Papers

The Weak Commutativity Group was defined by the first author in the early 1980's as $ \chi(H) = \langle H, H^\psi \mid [h,h^\psi] = 1, \ \forall h \in H \rangle $ where $H$ and $H^\psi$ are isomorphic groups under an isomorphism $\psi$, and proved that $\chi(-)$, seen as a group operator, preserved the classes of solvable groups, finite groups, finite $p$-groups, and perfect groups. The list has grown considerably since then (see references). It is clear that the iterated operator $\chi^n (-)$ also preserves the same classes. However, we observe that for $n \geq 2$ many of the $2^n$ $H$-generating copies of $\chi^n (H)$ fail to weakly commute among themselves. We introduce in this paper the "$n$-ary Weak Commutativity Group" generated by $n$ copies of $H$ which weakly commute among themselves; that is $ \chi_n (H) = \langle H, H^\psi, \dots, H^{\psi_{n-1}} \mid [h^{\psi_i}, h^{\psi_j}] = 1, \ \forall h \in H, \ 0 \leq i < j \leq n-1 \rangle $ (by definition, $H^{\psi_0} = H, H^{\psi_1} = H^\psi$). We study quotients of $\chi^n (H)$ which are also simultaneously quotients of $\chi_{2^n} (H)$; we refer to them as Hybrid $\chi^n (H)$ - $\chi_{2^n} (H)$ groups. For $n=2$ we determine the largest common quotient $Q(H)$ of $\chi^2(H)$ and $\chi_4(H)$, that is, the largest hybrid $\chi^2(H)$-$\chi_4(H)$ group. A related class of groups in our study is $\chi(n,H)$, a quotient of $\chi_n(H)$, previously introduced by the third named author in his 2007 doctoral thesis. We prove that this class of groups continues to preserve finiteness, nilpotency and perfectness, and we provide detailed information about their structure. In contrast, $\chi_n(H)$ behaves differently: $\chi_3(C_2 \times C_2)$ is infinite, while $\chi_3(C_{p^t} \times C_{p^u})$ is finite for $p$ odd.

We study list coloring of common independent sets of two matroids. We construct a graphic matroid $M_1$ and a partition matroid $M_2$ with common chromatic number two and common list chromatic number three, showing that the two parameters need not be equal. This resolves a question raised by Király, later stated as a conjecture by Aharoni, Berger, Guo, and Kotlar. We also show that if two strongly base-orderable matroids are each $2$-colorable, then their intersection is $2$-list-colorable.

We prove that entropy alone, with no invariance hypothesis, forces a Borel probability measure on the circle toward Lebesgue measure under every sublacunary set of multiplicative endomorphisms $x \mapsto mx$. Here, we refer to an infinite set of multipliers $m$ as sublacunary if its consecutive ratios tend to $1$. Examples of such sets include the primes following perfect cubes, the range of the partition function, the integers $\lfloor n^{\log n} \rfloor$ and the semigroup generated by $2$ and $3$ which is the classical case of Furstenberg's conjecture. Specifically, we show that every measure $\mu$ has a weak-star limit of endomorphs which dominates Lebesgue measure scaled by the upper entropy dimension of $\mu$. These theorems are effective at finite entropy resolutions, and we provide the associated bounds for the primes and the perfect powers. The semigroup case may be interpreted as an invariance-free form of the Rudolph-Johnson theorem. We also show that an analog of Lyons' conjecture at full entropy dimension for every sublacunary multiplier set while failing by an arbitrarily large factor for the lacunary semigroup of powers of $3$. The proofs combine Fourier analysis with the algebraic structure of the multiplicative endomorphisms.

The complex of injective words is a cell complex that arises in a number of different areas. It has applications to proving homological stability and to the study of group cohomology, and it is closely related to the random-to-random Markov chain. This complex was first studied by Farmer, who proved it has the homology of a wedge of top-dimensional spheres. Later, Björner and Wachs established its shellability, and Reiner and Webb uncovered its $S_n$-module structure, observing in the process that the rank of its top homology group is the $n$th derangement number. We introduce natural subcomplexes of the complex of injective words by fixing a permutation pattern $\sigma$ and considering only those injective words in the alphabet $\{1,2,\dots,n\} $ that avoid $\sigma$. We prove that such pattern-avoiding complexes are shellable if $\sigma$ begins or ends with its largest or smallest letter, and we construct homology bases for the complexes avoiding such patterns. For patterns of length 3, all of which have this property, we show that the rank of the top homology of the resulting complex is a Riordan number. All but four patterns of length 4 also have this property, and for two of the remaining four patterns, we establish shellability using a different method. We also introduce a technique to use enumerative combinatorics to prove shellability, and we apply it to the complex of separable injective words, thereby deducing shellability in this case. Along the way, we give a combinatorial formula for all of the $h$-numbers in the full complex of injective words as well as for each of the subcomplexes which we prove are shellable. Going in the other direction, we use shellability of complexes of pattern-avoiding injective words to deduce new refined counting formulas for pattern-avoiding permutations.

We prove McLeod's (1959) conjecture that $(h_{\ell+r}/h_\ell)^{1/r}$ is subadditive for all $\ell\ge0$ and $r\ge1$, where $h_k$ is the complete homogeneous symmetric polynomial of degree $k$ (J. B. McLeod, "On four inequalities in symmetric functions", Proc. Edinburgh Math. Soc., 1959). We first prove that the ratio $h_k/h_{k-1}$ is convex, and our proof takes an unexpected detour through queuing theory. The geometric mean of $r$ consecutive such ratios equals McLeod's conjectured function, but geometric means of convex functions need not be convex. Our second and decisive step is a tensorization trick that couples neighboring ratios tightly enough to make their geometric mean convex.

In this paper, we present a model-based Galerkin framework for output-feedback control of nonlinear systems with known dynamics. With a suitable actuator augmentation, the finite-dimensional realization preserves the actuator dynamics and the constant input matrix. We retain finite-order nonclosure as an explicit additive residual. The LTI part and this residual describe the lifted nonlinear dynamics exactly. The decomposition does not require an invariant subspace of observables and also applies to non-control-affine systems. We use regional bounds on the closure residual, actuator-realization defect, and output-reconstruction error in the standard $\mathcal{H}_\infty$ design. The proposed controller is a linear dynamic system driven only by the measured tracking error and requires no online lifting. Retaining the state coordinates gives direct bounds on the original state. We use these bounds to derive an a priori containment condition. Under this condition, the nonlinear closed loop is forward complete, and the state and tracking error satisfy explicit transient bounds and are uniformly ultimately bounded. The cart-pendulum example illustrates the nonlinear closed-loop guarantees and compares the controller with full-state backstepping. Although the output-feedback controller uses only the measured tracking error, it achieves almost the same nominal tracking RMSE as backstepping, with a lower peak tracking error.

We consider the problem of learning nonlinear operators $G_0: \mathcal X\to \mathcal Y$ from finitely many noisy observations. While finite-regularity classes such as Lipschitz- or $C^k$-operators suffer from the curse of sample complexity (i.e. algebraic sample complexity is impossible), algebraic convergence rates can be obtained for holomorphic operators. In this paper we study information-theoretic minimax rates for estimating holomorphic operators. We provide algebraic upper and lower bounds of the metric entropy of holomorphy classes (matching up to an arbitrarily small loss in the exponent) as well as statistical minimax lower bounds in both the random-design and a Gaussian white noise model. For the mean squared error, our novel lower bounds match our upper bounds (up to an arbitrarily small loss in the exponent) and coincide with the convergence rates obtained previously for related holomorphy classes (arXiv:2412.17582). Finally, we apply the theory to the parameter-to-solution map and the parameter-to-quantity-of-interest map arising from the elliptic Darcy flow equation.

We construct a deterministic Farey-boundary algorithm converting a cyclic prefix table for an element of $\mathsf T$ into a word in addressed boundary flips. The reduction uses rational boundary pairs and cyclic order, with a marked edge determining the residual modular correction. Its unmarked stage uses at most twice the minimum number of diagonal exchanges. For input size $s$, the complete output has $O(s)$ factors, encoded length $O(s^2)$, and running time $O(s^3)$. Adjoining shuffles extends the construction to $\mathsf V$ with polynomial bounds. We also give explicit boundary generating sets and their local flip–shuffle relations. Changes of generators in the Lochak–Schneps and Bleak–Quick presentations yield a two-flip presentation of $\mathsf T$ and a presentation of $\mathsf V$ with two flips, one shuffle, and nine relations.

Missing covariates are frequently encountered in supervised learning problems, and classical methods for estimation using such data use carefully chosen imputation schemes for missing data, or likelihood approximations that lead to nonconvex $M$-estimation problems. These methods and their relatives are suitable for scenarios in which the covariate distribution is known, and more broadly, have enjoyed tremendous success in linear models. But even in basic nonlinear problems such as logistic regression in moderate dimensions, such methods can experience drastic failure modes when the covariate distribution is unknown. Motivated by the need for reliable alternatives, we consider the problem of parameter estimation in logistic regression with missing covariates. Crucially, we operate in the assumption-lean setting where the covariate distribution is unknown (but bounded). We design a stochastic approximation method that is based on $Z$-estimation with a novel monotone operator, and establish that our algorithm is computationally efficient and achieves provable signal recovery at parametric rates under the hypothesis that covariates are missing completely at random. Our theory sharply characterizes the $\ell_2^2$ risk of the estimator in terms of the missingness profile, accommodating heterogeneous observation probabilities. Importantly, it shows that our method always outperforms the de facto “complete-case” estimator that ignores observations with any missing data. Even in the setting with homogeneous missingness (in which each covariate is observed independently with probability $q$), our bounds exhibit intricate and nonstandard dependence on $q$ that can yield significant improvements over using only complete cases. We complement our upper bounds with new information-theoretic lower bounds that show that this intricate dependence on $q$ is fundamental in a minimax sense.

Recent advances in penalty-based methods for stochastic bilevel optimization (SBO) have eliminated the need for second-order derivative oracles. However, for stochastic nonconvex-strongly convex bilevel problems, existing first-order methods typically rely on nested loops and/or large batch sizes for attaining $O(\epsilon^{-6})$ or $O(\epsilon^{-4})$ sample complexity under standard bounded-variance assumption or mean-square smoothness assumption. Achieving these rates with a single-loop penalty method and a constant batch size remains challenging due to a large penalty value needed for an accurate approximation. To address this challenge, we develop a stochastic SIngle-loop COnstant-Batch first-order penalty method (SICO) that combines two complementary ingredients. First, it performs one stochastic-gradient update per-iteration for both the original lower-level and penalized problems, with a projection that controls the separation between their iterates. Second, it applies an exponential moving average to stabilize the upper-level gradient estimator. We show that this combination achieves $ O(\epsilon^{-6}) $ sample complexity using only $O(1)$ stochastic-gradient samples per iteration under unbiased, bounded-variance stochastic gradients. Under the additional mean-square smoothness assumption on the lower-level stochastic gradients, the same algorithm improves the complexity to $O(\epsilon^{-4})$ also with $O(1)$ batch size. To the best of our knowledge, this is the first work to match the best-known convergence rate for fully first-order SBO methods using a single loop and a constant batch size. This result addresses an open problem posed in the literature.

We study embeddings of factorizations of the $\lambda$-fold complete $h$-uniform hypergraph $\lambda K_m^h$ in factorizations of $\lambda K_n^h$, where $m<n$ and both the source and target degrees may vary by color. From every embedding one can obtain another that minimizes the number of components in every target color without increasing its multiplicity spread. This change also does not increase the sum of any convex function of the edge multiplicities, simultaneously in all colors. For $h=2,3$ we give exact existence criteria for every $m<n$. For $h\ge4$, the necessary divisibility and degree-sum conditions are sufficient once $n\ge(h-1)m$, improving the previously known threshold $n\ge hm$ for color-dependent degrees. In this range we also give exact criteria for connected, simple, and equimultiple target factors. The proofs use a system of integer counts recording the number of added edges of each type. Every system satisfying these equations gives an embedding with the minimum multiplicity spread allowed by

We study an online trading problem where a trader, given a sequence of i.i.d. prices drawn from a known distribution $F$ on $[0,1]$, must make irrevocable buy, sell, or hold decisions subject to storage constraints. We investigate achievable algorithmic performance measured in terms of regret, the difference between the expected profit of the hindsight optimal policy that knows the entire price sequence and an online algorithm. We analyze finite atomic and continuous distributions characterized by their local behavior around the median which we capture using a parameter $\beta$. The parameter $\beta$ quantifies how the mass of prices accumulates around the distribution median. We identify a new driver of algorithmic performance, demonstrating that median gaps coupled with an initial inventory level of zero can force regret scaling of $\Omega(T^{(\beta + 1)/(2\beta+4)})$ — establishing a novel spectrum of fundamental limits on algorithmic performance. We then study STARS, short for Storage Trading by Averaging Repeatedly across multiple Scenarios, which simulates possible future price scenarios to approximate the value-to-go function and make buy/sell/hold decisions. We show that STARS obtain near-optimal algorithmic performance (upto poly-logarithmic factors) across a broad range of distributions. In particular, it achieves $O(\log T)$ regret for finite atomic prices, $\widetilde{O}(T^{\beta/(2\beta+2)})$ for continuous distributions without a median gap and $\widetilde{O}(T^{(\beta+1)/(2\beta+4)})$ for continuous distributions with a median gap for $\beta \geq 0$.

Let $W$ be an irreducible complex reflection group with reflection representation $V$. A $W$-stable, faithful homogeneous system of parameters $\Theta \subseteq \mathrm{Sym}(V^*)$ of common positive degree $p$ is called a posh hsop; $\Theta$ carries a $W$-representation $U$ if $\Theta \simeq U$ as ungraded $W$-modules. We classify the posh pairs $(p,U)$ for which a posh hsop of degree $p$ carrying $U$ exists. Every such $U$ is a Galois twist of $V^*$. Extending work of Ito and Okada, we deduce that the quotient $S/(\Theta)$ is a permutation module for $W$ if and only if $U \simeq V^*$.

We analyze the level lines of the Gaussian Free Field with logarithmic singularities in polygonal domains. We focus on the (none-zero) singular case and compare it to the singular-free setting. The law of the level lines for the field in singular case is absolutely continuous with respect to the law for the singular-free case, with a Radon-Nikodym derivative related to the reduced Green's energy which is introduced by F. Viklund and D. W. Nyström~[VN24]. These level lines admit a Brownian motion representation. In the singular-free case, such representation involves Poisson point processes of Brownian loops and excursions; while for the singular case, it additionally includes Poisson point processes of Brownian bubbles and bridges. The special cases of one or two singularities yield moment-generating functions of the conformal radius, which can be interpreted as one-point and two-point correlation functions of vertex operators in conformal field theory at the free-boson point. While the Radon-Nikodym derivative is bounded away from the singularities, the behavior of the level lines near the singularities is markedly different from the singular-free case, with different asymptotic probabilities for all level lines to remain close to the singularity.

In this note, we establish quantitative stability of aggregation-diffusion energy for compactly supported data. Our consideration focuses on the degenerate diffusion with local attraction that are asymptotically flat at infinity, such as Riesz kernels. We characterize the spectrum lower-bound of the Wasserstein Hessian at steady states. Assumptions are kept as minimal as that needed for uniqueness.

In this paper, we propose an alternating numerical scheme for constructing quasi-periodic solutions to a class of nonlinear partial differential equations (PDEs) with bounded perturbations, including the one-dimensional nonlinear Schrödinger (NLS) equations and nonlinear wave (NLW) equations under periodic and Dirichlet boundary conditions. A key observation concerning the resonant $Q$-equations allows us to establish directly a local diffeomorphism between the amplitudes and the drifted frequencies. This renders the proof considerably cleaner, since it avoids the Birkhoff normal form and the associated coordinate transformations required in classical KAM theory for PDEs.Unlike in the finite-dimensional setting, the lattice boxes must also include the spatial index, so that the multiscale induction has to be carried out in one additional dimension. This introduces two difficulties. First, the measure estimate for the excluded set of “bad” frequencies must account for the spatial index. For both the NLS and NLW equations, the linear frequencies and their differences are integers or close to integers; this allows the spatial dependence to be controlled through the modulus of the temporal index, so that the excluded measure remains under control. Second, a singular set may contain sites with the same temporal index but widely separated spatial indices, which obstructs the “inversion implies localization” process. For the one-dimensional NLS and NLW equations, however, such sites occur only in pairs with opposite spatial indices. They can therefore be separated by small boxes, after which the global inverse is obtained from the local inverses via the resolvent identity. Numerical experiments illustrate the scheme for nonlinear traveling and standing waves, exhibiting both the computed quasi-periodic solutions and the convergence of the iteration.

We prove that every smooth quartic double $n$-fold, $n\ge2$, over an algebraically closed field of characteristic zero is algebraically elliptic in Gromov's sense. In particular, every smooth quartic double solid is algebraically elliptic, which answers Question~4.28 in [M. Zaidenberg, Algebraic Gromov ellipticity: a brief survey, Taiwanese J. Math. 29 (2025), no. 6, 1681-1705]. We also prove that removing any closed subset of codimension at least two preserves algebraic ellipticity for smooth quartic double spaces and smooth cubic hypersurfaces of dimension at least two.

We construct self-similar solutions to a nonlinear convection–diffusion equation posed on the whole Euclidean space, for a class of homogeneous initial data. The corresponding self-similar profiles solve a nonlinear elliptic equation on $\mathbb{R}^N$. We obtain these profiles as limits of solutions to Dirichlet problems on bounded domains with truncated nonlinearities, using a priori estimates that are uniform with respect to both the domain size and the truncation level. Our approach is inspired by recent constructions of large self-similar solutions for the Navier–Stokes system. The convection–diffusion equation considered here shares some structural features with the Navier–Stokes equations, but its scalar parabolic nature allows us to exploit tools such as the maximum principle and thereby obtain a more direct construction of self-similar solutions.

We study a Poincaré-type inequality for Gaussian measures restricted to origin-symmetric convex sets. The inequality arises from the second variation at $t=0$ of the function $t\longmapsto \gamma_{\Sigma}\left(e^{tA}K\right)$, where $\Sigma$ and $A$ are symmetric matrices. The relevance of this function comes from Saroglou's reformulation of the log-Brunn–Minkowski conjecture. We show that $\Sigma$ and $A$ commute if and only if the inequality holds for every box of the form $UD B_\infty^n$, where $U$ is orthogonal and $D$ is diagonal. In the non-commuting case, this yields counterexamples for orthogonal boxes whose axes can be chosen arbitrarily close to the coordinate axes. We also show that the inequality holds when $K$ is unconditional with respect to an eigenbasis of $\Sigma$ and $A$ is an arbitrary symmetric matrix.

We introduce contact monopoles on contact metric $5$-manifolds and study their basic geometric properties. We construct natural examples from Boothby–Wang fibrations over Kähler surfaces and relate the equation to contact instantons on Sasakian $5$-manifolds. In particular, we show that anti-self-dual contact instantons with a parallel Higgs field provide contact monopoles. We also prove a compact rigidity result: on compact contact metric $5$-manifolds, every smooth contact monopole has parallel Higgs field and satisfies the reduced curvature equation $F_A\wedge\omega=0$. In the Sasakian case, this relates compact contact monopoles to anti-self-dual contact instantons under the usual transverse type condition. We also explain why the associated deformation complex is not elliptic in the usual sense. We then focus on unit tangent sphere bundles of oriented Riemannian $3$-manifolds, endowed with their canonical contact metric structures. In this setting, we derive a local form of the contact monopole equation with respect to the canonical adapted frame and prove that pure pullback configurations from the base are rigid. Finally, for the homogeneous Sasakian manifold $T_1S^3\cong SO(4)/SO(2)$, we use Wang's theorem to reduce the invariant contact monopole equation to finite-dimensional algebraic systems and classify the invariant solutions for homogeneous $U(1)$- and $SO(3)$-bundles. In the abelian case, the classification yields non-flat homogeneous $U(1)$ contact monopoles for nonzero topological charge. In the non-abelian case, all weights except $m=1$ reduce to the maximal torus, while the weight $m=1$ gives a genuinely non-abelian branch of homogeneous $SO(3)$ contact monopoles.

Let $\theta \to \mathcal G(\theta)$ be a map from a parameter to the solution of some non-linear partial differential equation modelling a dynamical system. We provide conditions to show posterior consistency for $\theta$ under observations of $\mathcal G(\theta)$ with dependent noise, modelling potential time-correlations of the sensors used to make the measurements. We provide two examples, namely the Navier-Stokes equations with unknown viscosity and initial condition, as well as a Reaction-Diffusion type equation with diffusivity coefficient that is varying (heterogeneous) in space, which requires novel stability estimates for a Darcy transport operator with accumulation in time.

Multi-agent coordination often requires strategic agents to share sensitive information about their states or objectives, creating a tension between performance and privacy. Our paper studies this tradeoff in stochastic linear-quadratic (LQ) dynamic games with heterogeneous agent objectives. In our framework, agents share noise-perturbed state and reference information with a cloud computer that computes feedback Nash equilibrium strategies, with the injected noise calibrated to provide differential privacy. We derive an analytical expression for each agent's infinite-horizon steady-state cost of privacy relative to the non-private game. Then, we prove that when agents' objectives are sufficiently aligned, the injection of privacy noise necessarily incurs a positive performance cost. In contrast, we characterize a class of games with sufficiently misaligned objectives across agents for which an agent's cost of privacy can be strictly negative. Thus, counterintuitively, noise can simultaneously protect privacy and improve the equilibrium performance of an agent when the objectives of interacting agents are sufficiently misaligned. Finally, we present numerical experiments which corroborate our theoretical contributions.

Learning constitutive laws of complex fluids from velocity data (i.e. indirect observations) is a PDE-constrained inverse problem in which expressive neural parameterizations risk breaking the well-posedness of the forward physics model. We address this tension by learning, rather than the constitutive law itself, the dissipation potential: a scalar function whose convexity, frame-indifference, and dissipativity propagate into the underlying continuum mechanics and guarantee the structural properties needed for a well-posed and generalizable forward PDE. We introduce ICNNE, an input-convex neural architecture that exactly enforces convexity, frame-indifference, and evenness in the second strain-rate invariant via symmetrization, alleviating the numerical instabilities that occur at small strain rates in standard formulations. By weakly enforcing zero gradients in the origin of the potential via penalization, ICNNE also satisfies the dissipativity property. The loss objective and its gradient are computed using a combination of finite element and neural network methods, coupling the Firedrake and PyTorch libraries. We evaluate on four problems spanning compressible and incompressible regimes: compressible Navier-Stokes, Herschel-Bulkley yield-stress flow, Hibler's viscous-plastic sea-ice model, and discrete-element-method data with no known constitutive law. We show that the learned potentials (i) recover ground-truth physics where it is known, (ii) transfer accurately to geometries unseen during training, and (iii) succeed in regimes where unstructured methods diverge. These results suggest that embedding mathematical structure into the parameterization, rather than into the loss, is a robust path to learning physics from indirect data.

We study a representative two-loop family of worldline quantum field theory integrals relevant for classical gravitational scattering with a massive mediator and investigate the geometric structure of its maximal cut. We find that, unlike the massless case, the massive loops already give rise to an elliptic maximal cut at two loops, which degenerates into a nodal curve in the massless limit and in the static limit. In the global Baikov representation, two successive integrations reduce the maximal cut to a period of an elliptic curve. We further establish five primary consequences of this study: (a) The integration-by-parts method generates five master integrals, which, under reflection, split into a three-dimensional elliptic block and a two-dimensional polylog block. We derive the differential equation in two ways, one by Laporta reduction and subsequently by intersection theory in two cocycle bases, and construct the rational map between them. (b) The period matrices in arbitrary dimension have been computed in terms of Appell and Beta functions. Furthermore, we derive twisted Riemann period relations that yield nine quadratic identities among them. (c) We study the global monodromy relation of the differential system that satisfies Riemann's bilinear relation. (d) We explicitly derive the function space of the iterated integrals in terms of elliptic multiple polylogs. (e) In impact-parameter space, the Fourier-transformed cut is a Bessel-twisted differential on the elliptic curve, which we show has extra involution symmetries. Thus, the parity of the differential further decides which quotient - rational or elliptic its periods live on. All results presented concern the maximal cut and, therefore, the homogeneous solutions, but they open an important pathway to tackling the full massive two-loop integral.

We obtain new and essentially optimal bounds on the number of solutions of diagonal systems of $k$ polynomial equations in $s$ variables from an arbitrary finite subset of a finitely generated group in a field of characteristic zero. Our bounds generalise and sometimes improve previous results of J. Bourgain, M. Z. Garaev, S. V. Konyagin, and I. E. Shparlinski (2014), for $k=1$, and of A. Manning, A. Ostafe and I. E. Shparlinski (2026), for $k=2$. In turn, this improvement leads to sharpening a result of A. Manning, A. Ostafe and I. E. Shparlinski (2026) concerning arithmetic statistics of certain matrices. We also obtain similar results for $s$-tuples of linear recurrence sequences, in the case of linear equations and quadratic equations.

We establish well posedness of the Dirichlet problem in weak local John domains $\Omega\subset\R^\dmn$ with Ahlfors regular boundaries of possibly non integer dimension $0<\bdmn<\dmn$, for linear second order elliptic equations with real coefficients, and with data in Besov spaces with a very broad range of acceptable parameters.

A conjecture of Erdős asserts that every odd integer $n > 1$ can be written as $n = 2^a + m$, where $a \ge 0$ is an integer and $m$ is a squarefree positive integer. We study this conjecture for primes $p \equiv 7 \pmod{36}$. Writing $x = (p-1)/6$, the candidate remainders take the shapes $p - 1 = 6x$, $p-2 = 6x-1$, $p-4 = 3(2x-1)$, and $p-8 = 6x-7$, in which the primes $2$ and $3$ are completely controlled: every obstruction to squarefreeness comes from a prime $q \ge 5$. For such $q$ a separation principle applies — $q^2$ divides at most one of the remainders, since two of them differ by at most $7 < 25$ — and it persists across any window of exponents of width less than $20$. Separation collapses the square-divisor problem for the quartic $x(6x-1)(2x-1)(6x-7)$ into independent linear problems, and we deduce unconditionally that the primes of the class representable with $a \le 3$ have relative density $1 - \rho_4$, where $\rho_4 = 1.8637 \times 10^{-5}$ is an explicit Euler-type constant; in particular, infinitely many primes of the class are not so representable. Computationally, we verify the conjecture for all primes of the class below $10^8$, always with $a \le 5$, and we exhibit the eleven primes below $10^8$ requiring $a = 5$. An appendix corrects the error bound in a lemma of Hercher, whose conclusions are unaffected.

Systems of equations arise in a wide range of scientific and engineering applications. The present work focuses on solvers for general systems of equations, including but not limited to those arising from partial differential equations. These systems can be broadly categorized into linear and nonlinear problems. For large linear systems, iterative solvers are generally preferred over direct methods due to the latter's superlinear growth of computational costs. Although convergence theory is well-developed under certain assumptions on the coefficient matrix, many classes of systems still pose open challenges. These difficulties become even more severe for systems of nonlinear equations, where nonlinear solvers typically rely on repeated linearization. For example, Newton's method may even converge quadratically near the solution; it can also converge slowly or diverge when the initial guess is not chosen appropriately. A wide range of solvers with diverse variants and hyperparameter settings exists, and the development of efficient and robust iterative methods remains an active area of research. Recently, machine learning (ML) techniques have been applied to enhance the efficiency of classical iterative methods while preserving their interpretability and reliability. We refer to these ML-enhanced iterative methods as hybrid iterative methods, in the sense that they combine classical iterative methods with ML. This paper provides a comprehensive overview of state-of-the-art approaches to constructing hybrid iterative methods for systems of both linear and nonlinear equations, while also discussing open challenges and outlining potential directions for future research.

Consider a system of equations $A(x)\cdot F(x) = B(x)$ on a subset $X$ of $\mathbb R^n$, where $A(x)$ and $B(x)$ are matrix- and vector-valued semialgebraic functions on $X$, and the unknown $F(x)$ is a field of vector-valued $m$-jets. We assume there is a solution which is the field of Taylor polynomials of order $m$ on $X$ of a $\mathcal C^m$ vector-valued function $f$ on $\mathbb R^n$, and ask whether we can find a $\mathcal C^m$ semialgebraic solution $f$. Our main result is a positive answer in the case $\dim X = 1$. The methods are based on an article of Fefferman and Luli for $X \subset \mathbb R^2$, and a secondary goal is to show that their approach applies in a much simpler way in the case $\dim X =1$, or in the case of the semialgebraic Whitney extension problem in $\mathbb R^2$. Our results hold more generally for functions definable in an o-minimal expansion of $\mathbb R$.

We prove that the colored Jones polynomial of a knot does not determine the Links–Gould polynomial (associated with the 4-dimensional representation of the Lie superalgebra $\mathfrak{sl}(2|1)$) with two different approaches: the first is to find a concrete genus 2 mutation pair with different Links–Gould polynomials, and the second follows from the fact that the colored Jones polynomial is invariant under Conway mutation whereas the colored Links–Gould polynomial is not. The first approach also applies to a related family of polynomial invariants of Kashaev–Mangazeev, which are conjectured to be the $\mathfrak{sl}(n|1)$-knot invariants colored by the $2^n$-dimensional irreducible representation.