Papers

We prove the rectangular cyclotomic expansion conjecture of Kameyama, Nawata, Tao, and Zhang for reduced colored HOMFLY-PT polynomials of arbitrary zero-framed knots. We also prove their explicit double-twist formula for every rectangular color and all pairs of nonzero integer twist parameters. For each knot, we construct a double Newton expansion with coefficients in $\mathbb Z[A^{\pm1},q^{\pm1}]$. These coefficients are uniquely determined and independent of the width and height of the rectangular color. The construction uses finite-rank representation symmetries and integral interpolation. The double-twist formula follows from a simultaneous exterior-power comparison of the two four-point Casimir operators and the known one-row formula. Finite transition formulas express the double Newton coefficients of double-twist knots in terms of the prescribed twist coefficients.

In this paper, we construct the matrix nth root of a multivariate polynomial with coefficients in a field K which possesses an nth root of unity. Moreover, we explain why this concept is interesting. Finally, we give several examples to illustrate our construction.

I prove that all Mellin residues are determined exactly by integer samples known up to errors smaller than every algebraic order, for an explicit class of functions consisting of a slow component with a complete inverse-power expansion and countably many carriers $\exp(2\pi i\sigma 2^k a^x)$ with admissible envelopes. A single exceptional null set of parameters $a>1$ works for the entire function class. The key estimate is an eventual uniform lower bound for a polynomially weighted sampling Gram matrix on windows of length $N^{1/2}$, with $O(N^{1/4})$ active modes. I derive this bound from parameter-averaged oscillatory integrals and a finite trace-moment comparison with product Haar measure, retaining the exact resonances among dyadic frequencies. Summed Taylor estimates then identify every coefficient of the slow asymptotic expansion, and a Mellin continuation argument identifies these coefficients with the actual residues. I also give an alternative derivation from existing higher-correlation estimates, an infinite-mode example, and exact counterexamples at integer parameters. The result supplies an interpolation-invariance statement for a model class motivated by Mellin averaging in quantum gravity.

We prove the strong Artin conjecture for the largest solvable Artin representations in three dimensions, whose projective image is isomorphic to the affine special linear group $C_3^2\rtimes \mathrm{SL}(2, 3)$. It is also the last solvable case in three dimensions. This is achieved with a new case of base change and automorphic induction for non-Galois quartic extensions with no intermediate fields. We additionally deduce the strong Artin conjecture for all 6-dimensional primitive solvable Artin representations.

We study the short-maturity implied volatility and the skew stickiness ratio for baskets of assets with continuous, possibly rough, stochastic volatility. The fluctuation of the instantaneous basket variance has two sources: fluctuations of the constituent variances and fluctuations of the basket weights. We derive a near-the-money implied volatility expansion that separates these contributions. We then specialize the result to volatility models given by general functions of Gaussian Volterra factors and obtain an explicit basket skew coefficient in terms of the short-time kernel asymptotics, the factor sensitivities, and the return-factor correlations. A density expansion justifies differentiation of the near-the-money expansion at the money. Finally, using a Malliavin representation of the dynamics of total implied variance, we prove that the short-maturity skew stickiness ratio converges to the universal limit $H + 3/2$ for Gaussian factor basket models with $H \in (0, 1/2]$.

We construct Komlós signing laws with discrepancy below $6.84$, independent Gaussian reference blocks and exponentially many balanced signings. One law preserves hard constraints and exact conditional means while its reference controls every joint convex cost. We resolve both Reis–Rothvoss Schatten conjectures. For $n$ symmetric $n\times n$ matrices, one prescribed-mean signing law bounds all centered Schatten discrepancies with sharp powers of $n$ and a universal constant. One Gaussian event of measure $e^{-O(n)}$ controls every unitarily invariant norm at the square-function radius. We prove Reis's volume-ratio conjecture: every $r$-dimensional quotient of the $a\times b$ trace-class unit ball has volume ratio $O(\sqrt{1+\min(a,b)/r})$. We resolve Nutz–Wang–Zhang's directional and supermartingale conjectures and the higher-dimensional Bass-flow conjecture. For square-integrable laws, full convex reference support is the universal finite-potential criterion in Tschiderer's $q$-Bass problem. Every driver, even atomic, admits one extended potential and shift for all maximal-covariance martingale optimizers of each irreducible pair with full terminal affine span. For Talagrand's problem, five copies of a compact balanced Gaussian half-mass set contain compact convex output of mass at least $3/4$, also in Wiener space. Cheeger's bound is asymptotically exact throughout subexponential Dirichlet spectral ranges of mixed weighted products. For standard Borel sources, a Blackwell-greatest perfectly private full-data release exists exactly when, after discarding conditional point masses, the rest coincide or share a two-point support. Under conditional atomlessness, existence is equivalent to independence. Covariance-matched analytic uniformly log-concave sources can have a greatest useful-data-only private release with infinite information, while no greatest full-data release exists.

Pan and Zhang proved that a positive proportion of positive integers can be represented in the form $a^2+b^2+2^{c^2}$, where $(a,b,c)$ are nonnegative integers. In this note, we give a short proof of their result. The proof uses Romanoff's method and an elementary divisor argument to control the relevant second moment.

We estimate the number of integers up to $x$ whose $y$-friable component exceeds $z$. Uniformly for $\max\{2,\log z\}\leqslant y\leqslant z\leqslant x/2$, we determine the logarithm of this counting function up to an error $O(\log z/\log y)$. This sharpens an estimate arising from a problem in Ford's sieve methods lecture notes. The proof combines classical friable-number estimates with an elementary argument for the lower bound.

In this paper, the detailed behavior of solutions of both the Stokes and the Navier–Stokes system in the half space is investigated when the initial data belongs to $L^1$. We first give a detailed asymptotic expansion of the solution to the Stokes system supplemented with $L^1$ initial data. With the aid of this expansion, we provide two sufficient conditions on the initial data so that the associated solutions of the Navier–Stokes system do not belong to $L^1$, corresponding to long time and short time behaviors, respectively. Moreover, for the $n$- dimensional Navier–Stokes system, a pointwise spatial lower bound of order $n$ is derived in a conic neighborhood of the $x_n$-axis, provided that the initial data decays faster than order $n$. One of the main difficulties to get precise form of leading order term is the non-commutativity of the Leray projection operator with the Laplacian. Our strategy is to employ the special structure of the Green tensor observed by Han and a cancellation property of the initial data. This also yields the sufficient conditions on the initial data to get precise asymptotic behavior via comparing the linear and nonlinear parts.

Analyses of dual lattice attacks have often assumed that the individual scores associated with short dual vectors are mutually independent. Laarhoven-Walter used this heuristic to derive explicit trade-offs between the target radius and query time for bounded distance decoding (BDD) with preprocessing. Ducas-Pulles subsequently demonstrated theoretical and experimental failures of this heuristic and proposed an alternative model conditioned on the target norm. In this note, we prove an explicit asymptotic trade-off for (decision-)BDD with preprocessing in the Haar-random lattice model, without heuristic assumptions. Using moment identities of Siegel and Rogers, we analyze cosine scores over complete dual balls and bound both error probabilities when distinguishing targets planted at a prescribed radius from uniform targets modulo the lattice. Optimizing the dual radius yields a trade-off between target radius and query time that matches the asymptotic prediction from the conditional model of Ducas-Pulles.

We prove that below the multicritical (Nishimori) point the ferromagnetic boundary of a biased Ising spin glass, argued by Nishimori (1986) to be exactly vertical, can be vertical and can also be reentrant, so that cooling at fixed disorder destroys order. It is vertical in the Sherrington-Kirkpatrick (SK) model with a Curie-Weiss bias $J_0$, and reentrant near the triple point of fully connected $p$-spin glasses and, at one disorder strength, on a decorated planar $\mathbb{Z}^2$-periodic lattice. For every $T<J$ the SK magnetization vanishes for $J_0\le J$, continuously at $J_0=J$, and is macroscopic for $J_0>J$. For every integer $p\ge3$, at each bias slightly larger than the triple-point bias, the $p$-spin magnetization is macroscopic at the Nishimori temperature and vanishes at an explicit lower temperature; this is Nishimori's recent replica prediction in two-temperature form. On the lattice ($\mathbb{Z}^2$ with each bond replaced by a fixed series-parallel gadget; maximum degree four), with iid $\pm J$ couplings at $\mathbb{P}(J_e=-1)=9/10000$, the Gibbs state is almost surely unique at high temperature, non-unique in a window containing the Nishimori temperature, and unique again at low temperature. As Dey and Kang observed, the SK boundary is set by the zero-field susceptibility together with an envelope bound on the free-energy gain in a field, which follows from the gauge inequality. We prove that this susceptibility is the curvature at the origin of the zero-field Parisi PDE solution, which Lopatto evaluated for every $\beta>1$ via his theorem that the support of the Parisi measure accumulates at the origin; we give a short alternative proof of that theorem. The $p$-spin and lattice results and the alternative proof are computer-assisted. The models are proxies for the nearest-neighbour $\pm J$ model on $\mathbb{Z}^2$ and say nothing about $\mathbb{Z}^d$.

Call a set $A$ of positive integers admissible if, whenever $a_1\cdots a_r=b_1\cdots b_s$ with $a_1,\dots,a_r$ distinct elements of $A$ and $b_1,\dots,b_s$ distinct elements of $A$, necessarily $r=s$. Erdős asked whether admissible sets can have density $1-\varepsilon$ for every $\varepsilon>0$, and whether $\{1,\dots,N\}$ always contains an admissible subset of size $(1-o(1))N$. For the variant in which repetitions are allowed both questions were answered negatively by Erdős, Ruzsa and Sárközy and by Granville and Soundararajan; for products of distinct elements, the first question was answered only recently (with density bound $7/8$), and the second has remained open. We show that every admissible $A\subseteq\{1,\dots,N\}$ satisfies $\sum_{a\in A}1/a\le\tfrac12\log N+(\log\log N+2)^2$, and that there is an absolute constant $\eta>0$ such that every admissible $A\subseteq\{1,\dots,N\}$ has $|A|<(1-\eta)N$ for all large $N$. Both questions therefore have negative answers. The proofs are elementary; the second rests on a coupling that replaces the largest divisor of an integer composed of small primes, which avoids the divisor-function losses inherent in counting quotients along a multiplication table. Both negative answers are formally verified in Lean 4 against the statements of the Formal Conjectures project.

Chollet's permanent conjecture asks whether per(A o B) <= per(A) per(B), where o denotes the entrywise (Hadamard) product, for complex Hermitian positive semidefinite matrices. We present computer-assisted proofs of two restricted forms with explicit constants strictly smaller than one. For every integer n >= 10 and every such matrix A of rank at most four, the self-conjugate ratio per(A o conjugate(A)) / per(A)^2 is bounded by 999991742359 / 10^12 when the denominator is nonzero. For order nine, we obtain the bound 999815240367 / 10^12 for correlation matrices whose four nonzero eigenvalues all equal 9/4. The first argument combines complex-sphere integrals, spectral subspace tilts, projection bounds, exact finite covers, and an analytic infinite tail. The second uses a quadratic relation among nine Gram vectors, a positive operator on the ten-dimensional space of quadratic forms, and rigorously bounded entropy. All decisive finite calculations use rational arithmetic and full closed-domain certificates. The unrestricted conjecture, including general non-tight order-nine rank-four matrices, is outside these results.

Salikhov (2008), Zeilberger-Zudilin (2020) and Bai (2026) bound the irrationality measure of pi with the same shape of complex integral. This note gives one closed-form law mu(a0,a1,b) for the whole family, shows that it reproduces all three published records, proves that it is exactly scale-invariant, and gives evidence that Bai's parameter point is the global optimum of the family - a statement Bai explicitly declines to make about her own result. The admissible configurations are forced by a Machin condition: zeros at 0, +-w, +-wbar and poles at +-P give a linear form in 1 and pi if and only if P^2 - N(w) = 2P Im(w). The law depends only on the two ratios a0/a1 and b/a1, so the infinite three-parameter lattice is exactly two-dimensional. The 2-adic saving is exactly 5(2a1-b)/2 per n, in two regimes, and no further saving exists. The decay rate is governed by the min-max value of the steepest-descent method, the lowest level at which the two endpoints of the integral become connected within a sublevel set of the integrand; the value obtained from the least critical level is a lower bound for mu and serves as a screening. Among the sixteen admissible configurations with P < 100, exactly one carries a triple that yields a bound, namely Salikhov's (P = 5, w = 1+2i), and the least mu over all sixteen and over all primitive triples with a0, a1 <= 10 and b <= 22 is 7.103205334137 at (2,2,3). Within that configuration the minimum is 7.101862832356 at (1857,1857,2785), which is Bai's point, the vertex sitting exactly at Q = 2a/(3a-2b) = 3714. Each statement carries an explicit label, PROVED or VERIFIED or CONJECTURE, and the reservations of scope are stated in the note. An ancillary file re-derives every numerical claim from scratch, with no external dependency.

We prove that the Benjamin-Bona-Mahony equation is almost surely globally well-posed with respect to random Gaussian initial data of negative Sobolev regularity in $H^{s}(\mathbb{T})$ for any $s>-\frac 14$. This result is sharp in view of the mild probabilistic ill-posedness due to Oh-Tzvetkov (2026). This also improves on a previous result of the author which established this only in a logarithmically negative regularity. To break through this logarithmic regularity barrier, we combine the $I$-method with the low-high argument from Bona-Tzvetkov (2007), which motivates a refined first-order expansion for solutions.

Let $G$ be a group acting continuously on a compact metric space $X$. This naturally induces an action of $G$ on the space $\mathcal{K}(X)$ of nonempty compact subsets of $X$. In [KS26], Kra and Schmieding define a system $(X,G)$ to be IC-rigid if the union of the supports of the $G$-invariant Borel probability measures on $\mathcal{K}(X)$ is as small as possible. They leave open the question of existence of IC-rigid abelian systems. We show the existence of IC-rigid actions for three classes of abelian groups: mixing rank-one $\mathbb{Z}$-actions, horocycle flows, and certain mixing actions on a Cantor space of $G=\bigoplus_{n=1}^{\infty}G_n$, where $(G_n)_n$ is any sequence of nontrivial finite abelian groups. In the course of proving these results, we introduce the notion of weakly asymptotic pairs and establish its close connection to topological minimal self-joinings.

We define a class of classes of complete first-order theories, denoted by $\mathfrak{D}_{n\text{-var}}$, using positive consistency-inconsistency configurations and generalized indiscernibles. It contains the classes of stable, simple, NIP, NTP$_1$, NTP$_2$, NATP, NCTP, NBTP, NWP, NGP, and NPM$^{(k)}$ theories. For any dividing line $D\in\mathfrak{D}_{n\text{-var}}$ and any complete first-order theory $T\notin D$, we can always find a formula $\varphi(x,y)$ witnessing $T\notin D$ with an indexed set of parameters such that the set of its instances required to be consistent has a realization whose algebraic dimension over the whole set of parameters is $|x|$. We call statements of this form $n$-variable theorems, as they may be regarded as weak versions of one-variable theorems. Under the appropriate hypotheses on $T$, we prove that $T\in D$ if and only if the corresponding theory belongs to $D$ for (i) $T^{gt}$, (ii) $T_P$, (iii) $T^{ind}$, (iv) $T^G_K$, (v) ACF$_T$, and (vi) any completion of $T^{\bar\delta,nc}_g$. These are, respectively, the theories of generic trivializations, lovely pair expansions, $H$-structure expansions, vector spaces with a dense-codense generic $K$-subspace, algebraically closed fields with a distinguished subfield, and generic non-commutative derivations of algebraically bounded fields. The $n$-variable theorem is essential in proving (i)–(v). We also prove preservation for a special case of Mordell-Lang theories. We extend the framework to a larger class $\mathfrak{D}^h_{n\text{-var}}\supseteq\mathfrak{D}_{n\text{-var}}$, capturing higher-arity dividing lines such as NOP$_k$, NFOP$_k$, and NIP$_k$ for $1<k<\omega$. The $n$-variable theorem and preservation results (ii), (iii), (iv), and (vi) also hold for this larger class, while (i) holds assuming ${\rm acl=dcl}$.

Offline reinforcement learning seeks optimal decision rules from previously collected data. In some applications, a decision can be an entire function, such as a fluence map in radiation therapy or a smooth movement trajectory in robotics. In this paper, we study the finite-sample theory for fitted Q-iteration (FQI) with functional actions in a discounted infinite-horizon setting. Three major difficulties arise in this setting: first, the absence of a Lebesgue probability density for functional actions complicates coverage descriptions; second, conventional coverage requirements can be restrictive; and third, the large functional action space makes greedy optimization in FQI challenging. To address these difficulties, we study smoothness-regularized policy search under a critic-relative coverage condition. This condition measures how well logged data distinguish relevant action-value differences without requiring an action density. Our main theorem gives finite-sample guarantees for learned-policy regret relative to the best value within a fixed smooth class of functional-action policies. The results allow trajectory lengths to be either bounded or growing and Q-functions to be fitted by either functional-input kernel ridge regression or adaptive functional neural networks. For a few examples, we can obtain polynomially decaying regret bounds in the number of logged transitions, up to logarithmic factors, with logarithmically many FQI iterations. Numerical experiments show gains of learned functional-action policies over constant-action policies and support our adoption of a critic-relative coverage condition.

Binomial coefficients for multidimensional arrays count occurrences of particular configurations. For the sake of presentation, the emphasis is put on column-binomial coefficients for two-dimensional finite arrays. In this article, we first show that these coefficients can be computed through some Magnus transform. To get structural and combinatorial information on arrays, we then define \((k,\ell)\)-binomial equivalence for finite arrays and, from it, the \((k,\ell)\)-binomial complexity function of an infinite array. Roughly, two finite arrays are \((k,\ell)\)-binomially equivalent when they share the same number of subarrays of size at most $k\times \ell$. We obtain general results on the \((k,\ell)\)-binomial complexity of infinite arrays coding direct products of infinite words. Our main theorem gives an exact formula for the \((k,\ell)\)-binomial complexity of the two-dimensional Thue–Morse array. To that end, we closely examine the action of the bit-wise complement on the \(k\)-binomial equivalence classes of the factors of the Thue–Morse word.

We develop a fully relativistic extension of the energy-conserving, unstaggered electromagnetic-potential particle-in-cell method introduced in our earlier nonrelativistic work. As in the nonrelativistic generalized-momentum formulation, the fields are advanced by a Crank–Nicolson discretization of the Lorenz-gauge potential equations, the current is deposited from particle orbits, and charge is advanced through the discrete continuity equation. The new ingredient is a generalized-momentum adaptation of the Higuera–Cary relativistic particle pusher. The vector-potential force is decomposed into an orbit-discrete-gradient part, which enforces the same finite-difference chain rule used in our earlier nonrelativistic formulation, and a skew part, which generates a Higuera–Cary magnetic rotation in mechanical momentum. For prescribed fields and frozen orbit data, this generalized-momentum Higuera–Cary map is a unit-Jacobian conjugacy of the mechanical Higuera–Cary map and preserves particle phase volume in $(\xx,\PP)$. For the fully coupled implicit step, the same orbit data define the current deposit, mesh-to-particle gather, vector-potential chain rule, and relativistic kinetic-energy secant velocity. The resulting particle work cancels the Crank–Nicolson field-energy balance, yielding exact relativistic total-energy conservation up to nonlinear solver tolerance, orbit-quadrature error, and roundoff. A self-adjoint Fourier projection removes modes on even-grid Nyquist planes while preserving the particle–grid work identity. Numerical tests are organized around cold relativistic two-stream and Weibel/filamentation instabilities, with diagnostics for energy drift, Gauss's law, the Lorenz gauge, particle–grid work, and the orbit chain rule.

We present Suêtes, a fully differentiable, non-hydrostatic, limited-area atmospheric dynamical core implemented in JAX. Suêtes includes both a semi-implicit semi-Lagrangian scheme and a Split-Explicit Runge-Kutta scheme with acoustic substepping, enabling applications from coarse regional simulations to convection-permitting dynamics. Its terrain-following geometry, spatial discretization, time integration, initial conditions, lateral boundary treatment, and included physical parameterizations are all compatible with reverse-mode automatic differentiation. A single reverse-mode pass can therefore compute sensitivities of forecast diagnostics to selected initial and boundary fields, physical parameters, topography, and computational geometry. We provide a series of numerical experiments—including a tracer-source inverse problem, a sensitivity analysis of a three-dimensional squall-line, adversarial perturbation construction an ERA5-driven downslope hurricane force wind event, topography optimization, and terrain-following coordinate optimization—demonstrating the versatility of Suêtes as a framework for gradient-based diagnosis and model development at regional and convection-permitting scale.

In a previous paper, the authors introduced, by analogy with Stein manifolds, a class of complex manifolds called $\mathscr{M}$-manifolds. This class contains all complex projective manifolds. Moreover, every compact $\mathscr{M}$-manifold is Moishezon. In this note, we construct examples showing that the class of compact $\mathscr{M}$-manifolds sits properly between the classes of projective and Moishezon manifolds.

It is well-known that Douady and Hubbard proved that the Mandelbrot set is connected by constructing a uniformizing holomorphic map for its complement from the Böttcher coordinates. This strategy has a clear analogue in the setting of quasiregular Mandelbrot sets. In this paper, we show that the natural generalization of the Douady-Hubbard uniformizing map is quasiconformal in a neighbourhood of infinity, as well as, in fact, outside a certain neighbourhood of the quasiregular Mandelbrot set. On the other hand, we also show that this uniformizing map can reverse orientation at some parameters, which forces it to have folds and, in particular, prevents it from being injective. This feature cannot happen in the holomorphic setting. The connectivity of the quasiregular Mandelbrot sets therefore cannot be established along the Douady-Hubbard route, and remains open.

We construct a sheaf of $\frac{1}{2}\mathbb{Z}$-graded meromorphic open-string vertex algebras from a transitive Courant algebroid $E$ equipped with a generalized metric and a chosen generalized Levi-Civita connection on a smooth manifold. Using a holonomy principle for transitive anchored bundles, we realize these algebras as parallel sections of a tensor algebra bundle built from bosonic-fermionic affinizations of $E$. For even-rank $E$ with suitable Clifford module bundle data, we construct a sheaf of canonically twisted modules containing the canonical weighted spinor bundle in weight zero. The associative algebra of parallel tensors acts on spinor sections through iterates of an $E$-connection combining covariant differentiation and Clifford multiplication, with mixed terms incorporating the action of the bosonic connection on fermionic tensors. Distinguished states built from the inverse Courant pairing and inverse generalized metric give vertex-operator components, acting on embedded spinor sections as the canonical Dirac generating operator, its formal adjoint, and their anticommutator, the generalized Hodge Laplacian. In the exact Courant case with three-form $H$ and the exterior algebra Clifford module, the operators specialize to $d-H\wedge$, its adjoint, and the twisted Hodge Laplacian, reducing to the ordinary de Rham and Hodge operators when $H=0$. This extends the vertex-operator realization of geometric differential operators from functions to weighted spinors and differential forms, within the program of studying two-dimensional supersymmetric nonlinear sigma models.

In spatial matching markets, a supply unit's flexibility is measured by its service radius, the maximum distance at which it can serve demand. In dimensions $k \geq 2$, we study how a platform should allocate service radii among the supply nodes subject to a budget on their sum. The platform makes this choice before observing supply and demand locations, with the objective of maximizing the expected fulfilled demand. We show that the shape of a preferred allocation depends on the total budget: under suitable conditions, large budgets favor allocations that are more uniform in the sense of majorization, while small budgets favor concentration. We also characterize a non-uniform allocation that is asymptotically optimal for a very-sparse regime, and show that the uniform allocation is suboptimal in this regime. Our results provide theoretical explanations for the radius allocation questions raised by the numerical experiments in [ASY26b].

We study a fractional logistic-type elliptic problem on a bounded domain with homogeneous exterior conditions and linear and nonlinear integral interactions. The interaction kernels may be nonsymmetric and sign-changing, so the problem need not admit a variational formulation. In the symmetric competitive regime, coercivity yields a weak solution by global minimization. For arbitrary fixed values of the interaction parameter, explicit smallness conditions allow us to apply Schauder's fixed-point theorem without symmetry or sign restrictions on the kernels. Additional sign assumptions give nonnegative solutions. When the nonlinear exponent satisfies $q\geq2$, a contraction condition yields uniqueness in an invariant ball and convergence of the Picard iteration. Finally, as the fractional order approaches one, we prove subsequential convergence to a solution of the local Dirichlet problem, together with convergence of the fractional energies to the Dirichlet energy. Uniqueness of the local solution in the limiting ball gives convergence of the entire family.

We show that all normal covers of a fixed finite graph $G$ sitting between the universal and maximal abelian covers share the same atomic part of their spectral measures. We give an explicit double formula for the mass of atoms which generalises and unifies known results on universal covering trees and maximal abelian covers. It also proves an extension of a conjecture in physics literature on hyperbolic lattices. Moreover, we also establish the logarithmic Hölder regularity of the continuous part of the spectral measures. Our proof combines techniques from von Neumann algebras, matching theory and the monotone labelling method. In an appendix, we prove a converse generalised Gallai-Edmonds structural theorem which might be of independent interest.

We introduce the T-Adaptive Segment Routing, a multi-period optimization problem that emerges in the traffic engineering of core IP/MPLS networks when scheduled maintenance operations are involved. The problem seeks a sequence of Segment Routing (SR) configurations that can adapt to a multi-period scheduled maintenance, while allowing limited number of path reconfigurations between successive time steps. Instead of only minimizing the classical Maximum Link Utilization (MLU) criteria, we propose a more refined lexicographic objective that minimizes the sorted vector of link loads in its entirety, thereby enabling a more efficient resource utilization across all links. To tackle this problem, we develop three exact formulations for solving the resulting lexicographic optimization problem. Among these, the Stela and Carla formulations provide the most favorable computational basis for trajectory-based optimization. We thus develop a trajectory column-generation scheme for these formulations, using pricing oracles that vary from heuristic to exact. Exact pricing gives an exact solution of the LP relaxation of the trajectory master, but it does not itself give a certificate of integer optimality: solving the integer master over the columns generated at the root node only certifies optimality within the generated column pool. To overcome this gap, we incorporate trajectory column generation into a Branch-and-Price framework. The branching rule operates on aggregate variables representing segment usage, which correspond to the original compact routing variables. Thus, Branch-and-Price guarantees global integer optimality for every Stela and Carla rank, and when all ranks are solved to exactness, it guarantees lexicographic optimality of the final solution.

While nonlinear systems can be generally described as infinite-dimensional linear systems via their Koopman semigroups, the direct synthesis of state observers on the premise of learned Koopman operator models, in an analogous manner to finite-dimensional linear systems, remains an open problem. In this paper, the concepts of Koopman observability (exact and approximate), Koopman detectability (exponential and strong), and Koopman–Luenberger observer of nonlinear systems are provided. By defining the Koopman semigroup on a reproducing kernel Hilbert space (RKHS), the conditions for Koopman exponential detectability are established, which involves solving an operator Lyapunov inequality that only implicitly involves the the infinitesimal generator of the Koopman semigroup. Through the use of a finite-horizon Yosida approximation of the Koopman generator and an empirical sampling operator, the data-driven solution of the observer gain can be reduced to a finite-dimensional subspace specified by the sample and hence fulfilled by convex semidefinite programming. The state observation error is guaranteed to be bounded in mean squares, under closed-loop dissipation conditions on the sampling operator and its empirical operator.

The order $m$ antisymmetrised matrix product ensemble is represented by the product $X_1^T\cdots X_m^TJX_m\cdots X_1$, where $X_1,\ldots,X_m$ are independent real Ginibre matrices and $J$ is the elementary antisymmetric matrix, while the order $m$ Hermitised matrix product ensemble is represented by $X_1^\dagger\cdots X_m^\dagger HX_1\cdots X_m$, where $X_1,\ldots,X_m$ are now independent complex Ginibre matrices and $H$ is a Hermitian matrix drawn from the Gaussian unitary ensemble. These ensembles have recently been shown to be related to certain Muttalib–Borodin ensembles and integrals of Harish-Chandra–Itzykson–Zuber type, thereby motivating further investigation into their eigenvalue statistics. In this work, we construct ribbon graphs and constellations that are enumerated by the mixed cumulants of these ensembles and give loop equation characterisations for the generating functions of said cumulants when $m=1$.