of the domain. It also answers negatively the corresponding question for the positive weak maximizing property and closed sublattices of reflexive Banach lattices. Canonical lattice complexification yields a counterexample to the original question over the complex field as well.

Questions concerning representations of integers as the sum of a prime and a power of two go back to the correspondence between Euler and Goldbach in 1752 and to de Polignac's work of 1849. Romanoff proved that the set of integers admitting such a representation has positive lower density. Following the complementary direction studied by Kalmynin and Konyagin, we consider long intervals containing no Romanoff numbers. We use truncated products of divisor sums to derive the bound $G_{\mathcal{R}}(X)\gg\log\log X$ for the longest block in $[1,X]$ containing no integer of the form $p+2^n$, with $p$ prime and $n\ge1$. This improves the bound $G_{\mathcal{R}}(X)\gg\log\log X/\log\log\log X$ of Kalmynin and Konyagin. The key step extends a translate lemma to sets of $O(\log X)$ integer shifts of absolute value at most $X$: auxiliary primes dividing differences of shifts are removed at negligible cost. After preliminary sieving, this allows all remaining shifts to be covered simultaneously. The argument is unconditional and also applies to $p+a^n$ for every fixed integer $a\ge2$.

Homogeneous coordinate rings of Grassmannians are among the most fundamental instances of cluster algebras arising in “nature.” Despite their importance, however, the cluster variables of these cluster algebras are not well-understood, except in special cases. For the Grassmannian of 3-planes in n-space, Fomin and Pylyavskyy conjectured [FP16] that the cluster and frozen variables are exactly the indecomposable, non-elliptic web invariants which are arborizable. They proved that in particular, if an SL3 tensor diagram is a planar tree, its web invariant is a cluster or frozen variable. In this article, we give a new proof of their result for Gr(3, n) planar tree webs, and we generalize the result to Gr(4, n): if an SL4 tensor diagram for Gr(4, n) is a planar tree web, its web invariant is a cluster or frozen variable. Our proof uses some of the cluster quasihomomorphisms from [EZLP+23] and [EZPT+25] (namely, upper promotion and spurion promotion), as well as a new 6-leg pattern map.

We study whether every infinite-dimensional weakly compactly generated (WCG) Banach space fails the Lotz property. The answer is affirmative in the nonseparable case: a classical decomposition theorem for WCG spaces provides a Schauder decomposition, and an explicit diagonal $C_0$-semigroup has an unbounded generator. The unrestricted assertion is false. Our main abstract result is a Calkin-algebra criterion: if $X$ fails the bounded compact approximation property and, for some fixed $m$, every noninvertible element of $\Cal(X)=\Lop(X)/\Kop(X)$ has $m$th power zero, then $X$ has the Lotz property. In particular, this applies when every operator is scalar-plus-strictly-singular and one fixed power of every strictly singular operator is compact. Combining this criterion with the Maurey–Pisier–Szankowski extraction theorem and constructions of Argyros and Motakis yields separable hereditarily indecomposable Lotz spaces over both scalar fields: reflexive examples, and, over the complex field, an example containing no infinite-dimensional reflexive subspace. Moreover, the ambient Argyros–Motakis spaces considered here are Lotz-saturated, although they themselves are non-Lotz. Thus every nonseparable WCG space is non-Lotz, while the separable WCG class contains both Lotz and non-Lotz spaces.

Stochastic differential equations (SDEs) can be studied via Itô calculus and rough path theory. For stochastic optimal control, these two frameworks give distinct Pontryagin Maximum Principle (PMP) optimality conditions with forward-backward SDEs (FBSDEs) or rough differential equations. We show that the adjoint equations of the Itô and rough PMPs are connected via the conditional expectation $p_t^{\text{It\^{o}}}=\mathbb{E}[p_t^{\text{rough}} \mid \mathcal{F}_t]$, where $\mathcal{F}_t$ represents information available at time $t$. First, we derive a rough stochastic PMP for problems with adapted controls that does not use FBSDEs. Its proof extends the rough stochastic PMP over deterministic controls by considering stochastic needle variations. Second, we derive a unified PMP connecting the Itô and rough PMPs, using Itô-Stratonovich conversion formulas and duality identities between the forward tangent and backward adjoint SDEs. As a first application, we rederive the adjoint matching method for fine-tuning generative models. As a second application, we propose an indirect shooting method for a class of feedback problems. Overall, these results give a new conditional bridge connecting two popular frameworks for stochastic optimal control.

We prove that the intersection of $2d$ halfspaces containing the Euclidean unit ball in $\mathbb{R}^d$ contains a point of norm at least $\sqrt{d}$. Equality characterizes the cube. The proof gives a lower bound for the average squared norm of the vertices, weighted by the Gaussian measures of their normal cones. The main step is an inequality for Gaussian orthant probabilities, proved by a one-dimensional comparison of mass, centroid, and boundary density. We also discuss a quadratic extension and give an independent spectral estimate.

The finite-time reachability problem - whether two probability vectors can be transformed in a time t through a generator from a given set - is a key question in Markovian dynamics. We address this problem for normalized Kolmogorov generators with tools from differential geometry. The shortest time T connecting an ordered pair of states defines an (asymmetric) quasi-distance in the probability simplex, which distinguishes between outgoing and incoming reachable sets, and captures the inherent irreversibility of Markov dynamics. The Finsler metric corresponding to normalized generators with bounded diagonal entries is characterized, and time-independent evolutions that saturate speed limits are presented. Analytical boundaries of both reachable sets are derived and the ratio of their corresponding volumes assess the bounds obtained. The asymmetry with respect to the uniform probability vector is shown to be related to the Kullback-Leibler relative entropy.

This paper develops a high-order cut energy-based discontinuous Galerkin (CutEDG) method for second-order acoustic wave equations on complex domains and across stationary material interfaces. The method employs unfitted Carte- sian meshes and addresses the small-cut-cell problem by introducing ghost-penalty stabilization directly into the local bilinear forms defining the EDG energy, while leaving the original spatial coupling and numerical-flux structure unchanged. We establish semidiscrete energy stability and a priori error estimates showing that high-order accuracy is retained. The formulation is further extended to interface problems with discontinuous wave speeds. Numerical experiments confirm the pre- dicted convergence rates and demonstrate robustness with respect to arbitrarily small cut cells. In particular, the explicit time-step restriction is governed by the background mesh size rather than the smallest physical cut cell, and the stabilized energy Gram matrices exhibit cut-independent conditioning.

Given a vertex operator algebra $V$ over $\mathbb{C}$, it is known how to define so-called sheaves of coinvariants and conformal blocks on the moduli space $\overline{\mathcal{M}}_{g,n}$ of stable, $n$-pointed, genus $g$ curves. We extend this theory to vertex algebras equipped with a notion of change of coordinates over any algebraically closed field. As a by-product of our arguments, we also show that a number of standard assumptions in these constructions can be removed, such as being $C_1$-cofinite or CFT-type.

Can a constant number of linear minimizations per round improve on the $T^{3/4}$ regret rate of online Frank-Wolfe on general convex sets? Weibel et al. conjectured that fixed-coefficient methods cannot. We prove their conjecture and extend the lower bound to every deterministic learner in an oracle-only model. The learner receives an initial feasible point and a diameter bound, and must remain feasible on every domain consistent with its oracle replies. For $T$ rounds, at most $b$ calls between decisions, diameter bound $D$, and gradient norm bound $L$, we construct an instance in dimension $d=2b(T-1)+1$ with regret at least $2^{-1/4}LDb^{-1/4}T^{3/4}$. The adversary fixes the domain, initial point, deterministic tie rule and linear losses before play. The vertices form a path on which every point available before a decision has zero current loss, while the final vertex has negative loss on every round. For constant $b$, the result matches the known upper rate for dimension-independent guarantees. For one-call fixed schedules with a nonzero coefficient on the newest gradient, a second construction gives regret at least $3LDT^{3/4}/4$ with unique minimizers at every issued query. Exact-arithmetic certificates for the tuned schedule of Weibel et al. closely match their finite-horizon numerical worst cases, with unique oracle replies.

This paper presents approximate computational methods for modeling surface plasmon propagation on a graphene sheet with time- and space-dependent material properties. Starting from Maxwell's equations, the evolution of current density under a spatially and temporally modulated Drude weight is formulated as a partial integro-differential equation (PIDE). Assuming small-amplitude perturbations of the Drude weight, a regular perturbation expansion is developed. The solvability and Fourier invertibility of the resulting order-by-order integral equations are established in appropriate Sobolev spaces. A numerical solver combining a Volterra integral equation formulation with the fast Fourier transform (FFT) is introduced, along with pointwise error estimates and computational complexity analysis. Additionally, an analytical transform-based solution using contour integration and Laplace inversion is derived for traveling-wave Drude weight modulations. Numerical simulations demonstrate diverse wave phenomena, including traveling, standing, and exponentially growing plasmonic current densities.