Papers

Building on the point interaction method developed in our previous work, this paper studies inverse eigenvalue problems for regular Sturm-Liouville problems with Dirac weights. More precisely, we explicitly reconstruct the potential of regular Sturm-Liouville problems with the single-point and two-point Dirac weights from the solutions of a class of Cauchy problems which is completely determined by the first eigenvalues of a family of perturbed problems originating from moving point interaction models in quantum mechanics. Finally, the multi-point Dirac weighted case is also discussed.

We give an efficient algorithm with improved algorithmic guarantees for the (offline) Beck–Fiala problem when the sets have bounded size. Let $A$ be an arbitrary matrix $A\in\{0,1\}^{m\times n}$ with at most $d$ ones per column and at most $s$ ones per row. Let $\log^*$ denote the iterated logarithm and $\ell_j$ denote the $j$-fold composition of log. Assume $s\le\exp(O(\sqrt d))$. We provide an efficient algorithm that, for arbitrary sparsity $d$, gives $O(\sqrt d(1+\log^*n))$ discrepancy. Moreover, if $d\ge\ell_j(n)$ for a fixed integer $j\ge1$, the algorithm gives $O_j(\sqrt d)$ discrepancy. The proof is a bootstrapping scheme using the Bansal-Jiang algorithm.

We introduce the notion of a unitary disk-like $n$-category, which is a disk-like $n$-category equipped with a reflection structure and a sphere trace inducing positive-definite pairings. Since a finite unitary disk-like $n$-category is defined to be the local field data of a fully extended $(n+1)$D unitary TQFT, we propose a complete finite unitary disk-like $n$-category as the definition of a finite $(n+1)$-Hilbert space for all $n$. For $n=1$ and $n=2$ we verify this proposal, proving that complete finite unitary disk-like 1- and 2-categories are isometrically equivalent to finite 2- and 3-Hilbert spaces respectively, and that these equivalences are functorial. For $n=1$ we recover a unitary refinement of Schommer-Pries' classification of oriented $(1+1)$D TQFTs in terms of $\mathrm{H}^*$-Morita equivalence classes of $\mathrm{H}^*$-algebras, and for $n=2$ we categorify this to classify oriented $(2+1)$D unitary TQFTs by $\mathrm{H}^*$-Morita equivalence classes of $\mathrm{H}^*$-multifusion categories. Along the way, we investigate fully incomplete 3-Hilbert spaces, prove a strictification result for pivotal dagger 2-categories, and define functors and higher transformations between disk-like $n$-categories.

We study the convergence of the vanilla stochastic policy gradient method applied to the linear quadratic regulator (LQR) problem. The method is cheap in the following sense: (1) at each iteration only $\tilde{O}(1)$ interactions with the environment are needed, therefore allowing frequent policy improvement steps, and (2) to ensure stability throughout and convergence to an $\epsilon$-optimal policy with probability $1-\delta$, only $O(\mathtt{Polylog}(1/\delta)/\epsilon)$ interactions are needed. To the best of our knowledge, this appears to be the first time that a stochastic model-free policy optimization method for LQR converges with high probability with $\tilde{O}(1)$ per-iteration computation and polylogarithmic dependence on the confidence level. The convergence analysis presented here is agnostic to LQR specifics and hence could be potentially generalized to a broader class of problems.

We prove a positive mass theorem for one-ended asymptotically flat spin initial data with multiple time directions and a smooth compact boundary. Assuming the trace-norm dominant energy condition and pairwise commutativity of the second fundamental forms, a fixed auxiliary time direction on each boundary component determines a local chirality condition. We identify the surviving mixed boundary term and obtain the inequality $E\geq\|\mathcal P\|_{\mathrm{tr}}$ under the boundary condition $H+\operatorname{tr}_\Sigma k^T+\|\mathcal Q_T\|_{\mathrm{tr}}\leq0$. The proof constructs a Dirac–Witten harmonic spinor in an affine energy space and evaluates the mass identity by approximation. We also establish the pointwise sharpness of this boundary condition for the fixed projector and give two-time data with a nonzero mixed boundary term.

Several institutions test one family of hypotheses under family-wise error control but cannot pool their data, so each site releases only a quantized report of its p-value for each hypothesis, such as whether the p-value is significant. The report's resolution is the number of values it can take. We quantify the power lost to such reports relative to the most powerful centralized procedure given all p-values (the oracle), with reports and combining rule chosen optimally. A report from which the p-value can be recovered loses no power, so the loss is due to compression, not decentralization. Under the stated regularity and power-objective conditions, every finite resolution loses power, and the optimal loss decays as the inverse square of the resolution: equal-width intervals attain this order, and no partition into as many cells improves it. In two-site Beta examples, optimized one-bit reports are nearly lossless for a single hypothesis but lose several times as much power for two. At level 0.05 with two Beta(1,2) hypotheses, optimally combined three-bit equal-width reports retain 99.2% of oracle power. We give the power-optimal combining rule for given reports, and a rule using only their exactly known null distribution, valid under arbitrary dependence across hypotheses.

We study a Stackelberg model where two players compete in a sequential contest, and the player that successfully invests the most effort wins. Our formulation considers a probability that each player fails to invest their chosen level of effort, as captured by a failure function that increases with the amount of effort. The players each have their own cost of failure, as well as a maximum level of effort (capturing different abilities of the players). We prove three key results. First, we provide the optimal strategy for Player 1 (the follower), to maximize their utility given an observed effort exerted by Player 2 (the leader). Next, we characterize the optimal strategy to maximize the expected utility of Player 2, anticipating the optimal response by Player 1. Finally, we analyze the expected utilities of each player when both play their optimal strategy, and provide key insights into how heterogeneous player abilities and failure costs affect the outcomes of the game.

In this paper, we establish existence and pointwise Wolff potential bounds for nonnegative solutions of $-\Delta_{\mathbb{G},p}u=H_l(\alpha u^\beta)+\mu$ on Carnot groups, where $H_l(t)=e^t-\sum_{j=0}^{l-1}t^j/j!$ and $\mu$ is a nonnegative Radon measure. Under suitable conditions on the reaction parameters, we treat arbitrary open sets in the subcritical range $1<p<Q$ under a maximal potential condition, and bounded domains in the critical case $p=Q$ under a small total mass condition.

In this paper, we study polyhedral aspects on commutative and noncommutative Edmonds' problems for computing the rank of linear symbolic matrix $A = \sum_{k=1}^m A_k x_k$. We regard them as linear optimization over integral polytopes ${\cal P}(A)$ and ${\cal Q}(A)$, respectively, which are obtained by the convex hulls of exponent vectors of subdeterminants of~$A$ and its blow-ups $A^{\{d\}} = \sum_{k=1}^m A_k \otimes X_k$ $(d=1,2,\ldots)$. By extending previously known results on nc-rank, we establish a hierarchy of integral polytopes ${\cal P}(A) \subseteq {\cal P}^{\leq 2}(A) \subseteq {\cal P}^{\leq 3}(A) \subseteq \cdots = {\cal Q}(A)$ and show that the integrality gap of ${\cal Q}(A)$ relative to ${\cal P}(A)$ is at least $1/2$. Further, we show that if each $A_k$ is rank-2 skew-symmetric, then the above hierarchy terminates at the second level and the integrality gap is improved to $2/3$.

In this paper, we study the fixed angle scattering problems raised by Rakesh and Salo. We show that a compactly supported potential is uniquely determined by the far field pattern at a fixed angle, which is an open problem. Our method is based on a novel Carleman estimate and the ideas introduced by Bukhgeim and Klibanov on the use of Carleman estimates for inverse problems.

In our previous work we introduced the Reconstruction Dimension, an upper bound on the reconstructable degrees of freedom determined solely by the symmetry. In actual physical systems, however, the state quantities are generated through physical processes $P_o$ induced by the cause $O$. The state space is therefore often degenerate, and the upper bound determined by the representation-theoretic structure is not reached. This calls for a distinction between the upper bound determined solely by symmetry and the one realized after the physical process. We therefore introduce a three-level structure of upper bounds: the Structural, Effective, and Physical Reconstruction Dimensions (SRD, ERD, PRD), determined respectively by the representation-theoretic structure of the intermediate space, by the concrete design of the reconstruction map, and by the degeneracy of the state space under the physical process $P_o$. For each irreducible component they form the hierarchy $\mathrm{SRD}\ge\mathrm{ERD}\ge\mathrm{PRD}$, so that the two gaps identify at which level reconstructability is lost. The framework is illustrated through two contrasting systems: the orientation dynamics of flake-like particles, where the cause is a velocity gradinet, and the two-body problem, where the cause itself is defined through the symmetry. This paper provides a concrete framework for reconstruction in physical inverse problems.

In the present paper, we study finite subgroups of the outer automorphism groups of the absolute Galois groups of mixed-characteristic local fields from the point of view of mono-anabelian geometry. We shall refer to a finite subgroup of the outer automorphism group of the absolute Galois group of a mixed-characteristic local field as quasi-geometric if its inverse image in the automorphism group of the corresponding absolute Galois group is isomorphic to the absolute Galois group of some mixed-characteristic local field. It is well-known that the image of the natural injective homomorphism from the automorphism group of a mixed-characteristic local field to the outer automorphism group of the associated absolute Galois group yields a typical example of a quasi-geometric subgroup. The main result of the present paper asserts the existence of non-quasi-geometric finite subgroups of the outer automorphism groups of the absolute Galois groups of mixed-characteristic local fields. This result shows that the set of quasi-geometric subgroups of the outer automorphism group of the absolute Galois group of a mixed-characteristic local field cannot be characterized, in general, as the set of all finite subgroups of the outer automorphism group.

Perverse schobers are a categorification of perverse sheaves, originally proposed by Kapranov and Schechtman. The purpose of this paper is to initiate a microlocal study of perverse schobers. We first categorify $\operatorname{Perv}(\mathbb{C},R)/\operatorname{Loc}(\mathbb{C})$, the category of perverse sheaves on a complex line with singular points at $R$, modulo local systems. We use this to propose a general definition for microlocal perverse schobers supported on the open conormal to a germ of a hypersurface, and we conjecture that this is invariant under Radon transform. Here we make the key observation that while the analogous quotient of perverse schobers $\mathsf{2Perv}(\mathbb{C},R)/\mathsf{2Loc}(\mathbb{C})$ is a reasonable categorification, the resulting theory fails to be invariant under Radon transform. In fact, our proposed categorification can be recovered by correcting an instance of this failure in a universal manner. We prove Radon invariance when our hypersurface is the curve $y^m=x^n$ in $\mathbb{C}^2_{x,y}$. Along the way, we explain how our theory relates to Fourier transforms of perverse schobers, periodic SODs, and spherical monads.

It is shown that if $u$ solves the wave equation in $\mathbb{R}^{4+1}$ with initial data $u(\cdot,0) = u_0(\cdot) \in H^s$ and $\partial_tu(\cdot,0) = u_1(\cdot ) \in H^{s-1}$, where $0.5 < s \leq 0.55$, then $u(x,t) \to u_0(x)$ and $\partial_tu(x,t) \to u_1(x)$ pointwise as $t \to 0$, for all $x$ outside an exceptional set of Hausdorff dimension at most $6-4s$. In a very small range of $s$, this verifies a conjecture of Barceló, Bennett, Carbery, and Rogers. More generally, a partial improvement to the exceptional set bound in $\mathbb{R}^{n+1}$ is obtained for $n \geq 4$ and $1/2 < s < n/4$.

Large deviation theory lacks a geometric foundation for its rate functions and fluctuation symmetries. This paper develops a contact large deviation theory on stochastic vector bundles, in which the rate function, the scaled cumulant generating function, and a Gallavotti–Cohen-type fluctuation duality all follow from the contact 1-form. The constraint function acts as a generalized Lagrangian; the least constraint principle yields the dynamics, and the contact potential is built order by order from the master equation, producing a coupled Hamilton–Jacobi–transport system governed by the invariant density, drift, and fluctuation tensor. The contact path measure satisfies a large deviation principle with rate function given by the constraint action; the scaled cumulant generating function obeys a stationary eigenvalue equation with a Donsker–Varadhan variational characterization. The entropy production rate, the fluctuation–dissipation combination $e=\tfrac12 g^T Ag-\sigma$, is the physical observable; the time-reversal involution $J:(t,y,\phi)\mapsto(t,y,-\phi-\nabla\ln\rho)$ with reversed drift $v^{\mathrm{rev}}=-v+Ag$ yields a Gallavotti–Cohen-type duality $\lambda_{\mathrm{fwd}}(q)=\lambda_{\mathrm{rev}}(q+1)+\lambda_{\mathrm{fwd}}(-1)$. The classical one-dimensional GC symmetry is recovered in the reversible case $Ag=0$.

We prove that every surjective Fischer–Muszély map between positive cones of arbitrary JB-algebras is additive and positively homogeneous, and extends uniquely to a bounded positive linear surjection. In particular, this gives a complete affirmative answer to Molnár's problem for arbitrary $C^*$-algebras. The theorem includes nonunital and exceptional JB-algebras and assumes neither injectivity nor continuity. The proof connects the norm-valued functional equation to positive-cone rigidity through translation invariance of an induced pseudometric. For a surjective map satisfying the Fischer–Muszély identity with uniform error $\varepsilon$, we also construct a unique bounded positive linear map $L$ at uniform distance at most $3\varepsilon$. The set $L(A_+)$ is dense in $B_+$, although $L$ need not be surjective, even when the original map is continuous and bijective. For bijective FM maps, we obtain weighted Jordan representations on the biduals. We also extend the characterization of surjective norm-sum preservers to JB-algebras and characterize norm arithmetic-midpoint and harmonic-mean identities on positive invertible cones of unital JB-algebras.

Design of high-contrast freeform reflectors is challenging, as the presence of zero-intensity regions leads to degeneracy in the associated Monge-Ampere-type equation. A common remedy is to add a positive artificial background to the target intensity, which improves the regularity of the equation. However, this regularization inevitably reduces the achievable illumination contrast and introduces nonzero intensity into regions that are intended to remain dark. We propose a fast c-transform method based on optimal transport duality for high-contrast freeform reflector design in the far field without artificial background regularization. The method integrates repeated discrete c-transforms into a measure-based dual optimization scheme to maintain the c-concavity of the reflector potential throughout the iteration. This preserves the admissibility of the iterates and enables stable convergence for targets with zero-intensity regions. To efficiently compute the discrete c-transforms, we develop a localized search algorithm that exploits their contact structure and prove its exactness and linear complexity. Numerical experiments demonstrate that the proposed method accurately realizes high-contrast illumination targets with zero background.

For a connected graph $G$, let $\tau(G)$ denote the maximum number of pairwise edge-disjoint spanning trees, and let $\rho(G)$ be its adjacency spectral radius. For integers $n\ge1$, $k\ge2$, and $k\le\delta\le2k-1$, let $\mathcal{G}_{n,\delta}$ be the class of connected $n$-vertex graphs with minimum degree $\delta$, and let $\mathcal{L}_{\mathcal{H}}^{2}(n,k,\delta)\subseteq\mathcal{G}_{n,\delta}$ be the exceptional family introduced by Chang, Li, and Zhang. For sufficiently large $n$, the sharp adjacency-spectral threshold for $\tau(G)\ge k$ is determined by maximizing $\rho(G)$ over this family. Set $h=\delta-k$. For each fixed admissible pair $(k,\delta)$ and all sufficiently large $n$, we determine the maximum and minimum adjacency spectral radii in $\mathcal{L}_{\mathcal{H}}^{2}(n,k,\delta)$. In the core–placement representation, let $M$ be the missing-edge graph of the bounded core. For $h\ge2$, the unique maximizer, up to isomorphism, satisfies $M\cong K_{1,h}\cup(h+1)K_1$, with the exceptional edges nested on the large-clique side. Hence the matching configuration proposed in Conjecture~2 of Chang–Li–Zhang is not extremal. For every $h\ge1$, the unique minimizer satisfies $M\cong hK_2\cup2K_1$, with the $2h$ exceptional edges having distinct endpoints on the large-clique side. We also settle the cases $h=0,1$ and all equality cases. The proof combines an exact core–placement parametrization with a uniform Schur-complement resolvent expansion. The first candidate-dependent coefficient is affine in $\sum_{x\in V(M)}d_M(x)^2$, while the first placement-sensitive coefficient is a squared-load functional. Equitable quotient matrices yield the two extremal radii, and the maximizing graph gives the sharp global adjacency-spectral threshold.

A variety $\mathcal{V}$ is called a Schreier variety if every subalgebra of a $\mathcal{V}$-free algebra is a $\mathcal{V}$-free algebra. We use ideas from Tame Congruence Theory to classify locally finite Schreier varieties. One version of the classification theorem states that a locally finite variety $\mathcal{V}$ is a Schreier variety if and only if (i) every finite algebra in $\mathcal{V}$ is a $\langle 0,1\rangle$-minimal algebra and (ii) if $\mathcal{V}$ has a constant $1$-ary term operation, then $\mathcal{V}$ also has a constant $0$-ary term operation.

Let $D$ be an odd fundamental discriminant, with $D=1$ permitted, and let $r\geq 1$ be fixed. We prove that, for every sufficiently large integer $\ell$ satisfying $(-1)^\ell D>0$, the first $r$ traced diagonal Rankin–Cohen brackets \[ \mathrm{Tr}_1^{|D|}[G_{\ell-2e,D},G_{\ell-2e,D}]_{2e}, \qquad 1\leq e\leq r, \] are linearly independent in $S_{2\ell}(SL_2(\mathbb Z))$. Here $G_{k,D}$ is the Eisenstein series of weight $k$, level $|D|$, and nebentypus $\chi_D$. The Petersson formula of Kayath, Lane, Neifeld, Ni, and Xue then implies that at least $r$ normalized Hecke eigenforms $f\in S_{2\ell}(SL_2(\mathbb Z))$ satisfy $L(f\otimes\chi_D,\ell)\neq 0$. For $D=1$, this gives, for every fixed $r$ and every sufficiently large $K\equiv 0\pmod 4$, at least $r$ level-one eigenforms of weight $K$ with nonzero central value.

We prove the linear stability of an embedded mean curvature flow shrinker in $\mathbb R^4$ with the topology of the Möbius bundle. This provides a non-flat, non-spherical stable shrinker in higher codimension, in sharp contrast with the classification of stable hypersurface shrinkers. Topologically, the Möbius shrinker models the reversal of a real blow-up, suggesting that stable higher-codimension singularities may encode non-trivial topological operations under mean curvature flow.

Epidemic mitigation comes with a price in practice, and when budget is limited, its effective allocation becomes an important question on public policy. In this paper, we study epidemic control through a Stackelberg mean-field game in which a principal, representing the government, chooses social distancing guidelines and vaccination levels for a large population of rational minor agents subject to a budget-spending process. The principal anticipates the minor agents mean-field Nash equilibrium (MFNE) and selects intervention policies to minimize her own objective, inducing a bi-level optimal control problem. We define the Stackelberg mean-field game equilibrium (SMFE) and give its necessary condition by a forward-backward ordinary differential equation (FBODE) system, and provide existence and uniqueness results. We further propose an iterative fixed-point algorithm to solve the FBODE numerically and present computational observations that illustrate the resulting equilibrium behavior.

Recently, Ferroni constructed a family of counterexamples to the well-known conjecture in Ehrhart theory stating that the $h^*$-polynomial of a lattice polytope with the integer decomposition property is unimodal. This raises the question of whether the $h^*$-polynomial of a lattice simplex with the integer decomposition property remains unimodal. In this note, we prove that every lattice simplex with the integer decomposition property and prime normalized volume has a unimodal $h^*$-polynomial. Furthermore, we establish several sufficient conditions for the unimodality of the $h^*$-polynomial of such simplices.

For ordered graphs $H_1,\ldots,H_t$, let $\rt(H_1,\ldots,H_t)$ denote the least integer $N$ such that every $t$-coloring of the edges of the naturally ordered complete graph on $[N]$ contains an ordered copy of $H_i$ in color $i$ for some $i\in[t]$. We prove that a uniformly random ordered matching $M$ on $n$ vertices with interval chromatic number two asymptotically almost surely satisfies \[ \rt(K_3,M) =\Omega\left(\frac{n^{4/3}}{(\log n)^{1/3}}\right). \] This strengthens the lower bound $\Omega((n/\log n)^{5/4})$ of Balko and Poljak for such random matchings and improves the general existential lower bound of Conlon, Fox, Lee and Sudakov by a factor of $\log n$. The proof combines pseudorandom triangle-free graphs, a coarse encoding of order-preserving embeddings, and a permutation avoidance estimate derived from Brègman's inequality.

We prove that every countable meet-continuous lattice is Scott sober. For any family of such lattices, the Scott topology on the product equals the product of the factor Scott topologies, and the product Scott space is sober. These results answer affirmatively Questions 7.1 and 7.2 of Xu (arXiv:2609.18032v1), extending the countable-frame results beyond finite distributivity. A diagonal argument on meet coordinates extracts directed punctured intervals from nonprincipal ideals. Join translates of these intervals form a countable family detecting Scott openness and yield Scott open rectangles by successive finite choices. We also show that the cardinal spectrum of Scott non-sober meet-continuous lattices is upward closed and, under the Continuum Hypothesis, consists of all uncountable cardinals.

We study harmonic maps \(F:\D\to G\) into bounded domains in real Hilbert spaces, prescribing \(F(0)\) and \(dF_0(\R^2)\) when \(dF_0\) is nonzero and conformal. We prove a target-independent identity for the Möbius-weighted Hilbert pairing. After normalization, the pairing is affine in \(\|dF_0\|\) with positive slope, yielding an exact equivalence between the derivative extremal problem and a boundary-pairing problem. For a round affine section, this gives a necessary and sufficient integral criterion for the Möbius parametrization to be extremal. A supporting-hyperplane condition guarantees the required integral inequality and characterizes equality. Examples show that roundness alone is insufficient and that the supporting condition is not necessary. For unit balls of real Hilbert spaces of dimension at least two, including infinite-dimensional spaces, we obtain the sharp prescribed-value Schwarz-Pick estimate, all equality cases, and quantitative \(L^2\) boundary stability. We also prove a Cayley-Klein contraction under pointwise distortion. For \(K\geq1\), let \(M_K\) denote the supremum of \(L_F(0)\) over harmonic maps \(F:\D\to\B_H\) satisfying \(F(0)=0\) and \(0<\ell_F(0)\leq L_F(0)\leq K\ell_F(0)\), where \(L_F(0)\) and \(\ell_F(0)\) are the maximal and minimal stretchings at the origin, respectively. We determine \(M_K\), identify the unique optimizing parameter, and characterize all extremals. The function \(K\mapsto M_K\) is strictly increasing, with \(M_1=1\) and \(M_K\to4/\pi\) as \(K\to\infty\).

Let $\mathpzc K$ be an arbitrary commutative ring containing an invertible element $\varepsilon$. Let ${\mathcal H}_{{\!\vartriangle\!}}(r)_{\mathpzc K}$ be the extended affine Hecke algebra of type $A$ with Hecke parameter $\varepsilon$, let $\Omega_{\mathpzc K}^{\otimes r}$ be the affine tensor space, and let ${\mathcal S}_{{\!\vartriangle\!}}(n,r)_{\mathpzc K}$ be the corresponding affine quantum Schur algebra. We first prove that the natural right action of ${\mathcal H}_{{\!\vartriangle\!}}(r)_{\mathpzc K}$ on $\Omega_{\mathpzc K}^{\otimes r}$ is always faithful. Assume further that $\mathpzc K$ is a field of characteristic $0$ and that $\varepsilon$ is not a root of unity. We prove that, for any $n\geq 2$, the natural algebra homomorphism $\xi_r:{\mathcal H}_{{\!\vartriangle\!}}(r)_{\mathpzc K}\rightarrow\operatorname{End}_{{\mathcal S}_{{\!\vartriangle\!}}(n,r)_{\mathpzc K}}(\Omega_{\mathpzc K}^{\otimes r})^{\mathrm{op}}$ is an isomorphism. This proves Conjecture~3.8.8 of \cite{DDF}. As an application, we prove the conjecture formulated in \cite[5.2.4]{DDF} concerning the center of the affine quantum Schur algebra. We also prove that ${\mathcal S}_{{\!\vartriangle\!}}(n,r)_{\mathpzc K}$ is left and right Noetherian whenever $\mathpzc K$ is a Noetherian commutative ring, which verify a conjecture in \cite[Rem. 1.7]{DY}.

We study the Friedrichs Laplacian associated with the pullback of the round metric on \(\mb P^1\) by a nonconstant meromorphic function \(\varphi:X\to\mb P^1\) on a compact Riemann surface. For arbitrary ramification profiles, including several ramification points over the same branch value, we prove the local formula \[ \operatorname{Det}_{\zeta}(\Delta_{[\varphi],\mc F}) =C\,\det\operatorname{Im}B\,|\tau_B|^2 \prod_{k=1}^N\rho(z_k,\overline{z_k})^{c_k}. \] The zero eigenvalue is omitted. Here \(B\) is the period matrix, \(\tau_B\) is the local Bergman tau-function, \(z_k\) are the branch-value coordinates, \(\rho(z,\overline z)=4(1+|z|^2)^{-2}\), and \(c_k=\frac1{12}\sum_j(n_{kj}-n_{kj}^{-1})\), where \(n_{kj}\) are the ramification indices over the \(k\)-th branch value. The constant \(C>0\) is independent of the Hurwitz coordinates, and \(\det\operatorname{Im}B\) is taken to be \(1\) in genus zero. The proof combines smooth trivializations and trace-norm variation of resolvent powers with matrix comparison to spherical conic models. The Davies–Gaffney estimate provides the required uniform high-energy control, while the zero-energy terms are identified through the Schiffer bidifferential and Rauch variational formulas.

Human pairwise comparisons provide a reference for evaluating large language models (LLMs), but collecting sufficient judgments for each new release is costly and time-consuming. LLM judges offer a scalable alternative, although their comparisons may differ systematically from human preferences and across judges. We study the ranking of a new model that has received LLM-judge comparisons but no human comparisons. We propose ANCHOR (ANchored Comparisons for Human-reference inference with Orthogonal Riesz correction), which uses historical human and LLM comparisons to learn judge-specific sensitivities to human score differences and feature-dependent judge biases. These estimates are then used to infer the new model's human-reference score from its judge comparisons. The framework allows the feature distribution to change between historical and new-model comparisons. For inference, we construct a Neyman-orthogonal estimator through a joint Riesz correction that removes the first-order effects of estimating the historical human scores, judge sensitivities, and bias functions. We establish identification, convergence rates, and asymptotic normality with consistently estimable variance, and show that ANCHOR attains the semiparametric efficiency bound. Simulations demonstrate gains in score estimation and ranking accuracy. On Chatbot Arena, ANCHOR achieves the lowest score RMSE and insertion MAE among competing methods, with narrower score intervals on average.

In this article, we give various real-variable properties of local Muckenhoupt matrix weights $W$ and establish the quantitative boundedness of several operators on local matrix-weighted Lebesgue spaces $L^p(W)$, including local (fractional) maximal operators, local fractional integral operators, local Haar square functions, and Calderón–Zygmund operators with exponential decay. For the local maximal operators, we obtain the sharp quantitative bounds when $p\in(1,2]$, while, for local fractional integral operators, we obtain the quantitative bounds matching the global best known exponents, whose scalar case is known to be sharp. The key used strategies include giving a new extension property (which can clarify their relationships with global ones) and an optimal scale lifting property (which can balance the locality of weights and operators under consideration) of local matrix weights. As an application, we establish the quantitative boundedness on $L^p(W)$ of the Riesz transform associated with Schrödinger operators $-\Delta+m^2I$ with $m\in(0,\infty)$ being large enough, whose quantitative bound when $p=2$ is precisely $[W]_{\mathscr{A}^{\operatorname{loc}}_{2}(r)}^{\frac 32}$.