Papers

We study polynomial zero-avoidance over finite vector spaces by means of slice rank and partition rank. We first make the support-entropy method effective by showing how a finite dual certificate yields an explicit entropy gap whenever the coefficient support admits no probability distribution with uniform marginals. For the quadratic elementary symmetric polynomial over fields of characteristic three, the ternary structure of the coefficient support gives a certificate with optimal normalized margin and a uniform analytic bound for the corresponding higher-degree Erdős–Ginzburg–Ziv constant, avoiding a separate optimization for each field. We then use partition rank to handle solutions in pairwise distinct variables. Equality profiles are encoded by contracted local tensors, reducing the global problem to finitely many slice-rank estimates. Applying this reduction on the multiplicative torus gives restricted-alphabet zero-sum bounds with exponential base below the alphabet size. Coordinatewise inversion and support stratification then yield, to the best of our knowledge, the first nontrivial exponential bound for the higher-degree Erdős–Ginzburg–Ziv problem over $\mathbb{F}_5^n$ associated with the fourth elementary symmetric polynomial.

In this paper we study qualitative properties of the eigenvalues and eigenfunctions of $-\Delta$ with Dirichlet boundary condition in a smooth bounded domain $\Omega$ with a small circular hole. In the literature, this is known as a "singular perturbation", in contrast with the "regular perturbation" case. Denoting by $\Omega_\epsilon:=\Omega\setminus B(P,\epsilon)$ where $B(P,\epsilon)$ is the ball centered at $P$ and radius $\epsilon$, for $P\in\Omega$ and $\epsilon$ small enough we investigate 1) quantitative estimates for the eigenfunctions of $-\Delta$ in $\Omega_\epsilon$; 2) the simplicity of the eigenvalues of $-\Delta$ in $\Omega_\epsilon$; 3) the behavior of nodal sets of the eigenfunctions of $-\Delta$ in $\Omega_\epsilon$. A key ingredient in our analysis consists of pointwise estimates on the so-called $u$-capacitary potential firstly introduced in \cite{afhl}.

We study a class of Friedrichs Hamiltonians, that is, operators describing an excited state coupled to a bath through a rank-one perturbation, in the case where the bath Hamiltonian has purely discrete spectrum. We consider sequences of such Hamiltonians for which the spectral measures associated to the coupling functions by the bath Hamiltonians converge weakly to a limiting measure that is absolutely continuous with a Hölder continuous density near the excited energy. Under this assumption, we show that the survival probability of the excited state decays in an approximately exponential manner on suitable time scales and under favourable conditions, with a decay rate given by Fermi's golden rule for the limiting measure. The error is estimated in terms of the coupling strength and the Lévy distance between the discrete spectral measures and the limiting measure. As an application, we treat a two-level atom coupled to a massless bosonic field confined to a large cavity in the rotating-wave approximation.

Place the vertices of a finite graph at projective points. Each edge defines a span map to $\mathrm{Gr}(2,n)$; the pulled-back Chern classes generate a graded algebra $A_G^{(n)}$. This assignment is a covariant graph functor, so the abstract graded-algebra type is a graph invariant. In ambient dimension four, the Hilbert series is incomparable with the chromatic and Tutte polynomials. On five vertices in dimension three, the $34$ graph classes yield $33$ algebra types, strictly refining both classical polynomials. For every tree $T$, we obtain a closed Hilbert-series formula depending only on $|V(T)|$ and $n$, while $A_T^{(3)}$ determines $T$ up to isomorphism. More precisely, its cubic relation data is a complete tree invariant of polynomial size. Finally, the graphical configuration space embeds as a dense open in the picture variety, and picture spaces with equal additive homology can have nonisomorphic edge-Chern algebras.

Foraging is a universal animal behavior that has increasingly attracted the interest of both experimentalists and theorists. Most prior models assume an animal knows the distribution of resources in its environment, but this structure must be learned as the animal explores its environment. Foraging can thus be regarded as a hierarchical inference problem. We develop a normative Bayesian account of an agent learning a patchy environment while exploiting it, and show that resource depletion shapes the levels of that hierarchy differently. Within a patch, depletion accelerates rate learning, since successive encounters occur at falling rates whose spacing pins down the initial rate. Across patches, composition learning, inferring the fraction of patches that are high yield, is slow, set by the number of patches sampled rather than the time spent in each, and unaffected by depletion once the rates are known. Reward-maximizing and information-seeking strategies therefore diverge, a forager resolving the composition underharvesting rich patches because learning it requires departures, most sharply early in exposure. When a fixed set of patches replenishes between visits, the reward-maximizing policy collapses onto a stable orbit over the high-yield patches, and the replenishment rate sets whether the forager maps the whole environment or locks onto a rich subset. Which departure rule maximizes intake is itself set by how variable the patches are, switching from counting prey to timing the gaps between them once richness varies by more than about a quarter. Learning the environment thus buys significant intake over learning a rule from reward alone, as long as its assumptions about depletion leave room for the truth.

We propose the iterative SaMpling-And-Regression Technique (iSMART) for high-dimensional martingale-based partial differential equations (PDEs) in this paper. By leveraging the $L^2$-projection property of conditional expectation and adopting the stop-gradient technique, iSMART reformulates the continuous martingale condition derived from PDEs into a sequence of tractable sampling-regression problems within an iterative framework. This approach relies solely on standard SDE path simulation and plain squared-error loss minimization, completely bypassing the need for adversarial optimization or nested expectation estimation in previous methods. iSMART accommodates linear, semi-linear, and fully nonlinear martingale-based PDEs within a unified iterative procedure. In particular, for fully nonlinear Hamilton-Jacobi-Bellman (HJB) equations, a freezing-and-compensating technique is introduced to strategically shift a portion of the nonlinearity into the SDE drift, thereby improving the convergence behavior of the iterations. Numerous numerical experiments on linear reaction-diffusion equations with sharp gradients, semilinear Burgers-type equations, and fully nonlinear HJB equations demonstrate the accuracy, efficiency, and robustness of the proposed approach in various high dimensions.

Piecewise polynomial surface approximations used in surface finite element methods often seem to behave better than their standard approximation properties suggest if their polynomial order is even. We explain this superconvergence through cancellation of leading interpolation errors on suitably structured meshes that naturally arise in some refinement processes. This cancellation improves weighted integral estimates for functions, derivatives, and geometric quantities. Applications include estimates for surface normals, the Weingarten map, and Gaussian curvature. Numerical experiments reproduce the predicted parity-dependent behaviour and support the proposed explanation of superconvergent geometric consistency errors, while the corresponding pointwise errors retain their standard orders.

In this paper, we study the uniqueness of positive solutions to the semilinear elliptic Robin problem $$ \begin{cases} -\Delta u = u^p, & \text{in } \Omega,\\ u > 0, & \text{in } \Omega,\\ \frac{\partial u}{\partial \nu} + \beta u = 0, & \text{on } \partial \Omega, \end{cases} $$ where $\beta > 0$, $p$ is subcritical, and $\Omega$ is a bounded smooth domain. It is known that the uniqueness of the solution depends on the shape of the domain. Even if $\Omega$ is a ball, the problem is open for arbitrary $\beta>0$, since the method of moving planes does not work for Robin boundary conditions . By scaling arguments and a careful analysis of the linearized problem, we prove uniqueness for any $\beta>0$ provided that $p$ and $\Omega$ satisfy suitable conditions. Finally, we study the effects of concave and convex nonlinearities.

Let $K\subset\R^n$ be a convex body in isotropic position. We prove the optimal mean-width estimate \[ M^*(K)\leq C\sqrt{n\log n}. \] The main new ingredient is a geometric inequality relating the Gaussian mean of the support function to its mean under the uniform measure on $K$, obtained through a heat-flow argument. Combined with the Gaussian-log-concave comparison of Eldan and Lehec and the newly available dimension-free bound on the third-moment parameter $\kappa_n$, this yields the result. The boundedness of $\kappa_n$ also makes the mean-norm estimate of Bizeul and Klartag sharp. Combining both estimates yields \[ M(K)M^*(K)\leq C\log n, \] extending Pisier's $MM^*$ estimate to non-symmetric convex bodies and to the isotropic position.

Francoeur and Garrido recently provided the first explicit examples of finitely generated branch groups with maximal subgroups of infinite index, where one of their examples is the Grigorchuk-Erschler group. Here, using the Basilica operation constructed by Petschick and Rajeev, we show that the second Basilica group of the Grigorchuk-Erschler group also has maximal subgroups of infinite index.

We study extremizers for a trilinear Stein-Weiss inequality on $\mathbb{R}^n$. Within the known boundedness region, we prove attainment under two additional assumptions: all six weight exponents are nonnegative, and at least one pair of Lebesgue exponents is admissible. The proof combines symmetric decreasing rearrangement with a logarithmic radial reduction to a translation-invariant bilinear operator on $\mathbb{R}$ whose kernel belongs to $L^1\left(\mathbb{R}^2\right)$. A common-scale compactness argument rules out relative separation of the two arguments and yields norm attainment. We then derive the Euler-Lagrange system. In the fully symmetric case, every normalized nonnegative extremizing triple is diagonal. Finally, we establish the origin-centered Kelvin invariance of the resulting scalar equation at the scaling exponent and record the unweighted conformal example.

We prove the Penrose inequality with charge under the 2-convexity condition recently introduced by Dong. More precisely, given a complete, connected and asymptotically flat Einstein-Maxwell initial data set $(M,g,k; E,B)$ satisfying the charged dominant energy and the 2-convexity conditions, with divergence-free electromagnetic vector fields $(E,B)$ and a connected outermost past apparent horizon $\Sigma$ that satisfies $|\Sigma| \geq 4\pi q^2$ - where $q$ is the total charge - we show that the following inequality for the ADM mass $m$ holds: $m\geq \sqrt{\frac{|\Sigma|}{16\pi}} + q^2 \sqrt{\frac{\pi}{|\Sigma|}}$, with equality if and only if $k \equiv 0$ and $(M,g;E,B)$ is isometric to a canonical slice of sub-extremal Reissner-Nordström spacetime. Building on Dong's proof of the uncharged case, we use his $\mathbf{P}$-inverse mean curvature flow and its weak formulation, which only depends on $(g,\mathbf{P})$ and hence applies to the charged setting unchanged. The novelty of our work is the modification of the monotonicity formula to account for the additional charge term. For time-symmetric data ($k \equiv 0$), the flow reduces to the classical inverse mean curvature flow and our monotonicity formula to Jang's monotonicity of the charged Hawking mass, recovering the charged Riemannian Penrose inequality.

Many real-world systems rely on predictive models to inform decisions, and fairness concerns arise in both the prediction and decision stages. We introduce end-to-end fairness optimization (E2EFO) as a unifying framework that integrates fairness across the prediction-to-decision pipeline. We focus on resource allocation with group-based fairness: the prediction task estimates allocation impacts while limiting accuracy disparity across groups, and the decision task distributes those impacts equitably by optimizing a group-based alpha-fairness measure. Within this framework, we propose fair decision-focused learning (FDFL), a training paradigm that jointly accounts for prediction accuracy, prediction fairness, and decision regret – the loss in decision fairness due to imperfect predictions. FDFL trains the predictor by gradient descent, combining the objective gradients through multi-task learning techniques. The core computational challenge is the decision Jacobian with respect to the predictor parameters: we derive exact closed-form formulas for a tractable class of fair allocation and apply a differentiable optimization layer in the general case. We further establish a finite-sample generalization bound for the scalarized FDFL objective. Numerical experiments on a healthcare-based single resource allocation and a synthetic multiple resource allocation illustrate the value of jointly accounting for prediction fairness and decision fairness in prediction-informed decision-making.

Admission control is a widely used mechanism for regulating congestion in stochastic service systems, where restricting arrivals is expected to reduce workload and improve performance. In many settings, however, customers make decisions based on the observed system state, creating a feedback loop between control actions and future arrivals. This raises a fundamental question: does admission control necessarily reduce congestion under state-dependent behavior? We address this question using stochastic ordering of workload processes. We show that admission control may fail to reduce congestion due to endogenous feedback effects, and construct explicit counterexamples illustrating this phenomenon. We then identify a general condition under which admission control is effective over the initial busy period. In particular, independence between customers' types and service requirements eliminates the adverse feedback mechanism and restores stochastic dominance. These results highlight fundamental limitations of admission control and the importance of accounting for behavioral responses in the design of stochastic service systems.

We prove that, for either a Steinhaus random multiplicative function or a Rademacher random multiplicative function $f$, and every $\varepsilon>0$, almost surely $|\sum_{n\le x}f(n)|\ll_{\varepsilon,f}\sqrt{x}(\log\log x)^{1/4+\varepsilon}$. Together with Harper's almost sure lower bound, this determines the sharp logarithmic exponent in both models. This settles Harper's conjecture on the large fluctuations of random multiplicative functions.

We consider a class of anisotropic energy integrals including fast and slow growth under a unified approach, proving that the local minimizers are locally Lipschitz continuous.

In this work we study the high-temperature limit of the Schwarzian Field Theory probability measure. We show that on large scales this limit concentrates on jump processes, while its small-scale behaviour is governed by a process which we call Schwarzian Field Theory on the real line. This Schwarzian Theory on the real line can be viewed as the infinite-volume version of the Schwarzian Field Theory. In addition to showing this local convergence, we also provide a systematic treatment of the Schwarzian Theory on the real line. This includes the calculation of the correlation functions and a corresponding uniqueness theorem.

This survey gives an introductory review of the classical Hodge theory, arithmetic geometry and algebraic number theory that inspires p-adic Hodge theory. Basic results in this new area developed by Tate, Faltings and Scholze are presented with corresponding mathematical constructions. Some background in algebraic geometry and algebraic topology is required

We study a variant of the stochastic SIR model on graphs that has previously been introduced in the physics literature for modelling zombie outbreaks and here referred to as the Zombie Infection Model (ZIM). In this model, initially each node of a graph is either susceptible, infected or removed. As in the SIR model, a susceptible node becomes infected at rate $\lambda$ times the number of its infected neighbours. Moreover, in the ZIM, an infected node is removed at rate $1$ times the number of its susceptible neighbours. This process exhibits rich and sometimes counterintuitive behaviour. By combining various coupling techniques, we provide a rigorous mathematical analysis of the model, focusing on monotonicity properties and the probability of the infection spreading indefinitely. One of our main results is that this probability is monotone with respect to an increase of $\lambda$ for the process on trees, but that there are graphs of bounded degree for which it is continuous and yet not monotone. We also establish bounds on this probability for the process on general graphs, and derive more precise results for complete graphs, regular trees, and the $d$-dimensional integer lattice.

We propose and analyze a stochastic direct-search method for unconstrained zeroth-order minimization of locally Lipschitz, possibly nonsmooth, objectives. The method combines random polling directions with a stochastic extrapolating line search based on a sufficient-decrease test of order $p$. Under conditional accuracy assumptions on the stochastic estimates, we prove almost-sure convergence to Clarke stationary points. We further establish an expected iteration complexity bound. Specifically, using a supermartingale stopping-time argument, we prove that $\mathcal O\left( \max\left\{ r^{-p}, \varepsilon^{-p/(p-1)} \right\} \right) $ iterations are sufficient in expectation to reach an $(r,\varepsilon)$-Goldstein stationary point. Moreover, we derive a corresponding expected tested-point complexity bound of order $\mathcal O\bigl(\varepsilon^{1-n} \max\{r^{-p},\varepsilon^{-p/(p-1)}\}\bigr)$. To the best of our knowledge, this is the first convergence and expected-complexity analysis for an extrapolation-based direct-search method in a nonsmooth stochastic setting. Numerical experiments on a DFO benchmark suite highlight competitive performance against well-established stochastic direct-search methods.

System identification is an important step in modeling and evaluating vibrating structures, but many nonlinear system identification methods rely heavily on data-driven approaches that may not preserve physical consistency. This research extends the Energy-based Dual-phase Dynamics Identification (EDDI) method to multiple-degree-of-freedom (MDOF) mechanical structures undergoing nonlinear vibrations. The original EDDI framework was designed for single-degree-of-freedom (SDOF) systems and operates in two phases: the first identifies a model for internal nonconservative force, and the second captures internal conservative force. However, EDDI assumes that the potential energy is zero whenever the displacement is zero. For MDOF systems, this assumption requires all degrees of freedom (DOFs) to achieve zero displacement simultaneously, which simply occurs too infrequently in multimodal responses for direct application. To overcome this limitation, this work introduces Decomposition-based EDDI, which applies EDDI to decomposed response components to enable nonlinear system identification of MDOF systems. Wavelet-Bounded Empirical Mode Decomposition is used to extract nearly orthogonal, monochromatic intrinsic mode functions (IMFs) from the measured displacements. Each IMF is then treated as an individual SDOF oscillator and processed using EDDI to estimate its nonconservative and conservative internal forces. The IMF forces are then summed to reconstruct the total nonconservative and conservative forces acting on each physical DOF, which are used to identify the damping and stiffness models, respectively. The proposed EDDI framework is experimentally validated on a two-story tower structure with strong stiffness nonlinearity coupling the two floors. The results demonstrate the efficacy of EDDI in isolating and identifying complex, multi-modal nonlinear structural dynamics.

Helly, Carathéodory, and Radon numbers encode three kinds of finite certificates in a convexity space: for the emptiness of an intersection, for membership in a convex hull, and for the existence of intersecting hulls. We study exact versions of these certificates, in which a subfamily must preserve the whole intersection or a subset must preserve the whole hull. Our first main result shows that, for finite configurations in an arbitrary convexity space, five a priori different boundedness conditions are equivalent: VC-dimension, strong Helly number, strong Carathéodory number, comatching number, and strong Radon number (with the expected additive-one shift). We also obtain equivalent layered Tverberg-type decompositions and colorful consequences. The common mechanism is exposed by the bipartite incidence graph between points and a generating family. For finite spaces, the unique minimal generator yields a natural dual convexity space; we characterize double dualization and prove that the strong parameters are duality invariant. The same model gives a polynomial-size, $O(t^4)$, realization of Bukh's counterexample to the Calder-Eckhoff partition conjecture. Finally, we obtain the first Tverberg bound for separable convexity spaces that is simultaneously linear in the number of parts and polynomial in the Radon number. If an $S_3$-separable convexity space has Helly number $h$ and its halfspaces have VC-dimension $d$, then $r_t=O(dh\log h)\,t$; in particular, Radon number $r$ gives $r_t=O(r^2\log r)\,t$. The bound attains the weak-Eckhoff scale $O(rt)$ whenever the Helly number is bounded. For axis-parallel box convexity in $\mathbb{R}^k$, gives the optimal order $r_t=O(rt)$ uniformly in every dimension. This appears to be the first dimension-uniform estimate of weak-Eckhoff order for box convexity, whereas the previous direct theory was confined to dimension three.

We prove that every closed manifold of dimension $n\ge 4$ which admits a singular $2$-foliation (a foliation by closed surfaces whose quotient is a punctured $(n-2)$-torus) supports a volume-preserving homeomorphism with exponential decay of correlations for Hölder observables. The proof introduces a class of systems called relatively Anosov homeomorphisms, which fails differentiability only at the singular set. We apply a general dichotomy that bounds the decay of correlations of any homeomorphism preserving an invariant foliation by the maximum of the decay rates of the quotient dynamics and of the leafwise cycles. This dichotomy is of independent interest and yields, as immediate consequences, decay rates for product systems, skew products, partially hyperbolic diffeomorphisms with compact center leaves, finite covers, and extensions by expanding fibre maps. We then prove this gives a positive answer to a conjecture of Dolgopyat and Pesin regarding the realization problem for exponential decay of correlations over a large class of manifolds for which there were no known systems with exponential decay of correlations. In particular, we construct infinitely many pairwise non-homeomorphic closed $4$-manifolds which admit a volume-preserving homeomorphism with exponential decay of correlations, albeit not supporting either Anosov diffeomorphisms or strong partially hyperbolic diffeomorphisms.

We present an algorithm that finds an epsilon-approximate solution to a mixed integer quadratic programming (MIQP) problem, and that runs on a Turing machine in time polynomial in the size of the instance and in 1/epsilon, provided that the number of integer variables and the number of negative eigenvalues of the Hessian of the objective function are fixed. Unless P=NP, both restrictions are necessary, so this completes the characterization of the complexity of approximating MIQP in terms of the number of integer variables and the inertia of the Hessian; the result is new already in the purely continuous case. The main ingredient is a polynomial-time simultaneous diagonalization algorithm: it computes a rational change of basis that maps a given ellipsoid, presented in factored form, exactly to a ball, while making the objective function separable up to an arbitrarily small perturbation and preserving the inertia of its Hessian. The classical construction, in which the objective function is made exactly separable, requires a change of basis that is in general irrational, and cannot be carried out on a Turing machine; ours rests instead on rational Jacobi rotations, which we believe to be of independent interest.

We consider the initial–boundary value problem for the semilinear wave equation with fractional structural damping $$ u_{tt}+(-\Delta)^{\theta}u_{t}-\Delta u=|u|^{p-1}u $$ in a bounded domain, where the exponent $\theta\in[0,1]$ interpolates between external frictional damping ($\theta=0$) and internal Kelvin–Voigt viscoelastic damping ($\theta=1$), and therefore parametrises the frequency dependence of the dissipation mechanism. For initial data with energy below the depth of the potential well and negative Nehari functional we prove finite-time blow-up together with an explicit upper bound for the blow-up time. The bound comes from a single concavity functional in which the fractional dissipation cancels identically; it therefore has the same form for every $\theta\in[0,1]$, and it admits a variant that is uniform in $\theta$. Conversely, for $1<p\le\frac{n+2\theta}{n-2}$ when $n\ge3$, we establish an explicit lower bound for the blow-up time. Mechanically, the two bounds delimit a guaranteed interval of existence and a guaranteed failure time for the model. The admissible range of exponents in the lower bound widens linearly with $\theta$, which quantifies how the strength of the internal damping enlarges the class of nonlinear loads for which such a guarantee can be computed; we do not claim monotonicity in $\theta$ of the numerical value of the bound. Both theorems are proved for a class of energy solutions specified by a short list of requirements, so that they are independent of any particular local existence theorem, and they carry over unchanged to the elasticity and plate operators used in structural models. The two endpoint cases recover known results for frictional and strong damping.

We prove a conjecture formulated by Bolsinov, Konyaev and Matveev in [7] stating that, integrability of a system of hydrodynamic type ${\bf u}_t=A({\bf u}) {\bf u}_x$ with $\mathfrak{gl}$-regular $A$ at a point $p$ implies the vanishing of the Haantjes tensor of $A$ and of all its symmetries in a neighborhood of $p$. As a consequence, leveraging on the result of [8], in a neighbourhood of an algebraically generic point, any integrable system of hydrodynamic type defined by a $\mathfrak{gl}$-regular operator field can be written as ${\bf u}_t=X({\bf u})\circ {\bf u}_x$ where $X$ is a vector field and $\circ$ is a commutative associative product satisfying Hertling-Manin conditions.

We revisit Zeilberger's computational framework for enumerating polyomino tilings and demonstrate its use in counting tilings of $k\times n$ boards by $L$-tetrominoes, allowing rotations. Recently, Bělohoubek and Slavík gave a generating function for tilings of $2n\times 4$ boards by $L$-tetrominoes. Motivated by their work, we first verify the method by automatically recovering their generating function. We then apply the method to widths $5$ through $9$ and obtain explicit rational generating functions and new integer sequences that are not currently listed in the OEIS.

We investigate manifestations of the Nilpotent Heuristic, which posits that recurrence and convergence phenomena known for measure-preserving $\mathbb{Z}^D$-systems extend to nilpotent group actions. Our main results establish seminorm estimates and limiting formulas for multiple ergodic averages arising from actions of 2-step nilpotent groups. In particular, if $T_1,\ldots,T_\ell$ are totally ergodic and generate a 2-step nilpotent group, then \[ \lim_{N\to\infty}\frac{1}{N}\sum_{n=1}^N T_1^n f_1 \cdots T_\ell^{n^\ell}f_\ell = \prod_{j=1}^{\ell}\int f_j\,d\mu \] in the $L^{2}$ norm for all bounded functions $f_{1},\dots,f_{\ell}$; the same holds for any distinct-degree polynomial iterates. We also obtain popular-common-difference versions of the polynomial Szemeédi theorem in the same setting. In a different direction, our approach allows us to completely resolve the joint ergodicity conjecture for multidimensional polynomials and $\mathbb Z^D$-systems; we also present an example showing that, surprisingly enough, the 2-step nilpotent analog fails. We conclude with many open problems concerning joint ergodicity, seminorm estimates, and the structure theory of nilpotent systems.

A recent theorem of Madarasz characterises the parameter-free concepts of a finitely field-definable coordinate geometry by invariance under its affine automorphisms. This paper isolates the affine-geometric ingredient in that proof and extends the argument to include additional finite-relational coordinate geometries. The new hypothesis is "semilinear faithfulness": in every ultrapower, each automorphism of the induced geometry factors as an affine automorphism of the geometry followed by the componentwise action of an automorphism of the expanded base structure. Under this hypothesis, every parameter-free ambient-definable relation is a concept of the geometry exactly when it is preserved by the affine automorphism group. A corresponding dual inclusion theorem is obtained for concept sets. The framework recovers the original finitely field-definable theorem and applies beyond pure-field definability, including a named-scalar geometry and a frame-expanded geometry over an exponential field. A final section records the analogous factorisation principle for uniformly coded symmetry groups in arbitrary, including non-geometric, one-sorted structures.

We formalize a recursive format for representing a butterfly matrix. This new format naturally leads to a simple recursive algorithm for computing a quasi-optimal butterfly approximation to an arbitrary $N \times N$ matrix $A$. When the entries of $A$ are explicitly available, we show that the algorithm computes a butterfly matrix $B$ in $O(N^2)$ operations with approximation error $\|A - B\|_F$ at most a $O(\sqrt{\log(N)})$ factor away from that of the best possible approximation by a butterfly matrix. We also develop a matrix-free variant of the method, which uses $\widetilde{O}(\sqrt{N})$ matrix-vector products and $\widetilde{O}(N)$ working memory and, with high probability, returns a butterfly approximation with Frobenius norm error within a $O(N^{1/4})$-factor of the optimal error. We show that the algorithm is a reformulation of the hybrid butterfly factorization approach presented in [Liu et. al.; SISC, 43 (2021)]. Our paper therefore provides the first theoretical quasi-optimality guarantee for that algorithm.