Papers

Spectrum sensing (SS) is crucial for realising cognitive radio networks, where the secondary user (SU) needs to detect the presence of a primary user (PU) in order to utilise the spectrum. However, the ability of detection is influenced by unknown propagation environment factors such as multipath fading, correlated noise, transmission power of PU, etc. This paper investigates reconfigurable intelligent surfaces (RIS)-aided SS under correlated noise conditions using a generalised likelihood ratio test (GLRT) and energy detector (ED) frameworks. We first derive maximum likelihood estimates of the unknown channel state and transmit power, and employ these estimates to construct the GLRT-based test statistic using the signal received with an optimally configured RIS. The RIS phase shift matrix is optimally determined to maximise the gain of the estimated channel. Besides, the detection and false alarm probabilities of ED with optimally configured RIS are also derived. The numerical receiver output characteristics (ROC) demonstrate that the proposed GLRT achieves superior detection probability compared to ED, particularly under correlated noise and limited number of observations.

Let $k=\mathbb{F}_q$, $E=\mathbb{F}_{q^n}$ and $\mathrm{Tr}=\mathrm{Tr}_{E/k}$. For $r\ge 2$, $a\in k^{\times}$ and $x\in E^{\times}$, let $\mathrm{N}(E,r,x,a)$ be the number of $r$-tuples $(x_1,\cdots,x_r)$ in $(E^{\times})^r$ satisfying $x_1\cdots x_r=x$ and $\mathrm{Tr}(x_1+\cdots+x_r)=a$. We prove $\left|\mathrm{N}(E,r,x,a)-\left((q^n-1)^{r-1}+(-1)^r\right)/q\right|\le (r^n-1) q^{\frac{(r-1)n-1}{2}}$. This proves the square-root estimate predicted in Wan's conjecture and generalizes a previous result of Moisio and Wan. For a finite semisimple algebra $B=\prod\limits_{i=1}^s M_{d_i}(\mathbb{F}_{q^{n_i}})$ over $k$ and a regular element $x\in B^{\times}$, the same method combined with Zelingher's formula leads to analogous square-root estimates.

In this paper, we establish a quantitative Diophantine approximation result for complex multiplication (CM) points on Shimura curves. Specifically, we prove a lower bound for the distance between a sequence of CM points $P_n$ converging to a fixed CM point $P$ on a Shimura curve $X(D,1)$ in terms of the discriminant of the endomorphism rings of $P_n$. The proof exploits the complex geometry of the Fuchsian uniformization, the explicit matrix representation of the underlying quaternion algebra, and Liouville's inequality. We show that the distance between the corresponding fixed points $\tau_n$ and $\tau$ in the upper half-plane is bounded below by a positive constant times a negative power of the discriminant. This result provides a Shimura curve analogue of a result of Habegger on singular moduli and modular curves.

The classical Grünbaum inequality asserts that the proportion of the volume of a convex body cut off by a halfspace containing its barycenter is at least $1/e$. From its functional counterpart, for any log-concave random variable $X$, one has $\mathbb{P}(X\ge \mathbb{E}X)\ge 1/e$, with equality if and only if $X$ is exponential. Motivated by Grünbaum's inequality for convex bodies and its functional generalizations, we prove analogous inequalities for entropy, with characterizations of the equality cases. We show that if $X$ is a log-concave random variable on $\mathbb{R}$, then $$ h(X)-\frac{e}{e-1}H_2(1/e) \leq h(X|X \leq \mathbb{E}X) \leq h(X), $$ where $h$ is the differential entropy, $H_2(\cdot)$ is the binary entropy function and $X|X\leq \mathbb{E}X$ stands for the distribution of $X$ conditional on $X\leq \mathbb{E}X$. We generalize the upper bound for all Rényi entropies and the lower bound for min-entropy. Our inequalities are sharp and we characterize all equality cases. We discuss potential generalizations in high dimensions and give counterexamples in some directions. As an intermediate step for the proof of the lower bound, we establish a new inequality that we prove using a technique known as degrees of freedom, combined with a standard KKT-type optimization lemma. Along the way, we characterize the equality case in a known comparison inequality between differential and min-entropy, which may be of independent interest.

Gallai proved that every graph can be partitioned into two sets, each inducing a subgraph with all degrees even. We show that if a graph admits a bipartition in which every vertex has an odd number of neighbors in the opposite part, then it can be partitioned into two sets, each inducing a subgraph with all degrees odd. Consequently, every $n$-vertex graph without isolated vertices has an induced subgraph with all degrees odd on at least $n/5$ vertices, substantially improving the previously known universal lower bound.

We introduce projective logical geometry and prove that two algebraic structures are strongly bi-interpretable if and only if their categories of projective logical sets are equivalent relative to the class of interpretation functors, which is also equivalent to their categories of projective definable sets being equivalent relative to the class of translation functors. These constructions generalize two ideas of Boris Plotkin: the concept of geometric equivalence in universal algebraic geometry and the transition from universal algebraic geometry to logical geometry. Furthermore, our categorical approach offers a fresh perspective on the theory of interpretations, enabling us to establish a series of fundamental results using categorical methods.

In 2007, Zagier identified twelve families of candidates for rank-three modular Nahm sums and established the modularity of three of them. The remaining cases were subsequently confirmed by Wang. Zagier's duality observation indicates that the Nahm sums associated with the duals of these examples might still be modular. We confirm that this is indeed true. The modularity of the dual of the sixth example was previously established by Milas and Wang, while that of the dual of the eleventh example was partially established by the present authors. We settle all remaining cases, thereby establishing the modularity of every defined dual in Zagier's rank-three list. We achieve this by proving new Rogers–Ramanujan type identities that express the relevant Nahm sums as finite sums of infinite products. Along the way we discover two new families of rank-three Nahm sums. As applications, we prove some Nahm sum identities conjectured by Cao–Wang, Li and the present authors.

We introduce and study a pseudo-differential operator $W^\alpha$, defined via the $p$-adic Mellin transformation, acting on real- or complex-valued functions on the multiplicative group of the field of $p$-adic numbers. The operator $W^\alpha$ is maximally dissipative. An estimate of its heat kernel is given.

We establish an exact representation formula for the deficit in Hölder's inequality. Rather than estimating the deficit by auxiliary quantities, we show that it can be represented as an accumulated variance along the natural exponential interpolation connecting the two endpoint densities. More precisely, the logarithmic Hölder deficit is expressed as the integral of the variance of the logarithmic density ratio weighted by the Green kernel associated with the one-dimensional interpolation parameter. This identity reveals that Hölder's inequality is a consequence of the convexity of a logarithmic partition function and provides an intrinsic interpretation of the deficit as an interpolation energy. The representation suggests a broader framework for studying functional inequalities through variance identities.

On a locally finite connected graph $G=(V,E)$, using the first eigenvalue method introduced by Kaplan \cite{MR160044} and the discrete Phragmén-Lindelöf principle developed by Hu-Wang \cite{cvhuyuanyang}, we first establish the asymptotic behaviour of the lifespan of positive solutions to a semilinear heat equation with the power-logarithmic nonlinearity $u^p|\log u|^q$, provided that the initial datum is bounded below by a positive constant. These results extend those of Hu-Wang \cite{cvhuyuanyang} to equations with a power-logarithmic source term. Moreover, by means of a more direct argument, we show that analogous lifespan estimates remain valid for nonnegative initial datum $u(x,0)$, provided that $u(x_i,0)$ is suitably large at some vertex $x_i\in V$.

In this paper, we study the multi-target detection problem in a movable-antenna (MA)-enabled wireless sensing system with linear arrays, in which both the direct line-of-sight (LoS) paths and the first-order non-LoS (NLoS) paths are explicitly considered. Unlike conventional fixed-position antenna arrays, MAs provide additional design degrees of freedom by enabling adaptive antenna positioning to reconfigure the propagation geometry, offering great potential to enhance the sensing performance in complex multi-target multipath scenarios. Under this setup, we first develop a cross sparsity Markov mixture prior to derive the posterior probabilities of target locations, in which the structural correlation between the LoS and NLoS paths is effectively exploited to enhance the location estimation accuracy. Based on the derived posterior probabilities, we further analyze the angular-domain sensing performance for multi-target detection by proposing a new two-dimensional (2D) ambiguity function as the performance metric. Next, we optimize the MA positions to suppress the sidelobe levels and narrow the mainlobe width of the proposed ambiguity function. Although the resulting problem is highly non-convex, we develop a low-complexity Dykstra-based projected gradient descent algorithm to solve it efficiently. Finally, simulation results verify the accuracy of the proposed ambiguity function analysis, demonstrate the substantial performance gains enabled by the proposed prior model and MAs, and show that the proposed algorithm achieves performance comparable to existing methods with significantly lower computational complexity.

We study compact Kähler manifolds admitting Hermitian metrics with nonpositive holomorphic sectional curvature. We prove that the canonical bundle of such a manifold is nef, removing the pluriclosed assumption from the corresponding nefness result of Broder-Stanfield \cite{BroderStanfield}. In complex dimension two, we further show that negative holomorphic sectional curvature implies the ampleness of the canonical bundle. In complex dimension three, the same conclusion holds if the Hermitian metric is additionally assumed to be balanced. We also prove that vanishing holomorphic sectional curvature forces the first Chern class to vanish.

Let $E$ be an elliptic curve over $\mathbb{Q}$. We prove that the degree of the Shimura parametrization of $E$ arising from the optimal quotient $J^D(M)\to E$ is equal to the $D$-new modular degree of $E$, under mild assumptions on the local Galois representations of $E$. As an application, we prove a conjecture of Deines asserting the equality of the $D$-new modular degree, the $D$-new congruence number, and the Shimura degree of $E$. We introduce the Shimura congruence number into this framework and prove, under the same assumptions, that all four quantities coincide; moreover, the two congruence numbers are always equal. We establish new cases of multiplicity one for Shimura Jacobians and use them to prove the main theorems, following a strategy introduced by Agashe, Ribet, and Stein. Building on these results, we obtain new examples of the failure of multiplicity one. Finally, we give a negative answer to a question of Papikian and Rabinoff asking whether the functorial map between the component groups of the Néron models of $J^D(M)$ and $E$ is surjective when $D>1$, and we relate this phenomenon to the failure of multiplicity one.

A seminal work by Dacorogna and Moser introduced a way of constructing regular transport maps from a probability distribution on a bounded domain to another one. In this work, we extend this construction to the whole $\mathbb{R}^d$ for strictly asymptotically log-concave measures, a wide class of distributions that encompasses Lipschitz-perturbations of log-concave measures. We then leverage this construction to study geodesics in the space of probability measures on a product set with imposed marginal laws (couplings), for which we derive optimality conditions, answering an open question in a recent work by Conforti, Lacker and Pal. Taking inspiration from Brenier's variational model for incompressible fluids and its regularization, we further introduce an entropic regularization of the geodesic problem, which can be seen as the Schrödinger bridge problem on the space of couplings, for which we also derive optimality conditions. We eventually study convergence of minimizers as the regularization vanishes, and we prove convergence of the related Lagrange multipliers. Our approach involves proving uniform-in-time global regularity estimates on elliptic and parabolic equations on $\mathbb{R}^d$, by exploiting the structure of asymptotically log-concave measures using the probabilistic notion of reflection coupling.

This paper develops a unified kernel-based method for computing exact matrix elements of composite quantum operations by inserting resolutions of the identity after individual operators or between successive factors of an operator chain. A single insertion gives an exact scalar identity, and nested insertions between the factors give multiple integral identities for chains of closed operators under explicit Bochner-integrability conditions. The procedure is carried through on polynomial and analytic functions of the ladder operators, on one- and two-mode Gaussian unitaries, and on three composite operations: a Kerr phase followed by squeezing or by a cubic phase gate, a two-mode squeezer followed by a beam splitter, and a heralded cat-state circuit with two beam splitters. At rational Kerr phases the Kerr-squeezed amplitudes reduce to finite sums of squeezed coherent-state amplitudes weighted by quadratic Gauss sums, with an explicit error bound at the remaining phases; the two-mode network gives a bivariate Hermite polynomial; and the heralded circuit gives closed forms for the amplitudes, the heralding probabilities and their extension to an arbitrary passive network fed by one squeezed source. The examples are set against the reach of four established techniques: operator ordering, the Berezin symbol calculus, Lie disentangling and the symplectic formalism. Each technique reaches a subset of the examples directly, while the insertion procedure reaches all of them under the same hypotheses. A comparison with direct Fock summation identifies the regime in which the finite Kerr sums are efficient, and the choice of resolution at each boundary of a chain follows a selection rule set by the action of the adjacent factor. Each closed form agrees with an independent numerical computation.

The well-posedness theory of very weak solutions is a central topic in mathematical hydrodynamics, especially in the regularity theory for Navier-Stokes equations. It has been fully developed for incompressible fluid flows on bounded domains in R^3 of C^{2,1}-regularity. In this paper, based on the analytic theories in [D. Breit and A. Gaudin, ArXiv Preprint: 2511.19091 (2025)] and [V.G. Maz'ya and T.O. Shaposhnikova, Vol.337, Grundlehren der mathematischen Wissenschaften (2009)], we establish the well-posedness theory of very weak solutions to the Navier-Stokes equations on bounded Lipschitz domains whose boundary has local graphing functions with sufficiently small Sobolev multiplier norm, which contain the bounded Lipschitz domains with sufficiently small Lipschitz constants as a special case.

We advance the general theory of flows of tensorial $\mathrm{H}$-structures, focusing on non-isometric flows and on the case $\mathrm{H}=\mathrm{SU}(m)\subset\mathrm{SO}(2m)$. After developing the relevant $\mathrm{SU}(m)$ algebra, we compare two natural evolutions: the unrestricted negative gradient flow of the intrinsic-torsion energy and a Ricci-harmonic flow. We prove short-time existence and uniqueness for the Ricci-harmonic $\mathrm{H}$-flow, with arbitrary lower-order torsion-quadratic terms, for every closed subgroup $\mathrm{H}\subset\mathrm{SO}(n)$. For groups for which the projection to $\mathfrak{h}^\perp$ defines a $4$-form, including $\{1\}$, $\mathrm{SU}(2)$, $\mathrm{G}_2$, and $\mathrm{Spin}(7)$, we express the negative gradient flow in Ricci-harmonic form up to explicit lower-order torsion terms and prove short-time existence and uniqueness by a modified DeTurck argument. We treat the genuinely different $\mathrm{SU}(m)$ case by a separate principal-symbol computation, proving short-time existence and uniqueness for the unrestricted negative gradient flow of $\mathrm{SU}(m)$-structures. The same computation identifies the natural negative gradient flow of $\mathrm{U}(m)$-structures as a borderline case, which cannot be made strictly parabolic by first-order diffeomorphism gauges. For the modified Ricci-harmonic flow, we derive heat-type evolution equations for the intrinsic torsion, a doubling-time estimate and Shi-type derivative estimates for $(|\mathrm{Rm}|^2+|\nabla T|^2+|T|^4)^{1/2}$, and a finite-time continuation criterion. In dimension six, we translate the formalism into the standard torsion forms of an $\mathrm{SU}(3)$-structure and describe, to highest order, the corresponding family of second-order quasilinear $\mathrm{SU}(3)$-flows.

Let $\phi: G\to H$ be a continuous homomorphism between compact Lie groups, let $X$ be a $G$-space, and let $Y$ be an $H$-space. We formulate a filtered change-of-groups obstruction to the existence of a $\phi$-equivariant map $X\to Y$. We apply them to the inclusion of a finite cyclic group into the circle and to homomorphisms between tori acting coordinatewise on products of odd-dimensional spheres.

Near a crack tip or re-entrant corner, fractional radial powers can have unbounded derivatives and slow the convergence of high-order splines. A singular mapping grades the computational radius so that the pulled-back field is smoother without changing the physical domain. Classical maps require the singular exponent and grading in advance. Here the radial coordinate is trained from the mechanics problem. It is the normalized integral of a positive neural density, which fixes both radial boundaries and ensures r'(s)>0 away from the collapsed tip. The radial grading exponent and density-correction weights are inferred from Galerkin energies evaluated at discrete equilibrium, without exponent labels or exact interior fields. We test the mapping in scalar and plane-strain B-spline formulations and compare it with the identity map, radially graded knot vectors, prescribed power maps, and an adaptive enriched B-spline method. At 156 vector degrees of freedom, the mechanics-trained map reduces the relative energy-norm error by a factor of 33.42 compared with the identity map. The maximum error in the mixed-mode stress intensity factors recovered on 3 contours is 1.923 x 10^(-5). For the straight crack, the learned density correction vanishes, and the prescribed r=s^2 map gives the same improvement. For a nonlinear Robin family with test parameters outside the training interval, the density correction remains nonzero and gives a fixed-q incremental energy gain of at least 4.839. Thus, for the problems considered here, the neural correction is useful when the required coordinate is not represented by a prescribed power map.

A database instance on a small category may be represented as a set-valued functor or, equivalently, as a discrete opfibration. Its sections correspond to globally coherent choices of records. When no global section exists, we measure the failure of global coherence by the minimum number of subcategories on which coherent choices can be made. Regarding natural transformations as directed homotopies, we introduce right and left directed fibrations and relate them to Grothendieck opfibrations and fibrations. We define directed versions of Lusternik-Schnirelmann category and sectional category and establish their invariance and comparison properties. Every functor admits a Grothendieck opfibration model on which directed sectional category is computed by strict local sections, together with a canonical discrete approximation obtained from connected components of comma categories. For functorial databases, we study directed sectional category under decomposition, iteration, and data migration, and characterize initial objects of finite connected acyclic schemas through the existence of global sections of objectwise non-empty databases.

Despite the empirical success of score-based diffusion models, a complete theoretical understanding of how finite-sample learning, network parameterization, and numerical discretization jointly dictate generative quality remains underdeveloped. Existing sampling analyses often evaluate the generative performance conditional on an oracle score or a pre-specified error threshold. In this work, we establish a unified convergence and generalization framework for score-based diffusion models parameterized by practical ResNet-type architectures. We analyze the generalization and convergence properties from the practical finite-sample, discrete-time learning problem of the score function to the ideal continuous-time, population-level objective. Based on the generalization result of the learning problem of score function, we analyze the sampling process induced by the learned score function and provide an end-to-end total variation distance estimate for the generated terminal distribution. This estimate explicitly decomposes the overall generative error into four interpretable components: the truncation error of the forward process, the reverse-time discretization error, the generalization error incorporating both finite data and forward-time discretization, and the training optimization gap. Our results quantitatively characterize how the training sample size, temporal discretization grids, and optimization accuracy jointly control the final fidelity of samples generated by diffusion models.

Low-resolution quantization constrains the maximum achievable gains of multiple-input multiple-output (MIMO) systems. While the adverse effects and mitigation strategies have been thoroughly analyzed for communication systems, the impact of low-resolution quantization on integrated sensing and communication (ISAC) systems remains insufficiently explored in the existing literature. In this paper, we propose an analysis and design framework to investigate and mitigate the effects of 1-bit digital to analog converters (DACs) for ISAC systems. Firstly, an analytical sensing signal-to-noise ratio (SNR) expression is derived by using the Bussgang decomposition. Furthermore, two different methodologies are proposed to design a transmit waveform that satisfies both communication and sensing requirements simultaneously. The first method uses a separate constant modulus (CM) sensing signal since CM signals are known to be more robust to nonlinear distortion than orthogonal frequency division multiplexing (OFDM) modulated signals. The second method employs the squared-infinity norm Douglas-Rachford splitting (SQUID) approach to construct the transmit waveform using nonlinear quantized precoding. Finally, the performance of the proposed methods are validated via numerical simulations to indicate the complexity-performance tradeoff between two different methods.

In this paper, based on the $C^0$ interior penalty Galerkin ($C^0$IPG) discretization, we propose and analyze an a posteriori error estimator for eigenfunctions associated with multiple and clustered eigenvalues of the biharmonic operator. The proposed estimator may capture the local singularities of the target eigenfunctions efficiently and play an important role in adaptive procedures. We develop a rigorous cluster-projection-based analysis and introduce an auxiliary theoretical error estimator to connect the invariant subspace error with a computable residual estimator. Most importantly, the resulting reliability and efficiency bounds are robust with respect to the mesh size, the mesh level and the internal spectral gaps within the target cluster. Numerical experiments support the theoretical results and demonstrate the robustness of the proposed estimator.

Driven by the exponential growth of latency-sensitive applications, mobile edge computing (MEC) has emerged as a pivotal paradigm, yet mitigating its substantial energy consumption remains critical. This paper explores a movable-antenna (MA) assisted energy minimization scheme in an uncrewed aerial vehicle (UAV)-enabled MEC system, where a UAV equipped with an MA array serves as an edge server to process tasks offloaded from terrestrial consumer electronics (CE) devices. To minimize the total system energy consumption, we jointly optimize computation resource allocation, CE transmit power, receive beamforming, and MA positions. To tackle the resulting non-convex problem with coupled variables, a robust alternating optimization algorithm based on the block coordinate descent method is developed. The problem is iteratively decomposed into three subproblems. In particular, the subproblem of transmit power and receive beamforming is reformulated and optimized using the quadratic transform technique, while the MA array positions are optimized via the particle swarm optimization algorithm. Numerical simulations verify that the proposed scheme achieves substantial energy savings over conventional benchmarks.

Proper Orthogonal Decomposition (POD) provides low-dimensional surrogate models of evolution equations from solution snapshots. In parameterized problems, however, a basis computed at one parameter value need not remain accurate at another, while recomputing the basis for every parameter in a many-query study is costly. We develop a sensitivity analysis for POD reduced-order models of a class of parameterized semilinear evolution equations that includes the viscous G-equation and a viscous strain G-equation in their mean-free formulations. The analysis is based on a cross-space Lipschitz condition for the nonlinear operator, which accommodates the nonsmooth, gradient-dependent nonlinearities of these flame-propagation models, and on continuity moduli quantifying the parameter dependence of the bilinear form and of the nonlinearity. We prove that when a POD basis constructed at a reference parameter is applied to nearby query parameters, the resulting error variation is controlled by the corresponding parameter modulus, with constants independent of the POD dimension, the number of time steps, and the perturbation magnitude. Numerical experiments are consistent with the predicted modulus-dependent sensitivity behavior, and illustrate the practical robustness of basis reuse for nearby parameters.

Infinite Canons is an ongoing series of canons with infinite solutions. More specifically, each canon is based on a melodic line that can be combined in any number of voices, at any tempo ratios (rational or irrational), and with each voice moving either forward or in retrograde inversion, while maintaining harmonic consistency. This paper describes the structure of these maximally self-similar melodic lines based on two different constructions: 1) a discrete prime-factorization approach yielding self-similarity under all rational ratios, and 2) a continuous logarithmic approach extending this to irrational ratios. In both cases the vertical interval between voices in a tempo ratio of $\lambda_i / \lambda_j$ is given by a homomorphism $\phi(\lambda_i / \lambda_j)$, which is a constant independent of time. Furthermore, under these constructions retrograde inversion collapses to transposition, allowing for all manner of table canons. These structures are demonstrated with suggestive realizations of several different infinite canons. Future work includes a more complete mathematical treatment, musical applications, and an interactive program that allows users to explore an unlimited number of realizations of these pieces.

As 6G wireless networks transition toward sub-Terahertz (sub-THz) frequencies to satisfy extreme capacity demands, managing the trade-off between massive bandwidth and power consumption becomes a critical design challenge. In this paper, we investigate the fundamental energy efficiency (EE) limits of a dual-band base station site combining a coverage-oriented sub-6 GHz carrier with a capacity-oriented sub-THz carrier. By jointly optimizing hardware parameters and advanced sleep modes via activity factors, we identify four distinct operational regions that govern the EE-optimal behavior across the complete range of data rates. We derive closed-form analytical thresholds that dictate precisely when the sub-THz band should awaken from sleep and how to allocate traffic between the bands in that case. Our results demonstrate that utilizing the sub-THz band is EE-optimal when the power cost of the bandwidth-limited sub-6 GHz band surpasses the static power penalty of activating the sub-THz circuitry. Ultimately, this framework provides mathematically rigorous guidelines for power consumption minimization and sleep-mode management in future green networks.

We present a unified fixed-point construction of the elementary transcendental functions, encompassing the real exponential, the complex exponential (sine and cosine), and the natural logarithm. Each function is characterized as the unique solution of a duplication identity established through the Banach contraction principle. These foundational identities are $e(2x)=e^2(x)$ for the exponentials, and $\log(x^{2})=2\log x$ for the logarithm. Because a direct iteration of these identities is numerically unstable, owing to local expansiveness at the target, the central idea transfers the analysis to a residual function, on which the operator becomes a strict contraction with an explicit convergence rate. Beyond its theoretical economy, which dispenses with differential equations and power series, this characterization translates into efficient algorithms for the machine evaluation of elementary functions: the underlying framework yields floating-point kernels whose accuracy and iteration depth are governed by the theoretical contraction rate. We also present a computational study showing that, in a throughput-bound vectorized regime, these kernels are competitive with standard production libraries, and in favorable configurations exceed them, with the sine–cosine kernel faster at every tested iteration depth. These implementations operate without lookup tables or memory traffic, an architectural advantage for modern high-performance and energy-efficient computing.

We develop a data-driven approximation of the projected ambient connection Laplacian acting on differential forms over smooth Riemannian manifolds sampled by point clouds. The proposed construction extends the classical framework of diffusion maps and Vector Diffusion Maps from scalar functions and tangent vector fields to differential forms of arbitrary degree. Our approach is based on a novel representation of differential forms as alternating differential arrays obtained through an extension of the classical musical isomorphism. This representation enables the construction of a matrix-valued diffusion operator that approximates the projected ambient connection Laplacian directly from point cloud data without requiring a mesh or simplicial complex. The proposed discretization admits the asymptotically optimal kernel bandwidth scaling inherited from diffusion maps, leading to sharper convergence guarantees than previous data-driven approximations of the Hodge Laplacian. Building upon this operator, we derive a fully data-driven explicit Euler scheme for the heat equation on differential forms and validate the proposed methodology through numerical experiments on the unit sphere. The experiments confirm the predicted decay of the analytical solution and demonstrate the effectiveness of the proposed discretization. The proposed framework provides a natural generalization of Vector Diffusion Maps to differential forms of arbitrary degree and establishes a practical foundation for the numerical approximation of geometric partial differential equations directly from point cloud data.

This paper aims to provide a complete description of the spaces of local and 2-local automorphisms for the families of finite-dimensional totally graded complex filiform Lie algebras of maximum length, building upon established classification frameworks and algebraic-filtration methods. We systematically investigate six infinite sequences ($\mathfrak{m}_0(n)$, $\mathfrak{m}_2(n)$, $W^+(n)$, $\mathfrak{m}_{0,1}(n)$, $\mathfrak{m}_{0,2}(n)$, $\mathfrak{m}_{0,3}(n)$) and five one-parameter families ($\mathfrak{g}_{k,\alpha}$ for $k=7,\dots,11$). The analysis utilizes internal commutation boundaries and constructs non-linear, non-additive transformations on specialized parametric coordinate subspaces. We prove that for the structures $\mathfrak{m}_0(n)$ and $\mathfrak{m}_{0,1}(n)$, the space of local automorphisms strictly encapsulates the group of automorphisms, confirming the existence of pure local automorphisms. Conversely, for $\mathfrak{m}_2(n)$, $W^+(n)$, $\mathfrak{m}_{0,2}(n)$, $\mathfrak{m}_{0,3}(n)$, and $\mathfrak{g}_{k,\alpha}$, the local automorphisms are restricted to an invertible lower triangular matrix form due to rigid power constraints. Furthermore, the sequences $\mathfrak{m}_{0,1}(n)$, $\mathfrak{m}_{0,2}(n)$, and $\mathfrak{m}_{0,3}(n)$ are shown to possess pure non-linear 2-local automorphismsThe remaining investigated structures adhere strictly to linearity, forcing every 2-local automorphism to coincide with a genuine automorphism. This establishes a clear boundary between structures allowing non-linear transformations and those maintaining strict linearity within filiform Lie algebras of maximum length.