Papers

We consider a mixture of at most $k$ unit-covariance Gaussians in $\mathbb{R}^d$ whose means belong to a fixed-radius ball, with no separation or minimum-weight condition. Doss, Wu, Yang and Zhou (2023) proved that the minimax Hellinger risk is of order $\sqrt{d/n}\wedge 1$ and constructed a proper polynomial-time estimator with the slower general bound $(d/n)^{1/4}$; obtaining the sharp rate in polynomial time for fixed $k\geq 3$ was left open. We resolve this question. The key device is a moment-fiber range finder. A second-moment subspace controls the energy missed by projection. We then estimate finitely many one-free-index Hermite contractions. These vector-valued contractions recover every tensor component containing exactly one missed direction at the sharp $\sqrt{d/n}$ scale. Every remaining term contains at least two missed factors and is therefore controlled by the residual second-moment energy. The resulting subspace has dimension depending only on $k$. Exhaustive moment fitting in this constant-dimensional space produces a proper mixture and, together with the dimension-free moment characterization of Gaussian mixtures, achieves the optimal Hellinger rate in polynomial arithmetic time for every fixed $k$.

Motivated by sequential space-filling designs for computer experiments, we study algebraic constructions of extensible point sets in the $d$-dimensional unit cube whose two-dimensional coordinate projections are all quasi-uniform. Our two constructions share a common parametrization in terms of finite configurations of distinct rational directions on the projective line $\mathbb{P}^1(\mathbb{Q})$. First, using a cubic number field, we construct explicit Kronecker sequences for which the mesh ratios of all two-dimensional coordinate projections remain uniformly bounded over every initial segment of length $N\ge 2$. Second, using a real quadratic field, a split prime, and a compatible $p$-adic embedding, we construct nested rank-1 lattice designs with the same uniform projection property at every nesting level. The proofs combine algebraic norm estimates with transference principles between simultaneous and dual Diophantine approximation, yielding lower bounds for the separation radii and upper bounds for the covering radii, both of optimal order in the number of points, uniformly over all coordinate pairs. We also investigate how the choice of rational projective coefficients affects the resulting mesh ratios. This leads to a minimax problem for finite configurations on $\mathbb{P}^1(\mathbb{Q})$, in which one seeks to minimize the maximum mesh ratio over all two-dimensional coordinate projections. These constructions provide extensible point sets with uniformly controlled bivariate geometry.

Day-ahead residential tariffs must be posted before demand is observed. Aggressive high prices can cut peaks in simulation, but they can also raise customer bills. This paper measures how much peak reduction is lost when a tariff design problem is required to keep revenue near a flat-tariff baseline and to limit bill increases. Using half-hourly data from 5,566 Low Carbon London households (167.8 million validated readings), we estimate quasi-experimental price response against the standard-tariff comparison group, form bootstrap demand scenarios, and solve stochastic mixed-integer programs with HiGHS. On all 73 eligible held-out test days, a segment-protected stochastic tariff reduces simulated peak demand by 2.29% on average (95% day-bootstrap CI [2.11, 2.48]), with revenue change -2.26% and mean worst-segment bill increase 2.48%. Removing the segment bill cap raises peak reduction only to 2.37%, while the worst segment bill increase rises to 6.81%: in this sample, substantial average protection costs little peak-shaving performance. The same schedules leave a household 95th-percentile bill increase of 8.61% (CVaR95 14.76%), so segment-average caps do not bound household tails. We report the full price-of-protection frontier under consistent household simulation and compare segment protection with a representative-household (tail-aware) variant: the latter cuts household p95 from 8.61% to 7.05% while changing mean peak reduction only from 2.29% to 2.28%. All optimized outcomes are model-based counterfactuals under an opt-in trial; wholesale costs are unavailable, so we do not optimize profit.

We study volume and projection inequalities for zonoids through the multiaffine determinant polynomials that encode their volumes. We show that a log-submodularity conjecture for the volume of zonoids is equivalent to the Rayleigh property of zonotope volume polynomials, which we prove for the case when the degree or codegree is at most 3. This result is sharp in that when the degree and codegree are at least 4, we construct counterexamples using the existence of non-Rayleigh matroids in ranks at least four. Additionally, we provide unimodular or graphical counterexamples in dimensions four and higher to an equivalent projection inequality formulation of the conjecture. We also show that a stronger projection inequality fails already in dimension three. We next disprove Courtade's conjecture using a pair of orthogonal double bodies of revolution. Although Courtade's conjecture was originally formulated for general convex bodies, we show that it fails even for zonoids in every dimension at least three.

Let $P$ be a finite nonempty poset with $n$ elements, let $f:P\to\{1,\ldots,n\}$ be a uniformly random order-preserving bijection, and put $h_P(x)=\mathbb{E}[f(x)]$. Aires and Kahn (2025) introduced $\operatorname{gap}(P)$ as the largest difference between consecutive values in the ordered list consisting of $0$, $n+1$, and all the expected ranks $h_P(x)$. Write ${w}(P)$ for the largest size of a pairwise incomparable subset. We prove three results. First, we prove a weighted strengthening of an ideal inequality conjectured by Kahn and obtain the explicit gap-width bound $\operatorname{gap}(P)\le 2 {w}(P)-1$. Second, for every $L>0$ we construct a width-two poset such that the expected-rank list of every maximal chain has a gap of at least $L$, with $0$ and $|P|+1$ added as endpoints. Finally, for every $r\in\mathbb{N}$, we construct a poset $P_r$ for which the relative order induced on every nonempty selected set $X$ has base-two entropy below $3|X|$, while $\operatorname{gap}(P_r)\ge(3/2)^r$. Thus the gap can be arbitrarily large while the induced order on every selected set has relatively small entropy. The key ideas behind all three results were found by ChatGPT 5.6 Sol.

For $1\le s<t$ and any graph $G$, the weakened Gallai-Ramsey number $gr^t_s(G)$ is defined to be the least $p\in \mathbb{N}$ such that every Gallai $t$-coloring of the edges of $K_p$ (i.e., a $t$-coloring that lacks rainbow triangles) contains a subgraph isomorphic to $G$ whose edges use at most $s$ of the colors. In the case of a book graph $B_n:=K_2+nK_1$, Jakhar and Moun determined the values $gr^3_2(B_3)=6$ and $gr^3_2(B_4)=7$. In this paper, we extend their results to $t>3$ colors, and we determine the values of $gr^3_2(B_n)$ for $5\le n\le 15$. General lower bounds for $gr^t_2(B_n)$ are also given.

We investigate the long-time behavior of small solutions to the Dirac-Hartree equation in two spatial dimensions. This model describes the mean-field dynamics of relativistic fermions interacting through the three-dimensional Coulomb potential $|x|^{-1}$, which gives rise to long-range effects in the scattering dynamics. We prove global well-posedness and long-range scattering (modified scattering) for small initial data in weighted Sobolev spaces. In this setting, long-range scattering means that, unlike linear scattering, an additional logarithmic phase correction is required to describe the precise asymptotics of nonlinear solutions. Our approach relies on the space-time resonance method, combined with special null structures inherent in the equation. Compared to the three-dimensional case \cite{CKLY2022,cloos}, the novelty lies in overcoming the weaker time decay inherent to the two dimensional problem.

We introduce the generalized weighted core inverse with respect to two elements in a Banach *-algebra, extending the framework of existing generalized inverses. This new generalized inverse is characterized via the generalized weighted core decomposition, and its representations are established using the weighted g-Drazin inverse. We investigate the polar-like and principal ideal properties to reveal its structural behavior. Consequently, applications to complex matrices are derived. This finds encompasses both the core-EP and generalized core inverses, significantly broadening the theoretical framework.

In even characteristic, the first subconstituent of an orthogonal graph at a singular point is shown to be the complement of an affine polar graph. This identification yields the full automorphism group as an affine semisimilarity group and provides explicit extensions to the ambient graph. Maximum independent sets are affine cosets of maximal totally singular subspaces, giving the independence and chromatic numbers. In plus type, Delsarte cliques correspond bijectively to ovoids through the base point, leading to a sharp rank threshold. Over $\mathbb{F}_2$, we determine the exact maximum clique number in every dimension, prove that all maximum cliques form a single orbit, and obtain the core of every first subconstituent. A uniform Witt-extension argument gives a short proof of triple transitivity for the binary local graphs.

Khaleghi and Lugosi asked whether the sum of the $\phi$-mixing (uniform-mixing) coefficients of a real-valued discrete-time stationary ergodic process can be consistently estimated from a single sample path. We construct a deterministic sequence of Borel statistics that, for every such process, converges almost surely to the extended sum $\sum_{m\geq1}\phi(m)$, including divergence to $+\infty$ when the sum is infinite. The estimator combines finite dyadic cylinders with a vanishing empirical cutoff on conditioning-event frequencies. When the target is finite, $\alpha(m)\leq\phi(m)$ supplies summable covariance control for the growing finite classes, and a stable-division estimate yields the upper bound. Fixed positive-probability witnesses and Birkhoff's theorem give the lower bound, including the infinite-target case. No mixing rate or known positive lower bound on conditioning-event probabilities is required.

Let $G_D(n;r)$ be the random geometric graph generated by $n$ independent uniform points on the $D$-dimensional torus, with adjacency defined by torus $L^\infty$-distance at most $r$. We study first-order zero-one and convergence laws at fixed radius and in sparse regimes. At fixed radius, we determine the asymptotics of the expected number of adjacent twin pairs in every dimension. In dimension two, more generally, for each origin-symmetric convex connection body $K\subset(-1/2,1/2)^2$, the twin count converges to a Poisson variable with mean $\operatorname{area}(K^\circ)/16$; hence the zero-one law fails for all fixed $0<r<1/2$ in the $L^\infty$ and Euclidean models. At each critical component threshold $n^k r_n^{D(k-1)}\to a\in(0,\infty)$, the numbers of components of the feasible connected $k$-vertex types converge jointly to independent Poisson variables, yielding the complete first-order convergence law. Between consecutive thresholds a zero-one law holds. For $D\ge3$, we also construct a definable common-neighborhood configuration of probability order $1/n$.

We consider the uplink of a fronthaul-constrained cell-free massive MIMO system with a single multi-antenna user equipment (UE) transmitting multiple spatial streams to distributed access points (APs). Due to heterogeneous channel conditions and limited fronthaul capacity, uniform-resolution quantization across APs and streams is highly inefficient. To address this, we propose a stream- and AP-adaptive resolution control framework that jointly optimizes the transmit power and per-AP per-stream quantization resolutions. By leveraging a distributed implementation of singular-value decomposition (SVD)-based processing, the data streams are decoupled at the APs, enabling flexible fronthaul compression across both spatial streams and APs. The resulting non-convex optimization problem is tackled using a weighted minimum mean-squared error (WMMSE)-based block coordinate descent algorithm with semi-closed-form updates.

We study quantization-aware precoding for the downlink of cell-free massive MIMO systems with limited-resolution fronthaul. In such systems, the centrally designed precoder must be quantized before being conveyed to distributed access points (APs), creating a strong coupling between precoder design and fronthaul compression. To capture this effect, we develop an end-to-end signal model based on Bussgang decomposition, where the quantization distortion depends explicitly on the precoder coefficients. Building on this model, we formulate a joint precoding and bit allocation problem under per-AP power constraints and a total fronthaul bit budget. We propose an efficient block coordinate descent algorithm that iteratively updates the precoder, receive scaling, and distortion levels. Numerical results demonstrate that the proposed method significantly outperforms uniform bit allocation and quantization-unaware precoding schemes, particularly in fronthaul-limited regimes.

A recent conjecture states that a finite-dimensional complex evolution algebra is solvable if and only if it has no non-zero idempotents. We exhibit a three-dimensional counterexample over $\mathbb{C}$ whose isomorphism class already appears in the classification of three-dimensional complex evolution algebras. Since the conjecture is known in dimensions one and two, this counterexample has the smallest possible dimension. The algebra is defined over every field of characteristic different from $2$ and is the exceptional member of a one-parameter family of pairwise non-isomorphic non-solvable evolution algebras whose idempotents are determined explicitly. For the exceptional parameter, the derived series stabilises at a non-zero two-dimensional subalgebra whose only idempotent is zero. Direct sums with zero algebras give complex counterexamples in every dimension at least three. We also prove the conjectured equivalence whenever the stable term of the derived series is an evolution algebra.

We study vector-valued binary affine refinement operators with finitely supported matrix masks and compactly supported continuous piecewise linear input and forcing data. We prove that every finite refinement iterate admits an exact ReLU realization of fixed width and depth linear in the number of iterations. No separation of the forcing profile from the binary cell seams is required. The main mechanism is universal reflection doubling. Pairing each residual profile with its reflection replaces the two binary transition matrices by one fixed block matrix together with a fixed swap involution. The cell-seam identity makes the two branch candidates agree at the tent fold, while their swap-odd component is bounded linearly by the distance to the fold. This permits exact branch selection by a fixed continuous piecewise linear cone switch, without multiplication by a variable selector. The resulting primal recursion requires the residual orbit in reverse order. We obtain exact backward replay from the residual memory controller developed previously for affine refinement, interpreted here through the reflection quotient of circle doubling. The construction propagates the full vectorized profiles rather than decomposing the input and forcing into reference atoms. We also treat stage-dependent forcing from a fixed finite-dimensional family and show that genuine reflection equivariance reduces the doubled cascade to a single parity sector.

Large neuromorphic systems contain many processing tiles that may replicate a shared control parameter such as a threshold reference. If these copies diverge, identical inputs may be processed under different intended settings. We study how tiles can reduce this disagreement when communication carries only labeled spike times and up to \(f\) sender labels may be Byzantine. A raw event stream cannot supply the one-value-per-sender input required by classical approximate agreement because a faulty sender can remain silent, flood a receiver, or report different times to different receivers. We introduce the Spike-time Interface for Faults, or \SIF, which combines paced epochs, sender attribution, per-label \FirstSpike admission, bounded timing error, and a silence sentinel. For an affine one-spike code, midpoint decoding attains the exact deterministic minimax error \(\rho=\min\{1/2,\omega/L\}\), where \(\omega\) is the residual timing uncertainty and \(L\) is the usable encoding window. \SpikeTrim applies the classical mean-subsequence-reduced (MSR) rule to the sender-indexed decoded values. For \(n\ge3f+1\), it guarantees one-step robust validity, the tight noiseless contraction factor \(f/(n-2f)\) under direct updates, an explicit worst-case asymptotic disagreement bound, and finite recovery after transient agreement-state corruption. A closed-form test determines whether a validated timing budget meets a target disagreement. Simulations illustrate the fault threshold, timing dependence, flooding resistance, and recovery. A controlled spiking classifier experiment shows an association between faster control-state alignment and lower prediction disagreement under a finite maintenance budget.

In this paper, we investigate the semilinear parabolic equation $\partial_t f-\Delta f=a(x,t)f^p$ on a complete Riemannian manifold with Ricci curvature bounded from below. By means of Nash-Moser iteration, we establish Li-Yau type gradient estimates for positive solutions to this equation, where the coefficient function $a$ can be either strictly sign-definite or sign-changing. As an application, we derive Liouville theorems for ancient and eternal solutions on manifolds with nonnegative Ricci curvature, generalizing a number of classical results. Our proof features the incorporation and tuning of two parameters in the auxiliary quantities to accommodate the variable coefficient $a$ and to extend the admissible range of $p$.

We compare several "pathological" notions of continuity that make every additive function continuous. Our goal is to determine which of these notions is best behaved and deserves to be called Cauchy-Hamel continuity. The conclusion is that a bilateral version of a previously introduced notion of Q-continuity seems the most promising candidate. Our work is relevant to the axiomatic foundations of mechanics, specifically to the problem of axiomatically characterizing the parallelogram rule for the composition of forces. It was noted by Darboux that a continuity assumption is needed for such a characterization of the parallelogram rule and that, absent such a continuity axiom, "exotic" physical models based on alternative composition rules may exist. An intriguing question is whether such exotic physical models still possess some weak, residual notion of continuity. The concepts studied in this paper offer a first hint of a response to this question.

The Casas–Alvero conjecture asserts that a monic polynomial $f$ of degree $d$ over a field of characteristic zero sharing a non-constant factor with each of $f',\dots,f^{(d-1)}$ is the $d$-th power of a linear polynomial. Two reductions are carried out. Once a root is translated to the origin, the condition at index $i$ is vacuous unless the coefficient $a_{d-i}$ is nonzero, so the Casas–Alvero locus stratifies by the support of the centred normal form; prescribing in addition which root realises each surviving condition makes the system unit lower triangular and eliminates the coefficients of $f$ entirely. When one root suffices, what remains is a single integer, and that integer is MacMahon's determinant: the number of permutations of $\{1,\dots,d\}$ with descent set the support of $f$. Since every subset is a descent set, no counterexample has fewer than three recycled roots; in characteristic $p$ a prime dividing one of these counts but none of the accompanying integers $z_{i}$ is a bad prime, the largest count in degree $d$ being the Euler zigzag number $A_{d}$, so that the irregular prime $691$ is bad for $d=11$. The stratification also yields short proofs of the two characteristic-$p$ propositions on which all known cases rest; supports of two elements are settled separately in every characteristic, whence a counterexample has at least four terms in centred normal form and a further Gröbner-free criterion for bad primes.

Random spatial tessellations generated by point processes provide fundamental models for proximity, space partitioning, and local geometry in stochastic systems. Poisson–Voronoi and Poisson–Delaunay tessellations induced by homogeneous Poisson point processes form a canonical dual pair used in stochastic geometry, computational geometry, spatial statistics, and wireless-network analysis. Their typical-cell volume distributions provide important geometric inputs for modeling coverage, traffic load, clustering, connectivity, and other system characteristics. Despite extensive study, the literature remains analytically asymmetric. For Poisson–Voronoi cell volumes, exact integral representations exist in certain planar settings, while a recent scale–shape factorization provides an exact general-dimensional representation with conditional Gamma structure. However, the normalized shape laws and unbounded facet-count mixture remain implicit, and tractable unconditional closed-form distributions are unavailable. Practical modeling therefore relies largely on simulation, moment characterizations, and empirical approximations. By contrast, Poisson–Delaunay simplex volumes admit dimension-explicit PDFs, CDFs, and moment formulas derived through Mellin-transform analysis and Meijer's \(G\)-function representations. Motivated by this contrast, this paper surveys typical-cell volume distributions in Poisson–Voronoi and Poisson–Delaunay tessellations. We review the main analytical methods, synthesize exact and approximate results, summarize emerging high-dimensional limits, and discuss wireless-network applications, including load modeling, cooperative transmission, and three-dimensional architectures. We also identify open problems concerning unconditional Poisson–Voronoi distributions, non-Poisson spatial models, data-driven geometric inference, and dimension-aware network modeling.

Parameter estimation is a fundamental challenge in the calibration of ordinary differential equation (ODE) models, where repeated numerical integration can lead to high computational cost. In this work, we investigate whether quantum algorithms can be leveraged to assist parameter estimation in nonlinear dynamical systems. We develop a hybrid classical-quantum framework that reformulates a data-assimilation-augmented parameter estimation problem as a combinatorial optimization task. Model dynamics and data assimilation are enforced entirely on the classical side, while the resulting parameter estimation cost functional is discretized and approximated by a quadratic unconstrained binary optimization (QUBO) surrogate. This surrogate is mapped to an Ising Hamiltonian, and quantum optimizers are used to search for low-energy configurations corresponding to candidate parameter estimates. We apply the framework to SIS and SIR epidemic models, the chaotic Lorenz-63 system, and a high-dimensional two-layer Lorenz-96 system. In this setting, the method is used to recover classical system parameters from partial state observations across steady-state, chaotic, and high-dimensional multiscale dynamical systems. Numerical experiments with synthetic data show that the proposed approach accurately recovers parameters while requiring data-assimilation solves only on a prescribed coarse grid. The framework avoids quantum state tomography, illustrating a viable pathway for integrating quantum optimization into data-driven parameter estimation for nonlinear dynamical systems.

Suppressive interference, high-dynamic motion of the receiving platform, and carrier-phase RTK lead satellite navigation to one fundamental question: when the receiver actively participates in observation formation through its array geometry, channel states, weights, and signal-processing rules, how can the resulting carrier observation continue to represent the same objective propagation process while preserving continuous phase and the integer-ambiguity structure? The physical world is not changed by a reconfiguration of the observer, but its complex representation in the observation world is. Without a strict separation between propagation-induced and observer-induced variations, anti-jamming, high dynamics, and precise carrier-phase positioning become structurally coupled at the observation-formation level. Finally, reproducible and falsifiable protocols are provided for static interference, dynamic rotation, inter-branch timing mismatch, near-null response, RF replay, and RTK integer-risk validation. The scoped novelty of SN-ASMO is the unification of observation formation, reference covariance, exact nuisance quotients, same-source transport, intrinsic statistical information, and preservation of the RTK integer structure in one mathematically closed and physically testable framework.

This work proposes a semantic environment for arithmetic geometry in which passage between presentations carries an explicit weight. Morphism costs induce a Lawvere generalized distance by taking infima over transports, and heights satisfy the corresponding transport bounds. A labeled quantitative $(2,1)$-category retains invertible $2$-morphisms as coherent comparisons between transports. Its vertical groupoids admit a transport interpretation inspired by the identity types of Voevodsky's C-systems. If a height factors through the underlying object, then a vertical comparison has equal endpoints, whatever its transport cost. We formulate an intensional Szpiro inequality for elliptic packages, where the discriminant height transports along a morphism with a controlled model defect, and we relate a bound of this form to the $abc$ conjecture. The defect remains visible between non-minimal and normalized presentations, while conductor complexity stays attached to the underlying curve. We then treat the Birch and Swinnerton-Dyer conjecture in its pre-modularity form, using strong point-count asymptotics with an explicit convergence requirement. The refined constant retains separately typed arithmetic factors. A finite differential correction makes the period independent of the chosen differential and links its normalization to model transport. Within the declared finite-descent theory, Tate–Shafarevich finiteness reduces to finitely many closed prime columns of Selmer shadows and a uniform cutoff. For $y^2=x^3-q^2x$ with $q\equiv3\pmod8$ prime, two-isogeny descent closes the two-primary column uniformly. The statements admit expression in $\mathsf{ACA}_0$ relative to certified descent and Mordell–Weil data, with certified Cauchy names for the period and regulator.

For every $g>0$, we construct a closed Haken 3-manifold $M_g$, whose only connected, incompressible surface is a non-separating surface of genus $g$ and discuss the torsion behavior of the Kauffman bracket skein module $\S(M_g,\Z[A^{\pm 1}])$. We also use properties of shared surgeries between knots to estimate (and often compute) the dimension of the Kauffman bracket skein module over $\Q(A)$, for families of hyperbolic 3-manifolds obtained by surgery on double twist knots. We have similar computations for some surgeries on several knot complements in the SnapPy census of cusped hyperbolic manifolds with triangulations up to 9 tetrahedra.

For solving nonconvex equality-constrained optimization problems, a recent Gradient-Eigenstep Algorithm by Goyens et al.~is an iteration-efficient approach, based on minimizing Fletcher's augmented Lagrangian function, for finding an approximate second-order stationary point from an arbitrary starting point. In this paper, the analysis of this algorithm is extended, offering a two-fold contribution. First, it is shown that a local-linear rate of convergence can be obtained by this method if it is initiated sufficiently close to a strong second-order stationary point and employs a sufficiently small step-size parameter and sufficiently large penalty parameter. In this case, the algorithm reduces to a gradient descent algorithm applied to minimize Fletcher's augmented Lagrangian. Second, as a particularly useful application of the first result, it is shown that the Gradient-Eigenstep algorithm can be used as an iteration-efficient subproblem solver in the context of a progressive sampling strategy for solving equality-constrained optimization problems when the objective and constraint functions are defined by large sample averages, ultimately offering an algorithm with an improved worst-case sample complexity when compared to an approach that solves a full-sample problem directly.

We prove that, for every constant $\rho>1$, the Euclidean shortest vector problem is NP-hard to approximate within any constant factor $\rho$ under a deterministic polynomial-time many-one reduction. This extends our previous deterministic NP-hardness result from $\rho<\sqrt 2$ to arbitrary constants and gives a deterministic version of Khot's randomized arbitrary-constant theorem.

We define two notions of Morita equivalence for graded C*-algebras (graded Morita equivalence and homogeneous Morita equivalence) and provide Brown-Green-Rieffel Stabilization Type Theorems for both notions of graded equivalence. We apply our results to finite regular graphs by establishing an explicit connection between graded C*-algebras and coactions. Lastly, we incorporate Cartan subalgebras with totally disconnected spectra and obtain Brown-Green-Rieffel Stabilization Type Theorems for these cases.

An e-detector for a pre-change class $\mathcal P$ is a nonnegative process $M$ such that $\mathbb E_P[M_\tau] \leq \mathbb E_P[\tau]$ for all stopping times $\tau$ and all $P \in \mathcal P$. Thresholding e-detectors controls the average run length (ARL): declaring a change at the first time $T_b$ when $M$ crosses $b$ ensures that $\inf_{P \in \mathcal P}\mathbb E_P[T] \geq b$. But e-detectors do substantially more than control the ARL; they also satisfy a optional-horizon inequality: \[ P(T_b\leq\sigma)\leq \mathbb E_P[\sigma]/b \] for every data-dependent stopping time (monitoring horizon) \(\sigma\) and $P\in \mathcal P$. In particular, every e-detector-based procedure obeys $P(T\leq t)\leq t/b$ at each fixed $t$, thus avoiding early false alarms. Remarkably, the converse also holds: every stopping time $T$ that satisfies the optional-horizon inequality must in fact arise from thresholding an e-detector. We also derive a universal representation of stopping times that satisfy (only) ARL control. These are represented by weak e-detectors, that only require $\mathbb E_P[M_\tau] \leq \mathbb E_P[\tau]$ to hold at all threshold stopping times $T_b$. Appendices present universal representations for other (less common) change detection metrics.

The famous 15-Theorem and 290-Theorem fully characterise universal quadratic forms over $\mathbb{Q}$. Similar theorems exist for every totally real number field, but only over $\mathbb{Q}(\sqrt5)$ the criterion set is explicitly known. We study criterion sets both theoretically and computationally: We develop the method of escalation over number fields, thus providing a simple proof of finiteness of the criteria and, more importantly, a practical tool for computing them. We illustrate this by explicit computations for $\mathbb{Q}(\sqrt2)$ and $\mathbb{Q}(\sqrt3)$, obtaining a conjecture about the corresponding universality criteria; we also prove universality of many escalator lattices. Moreover, we develop the somewhat different theory of escalations for diagonal quadratic forms, obtaining the diagonal criterion set for $\mathbb{Q}(\sqrt5)$ and conjecturally for $\mathbb{Q}(\sqrt2)$ and $\mathbb{Q}(\sqrt3)$.

We establish lower bounds for the Smith–Thom deficiency of the Hilbert square of a maximal nonsingular real projective variety. These bounds recover the known surface case and provide new obstructions to Smith-Thom maximality in every dimension at least two. As applications, we prove that the Hilbert square of any real abelian variety of dimension at least two is not Smith-Thom maximal. We further show that the deficiencies of the Hilbert squares of maximal real abelian varieties grow exponentially with the dimension. We also obtain nonmaximality results for the Hilbert squares of certain Cartesian products.