Papers

A classic problem in additive number theory asks whether a given subset of integers has more sums or differences. This inquiry naturally extends to finite groups, where one compares the sizes of the product set $AA$ and the quotient set $AA^{-1}$ for a subset $A \subseteq G$. A finite group is considered product-dominant if it contains a higher proportion of subsets where $|AA| > |AA^{-1}|$, and quotient-dominant if the reverse is true. We establish statistical criteria to determine the dominance behavior of large finite groups by analyzing the moments of the random variable $d(A) := |AA| - |AA^{-1}|$. By translating product and quotient relationships into graph-theoretic constructions, we compute these moments through counting independent sets. Our main result demonstrates that for a growing sequence of finite groups $G_k$ with center size $Z_k$ and involution count $T_k$, the condition $Z_k = o(T_k)$ implies the groups are eventually product-dominant. Conversely, if $T_k \cdot \sqrt{3}^{R_k/Z_k} = o(Z_k)$ (where $R_k$ denotes the number of elements whose square is central), the groups are eventually quotient-dominant. Using these asymptotic conditions, we resolve the Miller–Vissuet conjecture by proving that all dihedral groups $D_n$ for $n \geq 3$ are product-dominant. Furthermore, we classify the dominance behavior of several major families of groups, proving that symmetric groups and non-abelian finite simple groups are product-dominant, while groups of odd order and certain abelian groups are quotient-dominant. Finally, for groups that fall outside our asymptotic conditions, we derive an explicit formula for the first moment utilizing the irreducible characters and conjugacy classes of the group.

A bipartite observable is stable when its idealized responses are continuous. We show that it is NIP exactly when they are of the first Baire class. They then form a Rosenthal compactum, and the dividing lines of Shelah in model theory, together with the classification of Rosenthal compacta of Todorčević, acquire a physical meaning. For observables, mixtures of states make every hereditarily separable compactum of idealized responses metrizable, and the Todorčević trichotomy becomes a classification into three classes. Each class contains a physical order: the comparison of photon numbers, the comparison of positions, and the causal order of spacetime, whose idealized responses are not first countable because a light cone can shrink to a point; the comparison of two phases on a circle joins it there. Each of the seven minimal families of the heptachotomy, which Argyros, Dodos and Kanellopoulos extracted from the methods of Todorčević, is realized by a tree of preparations of photons, of a particle in a box, or of a spin chain read by a probe; the spin chain also shows that NIP, unlike its uniform version, is not symmetric between the two parties. Finally, for the Ising model in a transverse field we show that, in the ordered phase, the responses of the two pure phases form split pairs whose midpoints, the responses of the symmetric state, form an uncountable discrete set; the proof uses a Lee–Yang theorem for this model.

Spacecraft stationkeeping is subject to various operational uncertainties such as missed thrust events, maneuver execution errors, and dynamics modeling errors. Missed thrust events are the most disruptive of these, and current practices mitigate that risk through Monte Carlo-driven heuristics and conservative margins that offer no formal safety guarantees and do not scale to multiple missed thrust events. This paper models these unplanned thruster outages through the Markov Jump System framework, and represents maneuver execution errors and dynamics linearization errors as bounded multiplicative noise. The resulting closed-loop system is a linear time-varying periodic Markov jump system with multiplicative noise, for which Lyapunov-like mean square stability and periodic set invariance conditions are derived. A linear matrix inequality-based optimal controller synthesis framework then designs a periodic, mode-dependent mean-square stable feedback policy that respects path constraints with probability one. The framework is demonstrated on the Earth-Moon 9:2 near-rectilinear halo orbit (NRHO), under both impulsive and low-thrust propulsion, and on a Saturn-Enceladus $L_1$ NRHO using impulsive propulsion. Nonlinear Monte Carlo simulations show that the spacecraft recovers from an arbitrary number of missed thrust events while respecting path constraints, and that the multiplicative noise model introduces additional robustness to maneuver execution error and linearization error.

Let $F_5$ denote the $3$-uniform hypergraph with edge set $\{abc,abd,cde\}$. For a hypergraph property $\mathcal P$, let $ex_{\mathcal P}(n,F_5)$ be the maximum number of edges in an $n$-vertex $F_5$-free $3$-uniform hypergraph satisfying $\mathcal P$. Let $\Delta_2(H)$ denote the maximum codegree of a pair of vertices in a $3$-uniform hypergraph $H$. We determine the exact value of $ex_{\Delta_2\ge m}(n,F_5)$ when $m$ is bounded away from $n/3$ by a fixed positive proportion. More precisely, for every constant $c\in (1/3,1)$, there exists $n_0$ such that for all $n\ge n_0$ and $m=cn$, \[ ex_{\Delta_2\ge m}(n,F_5) = m\left\lceil\frac{n-m}{2}\right\rceil \left\lfloor\frac{n-m}{2}\right\rfloor. \] The unique extremal hypergraph is the complete tripartite $3$-graph $K^3_{m,\lceil (n-m)/2\rceil,\lfloor (n-m)/2\rfloor}$. We also prove the corresponding exact result for the blow-ups $F_5^t$, together with stability theorems showing that every near-extremal configuration is close to the relevant tripartite construction.

For Gaussian observations, a deterministic multiple of Hotelling's $T^2$ statistic has an exact $F$ distribution. This paper investigates when this approximation remains valid for an independent and identically distributed (i.i.d.) data matrix $\mathbf{X}\in \mathbb{R}^{p\times n}$ with a fixed, possibly heavy-tailed entry law $\xi$. Under symmetry and $\limsup_{n\rightarrow \infty} p/n<1$, we prove that approximation by the $F$ distribution in Kolmogorov distance is equivalent to belonging to the domain of attraction of the normal law. Both conditions are equivalent to relative concentration of the diagonal entries of the null-space projection $\mathbf{P}=\mathbf{I}_n-\mathbf{X}^{\top}(\mathbf{X}\mathbf{X}^{\top})^{-1}\mathbf{X}$. The central quantity $R=\mathbf{1}^{\top}\mathbf{P}\mathbf{1}$ measures the geometric alignment between the averaging direction and the random row space of $\mathbf{X}$. For symmetric finite-variance entries, the sufficient approximation extends to every sequence $1\le p<n$, including fixed nullity $n-p$. As a consequence of the $F$ approximation, Hotelling's $T^2$ statistic exhibits different asymptotic distributions in different regimes, including the normal, chi-square, and inverse-chi-square limits depending on $p$ and $n$. Without symmetry, we establish a central limit theorem under the weak-$5/2$ tail condition $\lim_{x\to\infty}x^{5/2}\mathbb{P}(|\xi|>x)=0$ when $p/n\rightarrow \gamma\in (0,1)$, and obtain the $F$ approximation as a consequence. Centered standardized Pareto laws show that the exponent $5/2$ is sharp for this universal absolute-tail sufficient condition, although balanced-tail nonsymmetric laws can retain the approximation at the boundary, which reveals that the asymmetry of the tails matters for the $F$ approximation.

This is a lecture note to introduce the innovative concept of numerical smoothness and its application in the error analysis of numerical solutions of evolution equations. We start from the two fundamentally different error splittings. One splitting is popular, which brings us to local error defined along the original solution and error propagation by numerical schemes. The well know Lax Equivalence Theorem is based on this error splitting. The other splitting can only be seen in a few numerical ODE books, which brings us to local error defined along the numerical solution and error propagation by the solutions of the evolution equations. The innovative concept of numerical smoothness arises from the second error splitting. While there are many difficulties when we use numerical stability in error analysis of nonlinear evolution equations, we will show that numerical smoothness works naturally in error analysis/estimation/control with advantages in several ways. We will define temporal and spatial numerical smoothness indicators, prove the necessity of the boundedness of these indicators for having optimal convergence rate, and show computable error estimates as functions of these smoothness indicators. Moreover, these error estimates are optimal in the approximatons' convergence rates and also optimal in the error propagation of the evolution equation. In comparison with error analysis framework of the Lax Equivalence Theorem, from the necessity results and the error estimates proven in the numerical smoothness framework, we have a new equivalence theory: Consistency + Numerical Smoothness <=> optimal error estimates. Unlike the Lax Equivalence Theorem, the new equivalence theory works well for nonlinear problems, nonlinear schemes, and many other complications of problems.

Mixed-integer programming (MIP) solvers are widely used, yet their performance can vary substantially even under seemingly neutral perturbations. Such variability complicates benchmarking, and the interpretation of computational studies. We introduce the Weighted Variability Score (WVS), a robust multi-component metric that aggregates dispersion in normalized runtime, final optimality gap, LP iterations, and branch-and-bound nodes across repeated runs. On a large subset of MIPLIB 2017 instances, each solved in 100 randomized runs under a four-hour limit, WVS reveals a right-skewed distribution with a substantial upper tail: many instances are essentially stable, but some exhibit large variability that is not explained by basic properties alone. Notably, this experiment proved optimality for the open instance 'neos-5045105-creuse' in two SCIP runs. Using static descriptors and root-level features, a tree-based predictor with a gated tail uplift achieves a mean absolute error of 0.058 and Spearman rank correlation of 0.65 on a sealed test split. Feature-importance analysis highlights root degeneracy, matrix structure, coefficient ranges, and presolve reductions as the main drivers of high WVS. Because variability is systematic and partially predictable from inexpensive early-run features, our findings offer a practical tool for preemptive tuning and lay a foundation for fairer evaluation of new algorithms.

We study the asymptotic properties of the nonlinear stochastic GMM algorithm proposed by Chen, Lee, Liao, Seo, Shin and Song for recursively estimating parameters defined by possibly nonlinear and overidentified moment restrictions $\E[g(X,\beta^*)]=0$. Each observation updates estimates of the moment Jacobian, inverse covariance matrix and Gauss–Newton preconditioner at a cost independent of sample size. For linear and nonlinear instrumental-variable moments, we derive an exact rank-two Sherman–Morrison–Woodbury update. The algorithm combines preconditioned stochastic approximation with iterate averaging. Under a preconditioned monotonicity condition, satisfied globally in linear models and locally around a consistent pilot estimate in general, we establish almost sure convergence and rates, as well as a functional central limit theorem with the efficient GMM variance. These results require four moments of the moment function and its Jacobian when the step exponent exceeds $2/3$. The inverse Gauss–Newton matrix consistently estimates this variance, and a projected linearised statistic enables online testing of overidentifying restrictions. We extend the results to $L$-dependent data with long-run variance weights under stronger moment assumptions. Simulations show that the averaged estimator approaches offline two-step GMM in accuracy and coverage at $10^6$ observations, while requiring a fraction of its computing time and fixed memory at up to $10^7$ observations. The online test achieves correct size and comparable power. In an application estimating the effect of a third child on weeks worked using $254{,}654$ women from the 1980 US Census, single-pass estimates agree with offline GMM within a fraction of a standard error on average across data orderings, and both methods yield the same overidentification-test conclusion in most orderings.

Practical ship design requires localized refinement and performance evaluation. This paper presents a reward-directed diffusion framework for localized hull optimization. Parametric masking during dataset generation restricts geometric changes to selected variables, including the bulbous bow and forebody, while preserving full-hull continuity. Surrogate-assisted sequential Monte Carlo sampling targets reduced predicted calm-water resistance, with stability retained through training data and displacement matched during screening. An automated viscous free-surface CFD pipeline, validated against measurements for two benchmark container ships, evaluates a generated variant and GenHull E, a hull refined by a naval architect within the same representation. At matched KRISO Container Ship (KCS) displacement, the generated variant has comparable total resistance at $F_n=0.195$ but 45-50% higher resistance at $F_n=0.250$ and $0.260$, missed by empirical screening. GenHull E achieves 6-8% lower total resistance up to $F_n=0.240$ but higher resistance above $F_n=0.250$ as opposing friction and pressure trends reverse the gains. Aggregating resistance across this crossover can conceal tradeoffs that emerge when friction and pressure components are evaluated separately. The assessment reveals an inability to represent a faired bulbous bow and a reward lacking viscous pressure and stability terms. These findings show how high-fidelity assessment informs improved representations and objectives for physically realizable generative designs.

Let $\lambda$ be the Liouville function. In this paper we prove an average form of Chowla Conjecture for even $k $. that is, for every fixed even integer $k\geq 2$, we prove that \[ \lim_{\min\{M,N\}\to \infty}\frac{1}{MN}\sum_{m=1}^M\sum_{n=1}^N\lambda(m)\lambda(m+a_1n)\cdots\lambda(m+a_kn)=0. \] We construct a stationary model for both averaging variables, derive the finite conditional-entropy estimates. A comparison of overlapping intervals on the logarithmic axis then forces this function to vanish.

In 1970, P. R. Halmos raised ten open problems on operator theory. The third problem of Halmos is the following: if an intransitive operator has an inverse, is its inverse also intransitive? This question is reduced to the case of conjugate bi-geometric operators in arXiv: 2610.11202. In the framework of conjugate bi-geometric operators, using tools from complex geometry, operator theory, and function theory, this paper gives an affirmative answer to the third problem of Halmos in the following cases: (a) bundle-preserving operator;(b) subexponential growth of operator norms. These two cases cover a large class of operators.

We study half-line CMV matrices whose Verblunsky coefficients are decaying perturbations of periodic sequences. We quantify the stability of fine properties of the absolutely continuous spectral measure under such perturbations. More precisely, we establish Hölder continuity of the perturbed spectral measure and derive sharp time-averaged Fourier estimates governing the associated quantum dynamics. These results remain valid up to the edges of the absolutely continuous spectrum. Since the spectral measure of a half-line CMV matrix is precisely the orthogonality measure for the corresponding orthogonal polynomials on the unit circle (OPUC), our results also yield new statements for OPUC whose Verblunsky coefficients are decaying perturbations of periodic sequences.

Let $C$ be a smooth hyperelliptic curve of genus $g\ge2$ over an algebraically closed field of characteristic zero. We prove that the normal bundle of the embedding of $C$ in $\mathbb{P}^n$, $n\ge3$, by the complete linear series of a general line bundle of degree $n+g$ is semistable if and only if $n\ge3g+1$. By openness, general complete embeddings of general curves of genus $g$ also have semistable normal bundle when $n\ge3g+1$. We then show that a general nonspecial curve of genus two and degree $d$ in $\mathbb{P}^n$ has semistable normal bundle except when $(n,d)$ is $(3,5)$, $(4,6)$, $(5,7)$ or $(6,8)$. The degenerations used in genus two extend to every genus $g\ge3$: semistability in all nonspecial degrees and dimensions follows from the cases $0\le d-n-g\le4g-4$. Together with a separate estimate for large $n$, this reduces semistability in each fixed genus to finitely many cases. The proofs combine degenerations to reducible nodal curves, explicit bounds on the degrees of subsheaves, and, in genus two, finitely many exact computations, which are provided as ancillary files.

We derive an explicit zero-side formula for the imaginary part of a smooth truncation of the logarithmic derivative of the Riemann zeta-function. The resulting kernel is the odd companion of the Fejér kernel in Montgomery's pair-correlation identity. Motivated by the recent breakthrough of Anthropic's Claude \cite{Claude26}, we also give an expository account of how the real-part identity can be turned into a matrix problem, overcoming the clustering obstruction for zeros off the critical line. Finally, we discuss how the new imaginary-part identity may contain additional information about such clustering.

We prove a sharp Zhong-Yang type lower bound for the first nonzero eigenvalue of the $p$-Laplacian on closed manifolds with small integral Ricci curvature for $p \geq 2$. The proof combines a global elliptic estimate obtained by Sobolev truncation with strict perturbations of one-dimensional comparison profiles.

This paper investigates an abstract viscoelastic wave equation in a separable Hilbert space, in which the damping term depends on a power of the velocity norm. Depending on whether the power exponent is zero or nonzero, this damping corresponds to friction damping or nonlocal damping, respectively; the equation also contains a viscoelastic damping represented by an integral term with a relaxation function. We establish global well-posedness for initial data in \(\mathcal{H}_1\) by combining monotone operator theory with the Faedo–Galerkin method, and prove uniqueness for all exponents greater than \(-1\), with the singular range between \(-1\) and \(0\) handled through the monotonicity rather than the Lipschitz continuity of the nonlocal damping operator. We then study the long-time behavior of the energy via integral inequalities, multiplier techniques, and Nakao's method, and obtain explicit decay estimates showing that the energy decays exponentially in the friction case and polynomially in the nonlocal case. To the best of our knowledge, this is the first work to consider the combined effects of viscoelastic damping and nonlocal damping in such an abstract framework, and it is also the first to analyze nonlocal damping with negative exponents.

We prove comparison results about $\infty$-categories of derived commutative rings with extra algebraic structure. For example, we show that several notions of derived $\delta$-rings agree and obtain similar results for derived $\lambda$-rings, derived binomial rings, and derived Boolean rings. Our comparison results apply more generally to modules over a specific kind of monad on derived rings that we call derived plethory.

In this paper, we determine the propagation rates of the Fisher-KPP equation with free boundaries and fractional diffusion $(-\Delta)^s$ $(0<s<1)$, which was considered in \cite{CDN26} as a model for species spreading, where the density function $u(t,x)$ satisfies the fractional Fisher-KPP equation $u_t+(-\Delta)^su=f(u)$ over a one-dimensional spatial domain $(g(t), h(t))$, representing the expanding population range, with $x=g(t)$ and $x=h(t)$ the free boundaries. It was shown in \cite{CDN26} that the model is well-posed, and its long-time dynamics exhibits a spreading-vanishing dichotomy. We show in this paper that, when spreading occurs, \[ \begin{cases} \lim_{t\to\infty}\frac{h(t)}{t^{1/(2s)}} =\lim_{t\to\infty}\frac{-g(t)}{t^{1/(2s)}} =\left(\frac{2^{1-2s}\mu}{1-2s}\right)^{1/(2s)} & \mbox{ if $s\in(0,1/2)$,}\\[2mm] \lim_{t\to\infty}\frac{h(t)}{t\ln t} =\lim_{t\to\infty}\frac{-g(t)}{t\ln t} =\mu & \mbox{ if $s=1/2$,}\\[2mm] \lim_{t\to\infty}\frac{h(t)}{t} =\lim_{t\to\infty}\frac{-g(t)}{t} =c_0(s) & \mbox{ if $s\in(1/2,1)$}, \end{cases} \] where $c_0(s)\in (0,\infty)$ is uniquely determined by the associated semi-wave problem. The level sets of $u(t,x)$ propagate with the same rates as the free boundaries. The proof uses delicate upper and lower solutions constructed from a smooth profile function in the case $s\in (0, 1/2]$, and from the semi-wave profile when $s\in (1/2, 1)$.

For every $d\ge4$, every prescribed connectivity $r\ge d$, and every $\varepsilon>0$, we construct a simplicial $d$-polytope whose graph has connectivity exactly $r$, is nontraceable, and has toughness less than $\varepsilon$. Thus the classical theorems of Whitney (1931) and Tutte (1956), which guarantee hamiltonicity under sufficiently high connectivity in dimension three, have no higher-dimensional counterpart. This answers a question posed by Bojan Mohar in 2001 on whether sufficiently high connectivity guarantees hamiltonicity of $4$-polytopal graphs and the connectivity questions raised by Ikegami, Maezawa, and Zamfirescu in 2021. For fixed $d\ge4$, $r\ge d$, and integer $k\ge2$, the examples can be chosen to have no closed spanning walk with at most $k$ visits per vertex, no spanning tree with degrees at most $k$, and nontraceable prisms. For fixed $d$ and $r$, we obtain graphs on $n$ vertices whose longest paths and cycles contain $O(\sqrt n)$ vertices. Consequently, the family of simplicial $d$-polytopes with connectivity exactly $r$ has Grünbaum–Walther shortness coefficient zero and shortness exponent at most $1/2$.

Sobolev gradient flows for the Gross–Pitaevskii ground state problem are Riemannian gradient descent methods on the sphere under a metric induced by a discrete elliptic operator. We analyze their convergence through the condition number of the Riemannian Hessian at the ground state. Bounds of the Rayleigh quotient of the Riemannian Hessian give a uniform analysis of these methods for different metrics. The uniform analysis applies to the classical metrics, such as the shifted Laplacian and the energy-adaptive metrics. Inspired by this analysis, we also consider three new metrics. One of them is a Thomas–Fermi metric, whose condition number is bounded uniformly in the interaction strength for the one-dimensional harmonic trap and whose iteration count does not grow in the Thomas–Fermi regime in the numerical tests. The performance suggested by our analysis is consistent with three-dimensional numerical tests with up to one billion unknowns, for both Riemannian gradient descent and Riemannian conjugate gradient methods, whose practical performance is also related to the Hessian condition number. The Thomas–Fermi flow needs the fewest iterations, while the metrics inverted exactly by tensor-product solvers are the fastest in computational time on one GPU card.

Given an integer $k \geq 2$ and a prime $p$ large enough depending on $k$, we show that, if $A \subseteq \mathbb F^n_p$ does not contain a restricted polynomial progression of the form \begin{align*} x, x+\begin{pmatrix} y_1 \\ y_2 \\ \vdots \\ y_n \end{pmatrix}, x+\begin{pmatrix} y_1^2 \\ y_2^2 \\ \vdots \\ y_n^2 \end{pmatrix}, \ldots, x+\begin{pmatrix} y_1^{k-1} \\ y_2^{k-1} \\ \vdots \\ y_n^{k-1} \end{pmatrix}, \end{align*} where $x \in \mathbb F^n_p$ and $y \in \{0,2, 3, \ldots, M_k\}^n \backslash \{\vec{0} \}$, for some fixed $M_k$, then $A$ must have density bounded above by $\frac{C}{(\log^{(k^{O(k)})} n)^c}$, where $C, c>0$ only depend on $k$ and $p$. For $k\geq 4$, the best previously known bound came from the density Hales–Jewett theorem and was of inverse Ackermann type. The proof uses a recent inverse theorem for noise stability of Bhangale, Khot, Liu and Minzer, which arose in the study of constraint satisfaction problems. The main new ingredient is a proof that the underlying distribution counting restricted polynomial progressions does not admit a nontrivial abelian embedding.

Pseudospectral methods are a powerful framework for spacecraft trajectory optimisation that rely on fixed, problem-independent collocation nodes such as Legendre-Gauss-Lobatto (LGL) nodes. These nodes are located according to polynomial root distributions, which can result in insufficient resolution in regions of high dynamical activity, such as those encountered during collision avoidance manoeuvres. This paper presents an adaptive node distribution strategy for Birkhoff pseudospectral trajectory optimisation, where the collocation nodes are placed according to a probability density function derived from the thrust magnitude profile of the spacecraft. By concentrating nodes in regions of high dynamical variation, the proposed method simultaneously achieves feasibility preservation and improved convergence with fewer nodes in a collision avoidance case study when compared to the universal Birkhoff method, with Model Predictive Control serving as an independent validation comparator. The case study highlights the practical consequences of inter-node constraint violations and validates the adaptive node scheme as a robust and efficient alternative for fuel-optimal spacecraft manoeuvre design.

We construct a geometric theory on the brane space $S^2M=C^\infty(S^2,M)$ associated to a gerbe with connection on $M$. Normalized logarithmic holonomy around the latitude circles defines an average holonomy line bundle on the brane space, whose connection and averaged flux give an equivariantly flat superconnection. We prove that the completed periodic exotic twisted cohomology defined by this superconnection localizes to the twisted cohomology of $M$. This brane space localization happens in two stages: circle localization lands on the free path space of $M$, and contracting the paths, after a correction of the twisting form, brings us back to $M$. Parallel transport along the meridians lets us average gerbe module connections, after which the circular iterated-integral construction applies. The resulting twisted brane Bismut-Chern forms restrict to the twisted Chern forms on $M$; their variation is described by twisted brane Bismut-Chern-Simons forms. For a T-dual pair, transporting the classical Hori morphisms through localization yields graded genus-0 T-duality on brane cohomology. The composite of each morphism with its dual is minus the Euler operator in the level variable, so these morphisms are isomorphisms at every nonzero level. At level one these morphisms are compatible with the brane character and twisted $K$-theory T-duality.

The precorrected-FFT has been used to rapidly evaluate $N$-body sums arising in electromagnetics and acoustics settings. It has, however, been historically limited to kernels that satisfy a Green's identity. In this work, we demonstrate that, using an improved version of the method of proxy points, the precorrected-FFT can be used to efficiently compute $N$-body sums with a wide variety of kernels. This algorithm requires no analytic information beyond pointwise access to the kernel. Our examples include kernels from non-local PDEs, Matern kernels, anisotropic kernels, and the Green's functions of high order PDEs. We also provide a detailed algorithm for choosing the proxy points and spreading grid, as implemented in the new pcfft package. We show these algorithmic choices result in favorable scaling when compared to other kernel-independent fast summation methods. Finally, we use the algorithm to accelerate a wide variety of integral equations solvers involving non-standard kernels, with applications in surface scattering, geophysics, and microbial community formation.

Alon, Bishnoi, Das and Neri constructed sequences of strong blocking sets in $(k-1)$-dimension projective space $P({\bf F}_q, k-1)$ with sizes at most $Cqk$, where $k$ goes to the infinity and $C$ is a fixed positive constant not depending on $q$, in their paper published in Transactions AMS, vol. 277, pp. 5389-5410, 2024. This is equivalent to a sequence of minimal $[n_i,k_i]_q$ codes with positive rate $\frac{k_i}{n_i} \geq c$, where $k_i$ goes to the infinity and $c$ is a fixed positive constant only depending on $q$. They solved one of the main open problems on minimal codes. The Alon-Bishnoi-Das-Neri construction was based on asymptotically good sequences of algebraic geometric codes and expander graphs. However, the constant $C$ is large for small prime power $q$. In this paper, based on a simple approach using $(q-1)$-power intersecting algebraic geometric codes via concatenation to construct minimal codes, we construct explicit strong blocking sets in $(k-1)$-dimension projective space with sizes around $2(q^2-1)k$, equivalently, explicit minimal $[n,k]_q$ codes with length around $2(q^2-1)k$. This is an essential improvement on the previous result of Alon-Bishnoi-Das-Neri, when $q \leq 11$. For large $q$, strong blocking sets and minimal codes in our construction are worse than the previous construction of Alon-Bishnoi-Das-Neri. However, our approach is much simpler and does not depend on expander graphs.

We construct nonnegative bounded compactly supported functions on the real line that give the lower bound $\mathcal{A}\ge 0.4103779774321214877852987\ldots$ for the minimum-autocorrelation constant. This improves the explicitly certified bound $2378625/5958277=0.39921356\ldots$ reported by Russell. The construction combines a point mass with a monotone degree-ten polynomial density; a one-sided atom-removal argument then produces ordinary functions. Its numerical content reduces to exact rational polynomial identities and 20 Bernstein subinterval certificates. The finite certificate, actual overlap integrals, admissibility, mass formula, and ordinary-function witness bounds have been formalized in Lean 4.19.0; the pinned proof sources and recorded audit are supplied. The passage to the supremum constant is given separately as a mathematical argument. As applications, the same construction yields an upper bound below $1.5610183$ for the large-multiplicity generalized-difference-basis coefficient and a feasible squared mass below $2.436778$ for a continuous sparse-ruler relaxation. An appendix quantifies the loss under independent sampling followed by completion. No improved bound for the complete sparse ruler constant, or global optimality of the construction, is asserted.

Sierra and Townsend (ST) [1] showed that the “absorption spectrum” model of Connes [2] arises as the lowest-Landau-level (LLL) limit of a charged particle in crossed electric and magnetic fields, and suggested that the higher Landau levels supply the fluctuation part of the Riemann-zero counting function $N(E)$. Within the approximation used by ST we construct the eigenfunctions of an arbitrary Landau level $n$ and derive their asymptotics at the two box boundaries that ST identify. The boundary condition is the same as for $n=0$ up to an $E$-independent phase $(-\ii)^n$, so no $\ln E$ term arises for any fixed $n$. We also note that the $\ln E$ term in the semiclassical Eq.~(29) of ST is the first-order shift of the smooth LLL counting function, $E\to E+\gamma z^2$.

Using the spectral relation between irreducible cyclic codes (ICC) and generalized Paley (GP) graphs recently found by the authors, we study the weight distribution of irreducible cyclic codes of length $2^\ell$, $\ell \in \mathbb{N}$. These codes were previously studied by Sharma, Bakshi and Raka, who divided them into two families (each having two cases). First we characterize the GP-graphs spectrally related to these families of ICCs: they are complete, semiprimitive, Hamming, or Cartesian products of semiprimitive GP-graphs. We then compute the spectrum of these graphs and hence, by the mentioned spectral relation we finally obtain neat explicit expressions for the weight enumerators of all the irreducible cyclic codes of length $2^\ell$ with $\ell \in \mathbb{N}$.

Let $\Lambda_n$ be the set of all eigenvalues of standardized Laplacian matrices of order $n$. In 2005, a polygon $S_n\subseteq\Lambda_n$ was constructed, and it was conjectured that $\Lambda_n=S_n$. We observe that $S_n$ is a zonotope, which gives a short proof of $S_n\subseteq\Lambda_n$, and we describe $\Lambda_n$ as a union, over all vectors $v$, of intersections of zonotopes $Z_i(v)$ built from the ratios $v_j/v_i$. For a vector whose components are the $n$ distinct $n$th roots of unity, every $Z_i(v)$ equals $S_n$. We then show that the conjecture is false for $n=7$: the directed $6$-cycle with one vertex duplicated has an eigenvalue lying $2.1\cdot10^{-4}$ beyond one edge of $S_7$. For the analogous digraph of order $n$, the corresponding eigenvalue lies at signed distance $\delta_n\sim\frac{\pi^2}{2n^4}$ beyond the supporting line of the analogous edge of $S_n$, so the construction yields counterexamples for all sufficiently large $n$; numerically, it does so for every $7\le n\le300$. We also prove that $S_m\subseteq S_n$ whenever $m\mid n$, with $S_m\setminus\{0,1\}$ lying in the interior of $S_n$ for $n\ge3m$.

A geometric $(n_4)$ configuration is a set of $n$ points and $n$ lines in the real projective plane such that every point lies on exactly four of the lines and every line contains exactly four of the points. The question of which $n$ admit such a configuration was posed by Grünbaum in 2000, and successive work of Bokowski, Grünbaum, Schewe, Pilaud and Cuntz settled every case but $n=23$. We settle this last case by bounding the symmetry of such configurations and classifying the most symmetric ones exactly. We prove that for every prime $n\equiv3\pmod4$ the group of projective symmetries of a geometric $(n_4)$ configuration has order at most four, explaining why symmetric constructions failed at $n=23$, and the same argument bounds the symmetry of $(n_5)$ configurations for primes $n\not\equiv1\pmod5$. Reducing the Klein four-group case to 4021 bilinear polynomial systems and solving all of them exactly with Gröbner bases, we show that exactly two $(23_4)$ configurations reach this bound up to projective equivalence and that there are no others even over $\mathbb{C}$. Both are self-polar; the first has integer coordinates and the second is defined over $\mathbb{Q}(\sqrt{17})$ but not over $\mathbb{Q}$. Consequently geometric $(n_4)$ configurations exist if and only if $n\ge18$ and $n\ne19$. The same classification recovers Cuntz's $(22_4)$ and $(26_4)$ configurations and finds three new combinatorial types at $n=22$ and $94$ at $n=26$. Among self-polar $(23_4)$ configurations with a single involution we find $50$ more, defined over number fields of degree at most $7$. Four of them deform in one-parameter families, and computing three of these exactly, one birational to the modular curve $X_1(11)$, we prove that there are infinitely many geometric $(23_4)$ configurations, among them infinitely many with rational coordinates that are self-polar and have no projective symmetry.