Papers

Equivariance is increasingly used in machine learning and statistics, often without systematic justification. In a companion article, the equivariance criterion was applied to the normal linear model with a fixed design matrix (fixed-$X$), yielding the minimum risk equivariant (MRE) estimators of the coefficient vector and of the condensed diagonal covariance matrix under a multivariate invariant location–scale group. We extend these results to the random-$X$ case, with covariates sampled from a population. The extension hinges on a distinction vacuous for fixed-$X$ but fundamental for random-$X$: whether risk and unbiasedness are evaluated conditionally on the realized design or after averaging over the design distribution. Under conditional evaluation, the fixed-$X$ group applies given $X$: least squares remains the best equivariant estimator of the coefficient vector, and the MRE estimators of the population variances keep their fixed-$X$ forms with population sizes at the realized design. Under absolute evaluation with an i.i.d.\ design, the picture changes qualitatively: the natural scale group acting jointly on $(Y,X)$ fixes the coefficient vector, the induced parameter-space action is intransitive, equivariant risks are constant only along orbits indexed by the signal-to-noise ratio $\rho=\|\beta\|^2/\sigma^2$, and no uniformly minimum risk equivariant estimator exists. In the scalar case the optimal equivariant weight is the oracle shrinkage factor $w^*(\rho)=\rho/(\rho+E[T^{-1}])$, with least squares recovered as the infinite-signal limit $\rho\to\infty$—explaining and refining the known failure of the Gauss–Markov theorem with random regressors. For a centered design under location–scale transformations, least squares remains optimal within the natural invariant-contrast class, and the MRE estimator $S^2/(n-p+2)$ of the error variance is valid under both modes.

Protecting individual privacy has become a central and urgent concern in modern data analysis, given the vast quantities of data now generated and processed. In this paper, we develop a systematic theory of exact asymptotic efficiency for regular parametric estimation under central zero-concentrated differential privacy. In the privacy regime, the governing object is a diameter-constrained information region: the set of information matrices generated by statistics with diameter at most one. For weighted quadratic loss, we show that the exact local minimax risk is an inverse information variational functional over this region. More generally, a mixed information region yields a unified efficiency constant across different regimes, covering the privacy regime and classical Fisher efficiency. A matching estimator releases the empirical mean of a nearly optimal bounded statistic with Gaussian noise and locally inverts its population moment map. Our theory differs from classical efficiency theory in its set-valued information geometry and loss-dependent efficient estimator. As examples, we provide closed-form constants and optimal procedures for various concrete models, including one-dimensional regular families, Gaussian means, categorical probability vectors, and regression models among others. The theory also transfers to Gaussian differential privacy through an exact parameter rescaling.

In this paper, we establish an asymptotic formula of mixed moments of $L(1/2,f)$ and $L(1/2,\mathrm{sym}^2 f)$ for cusp forms $f$ of weight $k$, prime level $q$ and real primitive nebentypus $\chi$ modulo $q$ in the level aspect with a power-saving error term. As an application, we deduce the simultaneous non-vanishing of $L(1/2, f)$ and $L(1/2,\mathrm{sym}^2 f)$ for sufficiently large prime levels.

We prove that unrestricted static output-feedback stabilization is NP-hard, already in the single-input multiple-output case. More precisely, we give a polynomial-time many-one reduction from the NP-complete Betweenness problem. Given an instance with $n$ elements and $m$ betweenness constraints, the reduction constructs integer matrices $A\in\mathbb Z^{D\times D}$, $B\in\mathbb Z^{D\times1}$, and $C\in\mathbb Z^{n\times D}$ such that the instance is satisfiable if and only if there exists an unrestricted real row vector $K\in\mathbb{R}^{1\times n}$ for which $A+BKC$ is Hurwitz. The state dimension is $D=6(m+n)$, and every matrix entry has logarithmic bit length in the input size. The reduction uses one universal cubic source polynomial whose Routh determinant enforces the sign condition encoding betweenness, while fixed anchor values provide a mechanism for excluding spurious stabilizing gains of arbitrarily large magnitude. Frequency separation then compresses the simultaneous source conditions into one common affine polynomial, which is realized directly as the characteristic polynomial of $A+BKC$. As a by-product, deciding whether an affinely parameterized rational polynomial family contains a Hurwitz member on a rational box is NP-hard.

A novel semi-implicit second-order finite volume scheme integrating the scalar auxiliary variable (SAV) approach is proposed for solving the pseudo-time Eikonal equation in wall-distance computation. Unconditional energy stability under zero boundary conditions is rigorously proved, eliminating the dependence of the time step on the grid scale and enabling large-time-step computation. The scheme is a priori accuracy-preserving, and its discretization matrix forms an M-matrix, thereby guaranteeing strict non-negativity of the numerical solution inherently. The framework extends readily to any non-conservative scalar equation and, being independent of the specific finite-volume reconstruction, is compatible with schemes of arbitrary order of accuracy. A vanishing artificial viscosity is introduced to smooth the solution without compromising formal accuracy, and upwinding is incorporated through directional weighting in the weighted least-squares (WLS) reconstruction. Numerical experiments confirm that the scheme achieves the designed accuracy and permits stable computations with uniformly large time steps. For complex configurations such as a three-element airfoil and the three-dimensional ONERA M6 wing, accurate results are obtained on high-aspect-ratio grids, with relative errors in the computed wall distance below 3% relative to the search-based reference, except near geometric singularities.

We prove that the Subcomplete Forcing Axiom is consistent with $\diamondsuit^+$.

We introduce a novel online adaptive non-intrusive reduced-order modeling strategy for parameterized dynamical systems involving parameter and time-dependent reduced bases. The proposed framework is based on a unified Grassmann manifold formulation combining three key components: interpolation of local reduced subspaces for unseen parameters, geodesic online subspace updates driven by incoming high-fidelity snapshots, and a latent-space regression strategy relying on Grassmann-distance weighting and Procrustes alignment to consistently aggregate predictions from multiple local models. The adaptive reduced-order model is embedded in a partitioned fluid-structure interaction framework, where it predicts fluid interface forces to provide accurate initial guesses for the nonlinear coupling iterations, thus achieving computational speedups with no loss of accuracy. The reduced basis and the regression operators are adapted independently during the simulation and without requiring the storage of high-dimensional streaming data, preserving computational efficiency while substantially improving predictive capabilities. Numerical results on reference FSI test cases demonstrate superior accuracy with respect to static and global reduced-order models, leading to a significant reduction in the number of fixed-point iterations required for convergence. The proposed framework offers a flexible and fully non-intrusive approach for the efficient simulation of nonlinear parameter-dependent multiphysics problems.

We prove an $O(t^{1/4})$ bound on the support radius of nonnegative planar Euler vorticity with compactly supported $L^1$ initial data, removing the logarithmic factors from the classical confinement estimates. The result holds in the symmetrized vorticity formulation. A factorization of the interaction kernel yields a quadratic convolution inequality for high-order moments. Retaining this convolution allows an elementary comparison for finite sums of moments to control the full support. We also prove that the squared support radius is Hölder continuous in time with exponent $1/2$.

Increasingly severe and unpredictable extreme weather events-such as high-intensity forest firesthreaten long-term ecosystem service provision, driving a critical reassessment of current forest management planning. While planning frameworks exist, few incorporate optimization methods that account for the uncertainty and variability introduced by such extremes. Meanwhile, process-based models have advanced our understanding of fire impacts on forest dynamics and ecosystem services, yet their potential to inform strategic management decisions remains largely untapped. Focusing on intensive plantation forests, we propose a combined approach that uses fire simulation within a growth model to optimize forested landscape management under increasing fire risk. We demonstrate that incorporating fire behavior and its uncertainties significantly alters management outcomes, favoring diversified portfolios that balance high-yield but fire-prone regimes with lowerrisk alternatives. While diversification improves robustness for objectives like timber provision, it does not guarantee complementary benefits for carbon storage. Our method exposes the limitations of current management regimes and identifies key areas for improved planning to better mitigate fire risk and strengthen long-term resilience.

We study the effect of mixed precision on the numerical integration of the heat equation discretized with a Runge–Kutta method in time. A full space-time discretization is applied, which results in a very large and sparse linear system to be solved for the numerical approximations and the Runge–Kutta stages of all time steps. The linear system is solved by applying a suitable preconditioned iterative method that can be run in parallel. In order to speed up the solution process, the preconditioner is applied using a mixed precision framework. Sequential results show the robustness of the preconditioner, even when applied in mixed precision. Finally, we present numerical evidence of the improved performance of the mixed precision strategy when applied in a parallel environment, achieving up to a 50% reduction in CPU time.

Hall algebras can be associated with a broad class of combinatorial structures through the theory of 2-Segal sets. In this paper, we study the Hall algebras arising from the 2-Segal sets of rooted trees, undirected graphs, and directed graphs. In each case, we establish an analogue of Green's theorem giving a twisted bialgebra structure to the Hall algebra, describe the primitive elements, and derive a presentation by generators and relations. As an application, we realize the Hall algebra of an undirected graph as the cohomology ring of a topological space. In the case of directed graphs, we introduce the Hall polynomial, defined as the Poincaré polynomial of the space of primitive elements. The Hall polynomial is an invariant of the underlying undirected graph which we show to be closely related to the Tutte polynomial. We furthermore give examples of graphs with equal Tutte polynomials but distinct Hall polynomials showing thus that the Hall polynomial encodes different information from that of the Tutte polynomial.

We prove that $H_{\mathrm{cb}}^4(\Sp(4,\CC);\RR)=0$. Together with Blatz's secondary stability and the rank-one theorem of Bucher–Savini, this gives degree-four vanishing for all complex symplectic and odd complex orthogonal groups. In normalized symplectic Gram coordinates, we establish a bounded-primitive estimate and compute the measurable cohomology of the projective action. An explicit rational cocycle has divergent periods on a family of finite orbit cycles of uniformly bounded $\ell^1$-mass. A two-cone averaging construction extends the period estimate to measurable cochains and excludes bounded representatives of every nonzero degree-four measurable action class.

In this paper we explore the world of $t$-sumfree sequences, primarily for $t=2$ and $t=3$. Starting with $t$ increasing positive integers, build an infinite sequence by choosing as the next entry the least positive integer that is not the sum of $t$ distinct previous values. The main open problem concerns the question of the conjectured ultimate periodicity of these sequences. We formulate a sufficient condition to prove that a conjectured preperiod and period are correct. We also prove periodicity for a considerable subclass of 2-sumfree and of 3-sumfree sequences. Finally, we present the results of extensive computations on 3-sumfree sequences.

Let M be a plumbed integral homology sphere 3-manifold associated with a connected negative definite plumbing graph $\Gamma$. One of its most important invariants is its multivariable Poincaré series (or zeta function) $f_\Gamma(\mathbf {t})$. Several numerical invariants can be read from $f_\Gamma(\mathbf{t})$, or even from its `polynomial part' $ \operatorname{Pol}_{\Gamma}(\mathbf{t})$. For example, the `normalized' Casson's invariant equals $ \operatorname{Pol}_{\Gamma}(1)$. In this note we provide several splice (surgery) formulae for $f_\Gamma$ and $\operatorname{Pol}_\Gamma$ (reduced to the node variables, or to the node variables of connected sub-graphs). In this way, these global invariants can be recovered from a collection of certain smaller graphs. Recall that some (integral homology sphere) 3-manifold invariants are additive with respect to the splice decomposition (like the Casson's invariant). However, some invariants need some `splice correction terms'. In our formulae the correction terms are easily computable one variable Alexander polynomials.

The sequence reconstruction problem for $q$-ary deletion channels, introduced by Levenshtein in 2001, concerns the minimum number of channels required to uniquely recover a transmitted sequence when each channel introduces exactly $t$ deletions. Combinatorially, it is equivalent to determining $N_q(n,d,t)$, the maximum intersection size of two $t$-deletion balls with centers at Levenshtein distance at least $d$, for $q$-ary sequences of length $n$ over the alphabet \(\Sigma_q=\{0,1,\dots,q-1\}\). Levenshtein solved the uncoded case $N_q(n,1,t)$ for all $n\ge t$; subsequently, Gabrys and Yaakobi determined $N_2(n,2,t)$, and Wang et al. extended the result to $N_3(n,2,t)$. In this paper, we study the problem for \(q\)-ary sequences under minimum Levenshtein distance \(d=2\) with channels that introduce exactly \(t\) deletions. We determine the exact value of \(N_q(n,2,t)\) for all \(t\ge 2, q\geq 4\), and for sufficiently large \(n\), and construct explicit pairs of sequences attaining the maximum intersection. Furthermore, for each $q\ge3$, we characterize all extremal sequence pairs. In particular, if the intersection size matches the first two terms of \(N_q(n,2,t)\), then the two center sequences must contain, at the same positions, length-5 blocks of the forms \((a,b,c,a,b)\) and \((b,a,c,b,a)\) for some distinct \(a,b,c\in\Sigma_q\); for \(t\ge q+2\), the exact maximum \(N_q(n,2,t)\) is attained precisely by \(2q!\) unordered pairs of sequences with a specific block structure. Asymptotically, we prove that for \(q\ge 4\) and \(t\ge 2\), \[ N_q(n,2,t)=\frac{6}{(t-2)!}n^{t-2}-\frac{3t+13}{(t-3)!}n^{t-3}+\frac{3t^2+25t+64}{4(t-4)!}n^{t-4}+O(n^{t-5}). \] Moreover, \(N_q(n,2,t)\) and \(N_{q-1}(n,2,t)\) share their first \(q-1\) terms, and for \(t\ge q\) the coefficient of \(n^{t-q}\) in their difference is \(\frac{6t-6q+5}{(t-q)!}\).

We study binary quasi-cyclic LDPC codes whose parity-check matrices are full arrays of single circulant permutation matrices (CPMs), referred to here as CPM-LDPC codes. Their minimum distance is at most $(J+1)!$, where $J$ is the column weight. For every fixed pair of column and row weights $2\le J<L$, we show that this bound is attained for all sufficiently large integer lift sizes. First, we give one integer exponent matrix independent of the lift size $P$. Second, we show that independent uniform exponent choices attain the bound with probability $1-O_{J,L}(P^{-1})$. Both proofs use cycle conditions required by low-weight codewords and a lower bound on the number of terms in vectors satisfying polynomial check equations. Neither construction requires $P$ to be prime. We also give small-lift arrays attaining the bound 24 for $J=3$, $L=4,\ldots,8$, and arrays with distance at least 28 for $J=4$, $L=5,\ldots,8$, together with computational distance verification.

This paper proposes a movable waveguide (MW)-enabled pinching-antenna system (PASS), in which each waveguide is connected via a flexible cable and can be linearly moved by drivers. By simultaneously moving the MWs and the pinching antennas (PAs) on them, MW-enabled PASS can effectively track user locations and form flexible array geometries for efficient beamforming. We first examine the special case with a single user and derive the closed-form solutions for the optimal MW positions as well as an upper bound on the user rate. Furthermore, we develop a two-step optimization algorithm to maximize the achievable rate for the user, where the first step determines the optimal MW positions using the derived closed-form solutions, and the second step alternately optimizes the PA positions through a one-dimensional (1D) local search based on the user location. Then, for the general multi-user scenario, we derive the upper bounds on the minimum rate among all users. To maximize their minimum rate, we propose a low-complexity two-scale optimization algorithm, where the large-scale global search coarsely determines the MW and PA positions, followed by a small-scale local search to finely tune them. In addition, a two-timescale optimization scheme based on statistical channel information is investigated to reduce the mechanical movement overhead of the MWs. Simulation results demonstrate that the proposed scheme achieves performance close to the derived bounds. It also flexibly adapts to different user distributions compared with the conventional PASS employing dense or sparse fixed-position waveguides (FPWs), as well as fixed-position antenna (FPA) schemes.

Algebraic flux correction schemes for convection–diffusion–reaction equations rely on linearity preservation to avoid unnecessary artificial diffusion: the limiter is inactive whenever the discrete solution is affine. On quadrilateral and hexahedral meshes, this is insufficient, since the finite element space also contains bilinear functions that are reproduced exactly on axis-aligned meshes. We show that a mesh patch admits bilinearity preservation if and only if the central node lies in the convex hull of its neighbours after lifting each neighbour by the product of its coordinate differences, and we derive a sharp explicit value for the limiter parameter on such patches when the cell edges are parallel to the coordinate axes. We further characterize the meshes on which a mapped bilinear finite element space reproduces a bilinear function exactly, thereby establishing a fundamental limitation on what any limiter can achieve on general quadrilaterals. Numerical experiments confirm the theory: on axis-aligned grids, the proposed limiter reproduces a bilinear solution to the accuracy of the nonlinear iteration, whereas the linearity-preserving limiter does not. A sensitivity study shows that the parameter required for bilinearity preservation lies close to the upper end of the range for which the nonlinear problem can be solved by the fixed-point iteration considered here.

Penalizing a state constraint creates a singular localization problem: whole-space values may grow like the inverse penalty parameter, while numerical diffusion allows even an inward feedback to cross the boundary. For deterministic discounted optimal control, we show how the penalty itself supplies confinement that offsets this growth. With mesh size $h$ and penalty parameter $\varepsilon$, an inward barrier bounds the additional cost of numerical leakage by $O(h/\varepsilon)$ for a monotone centered-difference scheme with vanishing viscosity. Under $h\le\varepsilon$, an occupation estimate then yields $O(h)$ localization error with a sufficient box margin logarithmic in $1/h$ and independent of $\varepsilon$. We extend the result to bounded squared-distance penalties and combine it with residual-based evaluation bounds and discounted policy-error propagation. Under coupled refinement with vanishing penalization and discretization errors, explicit conditions on evaluation errors and greedy gaps ensure convergence of the neural value approximations to the constrained value, allowing measurable, nonunique greedy selectors. Reference calculations isolate leakage and localization across mesh and penalty scales, and distinguish evaluation, iteration, and approximation errors. An explicit cylindrical state-constraint solution provides a benchmark in arbitrary dimension, tested up to dimension twenty. Paired obstacle-navigation experiments illustrate why policy-value accuracy must be assessed alongside sampled residuals when comparing raw-residual and finite-grid-assisted neural evaluation.

We have observed that Hadamard matrices whether real or complex lead to a dense output of identical probabilities for any given single input state. Complex Hadamard matrices have unusual orders and introduce phase which prove useful when controlled phase transformation gates are used. By taking a superposed combinations of input states, we discovered that the Hadamards generate non-uniform probabilities which is practically significant towards amplitude amplification without the need for manual parameterization like Grover's operator~\cite{grover1996}. A variety of graph theoretic properties are applied to TSS graphs and their trends were explored. Additionally, graphs turn out to be nearly isomorphic for the same set of input states which has potential applications for developing Quantum Algorithms. Keywords: Quantum Algorithms, Hadamard Matrices, Unitary Matrix Approach, Topological Structure of Superpositions (TSS), Graph Theory

In this paper, we study a system of second-order integral-partial differential equations with interconnected obstacles. A particular case, is the Hamilton-Jacobi-Bellman (HJB for short) system associated with the optimal switching problem in the jump-diffusion model. Getting rid of the monotonicity condition on the generators with respect to the jump component, we construct a continuous viscosity solution of the system, which is unique in the class of continuous bounded functions. The Lévy measure $\lambda$(.) is of infinite activity. The main tool we use is the associated system of reflected backward stochastic differential equations with jumps and interconnected obstacles for which we also study existence and uniqueness of the solution.

We study large values of quadratic Hecke $L$-functions in the conductor aspect. Let $K$ be a fixed number field, and assume GRH for its finite-order Hecke $L$-functions. In a fixed ray class component with conductor norm comparable to $X$, we prove that \[ \max_\chi L\left(\frac12+\frac A{\log_2X},\chi\right) \geq\exp\left\{(e^{-A}+o(1)) \sqrt{\frac{\log X\log_3X}{\log_2X}}\right\} \] for every fixed $A\geq0$. Every fixed smaller constant is attained by at least $X^{1-o(1)}$ characters. The same count holds at a suitably slowly moving threshold approaching the displayed constant. The combinatorial input is a sparse squarefree Gál set of cardinality $N$, retaining the known leading constant $2$ and having square multiplicative energy $N^{2+o(1)}$. The energy bound reflects the low degree of the associated Boolean polynomial. Together with the resonance estimate, it yields the abundance bound. We also prove unconditional analogues for quadratic characters with prime conductor away from one fixed place over any global function field of odd characteristic. Finally, we give bounds in the fixed strip and at $s=1$, including the dependence on the residue of the Dedekind zeta function and the prescribed local factors.

We deduce a new inequality using the Hitchin-Thorpe inequality for noncompact complete four-dimensional manifolds with fibered geometry at infinity and Seiberg-Witten gauge-theoretic techniques. Fibered geometry at infinity is either fibered boundary metric or fibered cusp metric. We prove it for fibered boundary metric when the fiber at boundary is a single point.

We present a formalization in Lean 4 of Mouhot and Villani's theorem on nonlinear Landau damping in $\mathbb T^d$ in all Gevrey regularity indices $s > 1/3$ for small backgrounds.

We construct an adjunction relating two approaches to modelling the planar lambda calculus: semi-closed operads and planar lambda-models. We use this to obtain a planar version of Scott's representation theorem.

By studying the classical Minkowski's inequality, $4\pi A\leq M^{2}$, relating area and mean curvature, in the particular case of convex surfaces of revolution, one is led to a functional inequality that holds for general functions. We state and prove this inequality obtaining a new proof of Minkowski's inequality in this particular case and we establish a sharp inequality between the Minkowski deficit of the surface and the isoperimetric deficit of any meridian curve.

We classify the labelled final layer orders of ordinary orthogonal 2 x 2k maps whose mountain-valley assignments are periodic with period two in each crease family. For k >= 2, local flat-foldability leaves 16 of the 64 formal period words. Eight pleat words have a unique labelled state. The remaining eight words reduce, under labelled symmetries, to two constant-row families described by row pivots. For the canonical word MMMMVV, an ordered pivot pair (a,b) is realizable exactly when a = 1, a = m, or a = b, where m = 2k - 1. For MVMMMM, realizability is equivalent to BR(a) = BR(b). In both families each compatible ordered pivot pair determines a unique labelled state. The resulting state counts are 0, 1, 6k - 5, and 4k - 3. The case k = 1 is treated separately because two formal period bits have no physical crease carriers.

The ordered Ramsey number $r_<(G,H)$ of ordered $k$-graphs $G$ and $H$ is the least integer $N$ such that every red-blue edge-coloring of the naturally ordered complete $k$-graph on $[N]$ contains a blue ordered copy of $G$ or a red ordered copy of $H$. We prove that there is an absolute constant $c>0$ such that, for every integer $d\ge1$, there is a constant $C_d>0$ for which every weakly $d$-degenerate ordered $3$-graph $H$ on $t$ vertices satisfies \[ r_<\bigl(H,K_3^{(3)}(n)\bigr) \le t\,2^{C_d n^{2-c/d}} \] for every positive integer $n$. This resolves a problem posed by Balko and Vizer (SIAM J. Discrete Math., 2022) in a stronger form. Furthermore, we show that the weak-degeneracy hypothesis cannot be replaced by bounded standard degeneracy. In particular, for every sufficiently large $n$, there exists a $1$-degenerate ordered $3$-graph $F$ on at most $2^{O(n)}$ vertices such that $r_<\bigl(F,K_3^{(3)}(n)\bigr)>2^{\Omega(n^2)}.$

We study three plane-partition enumerators arising from Schreier-Aigner's quasi-symmetry classes. Their realizations as lattice-point enumerators, together with staircase translations of interior lattice points, yield factorizations by Ehrhart-Macdonald reciprocity. We determine the consecutive linear factors, the parity and degree of the residual polynomials, and explicit divisibility bounds for their coefficient denominators. The denominator argument includes the half-integral translation required by the second-kind classes. We also derive a corrected size-five formula for the symmetric second-kind class. Exact computations verify irreducibility of the quasi-symmetric residual polynomials for every size from $3$ to $24$.

Let $d\ge1$ and $V\in C(\mathbb{T}^d;\mathbb{C})$. We prove that the Bloch variety of $-\Delta+V$ is irreducible modulo periodicity. The proof combines holomorphic continuation of spectral band functions with integer translations of quasimomentum. Using analytic geometry and perturbation estimates along imaginary rays, we show that, after suitable integer translations, all irreducible components contain the same holomorphic spectral branch.