Papers

The $2 \rightarrow q$ norm of a matrix $X \in \mathbb{R}^{n \times d}$ is defined as $\lVert X \rVert_{2 \rightarrow q} = \sup_{\lVert v \rVert_2 = 1} \lVert Xv \rVert_q$. We give polynomial-time multiplicative approximation algorithms for this norm when $q > 2$ (i.e. in the hypercontractive setting). This problem either directly captures or is closely related to long-standing open problems in combinatorial optimization and hardness of approximation (e.g. Small Set Expansion), quantum information (e.g. Best Separable State), and algorithmic statistics. Very little is known about what approximation factors we can achieve for this problem in polynomial time, even though such approximations have significant downstream consequences. Barak, Brandão, Harrow, Kelner, Steurer, and Zhou showed that no polynomial-time algorithm can achieve an approximation factor better than $2^{\sqrt{\log n}}$, assuming the Exponential Time Hypothesis (FOCS'12). On the other hand, a simple spectral algorithm gives a $d^{1/4}$-approximation as a baseline. For the important special case of $q = 4$, prior work of Guth, Maldague, and Urschel (SIAM Matrix Analysis and Applications'25) can be combined with known sparsification techniques to give a $d^{1/6}$-approximation. We improve over these results by polynomial factors, giving a $d^{1/8}$-approximation. Moreover, we construct sum-of-squares certificates for the $2 \rightarrow q$ norm. This directly implies improved algorithms for robust mean and covariance estimation, robust regression, and clustering, when the data only satisfies a bound on its $q$-th moment.

Fastest arrival events, where the first among many diffusing particles reaches a target, are central in triggering signal initiation in molecular stochastic systems. Classical approaches to simulate such events rely on full trajectory generation of all particles, leading to prohibitive computational costs in the large particle number regime. In this work, we present a general simulation framework for efficiently generating order statistics of arrival times by exploiting asymptotic first-passage distributions. This framework applies to diffusion processes in bounded domains with localized absorbing targets, for which short-time first-passage asymptotics are available, such as Brownian motion in dimensions one, two, and three. Starting with the case of instantaneous emission, we derive and implement a recursive inverse transform algorithm to simulate the first $k$ arrivals without tracking particle trajectories. We extend this algorithm to time-dependent emission profiles via an iterative approach, enabling the simulation of extreme statistics in systems with temporal injection, ranging from rapid to prolonged emission. Additionally, we provide asymptotic estimates of the mean fastest arrival time. To conclude, the present acceleration algorithm which bypasses Brownian simulations of trajectories can be used for spatial reaction networks, rare event detection, or diffusion-controlled activation.

Let $\tau(z)=-1-z^{-1}$. We study the reduced rational maps $h_d:\mathbb{P}^1\to\mathbb{P}^1$ obtained by cancelling common factors in $H_d^{\rm raw}(z)=z^d(\tau(z)^d-1)/(z^d-1)$. These maps arise by Hilbert-90 descent from the trace-zero maps $X^{dq}-X^d$ on $\ker\operatorname{Tr}_{\mathbb{F}_{q^3}/\mathbb{F}_q}$, but the principal object is the resulting $\tau$-equivariant quotient-map family; nonconstant separable members are viewed as covers. We prove that cancellation is exactly a torsion-defect phenomenon. If $\ell(-)$ denotes scheme-theoretic length and $\boldsymbol{\mu}_d=\ker([d]:\mathbb{G}_m\to\mathbb{G}_m)$, then $\mathrm{deg}(h_d)=d-\ell((1+X+Y=0)\cap\boldsymbol{\mu}_d^2)$, and, in characteristic $p>0$ with $d=p^s d_0$ and $p\nmid d_0$, $h_d=\operatorname{Frob}_{p^s}\circ h_{d_0}$ and $\mathrm{deg}(h_d)=p^s\mathrm{deg}(h_{d_0})$. We classify the tame quotient strata of morphism degree at most one and exactly two; the maximal-defect stratum yields a characteristic-two Mersenne trace-zero permutation family. In characteristic zero we prove the main monodromy theorem: every non-linear quotient is Morse and has full symmetric geometric monodromy, $G_{h_d}=S_{\mathrm{deg}(h_d)}$; the proof rules out branch-value collisions via a cyclotomic cross-ratio equation. In positive characteristic we isolate Frobenius-sparse Kummer and Artin-Schreier quotients, a certificate-verified characteristic-19 Klein-four Galois quotient, and the first nonsparse Frobenius-lacunary tower up to its stated primitivity and wild-inertia boundary. A twisted off-diagonal fiber-square trace formula turns $2$-transitive monodromy into a uniform obstruction to $\tau$-twisted exceptionality.

In this article we describe a new inductive approach to compute the chromatic polynomial of simple graphs and the characteristic polynomial of central hyperplane arrangements.

Inverse optimization (IO) is used to estimate unknown parameters of an optimization model from observed decisions. In the data-driven context, the estimated parameters are inherently uncertain, yet quantifying this uncertainty has received limited attention in the literature, where existing methods return a point estimate. In this paper, we propose a hierarchical Bayesian framework for parameter uncertainty quantification in data-driven inverse optimization. Considering two data-generating processes, we develop two Markov chain Monte Carlo algorithms to estimate the posterior distribution of the unknown parameter vector, which is used to construct credible regions. We establish posterior consistency under standard identifiability conditions. Numerical experiments demonstrate near-nominal empirical coverage of the credible regions and show that the regions shrink as the number of observed decisions increases.

Classical atomistic simulations based on interatomic potentials resolve lattice instabilities, defect nucleation, and microstructure evolution with high fidelity, but their accessible system sizes remain far below those required for micrometer-scale structural analyses. We develop a two-scale atomistic-continuum framework that couples a nonlinear finite-element boundary-value problem at the microscale to periodic molecular-statics cell problems at quadrature points. The scale transition is formulated by computational homogenization in the sense of Hill-Mandel energy equivalence. Instead of prescribing a continuum constitutive law on the lower scale, the atomistic cell is driven directly by the continuum deformation and returns volume-averaged stresses in work-conjugate form together with effective tangent moduli. Numerical examples for single-crystalline copper show pronounced tension-compression asymmetry, abrupt instability-driven defect nucleation, rapid stabilization under reversed cyclic loading, and localized elastic-plastic transition in cantilever bending. In all these strongly nonlinear scenarios, the coarse-scale Newton solver remains robust and recovers near-quadratic convergence in its final iterations. The two-scale framework thus extends potential-based atomistic modeling to structural length scales that are inaccessible to direct atomistic simulation in the present quasi-static, athermal setting.

Given complex numbers $a, b, c$ and a non-negative continuous function $\varphi$ defined on $[0, +\infty)$, consider the $2 \times 2$ matrix $$ M_t = \begin{pmatrix} a & t \\ ct & b\varphi(t) \end{pmatrix}, \quad t \in [0, +\infty). $$ We establish conditions for the strict monotonicity of the norm function $t \mapsto \|M_t\|$. As an application, we characterize the norm attainment of the corresponding block operator matrix $$ T = \begin{pmatrix} aI_H & A \\ cA^* & b\varphi(|A|) \end{pmatrix},$$ where $I_H$ is the identity operator on a Hilbert space $H$ and $A$ is a bounded linear operator from another Hilbert space to $H$.

The multi-dimensional compressible Euler equations admit non-unique entropy solutions in turbulent regimes, and extreme-Mach astrophysical flows are a natural setting in which this breakdown of deterministic well-posedness becomes computationally visible. We compute statistical solutions of a Mach~2000 astrophysical jet, defined as the pushforward of an initial probability measure through a vectorial lattice Boltzmann method, by Monte Carlo sampling with $M=1000$ realizations on grids of up to $12.8$ million cells. Under mesh refinement the individual realizations diverge pathwise, while the statistical solution converges: Wasserstein distances of the one- and two-point marginals, the ensemble mean, and the ensemble standard deviation all exhibit stable positive convergence rates. A spatially resolved analysis along the jet axis traces this dichotomy to the structure of the one-point laws, which are numerically Dirac in the undisturbed core, skewed in the sheared turbulent regions, and intermittent two-state mixtures at the random leading front. We conclude that the computed statistical solution is non-Dirac and remains stable in the extreme compressible regime, in which no strong solution is expected to exist.

Understanding the statistics of level crossings in stochastic processes is crucial across many scientific disciplines. The traditional Kac-Rice formula gives the mean rate of level crossings and has found broad use. However, that mean rate captures only a coarse summary of the crossing process. It depends entirely on local properties of the stochastic process at a given instant and is therefore blind to the correlation structure of the process over time. To understand whether crossing events, such as neuronal spikes, tend to cluster in time, spread apart, or exhibit more complex temporal organization, one must go beyond the mean rate and study higher-order crossing statistics. Here we go beyond the mean by deriving the exact analytical formulae for the variance and Fano factor of arbitrary level crossings in smooth stationary Gaussian processes. Our exact solution reveals how the full temporal correlation structure dictates whether crossings cluster or become regular. In systems with oscillatory correlations, such as a stochastic damped harmonic oscillator, a recent crossing suppresses an immediate subsequent one, producing sub-Poissonian statistics. However, as damping increases and oscillations disappear, a large and slow excursion above the threshold can produce multiple closely spaced crossings, yielding super-Poissonian statistics. In purely relaxational, non-oscillatory systems, such as a mean-reverting process driven by Ornstein-Uhlenbeck noise, the competition between the timescales of the driving noise and system relaxation produces a richer landscape, including reentrant transitions between sub- and super-Poissonian statistics as the threshold level is varied. Taken together, the exact variance and Fano factor derived here complement the Kac-Rice mean rate, enabling more robust parameter estimation and model selection across any setting where Gaussian processes are used.

We show that the geometry of Hertling-Manin F-manifolds $(M,\circ,e)$ provide the appropriate theoretical framework for studying the integrability of quasilinear systems of first-order evolutionary partial differential equations of the form ${\bf u}_t=X\circ {\bf u}_x$ (F-systems) under the mild assumption that the unit vector field is cyclic with respect to the operator of multiplication by the vector field $X$. This approach is very general and allows us to treat even non-regular systems that were previously beyond the scope of existing techniques. Like in the regular case the information about integrability is contained in a torsionless connection associated with the system and the integrability condition reduces to a geometric condition involving the Riemann tensor of the connection and the structure functions of the product. We prove that a locally conservative F-system is integrable and, in the analytic setting, also the converse statement, thereby providing a full characterization of integrability. Moreover, in the analytic case, we prove the existence of a family of analytic symmetries providing, in principle, the unique local analytic solution of the Cauchy problem through the generalised hodograph method.

This paper discusses digital online mathematics examinations – a discussion ranging from high school to university level examinations. In particular, we consider the nature of mathematical writing, what is distinctive about mathematical writing, and how mathematics can be typed into a machine. This includes a review of features of notation and layout unique to mathematics and a survey of current technology for typed mathematics, including LaTeX and contemporary proof-checkers such as Lean. Artificial intelligence has already been highly successful for optical character recognition, generating text from hand writing and is even increasingly applied to assess students' work itself. A human-editable text-based format in the middle is important, but neglected. The design of digital online mathematics examinations, which take this text as the source of truth for students' work, will have a profound effect on how mathematics is perceived and how mathematical activity is mediated. Moving examinations on-screen effectively is an important design challenge and responsibility to future generations. The challenge is to design software tools which support mathematics, that is tools which recede into the background and support the generation of mathematical work. We argue for a human-editable text-based format, which includes semantic elements, at the heart of the process.

Consider a symmetric function $\mathcal{C}(x,y)$ on $[0,1]\times[0,1]$ which is twice continuously differentiable up to the boundary, and which satisfies $ \mathcal{C}(x,y)=\mathcal{C}(1-x,1-y)$. Let $A^{(n)} = \big(a^{(n)}_{i,j}\, :\, i,j \in [n]\big)$ be the matrix with entries $a^{(n)}_{i,j}\, =\, \exp(-\mathcal{C}(i/n,j/n))$. Soumik Pal conjectured the asymptotics $$\operatorname{perm}\big(A^{(n)}\big)/n!\sim \exp\big(n \Lambda[\mathcal{C}]\big)/ \sqrt{\mathcal{D}[\mathcal{C}]}$$ as $n \to \infty$ for known functionals that arise naturally in the context of entropy regularized optimal transport. The functional $\Lambda[\mathcal{C}]$ is the known large deviation rate function, already proved rigorously by Sumit Mukherjee. It is $\int_{0}^1 \int_0^{1} (\alpha(x)+\beta(y))\, dx\, dy$ where $\alpha(x)+\beta(y)$ is chosen such that $\rho(x,y) := \exp(-\mathcal{C}(x,y)-\alpha(x)-\beta(y))$ has uniform marginals. The algebraic term $\mathcal{D}[c]$ is given by Peter McCullagh's formula for doubly stochastic matrices: $\operatorname{det}_F(I+J-T^*T)$, the Fredholm determinant, where $I$ is the identity on $L^2([0,1])$, $Jf(x) \equiv \int_{0}^1 f(z)\, dz$ (for all $x$) and $Tf(x) = \int_0^1 \rho(x,y) f(y)\, dy$. We prove the conjecture for functions $\mathcal C$ that are constant on blocks, exploiting a well-known Ross Pinsky's combinatorial decomposition of permutations in blocks.

We study the symmetric polynomial $\prod_{\alpha\in A_{n,d}}\bigl(1+\alpha_1 x_1+\cdots+\alpha_n x_n\bigr)$ where $A_{n,d}:=\{\alpha\in\mathbb{Z}_{\ge 0}^n:|\alpha|=d\}$, which is the total Chern class of $\mathrm{Sym}^d(\mathbb{C}^n)$, viewed as a torus representation whose Chern roots are the weights $\alpha_1 x_1+\cdots+\alpha_n x_n$ for $\alpha\in A_{n,d}$. Its homogeneous degree-$k$ part $c_k(n,d)$ is the $k$-th Chern class of $\mathrm{Sym}^d(\mathbb{C}^n)$. These Chern classes, together with their coefficients in various symmetric function bases, play a central role in enumerative geometry. Despite their simple definition, general closed formulas for their coefficients are subtle, and many structural properties of these classes have remained poorly understood. In this paper we prove several conjectures concerning their structure, establish explicit formulas, and study log-concavity properties for both the Chern classes and their $K$-theoretic analogue. In rank two, passing to the Schur basis and expanding the Schur coefficients in the binomial basis of $d$, we uncover a new binomial log-concavity phenomenon and prove refined positivity results. The paper demonstrates a novel methodology: we combine several AI systems with human mathematical insight in a coordinated workflow, deploying each tool according to its strengths in experimental discovery, conjecture formation, symbolic proof construction, and verification. To our knowledge, this is one of the first detailed case studies of orchestrating multiple AI tools to make substantial progress on a coherent mathematical research project.

By the classic results of Fricke and Klein, for every word $w$ in the free group $F(a,b)$ there exists a unique integer \it{trace polynomial} $f_w(x,y,z)\in Z[x,y,z]$ such that $Tr(w(A,B))=f_w(Tr A,Tr B,Tr AB)$. for all $A,B\in SL(2,C)$. We study quantitative aspects of trace polynomials. We prove an exact formula for the leading homogeneous part of $f_w$ for every nontrivial cyclically reduced word $w\in F(a,b)$. In particular, if $w=u_1\cdots u_n$ is cyclically reduced over $\{a,a^{-1},b,b^{-1}\}$, and if $N_{rs}(w)$ is the number of cyclic occurrences of $rs$, then $deg f_w=n-N_{ab}(w)-N_{b^{-1}a^{-1}}(w)=n-\frac{1}{2}(N_{ab}(w)+N_{ba}(w)+N_{a^{-1}b^{-1}}(w)+N_{b^{-1}a^{-1}}(w)).$ We obtain sharp general bounds $\lceil n/2\rceil\le deg f_w\le n$ for $w\in F(a,b)$ with cyclically reduced length $n$. We also study $deg f_w$ for random positive words and for random freely reduced and random cyclically reduced words. We obtain explicit exponential upper bounds for the growth of the $\ell_1$ and $\ell_\infty$ norms of $f_w$ and exhibit examples with exponential coefficient growth at rate $\varphi^n$, where $\varphi$ is the golden ratio. We show that for random freely reduced, random cyclically reduced and random positive words $w_n$ of length $n$ in $F(a,b)$, the size of $supp(f_{w_n})$ grows at least quadratically in $n$ and the total bit-size of $f_{w_n}$ grows at least as $cn^3$. Hence, any algorithm computing $f_w$ in totally expanded form has worst-case time complexity as well as generic-case time complexity for the above models bounded below by $\Omega(n^3)$. We also give a deterministic algorithm which computes the fully expanded polynomial $f_w$ in time $O(n^5)$ and space $O(n^4)$, in terms of the input word length $n$.

For a split graph $S$, the combinatorics of 2-switches on $S$ is faithfully encoded by the factor graph $\Phi(S)$, a multigraph whose induced cycles have length at most $4$. In this paper we address the following question: for which $n \in \mathbb{N}$ is there a split graph $S$ whose factor graph contains an $n$-simple triangle, that is, a triangle all of whose edges have multiplicity $n$? We show that the answer is governed by a purely arithmetic condition, the $\Delta$ property, relating the differences and sums of complementary divisors of $n$, and thereby establish a two-way bridge between Graph Theory and Number Theory.

We report on a collection of open problems in commutative algebra and related areas that have been resolved (proved or disproved) using the Rethlas natural-language automated reasoning system. The problems are drawn from several published lists, including Open Problems in Commutative Ring Theory (Cahen-Fontana-Frisch-Glaz), Erman-Sam's survey of Boij-Söderberg theory. For each problem we record the precise statement and a self-contained proof produced (with no human intervention) by Rethlas and subsequently verified by human experts.

We study the algebraic curve over $\mathbb{F}_{q^2}$ defined by $y^{q+1} = x^n(x^n+1)$, where $n$ is a positive integer coprime to the characteristic. We first prove (when $q$ is odd) that the nonsingular model of this curve is $\mathbb{F}_{q^2}$-maximal if and only if $n \mid (q+1)$. Writing $n = \frac{q+1}{m}$, we obtain a family of maximal curves parameterized by the divisors $m$ of $q+1$, which extends the previously studied case $m=3$ corresponding to maximal curves with the third largest possible genus. For this family, we determine the Weierstrass semigroups at several classes of rational points, including those lying above the branch points of the natural projection. These semigroups are described explicitly in terms of $q$ and $m$, and exhibit different behaviors depending on the arithmetic properties of $m$. Moreover, we determine the full automorphism group of the curve under a mild condition on the characteristic. Our results extend an earlier work on the case $m=3$ and provide new insight into the structure of this family of maximal curves.

The boosted Frank-Wolfe algorithm accelerates the classical Frank-Wolfe algorithm by better aligning the update direction with the negative gradient. Its analysis, however, has been limited to deterministic convex problems, with step sizes that require either line search or knowledge of the Lipschitz constant of the gradient. We develop a novel step size strategy that does not depend on the Lipschitz constant of the gradient, which allows us to extend the boosted Frank-Wolfe algorithm to the stochastic setting. We prove that boosting with this step size strategy can be combined with many modern gradient estimators, including SAGA, L-SVRG, SAG, Heavy Ball momentum, and zeroth-order estimators, among others, while retaining the worst-case convergence rates of ordinary stochastic Frank-Wolfe. Our analysis also yields the first convergence rates for boosted Frank-Wolfe on nonconvex and quasar-convex objectives, results which are new even for deterministic problems. Experiments on sparse logistic regression and quantum process tomography show that stochastic boosted Frank-Wolfe achieves faster convergence per gradient oracle call (and on wall-clock) compared to the non-boosted baseline.

Let $S$ be a base scheme, assumed separated and Noetherian. We define adequate classes of morphisms of $S$-schemes by formalizing certain properties of homotopy equivalences of complex algebraic varieties. Other examples of adequate classes include morphisms of varieties over an algebraically closed field of positive characteristic which induce an isomorphism of étale cohomology, and weak equivalences in the cdh topology. An affine model for a Noetherian scheme $X$ over $S$ is an affine scheme $M_X$ equipped with an adequate morphism $M_X\to X$. In this paper we construct affine models for arbitrary separated schemes of finite type over $S$. Our construction can be viewed as a generalization of the Jouanolou trick. As an application, we construct a mixed Hodge structure on the Leray spectral sequence of an arbitrary proper morphism $f:X\to Y$ of complex algebraic varieties, generalizing an argument by Donu Arapura which assumed $Y$ quasi-projective and $f$ projective.

We present a theory for constructing optimal lower bounds for the discrete half-line $p$-Laplacian of higher order $\ell\in\mathbb{N}$ and general $p>1$. The abstract framework introduces higher-order monotonicity and asymptotic constraints on a parameter sequence that determines optimal weights. As a concrete application, we specialize the parameter sequence to deduce new optimal discrete $p$-Hardy ($\ell=1$), $p$-Rellich ($\ell=2$), and $p$-Birman ($\ell\geq 3$) inequalities.

The shortest path multiplicity, here denoted by $\mu$, is an important metric of complex networks. For real networks $\mu$ is high and it correlates with the network community structure. Since local network evolution induces network communities, it is possible that a high shortest path multiplicity is the natural expectation of local evolution rules. Here I demonstrate, by means of numerical simulations and exact results, that this is indeed the case. For random graphs with arbitrary degree distributions $p_k$, $\langle\mu\rangle\sim \langle k(k-1)\rangle / (\langle k\rangle e)$, growing with the network size when $p_k\sim k^{-\gamma}$ and $\gamma\leq3$. For networks generated by stochastic local rules, $\langle\mu\rangle$ increases with the network size and it does so faster than what is observed in their randomized versions. Furthermore, the number of communities increases with the network size and the correlation with $\langle \mu\rangle$ follows.

We give an attribution method for neural combinatorial-optimisation (CO) policies that (i) decomposes a decision by constraint families via LP-relaxation duals, (ii) certifies counterfactuals through a combinatorial feasibility model (implemented as a CSP feasibility-decision model), and (iii) bounds the size of a PAC-sufficient explanation with a Bonferroni-corrected Hoeffding sufficient-subset test along a greedy ordering. Across three CO problems and three seeds, our LP-anchored $\Lambda$-attribution matches the CF-derived signal at 96.5% on CVRPTW (n_cert=344) and 77.2% on the Orienteering Problem (n_cert=281) vs 75.0% and 35.2% for proxy gradient (paired diffs +0.215 and +0.420; McNemar exact $p \le 10^{-14}$). In the rank-aligned regime of the Flexible Job-Shop Scheduling Problem, both backends agree on every CSP-certified flip (n_cert=59), confirming the no-gain prediction. Bonferroni-PAC subsets average 5.0 nodes per step ($M=70$, $\varepsilon=\delta=0.2$, $k_{\max}=25$). Reference implementation: https://github.com/sohaibafifi/neuro-co-cax

Tonks' projection from the permutohedron to the associahedron and the Loday–Ronco map both send permutations to planar binary trees. We give a syntactic account of these maps in the equational calculus of the free non-symmetric, non-unital operad on one binary generator. The vertex restriction of Tonks' projection is obtained by evaluating the head-insertion encoding on the reversed permutation, while the Loday–Ronco map is obtained by evaluating the decreasing encoding. We also give a local operadic proof that Tonks' vertex map is order-preserving from the weak Bruhat order to the Tamari order.

We trace a conceptual genealogy from Abraham de Moivre's derivation of the normal curve (1733) to the modern distributional approach to statistics. De Moivre's Approximatio ad Summam Terminorum Binomii gave the first systematic derivation of the Gaussian density, its normalising constant (completed by Stirling's identification of $B = \sqrt{2\pi}$), and its tail probabilities computed to six decimal places – more than seventy years before Gauss. His method – extracting information from probability laws by evaluating sums against indicator probes – is recognisably an instance of the operational viewpoint that underlies distributional statistics. We identify a four-stage chain: coefficient extraction (De Moivre) $\to$ generating functions (Euler, Laplace) $\to$ characteristic functions (Fourier, Lévy) $\to$ distributional pairings $\langle T, \varphi \rangle$ (Schwartz). At each stage the probes become more flexible and the class of laws that can be studied grows wider. The distributional framework, in which a probability law is represented by a distribution–kernel pair $(T, \varphi) \in \mathcal{S}'(\mathbb{R}) \times \mathcal{S}(\mathbb{R})$, is the natural endpoint of this progression. We formulate and prove a distributional version of the De Moivre–Laplace theorem: the standardised binomial distribution converges to the Gaussian in $\mathcal{S}'(\mathbb{R})$, with De Moivre's original computation corresponding to the special case of indicator test functions. We also discuss the transversality framework, which provides a geometric explanation – via infinite codimension of degeneracy strata – for why pathologies such as moment indeterminacy, non-identifiability, and singular Fisher information are rarely encountered in parametric statistical models.

We give a simple computational approach to mathematical quasicrystals, combining cut-and-project methods with self-similarity. Starting with a Pisot unit $\beta$ and an iterated function system $g_k(z)=\beta z +w_k, \ k=1,...,m$ in a corresponding ring of algebraic integers, we take the attractor $A$ of the conjugate system as an acceptance window. This yields a unique cut-and-project model set $\Lambda$ in the complex plane which fulfils $\Lambda=\bigcup g_k(\Lambda) .$ We describe an algorithm which directly determines $\Lambda ,$ avoiding the difficulties with the fractal structure of $A.$ Classical constructions are based on tiles $A$ of different shape. The present study continues work on models with overlaps, as introduced by Gummelt (1996), Baake and Grimm (2013), Hejda and Pelantová (2016), and Hare, Masáková, and Vávra (2018). In our approach, the overlaps provide a natural decoration of $\Lambda .$ The method is illustrated with a variety of pentagonal examples.

Using the homotopy theory of polynomial monads developed by Batanin and Berger and extended to the $2$-categorical context by Weber, we prove the cofinality of a particular morphism of polynomial $2$-monads. We apply our result to give a new proof of the delooping of derived mapping spaces of infinitesimal bimodules due to Ducoulombier and Turchin.

Reservoir computing typically relies on large, randomly generated reservoirs, enabling simple, often linear readouts. Over the past two decades, most constructions have exploited the freedom to select the reservoir, constrained primarily by stability conditions based on state contraction or memory capacity. However, these designs are largely independent of the input data and learning objective, resulting in a trial-and-error methodology driven by randomness. In high dimensions, the reservoir acts as a random embedding of the input history, implicitly relying on Johnson–Lindenstrauss–type concentration phenomena to preserve information. In contrast, we develop reservoir design principles from a geometric perspective for inputs generated by deterministic dynamical systems. Rather than relying on random embeddings, we require reservoir state increments to align within a cone around an input-determined vector subspace, and prove that such a cone concentration reduces ridge-regression training error. When the cone angle is small, the variance of reservoir states concentrates in the input-determined subspace, improving conditioning of the empirical second-moment matrix and strengthening alignment between dominant covariance directions and the state-target cross-covariance. For echo state networks, we provide a constructive approach to reservoir design. The reservoir matrix is chosen so that associated Krylov-chain directions remain nearly closed within an input-determined subspace while permitting controlled mixing in its orthogonal complement. We also provide a spectral diagnostic for ridge regression training that identifies when reservoir geometry concentrates predictive information into a few dominant covariance modes and when “spectral pollution” inhibits forecasting. Numerical experiments demonstrate consistent performance gains over arbitrary reservoir constructions.

This paper presents a backstepping approach for the boundary control of first-order hyperbolic equations with spatially varying coefficients posed on domains of arbitrary dimension. The method is based on a change of variables induced by the characteristic flow of the time-invariant transport operator, transforming the original multidimensional system into a continuum of decoupled one-dimensional hyperbolic equations evolving along individual characteristic curves. A backstepping controller is then designed for each equation in the decomposition, and the resulting control laws are reassembled in the original coordinates to achieve finite-time stabilization of the full system. The framework relies on the existence of characteristic curves foliating the spatial domain, with uniformly bounded transit times (non-trapping).

We prove that any singular Kähler–Ricci shrinker $X$ arising as a noncollapsed limit of Kähler–Ricci flows admits a natural structure of a polarized Fano fibration. We also show that it is simply connected, has unique tangent cones at every point, and is a smooth orbifold outside a subset of complex codimension at least three. As an application, we prove a new long-time pseudolocality theorem for almost-selfsimilar Kähler–Ricci flows.

We extend Stein's method to include a prescribed dependence with respect to an auxiliary random variable, for conditional laws for which Stein characterizations do exist. This considerably extends the range of applications of Stein's method. The characterization we derive subsumes the case of independent random variables. Using the Malliavin calculus, we illustrate our extension on the example of a conditionally Gaussian/Beta-distributed random variable.