Papers

When exploring the Hankel determinant of the sequence $\mu_{k}=B_{k+1}/(k+1)$, where $B_{k}$ is the $k$-th Bernoulli number, we obtained two interesting results. The first one applies in general to all sequences $(c_{k})_{k\geq0}$ with all even-indexed term $0$, except for $c_{0}$. In this case, the coefficient of the second highest order of the corresponding monic orthogonal polynomials determines the Hankel determinants. Our second result shows, the corresponding J-fractions, obtained from the generating function of $\mu_{k}$, is exactly the same as in early work of Cao, on a faster sequence converging to the Euler–Mascheroni constant.

We study a coupled strongly damped wave–Moore–Gibson–Thompson (MGT) system with logarithmic source $f(u)=|u|^{\gamma-2}u\ln|u|$ in the full Sobolev-subcritical range $2<\gamma<2^*:=2n/(n-2)$. The main difficulty is that, in this full range, the logarithmic nonlinearity does not admit the standard compactness and difference estimates available in the lower subcritical regime. Using the augmented variable $w=v+\tau v_t$, we derive the exact energy-dissipation identity for the strongly damped system and exploit the associated coupled potential-well structure. For the local theory, a Faedo–Galerkin scheme combined with a closed nonlinear differential inequality and a spatial domain-splitting argument yields local existence. The strong damping provides the additional regularity needed to prove uniqueness throughout the full subcritical range and to establish a continuation principle. Below the well depth $d_\alpha$, we then obtain a sharp dynamical dichotomy: solutions with initial data in the stable set exist globally and decay exponentially, whereas solutions with initial data in the unstable set blow up in finite time. The blow-up argument reveals a structural cancellation specific to the augmented formulation: once the logarithmic contribution is reconstructed through the exact energy identity, the unfavorable MGT residual in Levine's concavity functional is absorbed algebraically. This allows the concavity method to close without imposing any additional sign condition on the initial velocities.

Fix a base $b\ge2$ and a digit set $\mathcal{D}\subset\{0,1,\dots,b-1\}$, and let $\mathcal C_k$ be the set of integers in $[0,b^k)$ whose $k$ base-$b$ digits all lie in $\mathcal D$. We study prime pairs and three-term arithmetic progressions in primes whose difference, or common difference, belongs to $\mathcal C_k$. Under a natural digit non-resonance condition, we establish a localized Fourier criterion guaranteeing asymptotic formulae with explicit leading constants for both problems. Using a transfer-operator argument, we verify this criterion in explicit parameter ranges for several natural families of digit sets.

For a finite group $G$ let $\psi(G)$ be the sum of the orders of its elements. The quotient $S_n/A_n$ has order two. Hence $\psi(S_n/A_n)=3$. The quantity to be compared with $\psi(S_n)$ is therefore $3\psi(A_n)$. Computation indicates that $3\psi(A_n)>\psi(S_n)$ for every $n\ge3$. We reduce this inequality to a question on the parity of square roots. Let $r(\beta)$ be the number of odd permutations $\sigma$ with $\sigma^{2}=\beta$. We show that the sum of the orders of the odd permutations equals $2\sum_{\beta\in A_n}r(\beta)o(\beta)$. We also show that $\sum_{\beta\in A_n}r(\beta)$ equals $|A_n|$. The inequality then says that $r$ and the order function are negatively correlated on $A_n$. It says equally that the average order of an odd permutation is less than twice the average order of an even one. We then rule out two natural approaches. No injection from the odd permutations into $A_n$ can halve the order at every point. The smallest failure occurs at $n=12$. A threshold argument through Landau's function fails as well. Computations up to $n=60$ are reported.

Let $(M^m,g)$, $m\geq2$, be a closed connected Riemannian manifold with $\secg_g\leq-1$, and let $(X,g_X)$ be its universal cover. Yau proved that $\hiso(X)\geq m-1$. We prove that equality is rigid: if $\hiso(X)=m-1$, then $(X,g_X)$ is isometric to $\Hh^m(-1)$.

Principal component analysis (PCA) can rotate away from its population target when a covariance matrix is estimated from limited data. We introduce diagonal attenuation, which preserves sample cross-covariances while reducing coordinatewise sample variances. The method is revealed exactly by averaging a linear full-output reconstruction loss over random input masks; studying the correction directly extends it beyond the range attainable by masking. We isolate the part of the random coupling between retained and omitted population directions that is contributed by sample-variance errors, and show how attenuation can reduce the resulting rotation. Under balanced marginal variances, we derive an explicit expected-risk theorem, uniform over the attenuation path for all sufficiently large finite samples, and obtain the asymptotically risk-minimizing strength. For general covariances, we characterize when attenuation leaves the population PCA subspace unchanged and give a risk theorem that also accounts for changing eigengaps and the population cost when the target moves. Simulations track this tradeoff from exact preservation back to PCA. Across local image patches, speech spectra, and smartphone acceleration, both mask-derived and direct attenuation improve PCA under two fitting-sample budgets, and one of them has the largest mean gain among seven methods in every data–budget cell. The full path selects strengths beyond the mask-derived boundary on $63\%$–$95\%$ of the subsamples.

The Lehmer transform encodes a positive dataset as a single analytic curve, the ratio of its power sums at consecutive orders read as a function of the order. As the order sweeps the real line the curve generates every location statistic between the sample extremes, and its logarithm is the unit increment of the free energy of an exponentially tilted ensemble; the statistical theory of the curve flows from that identity. The slope of the curve is a Jeffreys divergence, the arithmetic-geometric gap is a relative entropy, and multiplicativity over independent factors makes deconvolution of multiplicative noise a pointwise division. A quantitative rigidity theorem locates the frontier of identifiability at geometric growth of the curve, with resonant log-periodic families of distinct laws sharing one curve beyond it. The empirical curve is an exact re-coordinatization of the order statistic, hence minimal sufficient, and its influence function, exact leading bias and limit theory are developed in full. At heavy tails the curve undergoes a phase transition whose two knees, one unit apart, sit at the tail index and its unit shift, a whole-sample diagnostic complementing threshold-based estimators.

In earlier work, the authors gave a construction and description of an amalgamated free product of filtered Lie algebras within a fixed nilpotency class, based on an intricate induction rooted in the work of Maier and of Higman. In this paper, the authors give a new description of this amalgam, adopting the viewpoint and methods from the work of A. I. Shirshov on amalgamation of Lie algebras, substantially simplifying their previous approach. This new description yields both group-theoretic and descriptive set-theoretic applications. A $c$-nilpotent group is called UL-equivalent if its lower and upper central series coincide. We prove that every finite $c$-nilpotent group of prime exponent $p$ with $p>c$ embeds in a finite UL-equivalent $c$-nilpotent group of exponent $p$. This recovers a result due to Ivanov and Majcher that the Polish space of enumerated $c$-nilpotent groups of exponent $p>c$ has a comeager orbit. Our result also has the following consequences. First, the class of finite $c$-nilpotent groups of exponent $p>c$ has the cofinal amalgamation property (answering a question of Ivanov and Majcher, who showed that it has the weak amalgamation property). Second, the reduct of the Fraïssé limit of $c$-Lazard groups of exponent $p>c$ to the group language is generic in the space of enumerated groups. Finally, we prove analogues of these results also for torsion-free $c$-nilpotent groups.

For an $n\times n$ oscillatory matrix $A$, let $e_k(A)$ be the least positive integer $m$ for which every minor of order $k$ of $A^m$ is positive. We determine the profile $(e_1(A),\ldots,e_n(A))$. For every nonsingular totally nonnegative matrix, the column index sets of positive entries in each compound row form an interval in the componentwise order. The endpoint maps are order-preserving and compose under multiplication. This yields a two-corner criterion for positivity of all minors of a fixed order. Consequently, $e_k(A)$ is the larger of the first positivity times of two remote corner minors. We prove the sharp inequalities $e_k(A)\leq\max{k,n-k}$ for $1\leq k<n$ and $|e_{k+1}(A)-e_k(A)|\leq1$ for $1\leq k\leq n-2$. For $D=\operatorname{diag}(1,-1,1,-1,\ldots)$, the matrix $DA^{-1}D$ is oscillatory and satisfies $e_k(DA^{-1}D)=e_{n-k}(A)$ for $1\leq k<n$; the determinant exponent remains one. The ordered positions of the positive factors in an adjacent bidiagonal factorization determine the profile independently of their values. Each one-sided factor sequence can be represented by a single permutation, giving a finite characterization of all profiles and an integer unimodular realization of each. We also obtain a compatibility condition across minor orders: if $3\leq k\leq n-3$ and $e_k(A)\leq2$, then $e_j(A)\leq2+\lceil |j-k|/2\rceil$ for $2\leq j\leq n-2$. In particular, $e_k(A)\leq2$ implies $e_{k+2}(A)\leq3$ for $3\leq k\leq n-4$, and both the constant and this range are sharp.

The flatness of the under-actuated unmanned surface vessel with circular hull-shape (“the hovercraft") has long been known. We resolve the long-standing open problem: whether the generic under-actuated 6-degrees-of-freedom (dof) unmanned surface vessel (gUSV) admits a feedback linearization. Lie symmetries of the gUSV are used to construct a dynamic feedback linearization, explicit flat outputs and flat parametrization, for all hydrodynamic parameter values. In this generic case the dynamic compensator has order 6, in contrast to the hovercraft vessel whose dynamic compensator has order 4.

In this paper, we study thick subcategories of the (stable) module categories of finite-dimensional weakly symmetric algebras with radical cube zero. When the spectral radius of the Ext matrix is two, thick subcategories containing the algebra, equivalently thick subcategories of the stable category, are classified by Serre subcategories of a certain abelian category; when it is not two, only the trivial thick subcategories occur. As an application, this yields, to the best of our knowledge, the first examples of commutative Noetherian Gorenstein dominant local rings that are not complete intersections.

Is it possible to walk to infinity, avoiding three collinear points, using as steps only the standard unit basis vectors $(0,0,\ldots, 0, 1, 0, \ldots, 0)$? We prove this is possible in $16$ dimensions.

Elliptic homogenization is used to determine coarse-grained properties of materials with features on small scales. When these small scale features have rapid, periodic fluctuations, the solution field corresponding to a homogenized constitutive relation closely resembles the true solution based on the heterogeneous material. This homogenized behavior of the material is computed from a cell problem, where a cell is defined to be one period of the fluctuating material. In the context of linear elliptic partial differential equations, the homogenized constitutive relation is defined simply by a constant coefficient tensor, but for nonlinear problems, the homogenized response depends on the macroscopic state and/or its gradient, thus requiring solutions to parametric cell problems. When computing a numerical solution with the homogenized constitutive relation, it is useful to have a differentiable representation of the solution to the cell problem, as derivatives of the homogenized constitutive relation are required in Newton iterations for the macroscopic state field. In this work, we use the Deep Ritz method to solve the parametric cell problems that arise from nonlinear homogenization. First, we exploit the variational structure of the cell problem, then we discretize the dependence of the cell response on both space and the macroscopic state with a neural network. Enforcing boundary conditions on the cell response strongly, we next use the parametric Deep Ritz method to simultaneously solve the cell problem over a range of macroscopic states. We show that this method is accurate, efficient, and offers a continuous and differentiable representation of the cell response over the macroscopic state and gradient. We then show that our parametric representation of the cell response significantly expedites macroscale solutions when compared to a traditional $\text{FE}^2$ scheme.

In this paper, we present a Radon-Nikodym theorem for dominated fuzzy number multimeasures defined on $\Sigma$ and taking values in $\mathcal{F}_{cwk}(X)$, where $(\Omega, \Sigma, \mu)$ is a complete finite measure space and $\mathcal{F}_{cwk}(X)$ is the family of generalized fuzzy numbers on a Hausdorff locally convex topological vector space $X$, i.e., a map $u: X \to [0,1]$ is a member of $\mathcal{F}_{cwk}(X)$, if and only if for each $\alpha \in (0,1]$ the $\alpha$-cut $[u]^{\alpha}$ of $u$ is a member of the family $cwk(X)$ of all nonempty convex and weakly compact subsets of $X$, ($[u]^{\alpha} = \{x \in X: u(x) \geq \alpha \}$).

A question of Bourgain and Brezis, later restated as Open Problem 7.1 by Brezis, asks whether the $H^{1/2}+W^{1,1}$ norm of a mean-zero scalar function is controlled by the ambient $H^{-1/2}+L^1$ sum norm of its gradient. We show that this estimate fails in every dimension $N\geq 2$. More precisely, for every integer $K\geq 16$, we construct a smooth real-valued mean-zero function $\Phi_K$ on $\mathbb{T}^N$ such that $|\nabla\Phi_K|{H^{-1/2}(\mathbb{T}^N)+L^1(\mathbb{T}^N)}\leq C$ and $|\Phi_K|{H^{1/2}(\mathbb{T}^N)+W^{1,1}(\mathbb{T}^N)}\geq cK^{1/4}-C$, where the absolute constants $c,C>0$ are independent of $N$ and $K$. Equivalently, the ratio of the gradient-constrained decomposition cost to the ambient decomposition cost is unbounded. The construction applies the Hodge projection to a smooth approximation of a normal vector measure on a codimension-one sheet. Its potential records lacunary partial sums at geometrically separated distances from the sheet. A scale-path lower bound, combined with an odd critical Hardy inequality, excludes uniformly bounded $H^{1/2}+W^{1,1}$ decompositions.

Let $S_{26}$ denote the supremum of the sum of the radii of 26 pairwise interior-disjoint closed disks of positive radius contained in the unit square. We prove $2.635983084917607783\ldots \le S_{26}\le \sqrt{\frac{2577}{2500}+\frac{11}{\sqrt3}}=2.716919756099889566\ldots$. The lower bound is accompanied by an exact rational feasibility certificate. For the upper bound, we assign the disks their clipped power cells, convert cell area into a reciprocal exterior-angle cost, and combine this local inequality with Euler side counts. The extremal case for the resulting relaxation has four boundary transitions and a four-cell boundary cycle; its continuous angle allocation is reduced to a concave one-variable problem. All finite numerical inequalities are rigorously enclosed by a reproducible certificate. The argument does not determine $S_{26}$ exactly, but gives a nontrivial global upper bound for a benchmark whose best-known packing is asymmetric.

In this paper we state, prove, and discuss some new boundedness results concerning sequences of linear functionals of functions from the Lipschitz class \(Lip_1\) and general orthonormal systems. It is also proved that these results are, in a sense, the best possible.

Li, Ponomarenko and Zeman state their belief that every polyhedral graph (i.e. every $3$-connected planar graph) is Schurian: that its coherent closure $WL(X)$ coincides with the orbital configuration of $Aut(X)$. Combined with their Theorem 1.1, this would imply that every polyhedral graph has Weisfeiler-Leman dimension at most $2$. We report a computational verification of the Schurian property for all polyhedral graphs with at most $16$ vertices, a total of 413,024,847,068 graphs, none of which is non-Schurian. The computation combines the generator plantri with a one-sided $1$-WL discreteness filter and, for every graph surviving that filter, a full comparison of the stable $2$-WL partition of $\Omega^2$ with the orbit partition of $Aut(X)$. The computation is guarded by the published anchor counts, a seeded re-enumeration of the two largest stages, independent nauty cross-checks and control graphs with mathematically fixed verdicts; code, checkpoints and a verification script accompany the note as ancillary files. For $n\le 13$ the result reproduces what already follows from the published enumeration of coherent configurations, by an entirely different route; from $n=14$ on, deriving it from that side would require a step we could not find in the literature, and at $n=16$ the all-graphs statement fails outright - the Shrikhande graph is non-Schurian - so there the polyhedral restriction carries the result. The statement is established by exhaustive computation; with respect to the general conjecture it is instance evidence, not a proof. A companion note treats the torus side, where non-Schurian triangulation graphs do occur at $n = 16$.

On \[ S^5\times S^2,\qquad S^5\times S^3,\qquad S^5\times S^4,\qquad S^7\times S^2, \] we construct infinitely many pairwise non-isometric real-analytic Einstein metrics. After discarding finitely many members, none is isometric or homothetic to a Riemannian product or to a classical two-summand Böhm metric. The metrics have the form \[ dt^2+a(t)^2g_{S^p}+b(t)^2g_{S^p}+c(t)^2g_{S^q}, \qquad p,q\ge2,\qquad 2p+q\le8. \] The construction starts with solutions for which the two non-collapsing radii at one singular orbit are very small. After magnifying this region, the Einstein equations approach a Ricci-flat problem whose solutions tend to a cone by a decaying oscillation. The main point is to determine exactly which part of this oscillation can still be seen by the closing conditions at the other end of the manifold. Rather than following the full, increasingly oscillatory shooting map, we extract two response quantities from the Ricci-flat limit. One determines the common oscillatory phase; after that phase has been removed, the second closing condition is reduced to a scalar sign change. A uniform inner–outer matching argument transfers this limiting sign change to the exact Einstein equations. As the initial scale tends to zero, the phase makes arbitrarily many turns, and the closing conditions are therefore met infinitely many times.

In this paper, we study rooted spanning forests in the wheel graph $W_{N+1}$. For a root set $R$ consisting of rim vertices, we give an explicit formula, in terms of Fibonacci and Lucas numbers, for the number of rooted spanning forests in which each connected component contains exactly one vertex of $R$. When the central vertex is also included in the root set, we obtain a simple product formula involving only Fibonacci numbers. Furthermore, by using the correspondence between rooted spanning forests and vertex identification, we derive explicit formulas for the number of spanning trees of quotient graphs of wheel graphs obtained by identifying several vertices into one vertex. We then rewrite the obtained formulas combinatorially in terms of the numbers of matchings in path graphs and cycle graphs. In addition, for a fixed number $r$ of rim root vertices, we express the sum of the numbers of rooted spanning forests over all rim root sets of size $r$ as coefficients of generating functions, and we solve the extremal problem for the arrangement of roots.

The purpose of this paper is to rigorously quantify the statistical and learning-theoretic properties of adversarial training models for classification. Equivalently, we establish the statistical properties of empirical optimal partial transport. Precisely, first we provide two types of central limit theorems (CLT): CLT centered at the expected empirical value, and CLT centered at the population one with smoothing. These results are based on the uniqueness of optimal potential for various equivalent optimal transport formulations, and the empirical process theory argument. For the binary setting, we indeed prove the uniqueness of optimal potential by leveraging the connection between optimal partial transport and the derived multi-marginal optimal transport formula. As byproducts, we also obtain the stability of a saddle point of the adversarial training model, and the sample complexity and concentration probability of the generalization error.

We consider a fully connected gossip network of $n$ nodes that track a binary continuous-time Markov source through a strategic sender transmitting updates under a communication budget at a rate that depends on the source state. The receivers exchange packets through gossip and decide whether to follow the sender. We model this interaction as a Stackelberg game and analyze it through a stochastic hybrid systems (SHS) framework. We prove that the sender's budget constraint binds at every interior optimum, reducing its problem to a one-dimensional search on the budget line. When the sender pushes its preferred state at the higher rate, the receivers gossip at the highest available rate. Gossip has no direction of its own and works against the asymmetry in the sender's policy rather than reinforcing it. We prove that an optimistic Stackelberg equilibrium exists, and that it is unique and explicitly characterized whenever a policy on the strategic half of that line is feasible at the gossip cap. Monte Carlo simulations agree with the analytical recursion.

In this paper, we utilize the reflexive hull to unify the concepts of local and almost inner derivations. By investigating the algebraic structure of reflexive hulls, we prove the conjecture on the structure of local derivations proposed by Ayupov, Elduque, and Kudaybergenov (J. Pure Appl. Algebra, 2023), as well as the conjecture concerning the structure of almost inner derivations proposed by Burde, Dekimpe, and Verbeke (J. Algebra Appl., 2018).

It is known that any convex cone in a general vector space is the positive cone of a preorder compatible with the vector space, and that any convex set is an affine slice of a convex cone. We therefore treat convex sets order-theoretically, as affine slices of positive cones. We find that faces are sets of strong minima, and we generalize functionals to oriented maps valued in arbitrary ordered vector spaces. Oriented maps expose every face, and an oriented affine map $f$ separates $C$ from $D$ when $f(D) \le 0 \le f(C)$ element-wise. This generalization of hyperplane separation recovers mathematical objects that are "local", including the generated face at a point and the normal cone. The kernels of separating oriented maps form a lattice of separating affine flats whose bottom element is the joint supporting subspace (JSS), recently introduced in finite dimensions for qualification-free convex analysis results. We define the JSS in general vector spaces for two convex sets $C, D$ as the affine slice of the cross-lineality $\mathrm{lin}(K-G)$, where $K,G$ are the homogenization cones of $C, D$. We show that in finite dimensions, bilateral facial reduction corresponds to a descent in the lattice of separating affine flats, which we formalize as lexicographic products of oriented maps. The lexicographic characterization of faces follows as a corollary. By making no assumption other than convexity, our approach yields general results: we show that the face of a difference of two convex sets is a difference of faces, dropping assumptions of compactness, finite dimensionality, and exposedness. We also show that any face of an intersection of two convex sets is an intersection of faces, resolving open problem 5.10 by Weis in "A note on faces of convex sets", dropping the assumption of non-empty intrinsic core.

Families of sequences with low mutual aperiodic crosscorrelation assist the design of systems for multi-user asynchronous communications and multiple-input multiple-output radar. The crosscorrelation demerit factor of a pair of sequences is the sum of the squared magnitudes of their crosscorrelation values at every shift when the sequences are normalized to unit Euclidean norm, and the merit factor is the reciprocal of the demerit factor. For each positive integer $\ell$, we endow the $2^{2 \ell}$ pairs of binary sequences of length $\ell$ with uniform probability measure and study the distribution of their crosscorrelation demerit factors. Sarwate showed that the mean value is always $1$ regardless of length $\ell$. We develop a method for finding an exact formula for the $p$th central moment (for any positive integer $p$) as a function of $\ell$. Formulae for the variance and third central moment ($p=2$ and $3$) are then obtained by hand calculations, while the fourth through sixth central moments are obtained by computer-assisted calculations. Our theory also shows that all the central moments must be strictly positive for $p\geq 2$ and $\ell \geq 3$.

We study discrete-time $N$-dimensional mean-field systems on the torus subject to additive noise whose probability density is bounded away from zero. Given a Lipschitz one-particle map and interaction term, we prove that, if the lower bound on the noise density is sufficiently large, the $N$-dimensional transfer operator $\mathcal P_N$ preserves a dimension-independent class of sub-Gaussian probability measures. Moreover, $\mathcal P_N$ is a one-step contraction in the Dobrushin-Wasserstein distance and therefore admits a unique invariant measure $\rho_N$, which is itself sub-Gaussian. Under the same condition on the noise strength, we show that the associated self-consistent transfer operator is a one-step contraction in Wasserstein distance and hence admits a unique fixed point $\rho$. We further prove, using these contraction estimates, that $\mathcal P_N$ preserves an $O(N^{-1/2})$ neighbourhood of the product measure $\rho^{\otimes N}$ in the Dobrushin-Wasserstein metric, and in particular that $\rho_N$ belongs to this neighbourhood. Finally, we establish uniform-in-time propagation of chaos in the Dobrushin-Wasserstein metric and, under the additional assumption that the noise density is of bounded variation, in total variation for every fixed-dimensional marginal.

We prove the Second Main Theorem with truncation level one for holomorphic maps with Zariski-dense image intersecting a reduced effective Cartier divisor: Let $A$ be a semi-abelian variety with an equivariant compactification $A\subset\overline A$, and let $f:\mathbb C^p\to A$ be a holomorphic map with Zariski-dense image. Let $D$ be a reduced effective divisor on $A$ extending to a divisor $\overline D$ on $\overline A$. After possibly replacing $\overline A$ by another equivariant compactification depending only on $D$ and independent of $f$, for every $\varepsilon>0$, \[T_f(r,\overline D)\leq_{\mathrm{exc}}N_f^{[1]}(r,D)+\varepsilon T_f(r,\overline D).\]

We give a necessary and sufficient condition for the existence of power-one sequential tests in an i.i.d. composite testing problem. A level-\(\alpha\) test with power one against every alternative exists if and only if the alternatives are separated from the null by a countable family of finite-block events. We provide other equivalent conditions using randomized fixed-sample tests, bounded finite-block scores, e-processes, reduced-filtration test supermartingales, and a countable cover whose finite-block weak-$*$ closed convex hulls are positively separated in total variation. As a bonus, the constructive proof yields tests have pointwise expected sample size \(O_Q(\log(1/\alpha))\). Exactly the same conditions also characterize i.i.d.\ change detectability under optional-horizon average-run-length control: for every \(\eta>0\), they are equivalent to an alarm family \((T_\gamma)_{\gamma\ge1}\) satisfying \(\Prob_{P^\infty}(T_\gamma\le\sigma)\le \E_{P^\infty}\sigma/\gamma\) for every null law and every stopping time \(\sigma\). In fact, when these conditions hold, we can construct a single e-detector such that every null-law average run length lies between \(\gamma\) and \((1+\eta)\gamma+1\), and having robust Lorden delay \(O_Q(\log\gamma)\).

By a reflection group we mean an abstract group generated by nonidentity involutions. Knill proved that every reflection group of cardinality at most the continuum is natural, in the sense that some metric determines its group structure up to isomorphism among group structures whose right translations are isometries, and asked whether the cardinality hypothesis can be removed. We answer this question affirmatively. More precisely, every reflection group G admits a right invariant metric taking at most seven values for which the full isometry group is the right-regular action of G. The proof replaces distinct numerical labels on generators by a finite-valued encoding of a rigid graph on a well-ordered irredundant generating set.

We study the mixing time of the systematic scan dynamics for high-dimensional discrete distributions. This Markov chain updates coordinates sequentially according to a fixed predetermined order, in contrast to the Glauber dynamics that updates coordinates selected uniformly at random. The systematic scan is often favored in practice because it exhibits strong empirical performance, but its theoretical analysis remains far less developed than that of Glauber dynamics. We take a step toward addressing this imbalance by showing that two standard functional notions of weak dependence between the coordinates of the distribution provide strong convergence guarantees for the systematic scan dynamics. First, we show that approximate tensorization of entropy implies optimal $O(\log n)$ mixing time for every scan order under standard marginal, connectivity, and bounded interaction degree assumptions about the distribution. Second, we show that approximate tensorization of variance yields a constant-factor contraction of the variance functional per scan, which in turn implies an optimal $O(1)$ relaxation time for the natural additive and multiplicative reversibilizations of the systematic scan dynamics. Compared with our entropy result, the variance bound improves the dependence on the maximum interaction degree from exponential to quadratic and requires weaker assumptions on the distribution. As concrete applications of our results, we establish optimal $O(\log n)$ mixing of the systematic scan dynamics for bounded-degree antiferromagnetic two-spin systems in the tree-uniqueness region and for the ferromagnetic $q$-state Potts model on square boxes in $\mathbb{Z}^2$ throughout its subcritical regime.