Showing 1–50 of 1,228 results for author: Li, Z

The number of limit cycles of piecewise linear Liénard systems

Authors: Hebai Chen, Zhijie Li, Rui Zhang, Xiang Zhang

Abstract: For the planar Liénard differential system $\dot{x}=F(x)-y$, $\dot{y}=x$, where $F(x)$ is a piecewise linear function, Tonnelier (SIAM J. Appl. Math., 2002) conjectured that the maximum number of limit cycles of the system is $n$ when $F(x)$ has $n$ fold points and no jump points, and $2n$ when $F(x)$ has $n$ jump points and no fold points. This conjecture was confirmed by Llibre et al. (J. Nonlin… ▽ More For the planar Liénard differential system $\dot{x}=F(x)-y$, $\dot{y}=x$, where $F(x)$ is a piecewise linear function, Tonnelier (SIAM J. Appl. Math., 2002) conjectured that the maximum number of limit cycles of the system is $n$ when $F(x)$ has $n$ fold points and no jump points, and $2n$ when $F(x)$ has $n$ jump points and no fold points. This conjecture was confirmed by Llibre et al. (J. Nonlinear Sci., 2015) (resp. Chen et al. (J. London Math. Soc., 2026a)) when $F(x)$ has one fold point and no jump points (resp. two fold points). More recently, Chen et al. (J. London Math. Soc., 2026b) proved that the conjecture is correct when $F(x)$ has no fold points and one jump point. All other cases remain open. Here we verify that the lower bound for the maximum number of limit cycles of the system can be $n$ when $F(x)$ has only $n$ fold points, and $2n$ when $F(x)$ has only $n$ jump points, thereby confirming the lower bound part of Tonnelier's conjecture. Moreover, when $F(x)$ has $m$ jump points and $n-m$ fold points, $0\le m\le n$, we also show that the system can have $n+m=(n-m)+2m$ limit cycles. In addition, a complete classification of the {dynamics} near infinity for this class of systems is provided. △ Less

Submitted 19 August, 2026; originally announced August 2026.

An FPRAS for Antiferromagnetic Ising Models on Random Regular Bipartite Graphs

Authors: Zhidan Li, Kuan Yang

Abstract: We design randomized approximation schemes for the partition function of antiferromagnetic Ising models with uniform external field on random regular bipartite graphs. Our algorithm generalizes the approach of Kocurek, Oveis Gharan and Tjowasi (arXiv, 2026) for hard-core models on the same random graph model beyond the uniqueness threshold. We show that, as long as $λ$ is upper bounded by a consta… ▽ More We design randomized approximation schemes for the partition function of antiferromagnetic Ising models with uniform external field on random regular bipartite graphs. Our algorithm generalizes the approach of Kocurek, Oveis Gharan and Tjowasi (arXiv, 2026) for hard-core models on the same random graph model beyond the uniqueness threshold. We show that, as long as $λ$ is upper bounded by a constant and $λ(1 - β) \lesssim Δ^{-1/2}$, an efficient randomized algorithm approximates the partition function with high probability. The algorithm first truncates configurations that are large on either side of the bipartition and then samples from Gibbs distributions conditioned on fixed sizes on one or both sides. To choose an optimal truncation bound, we establish concentration properties of the Gibbs distribution on random regular bipartite graphs. Then we apply high-dimensional expansion and prove trickle-down theorems to obtain fast samplers for the conditioned distributions. △ Less

Submitted 19 August, 2026; originally announced August 2026.

Cyclicity of lips and center-focus problem near infinity

Authors: Hebai Chen, Dehong Dai, Vadim Kaloshin, Zhijie Li

Abstract: In this work, we discover an intriguing Liénard system which is integrable near infinity, integrable near zero and having Ilyashenko-Kotova lips in between. Moreover, we give a necessary and sufficient condition of polynomial Liénard system with center at infinity. In a different direction we improve a lower bound on Hilbert number for Liénard systems. The improvement is due to development of Brud… ▽ More In this work, we discover an intriguing Liénard system which is integrable near infinity, integrable near zero and having Ilyashenko-Kotova lips in between. Moreover, we give a necessary and sufficient condition of polynomial Liénard system with center at infinity. In a different direction we improve a lower bound on Hilbert number for Liénard systems. The improvement is due to development of Brudnyi method of calculation of cyclicity of centers near infinity and near zero. △ Less

Submitted 18 August, 2026; originally announced August 2026.

A concavity inequality and interior $C^2$ estimate for Hessian quotient equations

Authors: Zhisu Li, Ke Wu

Abstract: We establish a concavity inequality for the Hessian quotient operators $\frac{σ_k}{σ_l}$ in the cases $k-l\in\{1,2\}$, and then derive the corresponding Jacobi inequality. Combining this with the framework developed by Lu and Tsai, we obtain an interior Hessian estimate for convex solutions of $\frac{σ_k(D^2u)}{σ_l(D^2u)}=f$. As an application, we prove that any entire convex solution in… ▽ More We establish a concavity inequality for the Hessian quotient operators $\frac{σ_k}{σ_l}$ in the cases $k-l\in\{1,2\}$, and then derive the corresponding Jacobi inequality. Combining this with the framework developed by Lu and Tsai, we obtain an interior Hessian estimate for convex solutions of $\frac{σ_k(D^2u)}{σ_l(D^2u)}=f$. As an application, we prove that any entire convex solution in $\mathbb R^n$ with quadratic growth must be a quadratic polynomial. △ Less

Submitted 18 August, 2026; originally announced August 2026.

Comments: 15 pages

MSC Class: 35J60 (Primary); 35B45; 35B53; 35B65; 35J15 (Secondary)

Diffusion contrast induced dynamics in a time-periodic competition-diffusion system with equal total resources

Authors: Zhenzhen Li, Mingxin Wang

Abstract: This paper investigates a time-periodic, spatiotemporally heterogeneous competition-diffusion system, where two species have different intrinsic growth rates but equal total resources per period. Using asymptotic analysis and principal eigenvalue theory, we systematically examine its dynamics. When both diffusion rates are small, spatial heterogeneity of the difference in time-averaged resources l… ▽ More This paper investigates a time-periodic, spatiotemporally heterogeneous competition-diffusion system, where two species have different intrinsic growth rates but equal total resources per period. Using asymptotic analysis and principal eigenvalue theory, we systematically examine its dynamics. When both diffusion rates are small, spatial heterogeneity of the difference in time-averaged resources leads to uniform persistence, stable coexistence, and asymptotic spatial segregation; whereas its spatial uniformity produces multiple regimes--fast-diffuser selection, parameter-dependent coexistence, or slow-diffuser dominance--separated by smooth threshold curves under different structural conditions. In the regime where at least one diffusion rate is large, we obtain a detailed classification of the local dynamics, with sharp delineation of parameter regions for stable coexistence, competitive exclusion, and a novel bistable scenario, where both semi-trivial periodic solutions are linearly stable while an unstable coexistence state also exists. We further characterize the asymptotic profiles of positive solutions in the mixed-scale diffusion limit. Our results show that the interplay between temporal periodicity and spatial heterogeneity can overturn the classical "slower diffuser always prevails" principle, leading to a substantially richer range of ecological outcomes. △ Less

Submitted 16 August, 2026; originally announced August 2026.

Comments: 58 pages

Affine Anosov Maps on $\mathbb{R}^n$: Classification, Index Spectrum, and Stability at Infinity

Authors: Z. Li, A. Rojas, S. Romaña

Abstract: For $n\ge2$, we classify the affine diffeomorphisms $f_{A,v}(x)=Ax+v$ on $\mathbb R^n$ that admit a complete Riemannian metric with respect to which they are Anosov. Such a metric exists if and only if $A$ is hyperbolic or $f_{A,v}$ has no fixed point, equivalently $v\notin\operatorname{Im}(I-A)$. In the latter case, $f_{A,v}$ is smoothly conjugate to a translation when $\det A>0$ and to White's m… ▽ More For $n\ge2$, we classify the affine diffeomorphisms $f_{A,v}(x)=Ax+v$ on $\mathbb R^n$ that admit a complete Riemannian metric with respect to which they are Anosov. Such a metric exists if and only if $A$ is hyperbolic or $f_{A,v}$ has no fixed point, equivalently $v\notin\operatorname{Im}(I-A)$. In the latter case, $f_{A,v}$ is smoothly conjugate to a translation when $\det A>0$ and to White's map times the identity when $\det A<0$. We also determine the possible stable indices. In the hyperbolic case, the index is determined by the stable spectrum of $A$, whereas a map with no fixed point admits complete Anosov metrics of every stable index from $1$ to $n-1$. Along the $1$-eigenspace, every such metric must exhibit exponential growth of vector norms along one of the two half orbits. We then determine the interior and boundary of the Anosov-realizable locus in the affine parameter space and describe the corresponding change of the index spectrum near regular drift parameters. Finally, we show that Anosov-realizability is not open in the two-sided weak $C^1_{\mathrm{loc}}$ topology but is open in the two-sided strong Whitney $C^1$ topology. △ Less

Submitted 11 August, 2026; originally announced August 2026.

MSC Class: 37D05; 37D10; 37D75

Exterior Dirichlet Problems for Hessian Quotient Equations of Mixed Type

Authors: Yu Lei, Zhisu Li

Abstract: We study the exterior Dirichlet problem for the mixed Hessian quotient equation \[ \frac{σ_k(η(D^2 u))}{σ_l(η(D^2 u))} = 1, \] where $η(M) = (\operatorname{tr} M)I - M$. We establish existence and uniqueness of smooth admissible solutions with prescribed quadratic asymptotics at infinity, and obtain full derivative decay of the remainder. The proof relies on a three-stage subsolution… ▽ More We study the exterior Dirichlet problem for the mixed Hessian quotient equation \[ \frac{σ_k(η(D^2 u))}{σ_l(η(D^2 u))} = 1, \] where $η(M) = (\operatorname{tr} M)I - M$. We establish existence and uniqueness of smooth admissible solutions with prescribed quadratic asymptotics at infinity, and obtain full derivative decay of the remainder. The proof relies on a three-stage subsolution construction. △ Less

Submitted 11 August, 2026; originally announced August 2026.

Long-Time Trajectory Approximation via SA-NODEs: Model Predictive and Floquet Strategies

Authors: Ziqian Li, Nikolaos M. Matzakos

Abstract: We study the approximation of dynamical systems by semi-autonomous neural ordinary differential equations (SA-NODEs) over long time horizons. For a single network trained on the whole horizon, the available error bound deteriorates double exponentially in the horizon length. We develop two training strategies that avoid this barrier, each built on a reset of the state. The model predictive strateg… ▽ More We study the approximation of dynamical systems by semi-autonomous neural ordinary differential equations (SA-NODEs) over long time horizons. For a single network trained on the whole horizon, the available error bound deteriorates double exponentially in the horizon length. We develop two training strategies that avoid this barrier, each built on a reset of the state. The model predictive strategy partitions the horizon adaptively and restarts every window from observed data: when training meets a prescribed tolerance on every window, the composite model meets it uniformly in time, with a parameter budget linear in the horizon for targets with a bounded, uniformly regular reachable tube. The Floquet strategy addresses autonomous targets with a stable limit cycle and uses no data at deployment: a certified contraction of the learned return map confines the error to linear growth in the number of elapsed periods. For the time-periodic architecture we deploy, the scalar certificate degenerates; we prove instead a uniform-in-time orbital guarantee whose hypotheses are measured on the trained model, and an obstruction showing that, for an exactly periodic learned field, small one-period error and a contracting stroboscopic map cannot hold at once. Numerical experiments on four benchmarks confirm the predicted error laws and measure the hypotheses of every guarantee. △ Less

Submitted 11 August, 2026; originally announced August 2026.

MSC Class: 68T07; 93B45; 34C25; 65L05; 41A30

A proof of a weighted sum conjecture for finite multiple zeta values of level two

Authors: Zhonghua Li, Zhenlu Wang, Lihui Zhang

Abstract: M. Kaneko, T. Murakami and A. Yoshihara introduced finite multiple zeta values of level two and conjectured a weighted sum formula for indices whose components belong to $\{1,2\}$. In this paper, we use generating functions and linear recurrence relations to prove the conjecture. More precisely, we reduce the problem to a polynomial identity and solve the resulting second-order difference equation… ▽ More M. Kaneko, T. Murakami and A. Yoshihara introduced finite multiple zeta values of level two and conjectured a weighted sum formula for indices whose components belong to $\{1,2\}$. In this paper, we use generating functions and linear recurrence relations to prove the conjecture. More precisely, we reduce the problem to a polynomial identity and solve the resulting second-order difference equation by identifying its specialized solutions with $_4F_3$ hypergeometric polynomials of Racah-type. △ Less

Submitted 11 August, 2026; originally announced August 2026.

Global vs. Product Observables in Bipartite Quantum Systems: The Sharp Bound

Authors: Zhi Li, Xiaofei Shi

Abstract: To probe a bipartite quantum system, one may use arbitrary global operators or restrict to product operators acting separately on the two subsystems. We determine the sharp universal comparison between the resulting norms. For every $z\in M_n\otimes M_m$, we prove $\|z\|_1\leq\sqrt{2}\min\{n,m\}\|z\|_\varepsilon$, where $\|\cdot\|_1$ is the trace norm and $\|\cdot\|_\varepsilon$ is the injective t… ▽ More To probe a bipartite quantum system, one may use arbitrary global operators or restrict to product operators acting separately on the two subsystems. We determine the sharp universal comparison between the resulting norms. For every $z\in M_n\otimes M_m$, we prove $\|z\|_1\leq\sqrt{2}\min\{n,m\}\|z\|_\varepsilon$, where $\|\cdot\|_1$ is the trace norm and $\|\cdot\|_\varepsilon$ is the injective tensor norm associated with the trace norms on $M_n$ and $M_m$. To prove the upper bound, we establish an $L_1$ noncommutative Khintchine inequality whose random coefficients are the entries of a Haar unitary. We also show that the coefficient $\sqrt{2}$ is sharp. As applications, we show that the same sharp constant governs the gap between bipartite correlation measured in trace norm and that measured by a correlation function, and obtain an improved universal upper bound for quantum data hiding. The upper bound has also been formalized and machine-checked in Lean. △ Less

Submitted 6 August, 2026; originally announced August 2026.

Modulational spectrum of infinite-depth hydroelastic Stokes waves

Authors: Ting-Yang Hsiao, Zirui Li, Ye Zhang, Chengbin Zhu

Abstract: We determine the complete local Bloch spectrum bifurcating from the origin for small-amplitude periodic hydroelastic Stokes waves in infinite depth, under the combined effects of gravity, surface tension, and elastic bending. Away from the Wilton-type resonance set, we construct a real-analytic Stokes-wave branch and analyze the four eigenvalues emerging from the defective zero eigenvalue of the l… ▽ More We determine the complete local Bloch spectrum bifurcating from the origin for small-amplitude periodic hydroelastic Stokes waves in infinite depth, under the combined effects of gravity, surface tension, and elastic bending. Away from the Wilton-type resonance set, we construct a real-analytic Stokes-wave branch and analyze the four eigenvalues emerging from the defective zero eigenvalue of the linearized hydroelastic Euler system. Using analytic spectral perturbation theory and Hamiltonian-reversible reductions, we decouple them into a Benjamin--Feir pair and a long-wave pair. The long-wave pair remains purely imaginary and has the singular scale $\cO(\sqrt{|μ|})$, whereas the Benjamin--Feir pair is governed by an explicit discriminant whose leading sign yields a sharp criterion for modulational stability and instability. We derive the exact non-resonant phase diagram in the surface-tension-bending parameter plane and identify a bounded stability island generated by elastic bending. In the unstable region, and away from a drift degeneracy, the Benjamin--Feir branches form a local figure-eight curve. In the zero-bending limit, the reduced coefficients recover the known deep-water gravity and gravity-capillary results, while the change from the finite-depth $\cO(|μ|)$ long-wave scale to $\cO(\sqrt{|μ|})$ shows that the infinite-depth problem is singular. △ Less

Submitted 5 August, 2026; originally announced August 2026.

Comments: 51 pages, 2 figures

MSC Class: 76B15; 37K45; 74F10; 35B35

The spectrum of strict units of topological modular forms

Authors: Zhenpeng Li, Kiran Luecke

Abstract: In the theory of spectral algebraic geometry, few objects receive as much study as the spectrum of topological modular forms. In this paper, we compute the strict units of topological modular forms, defined as the connective cover of the mapping spectrum from $\mathbb{Z}$ to the units spectrum. In the theory of spectral algebraic geometry, few objects receive as much study as the spectrum of topological modular forms. In this paper, we compute the strict units of topological modular forms, defined as the connective cover of the mapping spectrum from $\mathbb{Z}$ to the units spectrum. △ Less

Submitted 11 August, 2026; v1 submitted 29 July, 2026; originally announced July 2026.

Comments: 23 pages; erroneous group cohomology calculation fixed and more details on $K(1)$-local cases added. Comments welcome!

MSC Class: 55P43 (Primary) 55P42; 55N34 (Secondary)

Irreducibility of the tensor product of Yangian \( \mathrm{Y}(\mathfrak{gl}_{m|n}) \) evaluation modules

Authors: Vyacheslav Futorny, Zheng Li, Jian Zhang

Abstract: The evaluation homomorphism from the super Yangian \( \mathrm{Y}(\mathfrak{gl}_{m|n}) \) to \( \mathrm{U}(\mathfrak{gl}_{m|n}) \) induces a \( \mathrm{Y}(\mathfrak{gl}_{m|n}) \)-module structure on any finite dimensional simple \( \mathrm{U}(\mathfrak{gl}_{m|n}) \)-module \( L(λ) \). In this paper, we give necessary and sufficient conditions for the tensor product of such evaluation \( \mathrm{Y}(… ▽ More The evaluation homomorphism from the super Yangian \( \mathrm{Y}(\mathfrak{gl}_{m|n}) \) to \( \mathrm{U}(\mathfrak{gl}_{m|n}) \) induces a \( \mathrm{Y}(\mathfrak{gl}_{m|n}) \)-module structure on any finite dimensional simple \( \mathrm{U}(\mathfrak{gl}_{m|n}) \)-module \( L(λ) \). In this paper, we give necessary and sufficient conditions for the tensor product of such evaluation \( \mathrm{Y}(\mathfrak{gl}_{m|n}) \)-modules, \( L_g(λ) \otimes L_h(γ) \), to be simple, provided each of \( λ\) and \( γ\) is either covariant tensor or essentially typical. Our proof is based on the existence of a Gelfand--Tsetlin basis for finite dimensional simple \( \mathrm{U}(\mathfrak{gl}_{m|n}) \)-modules with highest weights that belong to these two families: covariant tensor and essentially typical. The obtained result is a super analogue of the Molev's result for the classical Yangian \( \mathrm{Y}(\mathfrak{gl}_n) \). Combining this with the binary property of tensor products of covariant evaluation modules, we obtain an irreducibility criterion for arbitrary tensor products of covariant evaluation modules. △ Less

Submitted 27 July, 2026; originally announced July 2026.

MSC Class: 17B37

Hausdorff type Time-Trace Observability for Airy Equations on the Line and Point Observability on the Torus

Authors: Ze Li

Abstract: The main results of this paper are threefold. First, we prove an observability inequality for the Airy equation on the real line from Hausdorff-thick sets in a time-trace sense for every observation time $T>0$. The observation functional is a block supremum of $L^2(0,T)$ time traces over the Hausdorff-thick set. Second, we prove observability inequalities for the Airy equation on the real line wit… ▽ More The main results of this paper are threefold. First, we prove an observability inequality for the Airy equation on the real line from Hausdorff-thick sets in a time-trace sense for every observation time $T>0$. The observation functional is a block supremum of $L^2(0,T)$ time traces over the Hausdorff-thick set. Second, we prove observability inequalities for the Airy equation on the real line with observations on some periodic sets, which in particular yields observability on a class of spatial point sequences. Third, we give a necessary and sufficient condition for finite point observability for the Airy equation on the torus with a bounded real-valued potential. Indeed, for a finite observation set $F$, a Kalman rank condition on a finite-dimensional invariant subspace is found. As a corollary, we obtain sharp point observability results for the Airy and linear KdV equations on the torus. Moreover, for potentials with finite order regularity assumptions we prove that every observation set with an accumulation point, in particular any set of positive Hausdorff dimension, gives an observability inequality for each observation time $T>0$. Since the Airy equation has neither high frequency exponential decay nor pointwise smoothing effects, which are essential in recent works on Hausdorff type observation results on heat equations, we introduce several new ideas adapted to the Airy case. △ Less

Submitted 27 July, 2026; originally announced July 2026.

The Conclave Process

Authors: Itai Benjamini, Zhenhao Cai, Guanyi Chen, Shuyang Gong, Zhangsong Li

Abstract: We introduce a stochastic model for the papal conclave in which $n$ cardinals vote repeatedly among themselves until one cardinal receives all the votes. In each round, the probability that a cardinal votes for a given candidate is proportional to the $α$-th power of that candidate's vote count in the preceding round. For $α=1$, the model reduces to the Wright-Fisher model and is dual to Kingman's… ▽ More We introduce a stochastic model for the papal conclave in which $n$ cardinals vote repeatedly among themselves until one cardinal receives all the votes. In each round, the probability that a cardinal votes for a given candidate is proportional to the $α$-th power of that candidate's vote count in the preceding round. For $α=1$, the model reduces to the Wright-Fisher model and is dual to Kingman's n-coalescent. We reveal a sharp transition in the absorption time $\mathcal{T}$ at $α=1$. It was known that when $α=1$, $\mathcal{T}$ is typically of order $n$. We prove that for $α>1$, it drops to order $\textit{loglog n.}$ In contrast, for $α<1$, $\mathcal{T}$ is typically at least $\exp(Ω(n))$. We also prove a sharp phase transition in the identity of the winner when $α>1$. For every positive integer $k$, if $2^{1/k}<α<2^{1/(k-1)}$ (where we write $2^{1/0} = +\infty$), with probability tending to 1 as $n\to\infty$, the eventual winner is the unique leader after round $k$. These results show that reinforced voting processes reach consensus remarkably quickly even for large electorates. △ Less

Submitted 10 August, 2026; v1 submitted 24 July, 2026; originally announced July 2026.

Resolution of the ENO-TV conjecture: a parity dichotomy

Authors: Zhuoyun Li, Kailiang Wu

Abstract: We resolve the ENO--TV conjecture, a discrete coercivity problem in compactness theory for entropy-stable approximations of hyperbolic conservation laws. For order-$k$ essentially non-oscillatory (ENO) reconstruction from compactly supported cell averages, it asks whether the nonnegative ENO source times the $(k-1)$st power of the amplitude uniformly controls the $(k+1)$st absolute-jump moment. We… ▽ More We resolve the ENO--TV conjecture, a discrete coercivity problem in compactness theory for entropy-stable approximations of hyperbolic conservation laws. For order-$k$ essentially non-oscillatory (ENO) reconstruction from compactly supported cell averages, it asks whether the nonnegative ENO source times the $(k-1)$st power of the amplitude uniformly controls the $(k+1)$st absolute-jump moment. We prove a parity dichotomy: the estimate holds for odd $k\ge3$ and fails for even $k\ge4$; the known second-order case completes the classification. Localization gives a selection-independent finite-difference functional uniformly comparable to the source and reduces the conjecture to discrete interpolation. For odd orders, summation by parts reveals a hidden square; a discrete Gagliardo--Nirenberg inequality yields coercivity. For even orders, Euler-polynomial blocks from the functional's polynomial kernel yield counterexamples that persist under arbitrarily small perturbations making all affected ENO comparisons strict. We also prove two coercive estimates for every $k\ge2$: control of jumps larger than a fixed fraction of the amplitude and of local blocks modulo sampled polynomials of degree at most $k-2$. Via the Cayley--Sylvester decomposition, we compute the dimensions of homogeneous first-cohomology spaces for the lattice shift on polynomial jump profiles. At fourth order, for a cubic flux and a globally strictly convex entropy, a total-degree-seven component of a reduced entropy-flux mismatch represents a nonzero class on profiles of degree at most two and hence has no translation-invariant finite-stencil $C^7$ local primitive at the zero constant state. Odd-order coercivity persists on globally quasi-uniform meshes, whereas for each $k\ge2$ it fails on a fixed irregular mesh even though every interface contribution remains nonnegative. This failure is due to the mesh geometry. △ Less

Submitted 21 July, 2026; originally announced July 2026.

Spatial Dependence in Directed Preferential-Attachment Networks

Authors: Zihan Li, Tiandong Wang

Abstract: Spatially embedded directed networks, such as airline networks, often exhibit simultaneous high activity at nearby nodes. Preferential attachment (PA) explains hub dominance. We extend it to spatial co-movement through a directed PA model whose out- and in-node weights follow temporally persistent Gaussian-process lognormal fields. Under sublinear PA, out-degree proportions converge to explicit no… ▽ More Spatially embedded directed networks, such as airline networks, often exhibit simultaneous high activity at nearby nodes. Preferential attachment (PA) explains hub dominance. We extend it to spatial co-movement through a directed PA model whose out- and in-node weights follow temporally persistent Gaussian-process lognormal fields. Under sublinear PA, out-degree proportions converge to explicit normalized powered weights, whereas self-loop exclusion yields a coupled in-degree limit. We derive a strictly concave inverse that recovers the in-weights from terminal degree proportions. For ordered network histories, we develop a minorization-maximization (MM) weight estimator and profile likelihood for the PA exponent; temporal pre-whitening and a spatial quasi-likelihood estimate the latent covariance. Simulations verify transmission of distance-decaying dependence and show how random segment volume creates a distance-independent common mode in raw degrees. An analysis of U.S. domestic flights (2015-2019) separates network-wide volume variation from a short-range spatial component. An observed-volume reconstruction reproduces the raw-degree common mode, and the fitted field yields an exploratory co-exceedance transition scale of roughly 150 km. A per-carrier analysis of European air traffic also reveals the same decomposition. △ Less

Submitted 20 July, 2026; originally announced July 2026.

A Hadamard Formula for Equilibrium Envelopes under Parallel Deformation

Authors: Ziyu Li

Abstract: Let $(X,ω_0)$ be a compact Kähler manifold, and let $U\Subset X$ have $C^{3,1}$ uniformly strongly pseudoconvex boundary. For the equilibrium envelope $u_0$ associated with $X\setminus U$, the normalized Monge--Ampère measure vanishes on $U$ and equals the background measure $V^{-1}ω_0^n$ on $X\setminus\overline U$; its remaining component is a singular measure supported on $\partial U$, where… ▽ More Let $(X,ω_0)$ be a compact Kähler manifold, and let $U\Subset X$ have $C^{3,1}$ uniformly strongly pseudoconvex boundary. For the equilibrium envelope $u_0$ associated with $X\setminus U$, the normalized Monge--Ampère measure vanishes on $U$ and equals the background measure $V^{-1}ω_0^n$ on $X\setminus\overline U$; its remaining component is a singular measure supported on $\partial U$, where $V=\int_Xω_0^n$. We identify this boundary component as the negative outward Anzellotti trace of a divergence-measure flux current. We then prove a one-sided Hadamard formula for the normalized Monge--Ampère energy along the outward parallel family $U_\varepsilon=\{ρ<\varepsilon\}$. The nonlinear telescoping identity gives the mixed Bedford--Taylor boundary traces that sum up to the trace of the current \[ \mathcal J_{\mathrm{tot}} =\frac1V\mathrm{d}^c u_0\wedge\sum_{p=0}^{n-1}(p+1)ω_{u_0}^p\wedgeω_0^{n-1-p}, \qquad ω_{u_0}=ω_0+\mathrm{d}\mathrm{d}^c u_0. \] These results provide a local weak formulation of boundary flux and normal variation for regular interface problems related to Darcy/Hele--Shaw type problems and Monge--Ampère growth. △ Less

Submitted 15 August, 2026; v1 submitted 19 July, 2026; originally announced July 2026.

Comments: 29 pages, no figure. Comments are welcome!

MSC Class: 32W20 (Primary) 32U15; 32Q15 (Secondary)

Measures and generalizations of dual Littlewood identities

Authors: Zhongren Cai, Bin Jiang, Naihuan Jing, Zhijun Li, Qianyi Ye

Abstract: We introduce three families of vectors $|\underlineλ^{so}\rangle$, $|\underlineλ^{sp}\rangle$ and $|\underlineλ^{o}\rangle$ parametrized by partitions in the Fock space by using products of adjoint vertex operators. We show that the quotient space of the dual vacuum vector is spanned by the partition vectors indexed by a special family of partitions. The partition-indexed vectors also help us to d… ▽ More We introduce three families of vectors $|\underlineλ^{so}\rangle$, $|\underlineλ^{sp}\rangle$ and $|\underlineλ^{o}\rangle$ parametrized by partitions in the Fock space by using products of adjoint vertex operators. We show that the quotient space of the dual vacuum vector is spanned by the partition vectors indexed by a special family of partitions. The partition-indexed vectors also help us to derive the dual Littlewood identities of types B, C, and D in a new manner associated to the special family of partitions. As an application, we obtain a new free fermionic construction to show that the measures related to dual Littlewood identities introduced by Rains \cite[Section 7]{Rai2000} and Betea \cite[Section 3]{Be2020} are determinantal with repect to some explicit correlation kernels. Furthermore we establish a number of generalized Littlewood identities summed over certain restricted partitions by computing the inner products with elements indexed by one-column partitions {or generalized partitions $(0^m)$} in the complete dual Fock space. {In particular, for each positive integer $n$, we obtain generalized Littlewood identities for $(-n)$-asymmetric partitions. We show that these generalized Littlewood identities contain several well-known Littlewood-type identities as special cases. Consequently we also give a new proof of the generalized Littlewood identity \cite[(5.25)]{LSV2008} for Lie superalgebras. } %We also produce infinite generalized Littlewood identities by calculating the inner products between these elements and some elements indexed by one-column partitions in the complete dual Fock space. △ Less

Submitted 15 July, 2026; originally announced July 2026.

Comments: 22pp

MSC Class: Primary: 05E05; Secondary: 17B69; 17B37; 82B20; 05A17

Journal ref: Forum Math. Sigma 14 (2026), e106, 1-20

Ensemble Controlled-Flow Filtering for Implicit Data Assimilation

Authors: Zhuoyuan Li, Yue Zhao, Ming Li

Abstract: Data assimilation estimates the state of a dynamical system from model forecasts and incoming observations. Many observation mechanisms, however, are many-to-one, implicit, non-smooth, or accessible only through simulation, and need not provide the residual structures or likelihood guidance required by existing ensemble filters. We introduce implicit data assimilation, in which the analysis law is… ▽ More Data assimilation estimates the state of a dynamical system from model forecasts and incoming observations. Many observation mechanisms, however, are many-to-one, implicit, non-smooth, or accessible only through simulation, and need not provide the residual structures or likelihood guidance required by existing ensemble filters. We introduce implicit data assimilation, in which the analysis law is defined as an energy tilt of the forecast distribution. We then propose the Ensemble Controlled-flow Filter (EnCF), which realizes this update through a stochastic controlled flow and learns the observation-dependent control by adjoint matching from terminal energy gradients. For simulator-defined observations, EnCF-LF learns a surrogate conditional energy from samples and applies the same controlled-flow solver. We prove ideal exactness, derive a one-step error decomposition, and establish non-accumulation of local errors under filter stability. Numerical results show that Kalman-type filters remain preferable for smooth additive-Gaussian observations, while the proposed methods are better suited to non-Gaussian, many-to-one, multimodal, and implicit observation models. △ Less

Submitted 14 July, 2026; originally announced July 2026.

Comments: 26 pages

MSC Class: 65C30; 65C05; 62M20; 93E11; 93E20

Structured Preconditioning in Affine-Invariant Geometry: Projection, Certificates, and Kronecker Separation

Authors: Zavier Li

Abstract: Nearest structured approximation and best structured preconditioning solve different matrix optimization problems. We determine their exact relation for Kronecker positive-definite matrices under the affine-invariant Riemannian metric. The Kronecker family is closed and geodesically convex, so every full matrix has a unique affine-invariant projection. Its logarithmic residual satisfies partial-tr… ▽ More Nearest structured approximation and best structured preconditioning solve different matrix optimization problems. We determine their exact relation for Kronecker positive-definite matrices under the affine-invariant Riemannian metric. The Kronecker family is closed and geodesically convex, so every full matrix has a unique affine-invariant projection. Its logarithmic residual satisfies partial-trace normal equations and yields certified point and objective errors for an Armijo projection solver. Our central result shows that this unique projection is also a minimizer of the Hessian-relative condition number if and only if the extreme spectral states admit identical tensor marginals. A computable marginal-mismatch residual either vanishes at a condition-optimal projection or produces a strict descent direction. Two relative spectral levels always force projection optimality; more strongly, every $2\times 2$ Kronecker projection is condition-optimal. An explicit $2\times 3$ construction is therefore a dimension-minimal strict separation. Residual-calibrated bounds further bracket the best attainable Kronecker condition number and the suboptimality of the projection. Supporting results place classical diagonal and block Loewner sandwiches, fixed-basis primal--dual obstructions, and general log-spectral targets in the same certificate language. Given validated numerical enclosures and outward-rounded comparisons, an interval-safe corollary preserves the soundness of the full Kronecker tests. Deterministic small-matrix checks, including a multistart generic log-factor oracle independent of the partial-trace solver, verify the stated identities and bounds. △ Less

Submitted 13 July, 2026; originally announced July 2026.

Comments: 43 pages, 1 figure; includes appendices and deterministic numerical verification. arXiv admin note: substantial text overlap with arXiv:2607.07204

Central Elements and Determinantal Identities in the Elliptic Quantum Algebra \( \mathcal{A}_{q,p}(\widehat{\mathfrak{gl}}_N)\)

Authors: Yingjie Hu, Zheng Li, Jian Zhang

Abstract: Elliptic quantum algebra is the algebraic structure characterized by the elliptic solution of the Yang-Baxter equation. In this paper, we construct a family of central elements \( \mathfrak{z}(z) \) for the elliptic quantum algebra \(\mathcal{A}_{q,p}(\widehat{\mathfrak{gl}}_{N})\) and show that they can be expressed as quantum determinants, yielding an elliptic analogue of the Liouville formula.… ▽ More Elliptic quantum algebra is the algebraic structure characterized by the elliptic solution of the Yang-Baxter equation. In this paper, we construct a family of central elements \( \mathfrak{z}(z) \) for the elliptic quantum algebra \(\mathcal{A}_{q,p}(\widehat{\mathfrak{gl}}_{N})\) and show that they can be expressed as quantum determinants, yielding an elliptic analogue of the Liouville formula. In addition, we establish determinantal identities, including Jacobi's ratio theorem and Sylvester's theorem. △ Less

Submitted 13 July, 2026; originally announced July 2026.

MSC Class: 17B37

The Stability of the Backward Problem for Photoacoustic Imaging in Attenuating Media via Carleman Estimates

Authors: Qihang Chen, Zhiyuan Li, Song Xu

Abstract: This paper investigates the backward problem in time for photoacoustic tomography (PAT) in attenuating media. It is well-established that photoacoustic imaging in attenuating media can be accurately modeled by spatial fractional-order damping. This inverse problem is ill-posed in the sense of Hadamard. In this work, we construct a novel class of Carleman estimates independent of spatial variables,… ▽ More This paper investigates the backward problem in time for photoacoustic tomography (PAT) in attenuating media. It is well-established that photoacoustic imaging in attenuating media can be accurately modeled by spatial fractional-order damping. This inverse problem is ill-posed in the sense of Hadamard. In this work, we construct a novel class of Carleman estimates independent of spatial variables, and by virtue of these estimates, we establish conditional stability estimates for this problem for the first time. Building upon this, we propose a Tikhonov-type regularization functional and derive its associated adjoint system. Furthermore, leveraging the established conditional stability results, we derive the convergence rate of the proposed regularization approach. Finally, we validate the effectiveness of our theoretical findings through extensive numerical experiments. △ Less

Submitted 8 July, 2026; originally announced July 2026.

Causal Optimizer Interaction Calculus: Hidden Geometric Relaxation and Identifiable Interventions

Authors: Zavier Li

Abstract: Optimizer experiments observe responses to algorithmic configurations without uniquely revealing hidden mechanisms. We develop a causal optimizer interaction calculus that separates pathwise realization, Mobius decomposition, and experimental identification. Under a fixed innovation coupling, every finite-horizon innovation-driven optimizer admits a behaviorally minimal pathwise realization. For a… ▽ More Optimizer experiments observe responses to algorithmic configurations without uniquely revealing hidden mechanisms. We develop a causal optimizer interaction calculus that separates pathwise realization, Mobius decomposition, and experimental identification. Under a fixed innovation coupling, every finite-horizon innovation-driven optimizer admits a behaviorally minimal pathwise realization. For any finite effect support and intervention design, an incidence operator gives the complete observational gauge, exact identifiability, sharp quotient stability, held-out predictions, and exact noiseless configuration complexity. Smooth hidden relaxation generates interactions through inverse hidden-state stiffness. Building on this structural law, we prove an observable-readout transfer theorem: arbitrary smooth update or trace readouts inherit an explicit five-term interaction through first and second hidden responses. Unlike the reduced optimal value, a general readout has no universal interaction sign. Its Boolean effects remain exact integrals of continuous interaction curvature and can therefore be identified by factorial interventions. We also derive Gaussian quotient minimax risk, exact confidence sets and tests, misspecification decomposition, certified downstream decisions, and optimal replication. A controlled real-data experiment on a 65-dimensional strongly convex logistic model validates the complete reduced-value chain. Boolean effects and independently integrated curvature agree within 4.21e-11, while nine held-out continuous intensities agree within 8.88e-13. Gaussian campaigns attain the predicted coverage and power, and 4,500 real-minibatch observations reject an order-two interaction model. Neural trace audits provide complementary evidence that the declared response classes remain informative in nonconvex training. △ Less

Submitted 11 July, 2026; v1 submitted 8 July, 2026; originally announced July 2026.

Comments: 34 pages. Complete proofs, controlled real-data experiments, and reproducibility details are included

Restricted Dynamic Geometric Complexity: Path-Space Reduction and Möbius--Jacobi Response

Authors: Zavier Li

Abstract: Structured preconditioners restrict optimization to a small family of positive metrics, but endpoint condition-number reachability does not measure the geometric effort required to reach a useful metric. We formulate this effort as a path-space value problem. Restricted dynamic geometric complexity is the least affine-invariant length of an admissible metric path whose endpoint reaches a Hessian-r… ▽ More Structured preconditioners restrict optimization to a small family of positive metrics, but endpoint condition-number reachability does not measure the geometric effort required to reach a useful metric. We formulate this effort as a path-space value problem. Restricted dynamic geometric complexity is the least affine-invariant length of an admissible metric path whose endpoint reaches a Hessian-relative generalized-eigenvalue condition target. Path elimination gives an exact min-plus semigroup and Bellman principle, while fixed-horizon kinetic energy is exactly squared complexity divided by twice the horizon. The main response result is global on a Hadamard state space: geodesic convexity produces a smooth intervention-cube path branch and a uniformly coercive Jacobi form, while one Green inverse generates the value Hessian, two-sided force-to-curvature bounds, exact Möbius effects, and arbitrary prescribed finite-order responses. For the hard condition target, a bordered Jacobi--KKT theorem differentiates the moving projection endpoint and multiplier on every regular active spectral stratum; its indefinite inverse also explains why hard-target interactions need not share the unconstrained sign. The theory specializes to affine-invariant positive-definite geometry. A determinant-one two-dimensional diagonal model has an exact target interval, a closed-form forced path, and a strictly negative-definite interaction matrix. A moving diagonal Hessian gives a closed-form hard-target projection, multiplier, and pair effects of either sign, while a coordinate-sequential three-dimensional protocol yields an exact path metric strictly larger than the ambient projection distance. Thus the global Green and bordered hard-target responses are explicit laws of restricted metric-path elimination built on Bellman composition. △ Less

Submitted 13 July, 2026; v1 submitted 8 July, 2026; originally announced July 2026.

Comments: 32 pages, 1 table

Optimization Geometrodynamics: Variational Reduction and Interaction Curvature

Authors: Zavier Li

Abstract: Adaptive optimizers carry hidden states that change how visible gradients become parameter motion. We develop optimization geometrodynamics as a variational theory of this hidden geometry. Infimal pushforward eliminates all hidden states realizing the same action and composes across optimizer hierarchies. Under smooth nondegeneracy, it yields hidden susceptibility and the Schur-complement curvatur… ▽ More Adaptive optimizers carry hidden states that change how visible gradients become parameter motion. We develop optimization geometrodynamics as a variational theory of this hidden geometry. Infimal pushforward eliminates all hidden states realizing the same action and composes across optimizer hierarchies. Under smooth nondegeneracy, it yields hidden susceptibility and the Schur-complement curvature seen after relaxation. For affine pre-reduction perturbations, the induced interaction curvature is the negative-semidefinite operator $-G^*H^{-1}G$, whose mixed entries integrate to finite mechanism contrasts. Our main realization is the determinant-one affine-invariant SPD action map $P\mapsto PA$. We prove a global analytic bundle with closed totally geodesic fibers and a unique analytic nearest-controller section. A strongly convex fiber theorem and an explicit logarithmic action residual give a globally linearly convergent solver from every feasible initializer, together with nonasymptotic value, controller-distance, and residual bounds and observable posterior stopping certificates. A conditional inexact result propagates supplied rigorous residual-error and radius majorants. The dense spectral kernel is confined to an active subspace of dimension $r\le 2m$, yielding an explicit spectral-arithmetic operation bound and a strict dimensional reduction when $r<d$. For nested shape-normalized quadratic actions, canonical multi-secant projections satisfy an exact CAT(0) Pythagorean decrease and recover the determinant-one inverse Hessian shape at the sharp rank threshold $d-1$, provided the scalar gauge $c_H=(\det H)^{1/d}$ is known. These results turn the action bundle into an exact iterative computation with posterior certificates and a finite-identification theory. △ Less

Submitted 12 July, 2026; v1 submitted 7 July, 2026; originally announced July 2026.

Comments: 25 pages, no figures

Exploiting Variable Implications in Presolve for Mixed Integer Programming

Authors: Wei-Kun Chen, Chang-Long Li, Zhao-Wei Wang, Yu-Hong Dai, Zi-Shuo Li, Meng Lu

Abstract: Presolve for mixed integer programming (MIP) problems aims to eliminate redundant information, strengthen the formulation, and extract useful structural information for the subsequent branch-and-cut process. An important type of such structural information is the variable implications (VIs), which describe how a bound on a variable depends on a bound of a binary variable. In this paper, we develop… ▽ More Presolve for mixed integer programming (MIP) problems aims to eliminate redundant information, strengthen the formulation, and extract useful structural information for the subsequent branch-and-cut process. An important type of such structural information is the variable implications (VIs), which describe how a bound on a variable depends on a bound of a binary variable. In this paper, we develop two new presolve techniques that exploit VIs to derive reductions for MIP problems. The first technique, called VI aggregation, aggregates multiple VIs into a single inequality by using implications between a variable and a set of binary variables that form a clique. This aggregation can reduce the number of constraints and tighten the linear programming relaxation. The second technique, called VI-aware linear constraint propagation (LCP), builds on the standard LCP but incorporates VIs associated with the variable being tightened to derive more reductions and can derive tighter variable bounds. We show that although VI information is additionally considered, the tightest lower or upper bound of a variable can still be derived in linear time. Moreover, compared with a state-of-the-art approach in the literature, the proposed VI-aware LCP can derive tighter variable bounds. Computational results on MIPLIB 2017 benchmark instances demonstrate the effectiveness of VI aggregation and VI-aware LCP in improving the performance of the open-source MIP solver HiGHS. In particular, using the two proposed presolve techniques, a reduction of 4% in solving time and 6% in node number on HiGHS can be achieved. △ Less

Submitted 5 July, 2026; originally announced July 2026.

Comments: 24 pages, 1 figure, 6 tables

Augmenting airline networks using airside-to-airside buses to strengthen system resilience under disruptions

Authors: Micah M. Borrero, Max Z. Li

Abstract: Each year, disruptions in the air transportation network strand millions of passengers and cost airlines billions in revenue. Airline networks prioritize operational and cost efficiency through hub-and-spoke structures that maximize revenue; however, these hubs also act as critical choke points during disruptions. Previous studies have focused on reactionary measures in response to air transportat… ▽ More Each year, disruptions in the air transportation network strand millions of passengers and cost airlines billions in revenue. Airline networks prioritize operational and cost efficiency through hub-and-spoke structures that maximize revenue; however, these hubs also act as critical choke points during disruptions. Previous studies have focused on reactionary measures in response to air transportation network disruptions, whereas this work proposes a proactive strategy to improve resilience by reconfiguring the network's topology. Specifically, we consider airside-to-airside bus lines as a low-cost, frequent alternative to short, regional flights, offering service that can circumvent air traffic-related delays. We develop a network construction model that augments the existing air transportation network with these bus lines. The augmented networks are analyzed through an agent-based simulation, where increased resilience is measured in terms of decreased average hourly passenger delays under both nominal and disrupted conditions. Our results demonstrate that converting 10 regional routes from air service to airside-to-airside bus service, for a baseline scenario that is constrained by a $10 million investment budget, can reduce passenger delays by an average of 8% on disrupted days and 6% on nominal days. Furthermore, through a sensitivity analysis, we show that while augmenting the system using these buses decreases operational costs compared to the historic air-only network, continuously expanding bus parameters (i.e., range and investment budget) yields diminishing returns in delay mitigation. Finally, we discuss real-world precedents alongside regulatory and political hurdles to implementation. The proposed framework offers airlines, airports, and regulators a decision-support tool for integrating multimodal strategies into future disruption management policies. △ Less

Submitted 30 June, 2026; originally announced June 2026.

All limit points of the largest roots of matching polynomials are determined

Authors: Zhaoxi Li, Shi-Mei Ma, Jianfeng Wang, Weifan Wang

Abstract: The largest matching root $μ(G)$ of a graph $G$ is that of its matching polynomial. In this paper, all limit points of the largest matching roots of graphs are determined. More precisely, we identify the limit points of the largest matching roots of graphs less than $τ^{\frac{1}{2}}+τ^{-\frac{1}{2}}$. For any $γ\geq τ^{\frac{1}{2}}+τ^{-\frac{1}{2}}$ with $τ=\frac{\sqrt{5}+1}{2}$, there exists a gr… ▽ More The largest matching root $μ(G)$ of a graph $G$ is that of its matching polynomial. In this paper, all limit points of the largest matching roots of graphs are determined. More precisely, we identify the limit points of the largest matching roots of graphs less than $τ^{\frac{1}{2}}+τ^{-\frac{1}{2}}$. For any $γ\geq τ^{\frac{1}{2}}+τ^{-\frac{1}{2}}$ with $τ=\frac{\sqrt{5}+1}{2}$, there exists a graph sequence $\{G_i\, |\, i\in \mathbb{N}\}$ such that $\lim\limits_{i \rightarrow \infty}μ(G_i)=γ$. △ Less

Submitted 26 June, 2026; originally announced June 2026.

MSC Class: 05C31; 05C70; 05C50

Extreme-Case Distorted Utility under Moment Ambiguity

Authors: Zehao Li, Yijie Peng, Hui Shao, Chung-Piaw Teo

Abstract: Many operations decisions under distributional ambiguity, from pricing and inventory to capacity and contracting, evaluate an action through a tail-sensitive distorted utility of an uncertain payoff and hedge against the least favorable distribution consistent with a few known moments; the resulting worst-case evaluation is the inner problem of a moment-based distributionally robust decision. We s… ▽ More Many operations decisions under distributional ambiguity, from pricing and inventory to capacity and contracting, evaluate an action through a tail-sensitive distorted utility of an uncertain payoff and hedge against the least favorable distribution consistent with a few known moments; the resulting worst-case evaluation is the inner problem of a moment-based distributionally robust decision. We study this inner problem, the extreme-case distorted utility under moment constraints, for a locally Lipschitz utility that may be nonsmooth and neither convex nor concave together with a general, possibly atomic, distortion. Recasting the problem in the quantile domain, we develop a unified method that yields exact first-order optimality conditions and closed-form extremal values and distributions for both the worst and best cases, drawing on nonsmooth variational analysis. A central step treats the monotonicity constraint by isotonic projection onto the monotone cone, turning an abstract infinite-dimensional restriction into an inexpensive inner solve that scales linearly in the discretization. The method recovers and extends classical moment bounds through three examples: a range value-at-risk extension of the Scarf bound, GlueVaR distortions with a reward--penalty utility, and a capped incentive contract under conditional value-at-risk. As the inner oracle of a robust min-max decision, the characterization embeds directly in outer robust optimization, illustrated on a real capacity-provisioning problem for generative artificial intelligence inference where accounting for moment ambiguity lowers required capacity while preserving service compliance. △ Less

Submitted 23 June, 2026; originally announced June 2026.

Stable Image Reconstruction via Two-Parameter Power-Scale Variation Minimization

Authors: Ziwei Li, Wengu Chen, Huanmin Ge, Limei Huo, Dachun Yang

Abstract: In this article, we introduce a power-scale variation (PSV$_{a,p}$) with two tunable parameters: the sparsity-inducing exponent $p\in(0,1]$ and the scaling factor $a\in(0,\infty)$. By minimizing the PSV$_{a,p}$, we establish stable reconstructions in both the gradient and the image domains under the restricted isometry property (RIP) framework. Furthermore, we design an iteratively re-weighted lea… ▽ More In this article, we introduce a power-scale variation (PSV$_{a,p}$) with two tunable parameters: the sparsity-inducing exponent $p\in(0,1]$ and the scaling factor $a\in(0,\infty)$. By minimizing the PSV$_{a,p}$, we establish stable reconstructions in both the gradient and the image domains under the restricted isometry property (RIP) framework. Furthermore, we design an iteratively re-weighted least squares algorithm IRLSPSV to solve the unconstrained PSV$_{a,p}$ minimization. Numerical experiments demonstrate its superior performance and broad applicability. The main novelties are: (i) the PSV$_{a,p}$ minimization enjoys great flexibility and wide applicability due to its two tunable parameters $a$ and $p$, (ii) as $a\to\infty$, the PSV$_{a,p}$ minimization reduces to the $p$-th power total variation (TV$_p$) minimization and, even in this limiting case, the established RIP condition for image reconstruction is also new, (iii) the derived RIP upper bound $\overlineδ$ is proved to be asymptotically optimal in $a$ for gradient recovery, (iv) sensitivity analysis confirms the distinct roles of $a$ and $p$, thereby motivating a practical parameter tuning scheme for the proposed model. △ Less

Submitted 22 June, 2026; originally announced June 2026.

MSC Class: Primary 94A08; Secondary 90C31; 90C26; 94A15

Partial-twuality polynomial interpolation for binary delta-matroids

Authors: Zhuo Li, Qi Yan, Xian'an Jin

Abstract: Gross, Mansour and Tucker introduced the partial-twuality polynomials for ribbon graphs and investigated the interpolation property of these polynomials. The ribbon group generated by $δ$ and $τ$ acts on set systems as twist $\ast$ and loop complementation $\times$, yielding five nontrivial twuality operators: $ \{\ast,\times,\ast\times ,\times\ast ,\ast\times\ast \}.$ Yan and Jin extended partial… ▽ More Gross, Mansour and Tucker introduced the partial-twuality polynomials for ribbon graphs and investigated the interpolation property of these polynomials. The ribbon group generated by $δ$ and $τ$ acts on set systems as twist $\ast$ and loop complementation $\times$, yielding five nontrivial twuality operators: $ \{\ast,\times,\ast\times ,\times\ast ,\ast\times\ast \}.$ Yan and Jin extended partial-twuality polynomials to set systems, yielding partial-$\bullet$ polynomials with $\bullet\in\{\ast,\times,\ast\times ,\times\ast ,\ast\times\ast\}.$ For partial-$\ast$ polynomials, Zhao and Yan proved that this polynomial is either even, odd, or both even-interpolating and odd-interpolating for every binary delta-matroid. In this paper, we extend this interpolation property to all the remaining nontrivial partial-twualities of binary delta-matroids. Consequently, for every binary delta-matroid and every $\bullet\in\{\ast,\times,\ast \times ,\times\ast ,\ast \times \ast \}$, the partial-$\bullet$ polynomial is either even, odd, or both even-interpolating and odd-interpolating. We also provide examples to show that the binary assumption is essential. △ Less

Submitted 22 June, 2026; originally announced June 2026.

Comments: 23 pages

Bloch's conjecture for equivalences between twisted abelian surfaces and applications

Authors: Zaiyuan Chen, Zhiyuan Li, Ruxuan Zhang

Abstract: The Beauville--Voisin conjecture predicts a canonical descending filtration on the Chow group of zero-cycles of a hyperkähler variety, opposite to the conjectural Bloch--Beilinson filtration. A basic test for such filtrations is a Bloch-type principle: the action on zero-cycles should be governed by the action on the holomorphic symplectic form. While this principle has been verified in several ca… ▽ More The Beauville--Voisin conjecture predicts a canonical descending filtration on the Chow group of zero-cycles of a hyperkähler variety, opposite to the conjectural Bloch--Beilinson filtration. A basic test for such filtrations is a Bloch-type principle: the action on zero-cycles should be governed by the action on the holomorphic symplectic form. While this principle has been verified in several cases of hyperkähler varieties of $\mathrm{K3}^{[n]}$-type, the $\mathrm{Kum}_n$-type case remains much less understood. In this paper, we study this problem through twisted abelian surfaces and their associated $\mathrm{Kum}_n$-type varieties. We first construct a natural action of autoequivalences of twisted abelian surfaces on the Albanese kernel and prove Bloch's conjecture for all (anti-)symplectic autoequivalences. As an application, we prove the corresponding Bloch conjecture for symplectic birational automorphisms of twisted modular $\mathrm{Kum}_n$-type varieties; in particular, this applies to those admitting a birational Lagrangian fibration. Finally, we introduce and study a Shen--Yin--Zhao type filtration on twisted modular varieties and compare it with Voisin's filtration in the sixfold case. We also establish the anti-symplectic Bloch conjecture for twisted modular $\mathrm{Kum}_3$-type varieties. △ Less

Submitted 29 June, 2026; v1 submitted 20 June, 2026; originally announced June 2026.

Comments: 44 pages, slight modification

MSC Class: 14C25; 14F08; 14J42; 14K05

Weighted sum formulas for finite multiple mixed values

Authors: Zhonghua Li, Zhenlu Wang

Abstract: In this paper, we employ the iterated integral expression of multiple polylogarithms to establish a weighted sum formula for finite multiple mixed values. As applications, we derive various relations among level-two variants of finite multiple zeta values. In this paper, we employ the iterated integral expression of multiple polylogarithms to establish a weighted sum formula for finite multiple mixed values. As applications, we derive various relations among level-two variants of finite multiple zeta values. △ Less

Submitted 20 June, 2026; originally announced June 2026.

Counting and Sampling Anti-ferromagnetic Potts Models on Random Regular Bipartite Graphs in the Non-uniqueness Regime

Authors: Zhidan Li, Siyu Liu, Kuan Yang

Abstract: The anti-ferromagnetic multi-state Potts model, a generalization of the Ising model, is one of the most fundamental models in statistical physics. It was conjectured by Kotecký (Phys. Rev. B, 1985) that the model undergoes a phase transition from a disordered phase at infinite temperature to an ordered phase at sufficiently low temperature on lattices. Such phase transitions are believed to play a… ▽ More The anti-ferromagnetic multi-state Potts model, a generalization of the Ising model, is one of the most fundamental models in statistical physics. It was conjectured by Kotecký (Phys. Rev. B, 1985) that the model undergoes a phase transition from a disordered phase at infinite temperature to an ordered phase at sufficiently low temperature on lattices. Such phase transitions are believed to play an important role in computational complexity theory and remain closely connected to the problem of approximating the partition function of the system. For proper three-coloring models (corresponding to the zero-temperature), torpid mixing of a family of local-update Markov chains on lattices was established by Galvin, Kahn, Randall and Sorkin (SIDMA, 2015), coinciding with the presence of phase coexistence following shown by Feldheim and Spinka (J. Eur. Math. Soc., 2019). In this work, we study approximating the partition function of the anti-ferromagnetic multi-state Potts model at low temperature on random regular bipartite graphs, which are with high probability good bipartite expanders. On the negative side, we generalize the result by Geisler, Kang, Sarantis and Wdowinski (arXiv, 2026) for anti-ferromagnetic Ising models to show that when the temperature is sufficiently low relative to the degree of the underlying graph, the celebrated single-site Glauber dynamics has exponentially slow mixing time. On the positive side, we design a deterministic algorithm that yields an approximation to the partition function of the model via the framework of abstract polymer models as Jenssen, Keevash and Perkins (SICOMP, 2020), Liao, Lin, Lu and Mao (Theor. Comput. Sci., 2022), Galanis, Goldberg and Stewart (TOCT, 2021) and Geisler, Kang, Sarantis and Wdowinski (arXiv, 2026). △ Less

Submitted 2 August, 2026; v1 submitted 19 June, 2026; originally announced June 2026.

Comments: The tail bound Lemma 3.7 is improved under the help of AI, which leads to a better bound of our main results

Optimizing Agricultural Drone Operations: From Launch and Recovery Siting to Tiered Routing Strategies

Authors: Ethan Kolby, Josh Noble, Max Z. Li

Abstract: Drones are increasingly used in agriculture, where tight margins demand efficient planning. Current optimization tools suffer from exponential runtimes as problem sizes grow, necessitating practical heuristics for daily operations. This paper presents an operational framework and benchmarking analysis for drone spraying operations. We evaluate the trade-offs between facility siting methods and tie… ▽ More Drones are increasingly used in agriculture, where tight margins demand efficient planning. Current optimization tools suffer from exponential runtimes as problem sizes grow, necessitating practical heuristics for daily operations. This paper presents an operational framework and benchmarking analysis for drone spraying operations. We evaluate the trade-offs between facility siting methods and tiered routing parameters. For facility siting, comparing a Mixed-Integer Program (MIP) baseline against a $p$-Median heuristic shows that the heuristic reduces runtime by three orders of magnitude, from over 97 seconds to under 1.2 seconds, with only a 4\% reduction in serviced field area. For route planning, a tiered problem decomposition approach partitioning the target area into 6 to 8 spatial clusters reduces computation time by an order of magnitude with minimal degradation in serviced area. This framework achieves minute-scale planning on commodity hardware, demonstrating operational relevance. Future research will incorporate weather modeling, integrated optimization of facility location and routing, and validation across diverse field geometries. △ Less

Submitted 18 June, 2026; originally announced June 2026.

Comments: 33 pages, 4 tables, 10 figures, preprint submitted to Drone Systems & Applications

On Injectivity of Phase Retrieval

Authors: Zhangsong Li

Abstract: In this short note, we prove that if $A \in \mathbb C^{N \times M}$ with $N=4M-5$ has i.i.d.\ standard complex Gaussian entries, then the probability that the phase retrieval map generated by $A$ is not injective is positive. This proves Part (1) of a conjecture of Cynthia Vinzant, which was later restated by Afonso S. Bandeira in \cite{BDL+26}. The main result of this paper was obtained using gen… ▽ More In this short note, we prove that if $A \in \mathbb C^{N \times M}$ with $N=4M-5$ has i.i.d.\ standard complex Gaussian entries, then the probability that the phase retrieval map generated by $A$ is not injective is positive. This proves Part (1) of a conjecture of Cynthia Vinzant, which was later restated by Afonso S. Bandeira in \cite{BDL+26}. The main result of this paper was obtained using generative AI, in particular the Rethlas system. △ Less

Submitted 16 June, 2026; originally announced June 2026.

Comments: 4 pages; AI generated, human verified

Rotation Sets and Topological Entropy for Random Circle Endomorphisms

Authors: Zixu Li, Simon Lloyd, Sergio Romaña

Abstract: We study the topological dynamics of random circle endomorphisms of degree one over an ergodic measure-preserving dynamical system. Under an integrability assumption, we prove that the random rotation set is almost surely the compact interval whose endpoints are the mean random rotation numbers of the associated lower and upper random maps. We also show that the natural orbitwise versions of the r… ▽ More We study the topological dynamics of random circle endomorphisms of degree one over an ergodic measure-preserving dynamical system. Under an integrability assumption, we prove that the random rotation set is almost surely the compact interval whose endpoints are the mean random rotation numbers of the associated lower and upper random maps. We also show that the natural orbitwise versions of the random rotation set agree almost surely, and on the same full-measure set, every value in this interval is realised as the asymptotic average displacement along an individual orbit. In addition, every closed subinterval of the random rotation set is realised, on a full-measure set, as the set of accumulation values of the displacement averages along a single orbit. Finally, we prove that a positive length of the random rotation set implies a positive random topological entropy, in contrast to random monotone maps, which have zero random topological entropy. We illustrate the theory by computing the random rotation set and random topological entropy for a piecewise linear example. △ Less

Submitted 15 June, 2026; originally announced June 2026.

Comments: 28 pages, 1 figure

MSC Class: 37E10; 37H12; 37E45; 37B40

Logarithmic stability for determining the damping coefficient for the time-fractional damped wave equation

Authors: Kai Yu, Zhiyuan Li

Abstract: This paper investigates the inverse problem of determining the spatially dependent damping coefficient in a time-fractional damped wave equation, where the damping term is given by a Caputo derivative of order \(α\in(0,1)\). We first prove the well-posedness of the direct problem in exponentially weighted Sobolev spaces. Then, by means of the Fourier--Laplace transform in time, the nonstationary p… ▽ More This paper investigates the inverse problem of determining the spatially dependent damping coefficient in a time-fractional damped wave equation, where the damping term is given by a Caputo derivative of order \(α\in(0,1)\). We first prove the well-posedness of the direct problem in exponentially weighted Sobolev spaces. Then, by means of the Fourier--Laplace transform in time, the nonstationary problem is reduced to a family of stationary elliptic equations with complex frequencies. Based on complex geometrical optics solutions and a suitable integral identity, we estimate the Fourier transform of the difference of two damping coefficients in terms of the difference of their Dirichlet-to-Neumann maps. Combining low-frequency estimates with a high-frequency decay argument, we obtain a conditional logarithmic stability result. △ Less

Submitted 15 June, 2026; originally announced June 2026.

Benjamin-Feir spectrum of hydroelastic Stokes waves

Authors: Ting-Yang Hsiao, Zirui Li, Ye Zhang, Chengbin Zhu

Abstract: We determine the complete Benjamin-Feir spectrum near the origin for small-amplitude hydroelastic Stokes waves of the two-dimensional finite-depth irrotational Euler equations with surface tension and elastic bending. For the non-resonant Stokes branch and away from an intrinsic characteristic-collision surface $\mathfrak D$, we resolve all four Bloch eigenvalues bifurcating from the origin in the… ▽ More We determine the complete Benjamin-Feir spectrum near the origin for small-amplitude hydroelastic Stokes waves of the two-dimensional finite-depth irrotational Euler equations with surface tension and elastic bending. For the non-resonant Stokes branch and away from an intrinsic characteristic-collision surface $\mathfrak D$, we resolve all four Bloch eigenvalues bifurcating from the origin in the long-wave Floquet regime. Exploiting the Hamiltonian and reversible structure of the problem, we reduce the linearized Bloch operator to the four-dimensional spectral subspace bifurcating from the generalized kernel at the origin and conjugate the resulting matrix to the direct sum of a Benjamin-Feir block and a long-wave block. The long-wave pair remains purely imaginary, whereas the Benjamin-Feir pair is governed by an explicit closed-form instability index $\operatorname{Ind}(\mathtt{h},κ,b)$: a positive index produces a local figure-eight spectral curve with nonzero real part, while a negative index implies that all four small eigenvalues remain purely imaginary. Together with the Wilton-type resonance loci and the characteristic-collision surface $\mathfrak D$, this index yields a three-parameter spectral-stability diagram in the depth $\mathtt{h}$, surface tension $κ$, and bending rigidity $b$. The diagram recovers the classical pure-gravity critical-depth limit and, on the zero-bending boundary, the gravity--capillary stability diagram. It also reveals a genuinely hydroelastic phenomenon: all Wilton-type resonances disappear whenever $b\geq 1/14$ or $κ\geq 1/2$. This provides the first complete rigorous characterization of the local Benjamin-Feir spectrum for a hydroelastic free-boundary problem. △ Less

Submitted 8 June, 2026; originally announced June 2026.

Comments: 55 pages, 5 figures

MSC Class: 76B15; 37K45; 35B35; 76B15

Simultaneous recovery of multiple parameters in nonlocal diffusion equations from internal measurements

Authors: Kai Yu, Zhiyuan Li, Yikan Liu

Abstract: This paper is devoted to simultaneously recovering multiple parameters from internal measurements for nonlocal diffusion equations. The uniqueness of the inverse problem is established by employing the asymptotic behavior of solutions, analytic continuation, the Laplace transform, and properties of analytic functions. For numerical reconstruction, we apply the Levenberg-Marquardt method to obtain… ▽ More This paper is devoted to simultaneously recovering multiple parameters from internal measurements for nonlocal diffusion equations. The uniqueness of the inverse problem is established by employing the asymptotic behavior of solutions, analytic continuation, the Laplace transform, and properties of analytic functions. For numerical reconstruction, we apply the Levenberg-Marquardt method to obtain a stable approximate solution of the inverse problem. Numerical examples are provided to demonstrate the efficiency of the proposed algorithm and to validate our theoretical findings. △ Less

Submitted 7 June, 2026; originally announced June 2026.

Comments: 18 pages, 2 figures, 2 tables

MSC Class: 35R30; 35R11; 26A33

Quasi-isometric rigidity for random subsets in products of trees

Authors: Zhiqiang Li, Ranfeng Yu, Tianyi Zheng

Abstract: In this article, we prove a rigidity result for quasi-isometric embeddings from a random subset $D$ of the product $\mathbb{X}$ of two regular trees into $\mathbb{X}$ itself. This can be seen as an extension of Eskin's quasi-isometric rigidity of higher-rank nonuniform lattices to random subsets. As a consequence, we give a description of the self-quasi-isometric embeddings of a random sample. We… ▽ More In this article, we prove a rigidity result for quasi-isometric embeddings from a random subset $D$ of the product $\mathbb{X}$ of two regular trees into $\mathbb{X}$ itself. This can be seen as an extension of Eskin's quasi-isometric rigidity of higher-rank nonuniform lattices to random subsets. As a consequence, we give a description of the self-quasi-isometric embeddings of a random sample. We also show that two independent samples are almost surely non-quasi-isometric, confirming that such a phenomenon occurs in the higher-rank setting, as suggested by Abért. This result contrasts with the result on quasi-isometric equivalence between random sequences by Basu and Sly. △ Less

Submitted 3 June, 2026; originally announced June 2026.

MSC Class: Primary: 20F65; Secondary: 51F30; 60K35

A Quantified Two-projection Theorem for Nonlinear Projections

Authors: Zhangze Li, Krystal Taylor

Abstract: The classic Besicovitch projection theorem asserts that if a set is purely $1$-unrectifiable with finite length in $\mathbb{R}^2$, its orthogonal projection has Lebesgue measure zero in almost every direction. In the opposite direction, the two-projection theorem states that if a Borel set has zero measure under orthogonal projections onto two distinct non-antipodal directions, it must be purely… ▽ More The classic Besicovitch projection theorem asserts that if a set is purely $1$-unrectifiable with finite length in $\mathbb{R}^2$, its orthogonal projection has Lebesgue measure zero in almost every direction. In the opposite direction, the two-projection theorem states that if a Borel set has zero measure under orthogonal projections onto two distinct non-antipodal directions, it must be purely $1$-unrectifiable. We extend the two-projection theorem to certain families nonlinear projections and consider applications to pinned distance sets, radial projections, and curve projection operators. Further, we use a multiscale framework to obtain a quantitative version of our nonlinear two-projection theorem. Our arguments utilize methods introduced by Tao, who provided a quantitative treatment of the classic linear two-projection theorem. △ Less

Submitted 12 August, 2026; v1 submitted 29 May, 2026; originally announced June 2026.

Reconfiguration graphs of $K_{2,3}$-minor-free graphs

Authors: Ruijuan Gu, Hui Lei, Zhaoxiang Li, Yulai Ma, Susu Wang

Abstract: The $\ell$-reconfiguration graph of a graph $G$, denoted by $\mathcal{R}_{\ell}(G)$, is the graph whose vertices are the proper $\ell$-colorings of $G$, with an edge between two colorings if they differ in color on exactly one vertex. For any graph $G$ of treewidth at most $2$, Bousquet and Perarnau showed that $\mathcal{R}_\ell(G)$ has linear diameter for $\ell\geq 6$. This result was later exten… ▽ More The $\ell$-reconfiguration graph of a graph $G$, denoted by $\mathcal{R}_{\ell}(G)$, is the graph whose vertices are the proper $\ell$-colorings of $G$, with an edge between two colorings if they differ in color on exactly one vertex. For any graph $G$ of treewidth at most $2$, Bousquet and Perarnau showed that $\mathcal{R}_\ell(G)$ has linear diameter for $\ell\geq 6$. This result was later extended by Bartier, Bousquet, and Heinrich, who proved that $\mathcal{R}_5(G)$ also has linear diameter. In this paper, we show that for each $\ell\geq 5$, the $\ell$-reconfiguration graphs of $K_{2,3}$-minor-free graphs, some of which include graphs of treewidth $3$, have linear diameter. As a key step in our proof, we also establish that the $(\ell-1)$-reconfiguration graphs of cactus graphs have linear diameter. △ Less

Submitted 29 May, 2026; originally announced May 2026.

On the geometry of Certain Non-Basic Affine Deligne-Lusztig Varieties

Authors: Zhiming Li

Abstract: Let $F$ be a non-Archimedean local field, let $L=\breve F$, and let $G=\mathrm{GL}_n$. Let $M\subset G$ be a standard Levi subgroup and let $b\in M(L)$ be basic in $M$, but not necessarily basic in $G$. For a dominant cocharacter $μ$, we study the reduction-to-Levi morphism $β:X^G_μ(b)\to \bigsqcup_{μ_M\in S_M(μ,ν_b)}X^M_{μ_M}(b)$ for affine Deligne--Lusztig varieties in the affine Grassmannian. U… ▽ More Let $F$ be a non-Archimedean local field, let $L=\breve F$, and let $G=\mathrm{GL}_n$. Let $M\subset G$ be a standard Levi subgroup and let $b\in M(L)$ be basic in $M$, but not necessarily basic in $G$. For a dominant cocharacter $μ$, we study the reduction-to-Levi morphism $β:X^G_μ(b)\to \bigsqcup_{μ_M\in S_M(μ,ν_b)}X^M_{μ_M}(b)$ for affine Deligne--Lusztig varieties in the affine Grassmannian. Using an Iwasawa factorization relative to $P=MN$, we reduce the fiber condition to explicit Frobenius-twisted lattice equations in the off-block coordinates. In the Drinfeld case, where the base $X^M_{μ_M}(b)$ is zero-dimensional, we prove that $β$ is globally trivial with constant affine-space fiber in the non-basic cases considered. More generally, in the minuscule case we develop a nonzero-slope lattice-theoretic criterion which shows that the fibers are affine spaces and that $β$ is Zariski locally a trivial affine-space bundle in the non-basic cases considered. We also give examples in the non-minuscule setting where the fibers need not be affine spaces. △ Less

Submitted 15 June, 2026; v1 submitted 29 May, 2026; originally announced May 2026.

Comments: 37 pages; comments welcome

MSC Class: 14M15; 20G25; 14G35

Integrated Exploration-Aware UAV Route Optimization and Path Planning

Authors: Jimin Choi, Grant Stagg, Cameron K. Peterson, Max Z. Li

Abstract: Uncrewed aerial vehicles (UAVs) are increasingly used for exploration-driven monitoring in hazardous environments such as disaster zones, contaminated sites, wildfire areas, and damaged infrastructure, where limited flight endurance must be allocated between visiting reported locations and gathering new information. In these settings, prior information regarding hazards is often incomplete, spatia… ▽ More Uncrewed aerial vehicles (UAVs) are increasingly used for exploration-driven monitoring in hazardous environments such as disaster zones, contaminated sites, wildfire areas, and damaged infrastructure, where limited flight endurance must be allocated between visiting reported locations and gathering new information. In these settings, prior information regarding hazards is often incomplete, spatially imprecise, and subject to change during execution. For example, initial reports may identify a region where a hazard is likely to exist, but the actual hazard may be displaced, partially observed, or entirely unreported. We present an integrated exploration-aware UAV route optimization and path planning framework for hazard monitoring under uncertain and evolving prior information. The environment is represented as a spatial risk map, where each location has an associated belief of hazardous conditions. Reported hazards are modeled as uncertain regions of interest (ROIs) rather than confirmed target locations, requiring the UAV to inspect reported areas while also using its limited flight endurance to explore informative regions. The proposed method solves a vehicle routing problem over reported ROIs, augments the route with auxiliary pseudo-nodes to improve spatial coverage, allocates the remaining flight distance budget across route segments, and optimizes dynamically feasible B-spline trajectories for local exploration. During execution, UAV measurements update a grid-based belief map, and the remaining trajectory is replanned when new information and the remaining budget justify adaptation. Across 48 scenario configurations, online replanning improves average KL reduction by 15.9% over the offline optimized planner and 48.6% over straight-line traversal. △ Less

Submitted 18 June, 2026; v1 submitted 27 May, 2026; originally announced May 2026.

Positive Measure of Unions of Variable Surfaces

Authors: Alex Iosevich, Zhangze Li, Krystal Taylor

Abstract: Let $E \subset \mathbb R^d$, $d \ge 2$, be compact, and let $φ(x,y)$ be a smooth function satisfying the Phong--Stein rotational curvature condition on $\{φ(x,y)=1\}$. We prove that if $\dim_{\mathcal H}(E)>1$, then $$ \left|\bigcup_{x \in E} \{y : φ(x,y)=1\}\right|>0. $$ This extends the positivity theorem of Mitsis ($d\geq3$) and Wolff ($d=2$) for spheres to a general variable coefficient settin… ▽ More Let $E \subset \mathbb R^d$, $d \ge 2$, be compact, and let $φ(x,y)$ be a smooth function satisfying the Phong--Stein rotational curvature condition on $\{φ(x,y)=1\}$. We prove that if $\dim_{\mathcal H}(E)>1$, then $$ \left|\bigcup_{x \in E} \{y : φ(x,y)=1\}\right|>0. $$ This extends the positivity theorem of Mitsis ($d\geq3$) and Wolff ($d=2$) for spheres to a general variable coefficient setting via $L^2$ estimates for Fourier integral operators. The argument also shows that positivity is stable under finite-order degeneracies of the Monge--Ampère determinant through the weighted averaging theory of Sogge and Stein. We next consider variable level sets $$ Σ_x=\{y:φ(x,y)=t(x)\}, $$ where $t(x)$ is measurable. A maximal operator argument yields positivity under the condition $\dim_{\mathcal H}(E)>2$. We show that this loss reflects a genuine geometric obstruction related to Kakeya-type compression phenomena. In contrast, under a direct geometric intersection hypothesis controlling overlaps of the hypersurfaces $Σ_x$, we recover the full threshold $\dim_{\mathcal H}(E)>1$ for arbitrary measurable selections $t=t(x)$. At the endpoint $\dim_{\mathcal H}(E)=1$, we obtain positivity under the additional assumption that $E$ is $1$-rectifiable with $\mathcal H^1(E)>0$. We also show that positivity of Lebesgue measure does not in general imply interior regularity: even for large or rectifiable parameter sets, the resulting unions may have empty interior. Finally, we discuss extensions to higher co-dimension families and the role of geometric structure in preventing compression phenomena. △ Less

Submitted 26 May, 2026; originally announced May 2026.

On the axisymmetric Navier-Stokes flow passing a cone with the total-slip boundary condition

Authors: Zijin Li, Xin Yang, Qi S. Zhang

Abstract: (A) It is known that among the currently unresolved cases of the axially symmetric Navier-Stokes equations (ASNS), the most relatively tractable one is where the fluid passes the exterior of a cone. In this paper, we investigate this case with Navier total-slip boundary condition. We show that there exists an absolute constant $C_* > 0$ such that if \[ \sup_{x\in D}r|v_{0,θ}|\leq C_* \quad\tex… ▽ More (A) It is known that among the currently unresolved cases of the axially symmetric Navier-Stokes equations (ASNS), the most relatively tractable one is where the fluid passes the exterior of a cone. In this paper, we investigate this case with Navier total-slip boundary condition. We show that there exists an absolute constant $C_* > 0$ such that if \[ \sup_{x\in D}r|v_{0,θ}|\leq C_* \quad\text{and}\quad \int_{D} r v_{0,θ}(x) \mathrm{d} x = 0, \] then there exists a unique global bounded strong solution with finite energy. Note that, for the initial velocity, there is neither a size restriction on other components, nor a parity assumption. There are four key ingredients in the proof. (1) Three new good unknowns are introduced, and a self-closed energy estimate for them is derived. (2) An elliptic estimate for pressure is established to control boundary terms arising from the boundary condition. (3) A De Giorgi iteration scheme is applied to establish the boundedness of $rv_θ$. (4) A new anisotropic Hardy's inequality is derived for weighted mean-zero functions to overcome the lack of parity of $\boldsymbol{v}$. (B) Based on (A), we introduce and prove the so-called controlled regularity for the above problem, i.e. for suitable initial data without any smallness assumption, there exists an external force supported away from the axis of symmetry such that the corresponding problem admits a global strong solution. This seems to add a little weight to the regularity scenario for ASNS, since the force is supported away from the axis which is the only place regularity may break down. We also prove that if there exists a solution that blows up in finite time, an unstable blow-up solution must exist. △ Less

Submitted 23 July, 2026; v1 submitted 24 May, 2026; originally announced May 2026.

Comments: We have modified part of the argument in the controlled regularity section. All comments are welcome!

MSC Class: 35Q35; 76D05

Reachability-Augmented Dual Dynamic Programming for Optimal Path Parameterization

Authors: Yunan Wang, Jizhou Yan, Chuxiong Hu, Zeyang Li

Abstract: Optimal path parameterization (OPP) is a fundamental problem for planning trajectories along a prescribed geometric path under kinodynamic constraints and task-dependent objectives. While TOPP minimizes traversal time, its saturating states and controls may induce vibration and tracking errors, which can be mitigated by introducing smoothness objectives. However, a key capability gap remains in OP… ▽ More Optimal path parameterization (OPP) is a fundamental problem for planning trajectories along a prescribed geometric path under kinodynamic constraints and task-dependent objectives. While TOPP minimizes traversal time, its saturating states and controls may induce vibration and tracking errors, which can be mitigated by introducing smoothness objectives. However, a key capability gap remains in OPP: feasibility guarantees, general-objective optimality certificates, and computational efficiency are difficult to achieve simultaneously in a unified framework, especially for third-order OPP (OPP3) with non-convex constraints. This paper proposes reachability-augmented dual dynamic programming (RDDP), a state-grid-free and objective-aware DP framework for OPP. The key idea is to replace the relatively complete recourse assumption used in classical dual DP (DDP) with OPP-specific backward reachable sets, and then generate both value-function cuts and trial trajectories only inside these reachable sets. For convex and non-convex OPP, we prove global optimality and Karush-Kuhn-Tucker convergence of RDDP under OPP-specific conditions, respectively. Efficient instantiations are developed for OPP2 and OPP3. Experiments show that RDDP achieves objective values comparable to convex-optimization baselines while reducing computation time by 28.6 times for OPP2 and 5.8 times for OPP3. RDDP also achieves faster convergence than grid-based DP. Compared with reachability-analysis methods, RDDP retains the reachability mechanism while replacing local maximum-control propagation with value-function-guided control selection, thereby enabling objectives beyond traversal time. In summary, RDDP addresses a key capability gap in OPP by unifying certifiable general-objective optimization, reachability-based feasibility preservation, and online-compatible low-dimensional DP computation in a single OPP framework. △ Less

Submitted 18 May, 2026; originally announced May 2026.

Positivity of arbitrary-order P-recursive sequences with a unique dominant root

Authors: Zhongjie Li

Abstract: We establish a sufficient condition for the ultimate positivity of P-recursive sequences of arbitrary order with a unique dominant root. By additionally verifying finitely many initial terms, the positivity can also be resolved. As an application, we provide several examples of P-recursive sequences of order greater than two. We establish a sufficient condition for the ultimate positivity of P-recursive sequences of arbitrary order with a unique dominant root. By additionally verifying finitely many initial terms, the positivity can also be resolved. As an application, we provide several examples of P-recursive sequences of order greater than two. △ Less

Submitted 16 May, 2026; originally announced May 2026.