Papers

Joint frequency and chirp-rate estimation for a noisy chirp signal arises in radar, sonar, and burst satellite communications. Conventional estimators combine a coarse grid search with fine interpolation; accuracy degrades at the edges of the residual cell (the edge effect) and below the breakdown SNR (the threshold effect). We present a deterministic two-stage estimator that controls both failure modes uniformly over the residual cell. The estimator combines a time-centered, zero-padded dechirp-FFT acquisition bank with alternating selectable-$p$ amplitude-interpolation refinements on DTFT samples at fractional bins; in the centered frame, the frequency-chirp-rate cross-term of the Fisher information vanishes. The paper derives a mean-squared-error and threshold characterization over the full SNR range, in closed form except for one calibrated scalar (an effective cell count), to our knowledge the first for the joint problem: the breakdown threshold is governed by the cell count, and its cell-position dependence is dominated by the scalloping loss of the coarse FFT, which the padding bounds at 0.4 dB. An asymptotic uniformity analysis over the cell, including its corners, gives fixed-point variance ratios of $1.003$ and $0.998$, analytically free of the residual. A closed-form bias analysis under a cubic phase mismatch shows the centered chirp-rate estimate is insensitive to first order. Monte Carlo experiments at $N=256$ (validated at $N=32$-$512$) measure frequency- and chirp-rate-axis efficiencies with median $1.03$ and worst case $1.07$ over $144$ cell positions at $-5$ dB. Threshold predictions hold within $1.0$ dB on four configurations not used in the calibration. The dechirp-FFT bank is fully parallel, and each of the four refinement iterations evaluates three DTFT samples per axis; under fixed operating conditions, per-estimate latency is constant at $O(N\log N)$ cost.

For a finite group $G$, let $\omega(G)$ be the set of element orders of $G$ and let $h(G)$ be the number of pairwise nonisomorphic finite groups $H$ with $\omega(H)=\omega(G)$. We say that the recognition problem is solved for $G$ if the number $h(G)$ is known, and if it is finite, then all finite groups $H$ with $\omega(H)=\omega(G)$ are listed. We complete the solution of the recognition problem for all finite simple groups.

The partial Petrial polynomial of a bouquet can be computed from the coranks over GF(2) of matrices obtained by varying the diagonal entries of the adjacency matrix of its intersection graph. Motivated by this matrix formulation, we study the corresponding polynomial for simple graphs and determine when it has exactly two or three nonzero terms. A key tool is the interpolating property: the exponents of the nonzero terms are consecutive. Using this property together with local complementation minors of grafts, we extend the known characterization of the binomial case from connected circle graphs to all connected simple graphs, showing that such a graph has a binomial partial Petrial polynomial if and only if it is a path. Our main result characterizes the trinomial case for trees: a tree has a trinomial partial Petrial polynomial if and only if it is a T-shape tree or an H-shape tree. Here, a T-shape tree has maximum degree 3 and exactly one vertex of degree 3, whereas an H-shape tree has maximum degree 3 and exactly two vertices of degree 3, which are adjacent. We also derive explicit formulas for both families in terms of Jacobsthal numbers.

We study the Katz-Long-Moody (KLM) construction and classify exactly when the resulting braid group representations factor through Hecke algebras, for scalar braid part and semisimple free-group part, over an algebraically closed field of characteristic zero and at every convolution parameter lambda different from 1. We show that, except for one exceptional two-strand family, this property then depends only on the eigenvalues of the free-group part and is independent of lambda. Semisimplicity is a genuine hypothesis: we exhibit non-semisimple inputs, namely g = I + N with N nonzero and N^2 = 0, whose KLM quotient is nonetheless a Hecke module, realizing the permutation representation of the symmetric group. We also give an exact criterion for when the resulting representations factor through Temperley-Lieb algebras.

In this paper, we establish necessary and sufficient conditions for continuous embeddings between weighted local Morrey-type spaces and weighted complementary local Morrey-type spaces. Rather than relying on standard multidimensional approaches, our methodology is based on a one-dimensional reduction. Specifically, we reduce the multidimensional embedding problems to corresponding weighted inequalities involving one-dimensional Cesàro and Copson function spaces. Applying this reduction alongside recent results in the characterization of the embeddings between Cesàro and Copson spaces, we provide complete characterizations for the embeddings \[ \LM_{p_1,q_1}(v_1, w_1) \hookrightarrow \LM_{p_2,q_2}(v_2, w_2)\] and \[\dual\LM_{p_1,q_1}(v_1, w_1) \hookrightarrow \LM_{p_2,q_2}(v_2, w_2).\] Our results cover all possible finite cases of the parameters $0 < p_1, p_2, q_1, q_2 < \infty$.

We determine the sharp number and the complete finite spectrum of barycentric hyperplanes through the Tukey median. We further prove that every convex body in $\mathbb R^n$, $n\ge3$, contains an interior point incident with at least five barycentric hyperplanes. In particular, this settles Grünbaum's original conjecture in dimension four.

We introduce a Lovász-type semidefinite parameter for balanced colouring of signed graphs. Its homomorphism target is a unit sphere equipped with an orthogonal involution: the fixed and anti-fixed components play different roles, while applying the involution to one endpoint realizes switching. The resulting parameter admits a symmetric formulation with two positive semidefinite matrices and an equally symmetric dual. It is also one half of the strict vector chromatic number of the ordinary graph formed by the negative edges of the double switching graph. Our second main contribution is a hierarchy of convex corners. Starting from the balanced induced subgraph polytope, we define signed analogues of the stable-set, theta, and clique-inequalities relaxations in the original vertex space. The signed theta body has an intrinsic two-matrix description, its all-ones gauge is the new scalar parameter, and the ordinary stable-set hierarchy is recovered exactly from signed digon graphs. For all-negative signatures, the construction becomes a relaxation of the maximum induced bipartite subgraph problem and is related to the generalized theta number. Finally, we propose a notion of balanced perfectness and show that it is strictly weaker than perfectness of the associated double-cover graph.

Given a Riemann surface {\Sigma}, let {\Gamma} {\subseteq} Mod({\Sigma}) denote the monodromy subgroup of a family of complex curves homeomorphic to {\Sigma}, arising from a sufficently ample Lefschetz pencil. We establish that the group {\Gamma} acts with Zariski dense orbits or ergodically on certain character varieties for {\pi_1}({\Sigma}). This answers a version of a Conjecture of Katzarkov, Pantev, and Simpson appearing in [KPS03].

Let $K$ be a finite $k$-dimensional simplicial complex with $k\ge1$, and let $\Lup_{k-1}(K)$ be its $(k-1)$-dimensional up-Laplacian. We resolve the equality case of a conjecture of Duval and Reiner: for a pure complex, the nonzero spectrum of $\Lup_{k-1}(K)$ equals ${d^{\mathrm v}(K)}^{\T}$, the conjugate vertex-degree partition, if and only if the complex is isomorphic to a shifted complex. In fact, we prove more: equality of the second power sums alone already forces shiftedness, and the gap between them equals twice the number of failed elementary shifts of facets. We next characterize equality in the Duval–Reiner bound $\lambda_1(\Lup_{k-1}(K))\ge d_1(K)+k$, where $d_1(K)$ is the maximum upper degree of a ridge. We introduce the clean ridge, a higher-dimensional analogue of a dominating vertex, and combine it with a signed Sachs expansion for the associated signed facet-adjacency matrix to obtain a necessary and sufficient combinatorial criterion. When the signed facet-adjacency graph is connected and balanced—with balance equivalent to disorientability in dimension $k$—the criterion reduces to the simple condition that all facets share a common ridge. Finally, whenever $K$ has at least two facets, we prove the sharp bound \[ \lambda_2\bigl(\Lup_{k-1}(K)\bigr)\ge d_2(K)+k-1. \] This gives a higher-dimensional analogue of the second-index Brouwer–Haemers bound for graphs. The natural termwise extension to higher eigenvalue indices fails already at $m=3$, as a s

Given a growth function $\varphi:[0,\infty)\rightarrow [0,\infty)$, it is established that the commutators generated by sublinear operators and Orlicz-$\mathrm{BMO}$ function $b$ are bounded from $H_{b}^{\varphi}(\mathbb{R}^{n})$ to $L^{1}(\mathbb{R}^{n})$, and from $H^{\varphi}(\mathbb{R}^{n})$ to $L^{1,\,\infty}(\rn)$, where $H_{b}^{\varphi}(\mathbb{R}^{n})$ is a specific subspace of Orlicz-Hardy space $H^{\varphi}(\mathbb{R}^{n})$ and sublinear operators include Lusin area integral, g-function, Marcinkiewicz integral and Bochner-Riesz mean operator. Under the assumptions $T^*1=0$ and $T^*b=0$, it is shown that the Orlicz-$\mathrm{BMO}$ commutator associated with the Bochner-Riesz mean operator admits endpoint boundedness from $H_{b}^{\varphi}(\mathbb{R}^{n})$ to $H^{1}(\mathbb{R}^{n})$. However, the commutators corresponding to other operators discussed in this paper do not possess the aforementioned endpoint boundedness, and a counterexample is provided to illustrate this point.

The pencil diffeological space is a plane whose smooth structure at the origin is detected only along lines through it. We show that the Zariski cotangent space at the origin is entirely determined by the derivatives along these lines. Moreover, the derivatives along different lines form precisely a $\sigma$-continuous family, a weak form of continuity. We also compute the external and right tangent spaces at the origin from this description. This suggests that the local smooth structure at a singular point of a diffeological space can give rise to unexpected topological notions.

For every $n\geq 3$, we construct a nonconstant real-valued function $U\in C^\infty(\overline{\mathbb R^n_+})$, harmonic in the upper half-space, whose complete boundary jet vanishes on a compact nowhere dense subset of $\partial\mathbb R^n_+$ of positive $(n-1)$-dimensional measure. The essential construction takes place in two dimensions and yields the case $n=3$; higher-dimensional examples follow by cylindrical lifting. In every dimension, the exceptional set may occupy an arbitrarily large proportion of a fixed boundary cube. This resolves, in the negative, the smooth case of the boundary unique-continuation problem left open by Bourgain and Wolff in 1990 \cite[p.~260]{BourgainWolff1990}. In dimension three, a Möbius–Kelvin transfer gives the corresponding counterexample in the unit ball. It disproves Nadirashvili's smooth unit-ball conjecture on boundary singular sets \cite[Conjecture~4, p.~232]{Nadirashvili1997} and, a fortiori, disproves the gradient-only formulation subsequently recorded by Logunov and Malinnikova \cite[Section~7.4]{LogunovMalinnikova2020} and by Lin \cite[Conjecture~3, pp.~15–16]{Lin2020Current}. The proof uncovers a hidden flexibility principle for nonlocal elliptic equations: microscopic modifications can exert macroscopic control over exterior data. A quantitative correction mechanism for the half-Laplacian, iterated across scales, produces flat nonlocal Cauchy data on a set of positive measure. Thus nonlocality has a striking dual character: the same long-range interaction that drives unique-continuation rigidity can also furnish the flexibility through which that rigidity fails in the smooth category.

Through a careful analysis of the Yamabe equation on singular spaces, we establish a positive mass theorem for singular asymptotically hyperbolic manifolds with arbitrary ends. We also derive a rigidity result that is novel even in the smooth setting.

We develop a characteristic-uniform arithmetic theory for \[ \mathcal C_d:\quad (u^2+u)(v^2+v)=d. \] For \(d(1-16d)\ne0\), its smooth \((2,2)\)-completion has four rational boundary points forming \(\mathbb Z/4\mathbb Z\), an intrinsic \(D_8\)-action, the inverse \(-(u,v)=(u,-v-1)\), and the native Kummer map \(\kappa_d(u,v)=(u+1)\). Working natively, we derive complete full-point and differential laws, Kummer ladders and recovery, halving, tripling, \(2P+Q\), division polynomials, isogenies, CM endomorphisms, and pairings, with dedicated formulas in characteristics two and three. Classical models provide proof and optimization dictionaries while all endpoints remain native. For Cd25519, the Kummer line is exactly the X25519 line, and a native Segre recoding realizes the optimized complete \(a=-1\) Edwards full-point dependency graph. We further study \[ \begin{gathered} \mathcal C_{a,b,d}:(u^2+u+a)(v^2+v+b)=d,\quad \mathcal T_{a,d}:(u^2+u+a)(v^2+v)=d,\\ \mathcal R_{\tau,\sigma,\kappa}:(x^2-\tau)(y^2-\sigma)=\kappa xy,\quad \mathcal Q_{\alpha,\beta,\gamma}:x^2y^2+\alpha(x^2+y^2)+\beta xy+\gamma=0. \end{gathered} \] For these product, one-sided twisted, reciprocal, and QRT families, we determine their genus-one geometry, finite-field forms, arithmetic, and isogenies. On each smooth QRT fibre, the Vieta–McMillan map is a fixed elliptic translation. Marking \(D\) gives the state \(P\mapsto(\kappa(P),\kappa(P+D))\), with maps realizing \(n\mapsto mn+r\). This yields logarithmic ladders and an elliptic Lucas calculus with nonlinear addition, fast-index doubling, state-division polynomials, and bridges to elliptic divisibility sequences, sigma functions, and elliptic nets. In characteristic two, every ordinary pointed elliptic curve over a perfect field admits the binary state model. We further develop the \(\mathcal C_d\) platform for isogeny-based cryptography.

We study the partial-sum behavior of tempered linear processes \[ X_{N,n}=\sum_{j=1}^{\infty}e^{-\lambda_Nj}\frac{\ell(j)}{j}\varepsilon_{n-j}, \qquad \lambda_N\downarrow0, \] where $\ell$ is slowly varying and the innovations belong to the domain of attraction of an $\alpha$-stable law with $1<\alpha\leq2$. The filter $j^{-1}\ell(j)$ represents the logarithmic boundary between summable and power-law long-memory coefficients. Let \[ Q_N=\sum_{j=1}^{N}e^{-\lambda_Nj}\frac{\ell(j)}{j}, \qquad L(N)=\sum_{j=1}^{N}\frac{\ell(j)}{j}. \] We prove that the partial-sum process, normalized by $B_NQ_N$, converges to the stable Lévy motion associated with the innovations. Moreover, \[ Q_N\sim L(N)\quad\text{if }N\lambda_N=O(1), \qquad Q_N\sim L(1/\lambda_N)\quad\text{if }N\lambda_N\to\infty. \] Then weak, moderate, and strong tempering have the same first-order Lévy limit but different normalizations. In the weakly and moderately tempered regimes, the second-order remainder, normalized by $B_N\ell(N)$, converges in finite-dimensional distributions to a logarithmically tempered stable process; in the Gaussian finite-moment case the convergence is functional. These results extend the untempered logarithmic-boundary theorem and complement existing invariance principles for tempered linear processes.

We study how sparsely a nonzero discrete harmonic function on the standard lattice $\mathbb{Z}^d$ can be supported. Let $Q_n^{(d)}=\{-n,\cdots,n\}^d$, and let $m_d(n)$ denote the least possible value of $|\mathrm{supp}(u)\cap Q_n^{(d)}|$ among discrete harmonic functions $u:\mathbb{Z}^d\to\mathbb{C}$ with $u(0)\neq0$. For all $n\geq1$, we prove \begin{equation*} m_3(n)\asymp n^2, \quad c_dn^{d^2/(2d-1)} \leq m_d(n)\leq (2n+1)^{\lfloor d/2\rfloor+1} \quad d\ge 4. \end{equation*} These estimates extend the two-dimensional support estimate of Buhovsky, Logunov, Malinnikova, and Sodin [Duke Math. J. 171 (2022), 1349–1378] to higher dimensions and obtain sharpness in dimension three. For $d\geq4$, the lower exponent and the upper one differ by less than $3/4$ in even dimensions and $1/4$ in odd dimensions. The proof combines Hilbert functions of finite support sets with a position-translation uncertainty principle. The sharp three-dimensional bound additionally uses Cayley–Bacharach relations and rigidity of algebraic curves. Finally, for every nonzero lattice eigenfunction with eigenvalue $\lambda$, the Zariski closure of its full support has dimension at least $\lceil d/2\rceil$, and at least $\lfloor d/2\rfloor+1$ when $\lambda\neq0$. Both bounds are optimal. All proofs resulted from human-guided exploration by GPT-5.6 Sol in Ultra mode and checked by the author.

Erdős Problem 725 asks for an asymptotic formula for the number $L_{k,n}$ of ordered, labelled $k\times n$ Latin rectangles. Godsil and McKay proved that $L_{k,n}\sim (n!)^k((n)_k/n^k)^n(1-k/n)^{-n/2}e^{-k/2}$ for $k=o(n^{6/7})$. We provide a partial solution to Erdős Problem 725 by proving this asymptotic for every $k=o(n)$. More precisely, set $\widetilde A_{k,n}=(n!)^k((n)_k/n^k)^n\exp\{[n(H_n-H_{n-k})-k]/2\}$. For every $K(n)=o(n)$, uniformly for $0\leq k\leq K(n)$, we prove $\log(L_{k,n}/\widetilde A_{k,n})=O(k^2/n^2)$, with an absolute implied constant. The results of this paper have been formally verified in Lean.

For mirror descent generated by a Legendre kernel, perhaps one of the most basic question in optimization is this: must every accumulation point of a bounded mirror descent sequence be Karush–Kuhn–Tucker (KKT) stationary under proper stepsizes? We show that the answer is no. A longstanding obstacle to resolving this question is the boundary blow-up of the Legendre gradient: it keeps every mirror step in the interior, while at a boundary limit, the inverse entropy metric vanishes on active coordinates and can erase the dual-feasibility in the KKT system. We construct $C^\infty$ objectives and bounded sequences generated by the Shannon-entropic mirror descent on the nonnegative orthant $\R_+^n$, for every $n\geq 3$, and on the probability simplex $\Delta_n$, for every $n\geq 4$, such that, in each case, the set of accumulation points is a smooth boundary circle containing a nonempty relatively open arc of non-KKT points. The steps satisfy $\alpha_k\asymp k^{-\beta}$ with $\beta\in(1/2,1)$, the objective values are nonincreasing, and the objectives are entropy-relatively smooth. Hence the pathology stems from the degeneracy of the Bregman geometry at the boundary, rather than from failure of descent, or improper stepsizes. To the best of our knowledge, these provide the first counterexamples to KKT accumulation for bounded mirror descent sequences with nonincreasing objective values.

We analyze approximate message passing (AMP) with an independent Gaussian initialization for noiseless phase retrieval in the proportional asymptotic regime. A random initialization has overlap of order $d^{-1/2}$ with the signal, and AMP requires a growing number of iterations to attain non-vanishing overlap. Thus, its precise behavior cannot be characterized by classical fixed-time state evolution. We prove a Gaussian decomposition of the AMP trajectory and control its error over the horizons required for recovery. The resulting analysis shows that random initialization attains the weak-recovery threshold $\delta_{\rm weak}=1/2$. For $\delta\in(\delta_{\rm weak},\delta_{\rm str})$, where $\delta_{\rm str}\approx1.13$, the signal strength follows state evolution and approaches its stable finite fixed point uniformly for \(n^{1/3}/\operatorname{polylog}(n)\) iterations. For $\delta>\delta_{\rm str}$, AMP reaches any prescribed fixed recovery accuracy within $O_{\delta,\varepsilon}(\log n)$ iterations. The majority of our analysis applies more generally to generalized AMP for single-index models.

We formalize the algebraic structure of rational polynomial phase sequences (discrete chirps) over the integer lattice $\mathbb{Z}$. By constructing a convolution algebra generated by polynomial phases with coefficients in $\mathbb{Q}$, we establish a rigorous operational calculus for discrete difference equations. We prove an exact orthogonality relation for these sequences over finite periodic windows, yielding a generalized discrete Fourier isometry. Furthermore, we demonstrate that the continuum limit of this discrete framework can be rigorously defined via the inductive limit of finite-dimensional $C^*$-algebras (an AF-algebra). This framework provides a rigorous algebraic substrate for finite-mode lattice approximations in quantum mechanics and the construction of mutually unbiased bases in finite-dimensional quantum information theory.

Datasets in many settings naturally partition into clusters arising from sub-populations, batch effects, or aggregation across multiple sources. A common response to such heterogeneity is to ensemble learners trained on each cluster rather than fit a single model to the pooled data. Prior work motivating such approaches has typically considered settings in which both the covariate distribution and the conditional outcome model differ across clusters; the role of cluster-aware partitioning and ensembling based solely on the covariate distribution remains to be explored. We address this case for ridge-regularized least-squares regression under a linear outcome model and consider all ridge penalty values $\lambda \geq 0$, including the special case of the ridgeless predictor at $\lambda = 0$. By considering both fixed-effects and random-effects models, we argue that under random effects, an optimally tuned pooled ridge predictor always outperforms ensembles of individually optimally tuned predictors. For fixed effects, we derive a general formula for the pooled and ensembled predictors to characterize the role of both regression coefficients as well as the predictor distribution shifts. Together, these results generalize prior risk analyses of bagging and random-partition estimation using ridge and ridgeless regression predictors from the i.i.d. setting to encompass covariate shift and heterogeneity-aware partition structure.

We investigate a Cucker-Smale type flocking model with a biological memory term and conclude that memory makes flocking harder. The model introduces $h(t)$ to describe memory, with dynamics $\frac{dh}{dt}=x(t)+v(t)-\lambda h(t)$ reflecting path, velocity dependence and exponential decay. In microscopic level, the flocking phenomenon becomes conditional, requires communication kernel $\psi(t)$ has polynomial lower bound and memory deviation $H(t)$ decays exponentially. In mesoscopic level, flocking rates exhibit exponential or algebraic decay and the range of communication strength $\beta$ become smaller.

We obtain closed explicit formulas for the spectral determinants of the smooth hyperbolic Bolza surface and the Klein quartic. In each case, a multiplicative relation expresses the determinant of the surface in terms of determinants of singular quotient orbifolds of genera zero and one. The elliptic factors are evaluated by applying the singular Polyakov anomaly formula to explicit Belyi maps on CM elliptic curves of discriminants -8 and -7, while the genus-zero factors are evaluated by explicit determinant formulas for constant-curvature spheres with conical singularities. The same multiplicative relations hold fibrewise on the corresponding equisymmetric deformation strata and yield determinant and first-variation identities. We also prove that every compact quasiplatonic hyperbolic surface is a critical point of the spectral determinant on its Teichmüller space; in particular, this applies to the Bolza surface and the Klein quartic.

Let $\mathcal{H}$ be the Hurwitz quaternions, $p$ an odd prime, and $Q \in \mathcal{H}$ a prime of norm $q \neq p$. Metacommutation $PQ = Q'P'$ induces a permutation $\pi_Q$ of the $p+1$ left-associate classes of primes of norm $p$. Cohn and Kumar compute its sign and fixed-point count, and Leite and Machiavelo its full cycle structure, by formulas depending only on conjugation-invariant data of $Q$ (namely $q$ and $\mathrm{tr}\,Q$). We prove this is exactly the boundary of what such invariants can carry: no quantity $I(Q)$ invariant under unit conjugation determines $\pi_Q$ as a labelled permutation of the intrinsic class set, or even the image of a single specified class. The proof combines an equivariance identity $\pi_{uQu^{-1}} = \rho_u \pi_Q \rho_u^{-1}$ with a minimal, fully explicit witness at $(p,q) = (3,5)$: the four primes $2+i$, $2+j$, $2+k$, $2-i$ form a single unit-conjugacy orbit, hence agree under every conjugation-invariant function, yet induce four pairwise distinct $4$-cycles of the same four classes. We further observe that isomorphisms $\mathcal{H}/p\mathcal{H} \to M_2(\mathbb{F}_p)$ form a torsor under $\mathrm{PGL}_2(\mathbb{F}_p)$, so no projective labelling of the classes is canonical, and that by orbit-stabilizer the datum of one destination $\pi_Q(C)$ is exactly a coset $g_Q G_C$ in $\mathrm{PGL}_2(\mathbb{F}_p)/G_C$. All sixteen refactorizations in the witness are listed in the appendix and have been verified by machine along two independent routes.

An m-by-n real matrix A is said to be semipositive if there exists a vector x>0 such that Ax>0, where the inequalities are understood componentwise. Dorsey et al. conjectured that any invertible linear map that $L$ that leaves invariant the collection of all semipositive matrices is always in the standard form $A \mapsto XAY$ for some row positive matrix $X$ and inverse nonnegative matrix $Y$. This was settled in when $m \geq n$. Our aim in this paper is to settle the case when m<n of the above conjecture. In fact, our proof works for arbitrary positive integers m and n. The main ingredient is a classification of affine subspaces of the largest possible dimension contained in the set of semipositive matrices.

The Zarankiewicz number z(n;2) is the largest number of ones in an n x n zero-one matrix with no all-one 2x2 submatrix. At n=43, the Kovari-Sos-Turan bound, sharpened by Reiman, gives z(43;2) <= 301 with no rounding slack; equality would force a projective plane of order six, which Tarry ruled out in 1900. That fact alone says nothing about how far below 301 the truth lies. We prove z(43;2) <= 294. The argument is local and geometric. Writing D = 301 - E for the deficiency of a configuration with E ones, a crossing count paired with its dual forces every line to at most eight points and every point to at most eight lines whenever D <= 6. A deficiency count then produces a clean point of degree seven, collinear with all others, unless the configuration has both a line of eight points and a point of degree eight. The seven lines through a clean point partition the remaining 42 points into groups that every other line meets at most once, so the number of (line, missed group) incidences equals D. At D=6 this forces group sizes 6,6,6,6,6,6,6 or 7,6,6,6,6,6,5; the second forces a TD(4,6), and the first yields 32 blocks of a partial TD(5,6), whose refutation needs an unknown packing number. We determine it. pa(5;6) and pa(6;6) are listed as open in the Handbook of Combinatorial Designs at 30 <= N <= 34; a class holds at most six words, so at most one full class gives at most 6 + 5*5 = 31, and what was missing was refuting the two-full-class case, done here by exhaustive search. Hence pa(5;6) = pa(6;6) = 31. With no clean point, the configuration is rigid enough to complete to a projective plane of order six. With a known 290-one configuration, this gives 290 <= z(43;2) <= 294. We also derive z(43;2) <= 299 via Totten's classification, and rule out configurations with 290+ ones from deleting rows/columns of PG(2,7).

Topological $K$-theory supplies the decisive stable classification principle for gapped free-fermion phases in Euclidean crystals once symmetry and stabilization are fixed. In hyperbolic band theory, by contrast, it is less decisive for the full band problem: generalized momentum sectors vary through moduli spaces, but passing to ordinary $K$-classes collapses their geometric and holomorphic variation. We distinguish kinematical holomorphy, in which complex geometry organizes the sector spaces, from dynamical holomorphy, in which that geometry informs a Hamiltonian or projector, and formulate the resulting loss as an observable-factorization problem. For a compact hyperbolic surface $X$, we prove nonfactorization in three settings: fibrewise $K^0(X)$, occupied-state $K^0(B)$, and operator-algebraic $K$-theory. The affected quantities include spectra and Higgs spectral curves, the Berry holonomy and the quantum metric, the partially filled Hall response, and the Fermi surface and nodal geometries. We also identify the quantized pairings and local charges retained by topology. Up to an explicit area factor, the Kotani–Sunada bottom-band Hessian is the Hodge inner product on $H^1(X;\mathbb R)$. Together with the integral intersection form, it recovers the homology-marked principally polarized Jacobian and hence, by Torelli and uniformization, the underlying complex and hyperbolic surface, though not a full Teichmüller marking. We also separate finite-rank sectors from the thermodynamic bulk. Locally faithful covers reproduce polynomial traces exactly and control continuous spectral observables. In arithmetic congruence towers, analytic observables converge as $O(|G_n|^{-\eta})$ for some $\eta>0$, and $C^s$ observables as $O((\log |G_n|)^{-s})$, while established coherent large-rank limits recover bulk density-of-states moments.

Let $k$ be an algebraically closed field, and let $G$ be an algebraic $k$-group. We study finite abelian $k$-subgroups $A \subset G$ whose order is not divisible by the characteristic of $k$. This is a classical topic in the theory of algebraic groups going back to the work of Borel in the early 1960s. We sharpen previously known results on the structure of $A$. In particular, we show that there exists a maximal torus $T$ of $G$ such that the index $[A: (A \cap T)]$ divides the Grothendieck torsion index $t(G)$. We also show that there exists a maximal torus $T$ such that the quotient group $A/(A \cap T)$ is “small” in a suitable sense. As applications of these results, we (i) give a positive answer to a question of Totaro for $G$-torsors over fields $k_r = k((t_1))((t_2)) \ldots ((t_r))$ of iterated Laurent series, (ii) prove a variant of the “hypothèse optimiste” of Tits about splitting fields of $E_8$-torsors, and (iii) show that certain torsors over $k_r$ cannot be split by the function field of a genus $1$ curve.

In this paper, we prove a gradient estimate for holomorphic functions on complete Hermitian manifolds with nonnegative second Ricci curvature. As an application, we establish several Liouville-type theorems for holomorphic functions and mappings between complete Hermitian manifolds under suitable curvature assumptions. In particular, on a complete Hermitian manifold with nonnegative second Ricci curvature and Chern torsion of linear growth, any bounded holomorphic function must be constant. Moreover, we construct examples to show that both the gradient estimate and this Liouville theorem are sharp in a suitable sense. In the second part of the paper, we study holomorphic functions on simply connected complete Hermitian manifolds with vanishing Chern curvature and torsion norm of sublinear growth. Combining the gradient estimate with methods from sub-Riemannian geometry, we obtain a complete characterization of holomorphic functions of polynomial growth on such manifolds.

We propose method-of-moments estimators for the eigenvalues of variance component covariance matrices in multivariate mixed effects models. Assuming a parametric form for the eigenvalue distribution, we focus on the high-dimensional regime where the number of predictors is large and comparable to the number of realizations of each random effect. In this setting, we show that the empirical moments of sum-of-squares matrices (e.g., MANOVA estimators of the covariance matrices) can be closely approximated by deterministic functions of the underlying parameters. This relationship enables the construction of consistent and asymptotically normal estimators via moment matching. Our approach is motivated by applications in quantitative genetics, where estimating genetic covariance components across multiple phenotypic traits is of central interest. We implement our method in a new python package mlmm-mom, and demonstrate how our method adapts to several common experimental designs in this domain.