Papers

Given two finite essential adjacency matrices $A$ and $B$, Hazrat's graded classification conjectures posit that an order preserving $\mathbb{Z}[x,x^{-1}]$-module isomorphism of $K_0$ groups implies graded Morita equivalence of the Leavitt path algebras of $A$ and $B$, while the pointed version predicts a graded isomorphism of the Leavitt path algebras when the $K_0$ group isomorphism additionally preserves the class of the regular module. For any field $k$, we show that the Leavitt path algebras over $k$ of $A$ and $B$ are graded Morita equivalent if and only if $A$ and $B$ are strong shift equivalent. By appealing to counterexamples of Kim and Roush from symbolic dynamics, this shows that Hazrat's graded classification conjectures are false.

The 1-2-3 Conjecture states that for every graph without isolated edges, there exists an edge-weighting from $\{1,2,3\}$ such that adjacent vertices receive distinct sums of weights on their incident edges. In the sequence variant, adjacent vertices are to be distinguished by the sequences of weights on their incident edges. In this paper, we investigate whether, for every graph without isolated edges and for a fixed proper edge colouring (where colours are interpreted as weights), there exists a global total order of the edges such that the resulting sequences of incident weights distinguish adjacent vertices. For connected graphs, we prove that such an order exists whenever there exist two adjacent vertices that have distinct sets of incident weights. This yields a positive answer for every proper edge colouring of a connected non-regular graph or a connected graph of class two. Moreover, a probabilistic argument gives the same conclusion for connected regular graphs of degree at least six.

We study strong consistency and optimal tracking of the Astrom-Wittenmark self-tuning regulator with unknown input gain. Existing results establish consistency of all parameter estimates for the unmodified recursion under additional growth conditions on reference information. Other approaches obtain stability and optimal tracking by adjusting the estimates used in feedback or adding decaying probing signals. For a class of minimum-phase linear systems with martingale difference noise, ordinary least squares (LS) with certainty-equivalent control achieves stability and optimal tracking almost surely for each bounded reference, provided the recursion is well defined. No reference excitation, adjustment of feedback estimates or added probing is required. When at least two parameters are estimated, all parameter estimates are strongly consistent, with squared norm of the estimation error $O(\log\log\log n/\log n)$ almost surely. A law of the iterated logarithm under zero reference shows that this rate cannot in general be improved. We also derive a logarithm law for cumulative squared tracking residuals, with an explicit dependence of the leading term on reference energy. A chi-square limit yields asymptotically exact parameter confidence ellipsoids for each bounded reference. The proof combines a logarithmic lower bound on the cumulative squared difference between an auxiliary LS prediction of the noise and the reference with estimates from the LS recursion. If the gain estimate were inconsistent, these bounds would yield incompatible estimates for the same weighted sum of squares, establishing gain consistency without assuming stability.

Increasingly frequent and severe wildfires threaten ecosystems, public health, and infrastructure. Early detection is vital but limited by existing monitoring systems. Drones offer mobile, real-time coverage, but optimizing sensor placement and drone routing in dynamic fire zones remains challenging. To address this, we introduce WFDroneBench, an open-source Python benchmarking library for early wildfire detection that integrates machine-learned risk maps with optimization-based deployment strategies for sensors, charging stations, and drones. It evaluates risk maps, optimization strategies, and monitoring equipment using standardized metrics and realistic wildfire simulations. The framework supports benchmarking across predictive and decision-making components: machine learning researchers can assess risk models and compare routing strategies. WFDroneBench includes 7746 scenarios across 49 locations, built from historical ignitions, real-world wildfire risk maps, and simulated fire spread, along with two ground detector and three drone routing strategies. Our experiments show that the risk-aware strategy Max-Coverage significantly outperforms other baselines when risk maps are sufficiently accurate, achieving the fastest detection on the most difficult fires. We further find that risk-aware static infrastructure helps even under an imperfect risk map and drone-based detection outperforms ground sensors. Finally, our results reveal two key open challenges: (i) detecting small fires rapidly and reliably, and (ii) improving risk-map prediction, where the gap between ground-truth ignition patterns and available risk maps highlights a significant opportunity for ML innovation. We openly release all code, data, and documentation.

We study interview positioning in the secretary problem when an applicant is uncertain about their own absolute rank. The employer uses a given rejection threshold, and the applicant's self-rank distribution is truncated geometric. We show that the applicant need only compare the first eligible position with the final position and that, for finite $n$, a unique critical value exists precisely when $r(r+1)>n-1$. If $r_n$ is the employer's rejection threshold and $r_n/n\to\alpha\in(0,1)$ as $n\to\infty$, then \[ \lim_{n\to\infty}q_c(n,r_n) = q_c(\alpha) = \frac{1-\alpha}{1-\alpha+\alpha^2}. \] Thus uncertainty about absolute rank produces an explicit asymptotic phase transition in optimal interview positioning.

We prove that for every smooth well formed weighted complete intersection of general type of dimension $n>0$, the Hodge number $h^{0,n}(X)$ is positive; in other words, its Hodge level is maximal. We also obtain an explicit lower bound for the geometric genus $p_g(X)$. This implies that the only weighted complete intersection of general type that is not a numerical intersection with a linear cone with $p_g(X)=1$ is $X_{6,6}\subset\mathbb{P}(1,2,2,3,3)$. We also prove the Ambro–Kawamata effective non-vanishing conjecture for smooth well formed weighted complete intersections.

This article presents a data-driven approach to minimally adjust forecast-based steady-state generation dispatch in electrical transmission networks. The goal is to manage congestion caused by uncertainty in demand and renewable generation under a fixed automatic generation control (AGC) policy operating without saturation. Under this policy, we first show that the post-AGC residual injection distribution is invariant to the scheduled generation dispatch. This result separates line loading into a dispatch-dependent baseline and an uncertainty-induced residual component. Using historical forecast errors, we then apply split conformal prediction to construct transmission-line margins with distribution-free, finite-sample linewise or joint coverage guarantees. We also calibrate the aggregate pre-AGC mismatch to determine the generation headroom required for the prescribed AGC response. We incorporate these calibrated quantities into a convex secure re-dispatch problem that minimally adjusts the nominal schedule while reserving sufficient transmission capacity and AGC headroom. The resulting corrected dispatch provides finite-sample probabilistic guarantees on per-line security and unsaturated AGC operation. Case studies on the RTS-GMLC system demonstrate substantial reductions in out-of-sample line-limit violations with modest operating-cost increases.

We prove that every closed $m$-quasi-Einstein manifold $(M^n,g,X,\lambda)$ with constant $\lambda\leq0$ is trivial whenever $m\leq-2$. This establishes the Colling–Dunajski conjecture in the range $m\leq-2$ and, for $n\geq5$, extends the previously known range $m\leq2-n$. This result is placed within a broader study of closed generalized $m$-quasi-Einstein manifolds $(M^n,g,X,\lambda)$. We establish differential and integral identities for such manifolds. These identities yield criteria for conformality, the Killing condition, and triviality, and provide a unified framework for several rigidity phenomena. After this, we recast the integrated Bochner formula as a Witten-type Hodge-energy identity. The cancellation of its quartic term at $m=-2$ leads to a triviality theorem in the generalized setting under a natural sign condition on the integral of $\langle X,\nabla\lambda\rangle$. An identity for the drift Laplacian gives a new proof of the previously known triviality result for $m\leq-n$ and extends it to generalized $m$-quasi-Einstein manifolds under a pointwise sign condition on $\langle X,\nabla\lambda\rangle$. Finally, complementing our criterion characterizing when a conformal potential field is Killing, we exhibit in the appendix a closed generalized $m$-quasi-Einstein manifold whose potential field is conformal but non-Killing.

We study multi-product replenishment subject to aggregate operating limits imposed pathwise in place of expectation. On-hand and in-transit inventory, warehouse capacity and an emission allowance define a safe set, and the resulting multi-product lost-sales problem is posed as a safe stochastic optimal control problem. The Hamilton–Jacobi–Bellman (HJB) equation is represented as a system of forward–backward stochastic differential equations (FBSDEs) and solved by learning the value function and its gradient, with a quadratic program derived by control barrier function (CBF) is solved at each decision epoch. Because purchase orders enter the in-transit pipeline only, the in-transit constraints have relative degree one while the on-hand constraints have relative degree two. We extend the stochastic high-relative-degree construction of (Clark, 2021) from a single barrier to four coupled aggregate constraints of mixed relative degree, and we establish its initial-condition requirement by design.

Let $c\phi_k(n)$ denote the number of $k$-colored generalized Frobenius partitions of $n$. We prove that, for every $m\geq2$ and every $k\equiv2\pmod{2^m}$, \[ \sum_{n\geq0}c\phi_k(n)q^n\equiv\frac{\varphi(q)\,(q^2;q^2)_\infty}{(q;q)_\infty^2}\sum_{n\geq0}c\phi_{k/2}(n)q^{2n}\pmod{2^m}, \] where $\varphi(q)$ is the classical theta function. For $m=2$ this recovers a congruence of Chan, Wang, and Yang. Applied with $k=18$, we determine $c\phi_{18}(2n+1)$ modulo $16$ completely. In particular, we also prove \[ \sum_{n\geq0}c\phi_{18}(6n+1)q^n \equiv4\sum_{r\in\mathbb{Z}}q^{r(3r-1)/2}\pmod{16}, \] which proves the congruences $c\phi_{18}(30n+19)\equiv c\phi_{18}(30n+25)\equiv0\pmod{16}$ recently conjectured by Das, Nath, and Sarma (2026). It also yields further congruences modulo $16$ and a simple modulo-$8$ characterization that recovers and extends a recent congruence of those authors. As a second application we set $k=10$ and determine $c\phi_{10}(2n+1)$ modulo $8$.

We develop tools to compute moduli spaces of geometric functorial field theories as mapping spaces of equivariant simplicial presheaves. Given a d-dimensional geometric structure F, presented as a presheaf on the site of smooth families of d-manifolds, we define its Cartesian realization, which is an O(d)-equivariant simplicial presheaf on the site of Cartesian spaces. We use Cartesian realizations to present the moduli space of functorial field theories with geometric structure F as a mapping space between O(d)-equivariant simplicial presheaves. In a companion paper, we use this result to compute the moduli space of smooth one-dimensional oriented Riemannian functorial field theories valued in an arbitrary smooth symmetric monoidal infinity-category.

Let $\overline{po}_k(n)$ denote the number of $k$-colored overpartitions of $n$ into odd parts. For every odd number of colors, we establish infinite families of internal congruences modulo high powers of $2$. The proof proceeds by expressing the relevant difference generating functions as a fixed theta quotient multiplied by polynomials in an auxiliary function. Uniform lower bounds for the $2$-adic valuations of the polynomial coefficients then yield the stated internal congruences. These results generalize an internal congruence of Saikia, Sarma and Sellers (J. Ramanujan Math. Soc.~2025) on overpartition triples into odd parts.

We extend the sphere-summing construction of \cite{AZ} from torus sums of spheres to symplectic sums along surfaces of arbitrary genus. The relative surfaces may have arbitrary positive intersection number with the summing surface. We give formulas for the genus and self-intersection of the resulting surface, together with graph and simultaneous versions of the construction. The original elliptic-surface case is recovered as a special case, and higher-genus examples are obtained from hyperelliptic Lefschetz fibrations and braided symplectic surfaces in $\Sigma_h\times S^2$.

We prove that the uniform Turán density of $K_4^{(3)}$ is equal to $1/2$. Kielak, Král', Lamaison, Liu, Shu, and Wu have recently proved the same result using combinatorial methods, while our proof is Fourier-analytic. This result answers a question of Erdős and Sós from the founding 1982 paper on this topic.

We show that scalar local minimizers of the Allen-Cahn functional with the double-well potential and homogeneous Neumann boundary conditions on a bounded Lipschitz domain are isolated in the $L^1$-topology. This affirmatively answers a question raised by Kohn and Sternberg (Local minimisers and singular perturbations, Proc. Roy. Soc. Edinburgh Sect. A 111 (1989)). The proof implements an analytic Lyapunov-Schmidt reduction inspired by the theory of monotone dynamical systems in works of Jiang-Yu and Hirsch-Smith in an elliptic setting.

Banerjee, Chen, and Sivakumar asked at COLT 2015 whether a uniform small-ball condition on the rows of a random design matrix forces a restricted-eigenvalue lower bound whose sample complexity is governed by the ordinary Euclidean Gaussian width of an arbitrary spherical subset. We give a negative answer to the natural distribution-free formulation of that question. For every sample size $n$, we construct a centered, genuinely heavy-tailed row distribution in dimension $p=4^n+1$ and a set $A=C\cap\mathbb{S}^{p-1}$, where $C$ is a closed polyhedral convex cone, such that \[ \inf_{v\ne 0}\mathbb{P}\!\left( \left|\left\langle Z,v\right\rangle\right| \ge \frac{\left\lVert v\right\rVert_2}{\sqrt{2}} \right)\ge \frac{1}{12} \quad\text{and}\quad w(A)<2. \] Nevertheless, for the matrix $X$ with $n$ independent copies of $Z$ as rows, \[ \mathbb{P}\!\left( \inf_{u\in A}\left\lVert Xu\right\rVert_2^2=0 \right) \ge 1-\exp(-2^n). \] Thus no positive constants depending only on the fixed small-ball parameters can yield a lower bound of the proposed form $c_1n-c_2w(A)^2$ with high probability. The construction isolates the obstruction: a marginal small-ball lower bound controls every fixed direction, but does not control the distribution-dependent complexity of searching over many directions. We state the quantifiers explicitly and discuss why isotropic or upper-tail assumptions lead to a different, still meaningful problem.

We derive the second-order asymptotics of randomness extraction from memoryless sources with side information under security criteria based on a broad class of Csiszàr f-divergences, treating both a fixed reference side-information marginal and optimization over that marginal. The conditional varentropy decomposes into fluctuations of the conditional entropy across different values of the side information and the average variance of the conditional surprisal for each value. Without marginal optimization, these contributions yield a Gaussian-mixture second-order profile. With marginal optimization, they combine into the total conditional varentropy, yielding a single Gaussian profile. As corollaries, we obtain second-order expansions for Rényi-entropy criteria of all orders $\alpha \in (0,1)$ and recover the known expansion for total variation distance.

The Katz-Long-Moody construction associates to a representation of the semidirect product of a free group and a braid group, defined by the Artin action, and a nonzero parameter a new representation of the same group. On the pure braid group it corresponds to Haraoka's multiplicative middle convolution for KZ-type equations. For unitary input and a convolution parameter on the unit circle other than one, the construction equips the quotient representation with a canonical non-degenerate invariant Hermitian form. We give a closed formula for its signature in terms of the eigenangles of the input local monodromies, the eigenangles of their ordered product, and the convolution parameter. The formula accounts for the kernel of the form before passage to the quotient and for signature changes at resonant parameters. It determines precisely when the induced form is definite, answering the definiteness problem posed in the companion paper. Definiteness implies unitarizability of the output representation; the converse holds when that representation is irreducible. The proof uses elementary linear algebra: a determinant identity, explicit block-pivot formulas, and an inertia formula for sums of Cayley transforms of unitary matrices. As applications, we compare the rank-one construction explicitly with Haraoka's invariant form for Pochhammer systems, recover the Gauss case of the Beukers-Heckman interlacing criterion, and determine the definite parameter intervals for the Hecke and Temperley-Lieb specializations.

Predetermined stepsize schedules featuring carefully chosen long steps have recently been shown to accelerate gradient descent (GD) on smooth convex functions. A prominent class of such schedules is built through recursive composition. In this paper, we characterize the convergence of these optimized recursive schedules, revealing a nonconstant log-periodic modulation across prescribed horizons. Specifically, we prove that for every $N \geq 1$, the optimized ConPP/OBS-S schedules satisfy $$f(x_{N-1})-f_\ast \le \frac{1}{2N^p\Phi(\log_2N)-1} \frac{L}{2}\|x_0-x_\ast\|^2, \qquad p=\log_2(1+\sqrt2), $$ where $\Phi$ is a positive, Lipschitz, nonconstant $1$-periodic function. We derive this by proving that balanced splitting is optimal at every horizon for these constructions, resolving a conjecture of Zhang and Jiang. Furthermore, for their asymmetric counterparts (ConPD/OBS-F schedules), we show that although optimal splits are not necessarily balanced, the same Silver exponent asymptotically persists alongside a distinct log-periodic modulation.

Grimmer, Shu, and Wang asked whether a balanced constant schedule is $s$-composable at every horizon. More precisely, for an integer $n\geq 1$, let $\bar{h}=1+r$, where $r\in(0,1)$ is the unique solution of $$r^n\bigl(1+n(1+r)\bigr)=1,$$ and run gradient descent for $n$ steps with normalized stepsize $\bar{h}$. The cases $n=1,2$ were known, while the general case $n\geq3$ was left open. We prove the conjecture for every $n$. Our proof gives an explicit, dimension-free nonnegative linear combination of the smooth convex interpolation inequalities. The certificate is assembled from matrices supported on contiguous index intervals. Its off-diagonal entries are automatically positive, its quadratic part is diagonal, and its remaining multipliers reduce to two scalar families. We derive closed forms for those families and prove positivity using strict concavity and elementary rational inequalities. Consequently, for every $L$-smooth convex function, the constant schedule satisfies the sharp mixed terminal Lyapunov inequality conjectured in the original paper, together with the associated simultaneous objective-gap and gradient-norm bounds. All exceptional horizons and boundary indices are treated explicitly.

In recent years, simplicial complexes have gained considerable attention as a useful tool for constructing distance-optimal codes over finite fields. In this article, we construct four infinite families of linear codes over the ring $\mathcal{R}=\mathbb{F}_{q}+u\mathbb{F}_{q}$ with $u^2=0$ using simplicial complexes with one or two maximal elements, and completely determine their Lee weight distributions via exponential-sum techniques. By employing a Gray map on $\mathcal{R}$, we obtain infinite families of distance-optimal codes over $\mathbb{F}_{q}$, including a near-Griesmer family, and establish sufficient conditions for their minimality. Furthermore, we investigate the corresponding subfield codes and derive sufficient conditions for their distance-optimality and minimality, yielding infinite families of Griesmer and near-Griesmer codes.

In the present work we study for a given Banach space the quotient of the family of equivalent norms modulus the collinearity relationship and prove that under a Banach-Mazur type metric this family is a complete metric space, and we also prove the completeness of the family of constructible norms for the Banach space of bounded Lipschitzian mappings between convex sets of a Banach space.

We consider boundary integral methods for Stokes mobility and resistance problems involving smooth axisymmetric particles. A primary numerical challenge is the accurate and efficient evaluation of layer potentials at off-surface points close to particle surfaces. We present a tolerance-driven workflow for evaluating the Stokes double layer potential at such evaluation points (targets) to prescribed accuracy while avoiding unnecessary computational cost. For each target–particle interaction, a fast classifier selects the least costly option estimated to meet the tolerance among standard, upsampled, and special quadrature. Geometry-dependent unit-density error indicators are precomputed and tabulated in reduced cylindrical coordinates, then combined on the fly with a local layer-density modifier, making its cost negligible relative to evaluating the potential. Targets requiring special quadrature are treated using a stabilized version of singularity swap surface quadrature: the periodic azimuthal integral is evaluated first using translated singularity swap quadrature to prevent severe cancellation near the surface, followed by adaptive Gauss–Legendre quadrature in the polar direction guided by error indicators. We integrate this workflow into a boundary integral solver with precomputed quadrature by expansion for on-surface self-interactions and demonstrate the workflow's performance for challenging configurations of spheroidal particles. Numerical results show that target classification is highly accurate. The prescribed tolerance is met for nearly all target–particle interactions and the error remains within a modest factor of the tolerance in the few remaining cases. Although the experiments focus on the Stokes double layer potential for spheroids, the off-surface framework applies to general smooth axisymmetric surfaces and can be easily adapted to other Stokes layer potentials.

We prove a generalized circular rearrangement inequality: among all circular arrangements of a finite collection of positive numbers, the greedy arrangement simultaneously maximizes the sums of products of $k$ consecutive entries for every $k$. This resolves a 2023 conjecture of Holmes-Holroyd-Ramírez and extends the classical circular rearrangement inequality for products of adjacent entries. As an application, we show that the greedy ordering minimizes the speed of a cyclic birth-and-death chain, resolving another conjecture of the same authors.

We prove the existence of nonnegative radial ground states for a class of nonlinear biharmonic equations in $\mathbb R^N$, covering subcritical and critical nonlinearities. The solutions are obtained as limits of clamped ground states on expanding balls. The Cassani-Tarsia homogeneous maximum principle in [D. Cassani. A. Tarsia, Adv. Nonlinear Anal. 11 (2022)] makes the approximating states positive. A variational decomposition on the radial Nehari manifold then excludes every zero sphere of positive radius, so the limiting ground state is positive on $\mathbb R^N\setminus\{0\}$. When the fourth order operator factorises into two second-order operators with positive resolvents, the conclusion upgrades to $u>0$ throughout $\mathbb R^N$, including at the origin. We also give an explicit counterexample showing that the tempting extension of the homogeneous maximum principle to super-solutions with arbitrary nonhomogeneous boundary data is false, even with an arbitrarily large positive zeroth-order coefficient.

We discuss some theoretical results and their applications to specific practical problems from wireless communications. We assume that we know the noisy version of the signal, which is sparse with respect to a given system of elements (dictionary), at a finite number of points and we want to approximately recover it. This problem of recovery of a noisy signal is closely related to the problem of establishing the Lebesgue-type inequalities for the corresponding algorithms and it motivates us to prove such inequalities. Under certain conditions on a dictionary (RIP-type condition, coherence condition) we obtain different kinds of the Lebesgue-type inequalities for the OMP and its version WOMP algorithms. The most important feature of our new theoretical result is the assumption that the dictionary has the RIP-type property instead of the assumption that it is the Riesz basis, which was used in the previous results. Our approach allows us to treat redundant (overcomplete) systems, which is important in applications. We consider the setting of sparse recovery for highly-coherent dictionaries that appear in OFDM setting for wireless communication. Our main example is the oversampled Fourier dictionary and the recovery of the frequency response of a sparse channel. We develop a general approach to such problems and we prove that under certain conditions the OMP algorithm recovers all dictionary elements with large coefficients, and estimate its accuracy (in NMSE metric) in terms of signal-to-noise ratio.

In this paper, we develop a new approach to studying the Fourier transform of Gaussian multiplicative chaos. We determine the Fourier dimension of these random measures on tori for every smooth log-correlated covariance and in every dimension, and prove sharp bounds and exact formulas for chaos on natural classes of bounded sets and hypersurfaces in Euclidean space. These results highlight how boundary geometry and multifractal concentration jointly govern Fourier decay, including regimes in which the boundary strictly reduces the Fourier dimension. In the circle setting, this gives a substantially simpler new proof of a conjecture of Garban and Vargas.

Let $B\subset\mathbb{C}^n$ be a ball centred at the origin and $D=\{z^1=0\}$. Let $g$ be a Kähler metric of constant holomorphic sectional curvature (chsc) on $B\setminus D$, uniformly equivalent to the model cone metric of angle $2\pi\beta$, with $0<\beta<1$, and polyhomogeneous along $D$. We prove that, in suitable holomorphic coordinates $(w^1, \ldots, w^n)$ centred at the origin, the metric $g$ is the pullback of the corresponding complex space form by the map $(w^1,w^2,\dots,w^n)\longmapsto \bigl((w^1)^\beta,w^2,\dots,w^n\bigr)$.

Cartesian 2-fibrations provide a way to understand indexed categories, but their classical “straightening” construction requires several layers of weak coherence data. This paper develops a homotopical framework that replaces much of this bookkeeping with a fully strict model. By using marked 2-categories to record the cartesian morphisms and 2-cells, we construct a model structure whose fibrant objects are precisely the cartesian 2-fibrations over a fixed 2-category $\mathcal{C}$. We then show that the marked Grothendieck construction identifies these 2-fibrations, up to weak equivalence, with strict 2-functors from $\mathcal{C}$ into $2\mathrm{Cat}$. As an additional contribution, we construct localizations of 2-categories that simultaneously invert selected morphisms and 2-cells.

We develop a cluster representation for one-dimensional CBI processes with jumps. Using this representation, we establish total variation convergence under conditions on the branching Lévy measure. In the subcritical case, we obtain polynomial and exponential rates under distinct regularity assumptions, while the strong Feller property is also established. In the critical case, we derive an explicit bound involving the cumulant integral. Our proofs use a coupling method that has proved effective for establishing ergodicity of Ornstein–Uhlenbeck processes.