Papers

Following the approach of B.~Enriquez~\cite{E} we exhibit the analytical properties of the products of the currents in the Yangian doubles restricted to the category of the highest weight representations. We will demonstrate that the Serre relations for the simple root currents in the Drinfeld's 'new' realization of the Yangian doubles \cite{Dnew,KhT-DY,LP1} can be reformulated as quadratic commutation relations between composed currents for the Yangian doubles associated with Lie algebras of the classical series.

We study the minimality of the system of root functions associated with a Sturm–Liouville problem whose boundary condition depends linearly on the eigenparameter. Two different criteria for minimality were previously obtained using independent approaches. In this paper, we establish the equivalence of these criteria and provide a unified characterization of the exceptional cases in which the removal of certain associated functions fails to preserve minimality. The theoretical results are illustrated by several examples involving multiple eigenvalues, demonstrating the consistency of the two approaches and clarifying the structure of the corresponding root function systems.

We review the collaboration that led to the first rigorous proof of the stability of the hydrogen molecule within quantum mechanics and discuss several related issues concerning few-charge systems. Particular emphasis is placed on the role of symmetry breaking, the stability domains of Coulombic few-body systems, and some applications to exotic hadrons in the quark model.

We consider the calibration of probability forecasts. Several notions of calibration exist when the forecaster issues a single forecast for each of the observations that is to be predicted. We extend one of these notions, auto-calibration, to the common situation in which the forecaster issues a sequence of forecasts for each observation, repeatedly updating their prediction as they receive additional information. For observations that sit in any Borel space, we show that auto-calibration is equivalent to a certain sequence of random probability measures satisfying the martingale property, and we propose a simple, statistical approach to testing this property. This provides, for the first time, a way of testing the calibration of such sequences of probability forecasts.

Network intrusion detection systems (NIDS) are crucial in cybersecurity infrastructure, needing advanced techniques to detect hostile activity in network traffic. This research introduces a hybrid approach that combines Topological Data Analysis (TDA) with Long Short-Term Memory (LSTM) networks to improve anomaly detection in network security. Our multi-layered design combines TDA's persistent homology with LSTM networks to capture topological characteristics of network traffic patterns and simulate temporal sequences. We assessed our methodology using the CIC-IDS2017 dataset, which includes over 2.8 million labelled flows, 77 network variables, and 14 attack categories that reflect modern threat landscapes such as DDoS, brute force, web attacks, penetration, and botnet activities. Integrating Betti curves and persistence diagrams with deep learning architectures enhances feature extraction performance. Our hybrid TDA+LSTM model has an AUC of 1.000 and F1-score of 1.000, with 5-fold cross-validation producing a mean AUC of 1.000 $\pm$ 0.000 and mean F1 of 0.999 $\pm$ 0.001. An ablation research demonstrates the complimentary contributions of topological (F1=0.990) and temporal characteristics (F1=1.000). Comparative research shows that the suggested strategy beats TDA+Random Forest (F1=0.994) and Isolation Forest (F1=0.835) baselines in several attack categories.

In this paper, we establish a comparison principle for non-negative weak solutions to a class of doubly nonlinear parabolic fractional partial differential equations within a space-time cylinder $\Omega_T=\Omega\times(0,T)\subset\mathbb{R}^{n+1}$. For the two solutions considered, we assume that at least one of them is time-independent outside the spatial domain, i.e. in $\Omega^{c}=\mathbb{R}^n\setminus\Omega$. As an application of this result, we readily infer the uniqueness of a non-negative weak solution to the corresponding Cauchy-Dirichlet problem.

Let \[ \rho(q)=\sum_{m\geq 0}\frac{q^{2m(m+1)}}{(1+q+q^2)(1+q^3+q^6)\cdots(1+q^{2m+1}+q^{4m+2})} =\sum_{n\geq 0}r(n)q^n \] be Ramanujan's third order mock theta function. We prove the sign law \[ r(3m)>0,\qquad r(3m+1)\leq 0,\qquad r(3m+2)\leq 0, \] with equality precisely at $n=2,4,8,11,20$. Watson's identity \[ 2\rho(q)+\omega(q)=T(q) \] reduces the problem to comparing the mock theta function $\omega(q)$ with the eta quotient \[ T(q)=3\frac{(q^6;q^6)_\infty^4}{(q^3;q^3)_\infty^2(q^2;q^2)_\infty}. \] We prove effective root-of-unity estimates for this difference. The polar contributions at $q=1$ cancel, the contribution at $q=-1$ is polynomially bounded, and the first surviving exponential term occurs at the primitive cubic roots of unity. It has the sign pattern \[ \kappa_0=\frac13\cos\frac\pi{18}>0,\qquad \kappa_1=-\frac13\sin\frac{2\pi}{9}<0, \qquad \kappa_2=-\frac13\sin\frac\pi9<0. \] The resulting effective asymptotic proves the desired sign law for all sufficiently large $n$, and an exact integer-arithmetic verification completes the finite range. We conclude by indicating how the same root-of-unity method should lead to analogous sign laws for other third order mock theta functions, including $\phi(q)$ and $\chi(q)$.

Digital signature schemes are an important cryptographic tool to ensure data authenticity and integrity in many applications that must be resilient to attacks, including those facilitated by quantum computers. We consider the two digital signature schemes based on error-correcting codes that are second-round candidates in NIST's call for Additional Signature Schemes, which is part of the Post-Quantum Cryptography Standardization Process. Specifically, we provide an overview of the Codes and Restricted Objects Signature Scheme (CROSS) and the Linear Equivalence Signature Scheme (LESS). We describe their underlying problems of syndrome decoding from restricted errors and code equivalence. We review sigma protocols and how they can be transformed into digital signature schemes via the Fiat-Shamir transform. Finally, we explain how this procedure yields code-based digital signatures believed to be post-quantum secure.

Conformal prediction and its variants, including the split conformal prediction, provide a distribution-free framework for uncertainty quantification by constructing prediction intervals or sets with finite-sample coverage guarantees. The statistical efficiency of these intervals depends critically on how the data are split into training and calibration samples. Despite its practical importance, a principled characterization of the training-calibration split that minimizes prediction interval length while maintaining coverage has remained largely unresolved. In this paper, we develop a theoretical framework for optimal data splitting in split conformal prediction. We first analyze the problem in a general setting and derive analytical characterizations of the length-optimal split ratio under both symmetric and asymmetric regimes. We then show how the general results specialize to several commonly used regression settings, including linear regression, nonparametric regression, and neural networks, thereby demonstrating the scope of the framework. We also describe a data-based method for selecting the optimal proportion. Our analysis clarifies how model-related features govern the optimal allocation of samples between training and calibration and provides principled guidance for constructing shorter prediction intervals. Experiments on both synthetic and real-world datasets demonstrate the applicability of the proposed methodology across a variety of practical scenarios.

We consider a family of continuous-time Markov chains with finite strongly connected transition graph and rates $\left(r_N\right)_{N>0}$ depending on a parameter $N$, so that, when $N$ is large, transitions may happen on different time scales. Under suitable general assumptions on the asymptotic behavior of the rates, we give a recursive characterization of the limiting invariant measure. The recursion is encoded in a forest structure equivalent to the one recently developed in the analysis of dynamical aspects of metastability \cite{BL,LX}. Our proof is based on a combinatorial representation of the invariant measure, given by the Markov chain tree theorem. Basic steps are the reduction of the chain by a trace process, the introduction of an effective dynamics, and a careful analysis of the set of relevant arborescences in the expansion. In particular we use a factorization of fast arborescences. As a byproduct we obtain properties of the arborescences of generalized star-delta reductions of weighted digraphs.

Classical minimax lower bounds for testing are typically derived for fixed error probabilities, while high-confidence results often impose a common failure probability. We study prescribed-error testing, in which the level and the target type II error may be small and of different orders. Standard prior-based reductions generally aggregate the two errors into a single quantity and therefore do not capture their distinct roles. We develop a general lower-bound technique based on a binary reduction that preserves the separate roles of the two error targets. The reduction yields two directed Kullback-Leibler information requirements, corresponding respectively to the level and the target type II error. When both directed mixture divergences can be controlled, they combine into a binary Jeffreys divergence, leading to the logarithmic dependence on the level and the target type II error. Applying the framework to Gaussian sequence testing, multinomial uniformity testing, and continuous uniformity testing over Hölder balls, we obtain lower bounds that match corresponding high-confidence upper bounds and hence establish prescribed-error minimax rates sharp up to constant factors.

We study the symmetrical Dziobek configurations where, in $\mathbb{R}^{d}$, there are $d$ bodies with unit masses at the vertices of a regular $(d-1)$-dimensional simplex of unit edge length and two more bodies with nonzero masses $s,k$ are on the line passing through the center of the simplex and being orthogonal to it. In the case of logarithmic potential, the finiteness is proved for all $s,k\neq 0, d>1$, and we obtain the bifurcation surface in the $(s,k,d)$-space through Gröbner basis computation. Using cylindrical algebraic decompositions, we find $197232$ sample points in the complement of the bifurcation surface. We propose a method to reduce the number to only $202$. By Hermite's root counting theorem, we find that, generically, there can be $0,1,2,3$ or $4$ concave, $1,2,3, $ or $4$ convex, and in totality, $1,2,3,4$ or $5$ such configurations for all dimensions $d>1$. For positive $s$ and $k$, generically, there is a unique convex configuration, while the number of concave ones can be $0,2$ or $4$. All possible combinations for the numbers described above are realized when $d=2$. We obtain a set of generalized Laura-Andoyer equations equivalent to the central configurations equations for all fixed number of bodies $n=d+h$ and configuration dimension $d$. For homogeneous force law with exponent $a\in \mathbb{R}$, we use the action of permutation group $S_d$ in the Laura-Andoyer equations to reduce the equivalent $\binom{d+2}{2}\binom{d}{2}$ Laura-Andoyer equations to only two generalized polynomial algebraic equations for the studied class of symmetric configurations with two variables representing the positions of the two bodies not at the vertices of the simplex in four parameters $a,d,s,k$.

The feedback arc set problem on tournaments arises in a rich variety of applications, and has been studied extensively in several research fields over the past six decades. It is well known that this problem is $NP$-hard and admits a polynomial-time approximation scheme (PTAS) in general. A tournament $T=(V, A)$ is called cycle Mengerian (CM) if, for every nonnegative integral weight function defined on $A$, the minimum total weight of a feedback arc set is equal to the maximum size of a cycle packing. In 2020 Chen et al. obtained a structural characterization of all CM tournaments; however, their proof is not algorithmic in nature. In this paper we present combinatorial polynomial-time algorithms for finding both minimum feedback arc sets and maximum cycle packings in arc-weighted CM tournaments.

This paper investigates the long-times behavior of the Fisher-KPP equation with slowly decaying initial data in an almost periodic medium. We mainly focus on two classes of initial data: exponentially decaying initial data and inital data that decay more slowly than any exponential function. Employing the Hamilton-Jacobi approach, we provide a unified framwork for analyzing the Cauchy problem with initial data in both cases. We demonstrate that the level sets of the solution can be estimated by the generalized principal eigenvalue of the linearized operator and the decay rate of the initial data.

The Kaplansky radical of a field consists of the nonzero elements represented by every norm quadratic form in two variables. D. Kijima and M. Nishi conjectured that, for quadratic extensions, the Kaplansky radicals are related by the norm map in a manner analogous to Hilbert's Theorem 90. Although this H-conjecture was disproved by K.J. Becher and D.B. Leep, it is known to hold for several important classes of fields. We introduce a cohomological analogue of the Kaplansky radical for arbitrary profinite groups and primes $p$, defined as the orthogonal of $\mathrm{H}^1(G,\mathbb{F}_p)$ with respect to the cup product with itself. For absolute Galois groups, this recovers the classical Kaplansky radical when $p=2$ and the $p$-radical of Dario-Engler for arbitrary p. We also formulate a group-theoretic analogue of the H-conjecture, proving that, for fields, it is equivalent to the original conjectural property and depends only on the maximal pro-$2$ quotient of the absolute Galois group. We establish this property for broad classes of fields, including local and global fields, rational function fields, and all fields whose maximal pro-$p$ Galois group is of elementary type. Beyond its arithmetic origins, we investigate the property for general pro-$p$ groups, proving its stability under several natural group-theoretic constructions and obtaining new examples, including generalized right-angled Artin pro-$p$ groups and fundamental pro-$p$ groups of suitable graphs of groups, many of which cannot occur as maximal pro-$p$ Galois groups.

A real quadratic field satisfies Hammarhjelm's condition if its ring of integers has unique factorization, and the Minkowski lattice of its ring of integers contains no point in a certain rectangle determined by the fundamental unit. Such fields have recently appeared in the study of visible points in algebraic cut-and-project sets. We prove that there are exactly seven real quadratic fields satisfying Hammarhjelm's condition, namely those with discriminant 8, 5, 13, 29, 53, 173, 293. The proof is based on showing that for such fields, the fundamental unit is small relative to the discriminant, together with genus theory and Biro's classification of class number one fields in Yokoi's family.

In this paper, we prove estimates for the Stokes resolvent problem with no-slip boundary conditions on the half space in weighted $L^p$-spaces. The weights we consider are power weights both inside and outside the Muckenhoupt range. Our estimates imply that the corresponding Stokes operator is the generator of a bounded analytic $C_0$-semigroup. We furthermore show that it admits a bounded $H^\infty$-calculus of angle $0.$ This results seems to be new even within the Muckenhoupt range.

High-precision effective resistance computation is a cornerstone of Electronic Design Automation (EDA) sign-off, yet it remains a fundamental bottleneck in large-scale power grid analysis, spectral sparsification, and circuit reliability. Existing approaches face a prohibitive "precision-memory impasse": approximate methods lack the stringent accuracy required for high-stakes industrial sign-off, while exact methods either suffer from redundant query overheads or trigger $O(n^2)$ memory explosions. To resolve this, we propose PEERS, a Parallel and Exact Effective Resistance Solver powered by an implicit inverse computing model of the Cholesky factor. By integrating a state-inherited augmented depth-first search (DFS) with a dynamic query update mechanism, PEERS eliminates numerical redundancy and evaluates all-edge resistance queries in a single parallel sweep. We provide a rigorous Work-Span analysis, proving that for graphs satisfying an $O(n^\alpha)$ separator theorem, PEERS achieves a theoretically optimal parallel span of $O(n^\alpha)$ while strictly maintaining $O(nnz(L))$ space complexity. Numerical evaluations on industrial benchmarks demonstrate that PEERS achieves an average speedup of 83.3x over state-of-the-art parallel solvers under identical memory constraints. Notably, PEERS processes a 1-million-node industrial graph in just 18.8 seconds and scales to 17 million nodes in under an hour, providing the first computationally feasible path for exact all-edge resistance analysis in multi-million-gate designs.

In the Graph Scheduling problem we schedule a given multiset of edges on discrete time steps, such that at each step the set of edges forms a matching. The goal is to minimize the sum of weighted group completion times, where a group is a set of edges and it completes when the last edge has been scheduled. Two popular variants of this problem are Coflow Scheduling and Data Migration. Our main result is extending a recent iterated rounding approach from Coflow Scheduling, roughly corresponding to the bipartite case, to the general Graph Scheduling problem. This yields an essentially tight $(2+\epsilon)$-approximation for the asymptotic setting where OPT is assumed to be large. For this we rely on polyhedral techniques from general matching, namely odd-set inequalities, and graph theoretical results on edge colorings in multigraphs. The state-of-the-art approximation algorithm for Data Migration is a $(1 + \phi)$-approximation that improves when OPT is small. Taking the best of this and our main result, we obtain an improvement of the approximation rate for Data Migration in any regime.

This paper studies free stochastic differential equations driven by free Brownian motion. Under local operator Lipschitz and Lyapunov-type conditions on the coefficients, we prove the global well-posedness of solutions in the noncommutative probability setting using free Itô calculus. We further establish the existence and uniqueness of stationary solutions under appropriate dissipativity conditions. Our results extend classical theory to the free probability framework.

We report the exact value of the number of labeled partially ordered sets (equivalently, labeled $T_0$ topologies) on 19 points, P(19) = 646099441937791106493755218560442089979, a 39-digit integer extending OEIS A001035, whose largest previously computed term was P(18) (Brinkmann and McKay). By the Stirling transform we also obtain the number of labeled topologies on 19 points, A000798(19) = 689054943207246404281592791142107048261. Our route is the Erné-Stege moment reduction, which expresses P(19) through a few sums of antichain counts over the posets on at most 16 points. All of these are available from the posets on at most 15 points (whose number is catalogued, and which standard software generates on demand), except a single moment over the 16-point posets. That moment is obtained not by enumerating the 16-point posets but by inserting a single element into the 15-point ones, with a per-parent kernel that advances the sum at the cost of computing the parent's own antichain count. The result passes several independent checks, among them the residue predicted by the modular periodicity of A001035 and the recovery from the same sweep of the known count P(16) and the Erné-Stege moments G(16,1) and G(16,2). We also report the moments G(16,3) and G(16,4), the latter an input to the analogous computation for 20 points.

We present MINT, a dynamic-precision CNN inference accelerator based on left-to-right (LR) arithmetic. LR arithmetic computes in most-significant-digit-first manner and exposes useful partial results early so that the computation can be terminated once the desired precision is achieved. At the core, there is a MSDF serial-parallel inner-product unit, which uses redundant signed-digit representation to compute each convolution window. A budget-constrained greedy search profiles all convolution layers from INT2 to INT7 and selects the lowest precision per layer while constraining total accuracy loss to within 2% of the INT8 baseline for VGG-16 and ResNet-18 networks. The design is synthesized on a Xilinx Zynq-7020 at \SI{200}{\mega\hertz}, and uses 5.64 average bits for VGG-16 and 6.04 for ResNet-18, while achieving 19.86 GOPS and 29.51 GOPS/W on VGG-16, and 18.86 GOPS and 26.40 GOPS/W on ResNet-18. This corresponds to 32.6% and 26.0% higher throughput and 82.10% and 62.90% higher energy efficiency than INT8 with only 1.81% and 1.96% drops relative to the INT8 baseline. Compared with representative prior FPGA CNN accelerators considered in this study, MINT delivers the highest energy efficiency among the listed VGG-16 and ResNet-18 designs on Zynq-7020 platform.

In this paper, we study distribution-dependent stochastic differential equations on the domain $\mathcal O=\mathbb T^d$ or $\mathbb R^d$, $d\geq 2$, of the form \begin{align*} dX_t = v(t,X_t,\rho_t)\,dt + \sqrt{2}\, \sigma(t,X_t,\rho_t)\,dW_t, \qquad \rho_t:=\frac{d\mu_t}{dx}, \end{align*} where $\mu_t=\operatorname{Law}(X_t)$. Our main construction is carried out at the level of the associated nonlinear Fokker–Planck equations. We first build non-unique probability solutions to these PDEs and then use the superposition principle to obtain non-unique martingale solutions to the corresponding DDSDEs. We establish two main non-uniqueness results concerning stationary states, both on the torus and in the whole space, under the corresponding structural assumptions. First, we construct a divergence-free drift $v\in C_tL^{d-}$ such that the DDSDE admits infinitely many distinct solutions starting from the stationary initial density. This result lies at the natural critical regularity threshold: in several models, well-posedness is expected for drifts in $C_tL^{d+}$. Second, for $d\geq 3$ and every prescribed $N\in\mathbb{N}$, we construct a divergence-free drift for which the DDSDE admits at least $N$ distinct stationary martingale solutions. The resulting multiplicity of equilibrium states is reminiscent of multistability and phase-transition phenomena in physical systems.

This paper establishes Bismut-type formulas and gradient estimates for Feynman–Kac semigroups on Riemannian manifolds with boundary, under geometric conditions formulated in terms of Ricci curvature $\mathrm{Ric}_Z \geq K$ and second fundamental form $\mathrm{II} \geq \sigma$ for potentially unbounded functions $K$ and $\sigma$. We then apply these formulas to derive pointwise gradient estimates for the Neumann semigroup under variable, possibly unbounded, lower curvature bounds. Both convex and non-convex boundary cases are treated. In the non-convex case, the boundary contribution is controlled by a conformal change of metric and an exponential estimate for the boundary local time.

Adder trees are the computational backbone of delay-and-sum (DAS) ultrasound beamforming, where their implementation directly determines the energy, throughput, and area of a real-time imaging pipeline. Conventional parallel adder trees perform full-precision combinational reduction on every sample, leading to wide critical paths, high LUT consumption, and timing failures on small FPGA devices. This paper presents an alternative adder tree architecture based on left-to-right (LR) or most significant digit first (MSDF) arithmetic. We implement the proposed and conventional adder trees on a Xilinx Zynq XC7Z010 FPGA and evaluate them for DAS beamforming of a 64-channel ultrasound dataset. The proposed design uses 2.5$\times$ fewer LUTs than the smallest conventional tree, successfully meets the timing constraint, and consumes 23% less dynamic power than the most efficient conventional baseline. A key advantage of the proposed MSDF adder tree is that it can generate high-quality beamformed images without waiting for full-precision completion. This naturally enables dynamic precision at runtime with negligible control overhead, since precision selection is achieved simply by stopping the computation clock after the desired number of cycles. Such quality–energy scalability is fundamentally unavailable in conventional fixed-cycle adder trees. Iso-area replication enables up to 15 parallel instances on the XC7Z010, achieving 67 FPS, which is 80% higher throughput than the best conventional design.

We study constrained online convex optimization with adversarial losses and stochastic or adversarial constraints. For stochastic constraints, existing algorithms that achieve nearly optimal regret and constraint violation bounds typically rely on regularity assumptions such as Slater's condition, while adversarial-constraint algorithms avoid these assumptions by using a rather restrictive round-wise feasible comparator. We bridge this gap with an anytime primal-dual framework that incorporates an adaptive regularizer into the dual update. The regularizer stabilizes the dual process without relying on the negative drift induced by Slater's condition. For stochastic constraints and convex losses, our algorithm achieves $O(\sqrt{T})$ expected regret and $O(\sqrt{T}\log T)$ expected cumulative constraint violation. Furthermore, we show that our algorithm also admits high-probability bounds of the same order on regret and constraint violation. For strongly convex losses, the regret bound improves to $O(\log T)$ with a violation bound of the same order. With a minor modification, the framework also applies to adversarial constraints and provides guarantees for hard constraint violation.

This work investigates uncertainty-aware deep learning approaches for direction of arrival (DOA) estimation in automotive radar, focusing on probabilistic modeling and downstream integration. A circular-statistics-based von Mises (VM) ensemble (ENS) is compared with an evidential deep learning (EDL) framework based on a normal inverse gamma formulation, yielding a Student t predictive distribution in the Euclidean domain. The ENS framework produces angular predictions parameterized by (mu, kappa), enabling interpretable uncertainty aligned with directional geometry. Performance is evaluated under in distribution and multiple out-of-distribution conditions using risk coverage and ROC or AUROC analyses. Results indicate that ENS achieves lower uncertainty under nominal conditions and exhibits stronger sensitivity to severe perturbations, whereas EDL provides smoother uncertainty variation and slightly improved ranking consistency. Importantly, the ENS representation enables direct probabilistic integration into association modules via closed form VM likelihoods, facilitating a unified detection tracking pipeline. These findings highlight a trade-off between geometric consistency and statistical generality in uncertainty-aware DOA estimation.

Partial cubes are a fundamental class of graphs that admit isometric embeddings into hypercubes. Klavžar and Kovše [Ars Combin. 93 (2009), 77–86] observed that the opposite semicubes of every harmonic-even partial cube are pairwise isomorphic, and asked whether the converse is true, that is, whether a partial cube is harmonic-even if and only if its opposite semicubes are pairwise isomorphic. In this paper, we answer this question in the negative by constructing an infinite family of partial cubes with pairwise isomorphic opposite semicubes that are not harmonic-even. This establishes that pairwise opposite-semicube isomorphism is strictly weaker than harmonic-evenness and naturally leads to the question of what additional condition restores the equivalence. To address this question, we introduce the opposite-semicube Helly property and prove that a finite partial cube satisfying this property is antipodal, or equivalently harmonic-even by Polat's theorem, if and only if it has pairwise isomorphic opposite semicubes.

Non-fixed flexible antenna architectures, such as fluid antenna system (FAS), movable antenna (MA), and pinching antenna, have garnered significant interest in recent years. In this paper, we deploy a rotatable antenna (RA) array at the base station (BS) to improve the integrated sensing and communication (ISAC) performance by exploiting the additional spatial degrees of freedom (DoFs) introduced by antenna rotation. To enhance the sensing performance over an extended region containing a potential target while meeting the communication requirements of multiple users, we aim to maximize the minimum echo signal power within the sensing region, subject to required minimum communication rates of the users. For the special case of a single user and a point target, we develop a tractable approximation of the original problem, leading to independent per-RA subproblems. We characterize the geometric structure of these subproblems and derive closed-form solutions. Under the far-field approximation, the per-RA subproblems become approximately identical, allowing a common pointing vector to be adopted for all RAs. For the general multi-user and regional sensing case, we propose an alternating optimization (AO) algorithm that alternately optimizes the transmit beamforming for communication, the covariance matrix of the probing signal, and the pointing vectors of the RAs in an iterative manner. Simulation results demonstrate that the proposed RA-enabled ISAC system significantly outperforms various benchmark schemes with restricted antenna-orientation flexibility, while the evaluation of the Cramér-Rao bound (CRB) further verifies its benefits for target-position estimation accuracy.

Functional principal component analysis has traditionally focused on processes indexed by Euclidean domains, although functional observations increasingly arise on curved domains. We develop functional principal component analysis for real-valued processes observed at discrete locations on a general Riemannian manifold. We estimate the mean and covariance functions using intrinsic kernel smoothing based on geodesic distance and a normal-coordinate volume correction, with a general weighting scheme accommodating unequal numbers of observations across subjects. We establish $L^2$ and uniform convergence rates for these estimators and derive corresponding error bounds for the estimated eigenvalues and eigenfunctions. These results quantify how manifold dimension, sampling frequency and the weighting scheme affect estimation accuracy and characterize the phase transition between sparse and dense sampling regimes. Simulation studies on the circle and sphere, together with real-data applications, illustrate the finite-sample performance of the proposed methodology.