We investigate the thermal properties of the $Y(4500)$ state within the framework of thermal QCD sum rules, assuming a $D_s \bar{D}_{s1}$ molecular configuration with $J^{PC}=1^{--}$. The analysis is performed at both zero and finite temperatures, employing the operator product expansion up to dimension-5 condensates. The Borel window and continuum threshold are carefully selected to ensure OPE convergence and pole dominance. As the temperature approaches the deconfinement temperature $T_c$, the $Y(4500)$ undergoes significant medium modifications: its mass decreases by $29\%$ and its decay constant is suppressed by $94\%$ relative to their vacuum values, while the decay width increases by $35\%$, signaling the dissociation of the state in the medium. These results indicate that the $Y(4500)$ becomes unstable near $T_c \approx 155~\mathrm{MeV}$, consistent with its melting into the quark-gluon plasma. The obtained thermal spectral parameters may serve as signatures for identifying the $Y(4500)$ in heavy-ion experiments at RHIC and LHC, and provide predictions for sequential suppression patterns in the exotic hadron sector.

Post-quantum signature schemes such as ML-DSA-65 produce signatures of 3,309 bytes and public keys of 1,952 bytes over 50 times larger than classical Ed25519. In TLS-authenticated environments like Kubernetes control planes and 5G Core networks, where every inter-component connection is mutually authenticated, this overhead compounds across thousands of handshakes per second. Merkle Tree Certificates (MTC), currently under development at IETF, replace per-certificate issuer signatures with Merkle inclusion proofs and, in the landmark mode, eliminate on-wire signatures from certificate authentication entirely. We present MTC-based PKI architectures for Kubernetes and 3GPP 5G Service-Based Architecture. Starting from the infrastructure layer, we replace the Kubernetes cluster CA with an MTCA deployment that issues MTC certificates to control plane components, with cosigners and a DaemonSet-based landmark distributor. Building on this, we design a certificate lifecycle for 5G Network Functions deployed against QORE, a post-quantum 5G Core. We implement MTC proof construction and verification in Go crypto/tls and crypto/x509 packages. Our measurements on an Intel i9-12900 show MTC landmark verification completing in under 2 μs compared to 24 microseconds for ECDSA signature verification-with no measurable impact on TLS handshake time. We further propose a 6G-native architecture where the NRF serves as the MTCA and the SCP as witness cosigner, and discuss applicability to Non-Terrestrial Networks.

In this work, we obtained separation results via codimension-1 maps to generalized manifolds. More specifically, we proved results that allow us to estimate the number of connected components of the complement of the image of such maps.

Building on the work of E. L. Medeiros [1] and our previous study [2], we generalize the alpha-nucleus nonlocality effect to odd-A and odd-odd nuclei within the two-potential approach (TPA) framework. The coordinate-dependent parameters introduced by this nonlocality are optimized using an advanced gradient boosting regression model. This improved TPA is applied to calculate the $α$-decay half-lives of 599 nuclei with $Z = 52-118$, yielding a root-mean-square (RMS) deviation that is 74.8$\%$ lower than that of the original TPA. Subsequently, we employ the improved TPA, along with the DZR [3] and MUDL [4]models, to predict alpha-decay half-lives for 142 superheavy nuclei with $Z = 117-120$. The predictions from all three models are in close agreement, with the results from our improved TPA and the DZR model being nearly identical.

Given a weighted digraph $G$, a $(t,g,μ)$-DAG cover is a collection of $g$ dominating DAGs $D_1,\dots,D_g$ such that all distances are approximately preserved: for every pair $(u,v)$ of vertices, $\min_id_{D_i}(u,v)\le t\cdot d_{G}(u,v)$, and the total number of non-$G$ edges is bounded by $|(\cup_i D_i)\setminus G|\le μ$. Assadi, Hoppenworth, and Wein [STOC 25] and Filtser [SODA 26] studied DAG covers for general digraphs. This paper initiates the study of \emph{Steiner} DAG cover, where the DAGs are allowed to contain Steiner points. We obtain Steiner DAG covers on the important classes of planar digraphs and low-treewidth digraphs. Specifically, we show that any digraph with treewidth tw admits a $(1,2,\tilde{O}(n\cdot tw))$-Steiner DAG cover. For planar digraphs we provide a $(1+\varepsilon,2,\tilde{O}_\varepsilon(n))$-Steiner DAG cover. We also demonstrate a stark difference between Steiner and non-Steiner DAG covers. As a lower bound, we show that any non-Steiner DAG cover for graphs with treewidth $1$ with stretch $t<2$ and sub-quadratic number of extra edges requires $Ω(\log n)$ DAGs.

Dirichlet distributions are probability measures on the unit simplex. They are often used as prior distributions in modeling categorical data, such as in topic analysis of text data. Motivated by this application, we consider Monte Carlo estimation of expectations $\mathbb{E}[\exp(nH(θ))]$, where $θ$ has a Dirichlet distribution, $H$ is a real-valued function, and $n$ is a parameter. We develop variance reduction techniques particularly designed to work well for large $n$. Our analysis is guided by the Laplace method for approximating integrals, which we extend to fit our problem setting. We develop an importance sampling method that achieves a near-optimal asymptotic relative error. We use related ideas to select a provably effective control variate. We illustrate these results through their application in topic analysis.

An analytical framework for turbulent channel flow is developed based on the Alexeev hydrodynamic equations, focusing on the coupled behavior of streamwise and transverse velocity components. The mean streamwise velocity is represented as a superposition of a laminar (parabolic) component and a nonlinear turbulent contribution, yielding velocity profiles that agree with experimental data from channel and pipe flows over a wide range of Reynolds numbers, $3\times10^3 \le Re \le 3.5\times10^7$, with deviations of approximately $1\%$ at moderate Reynolds numbers and up to $3\%$ at the highest Reynolds numbers. The transverse velocity component is analyzed using a simplified form of the governing equations, leading to analytical expressions that capture its dominant spatial structure. The coupling between transverse velocity and streamwise momentum is then examined, revealing that the streamwise turbulent component admits a family of kink-type solutions. These solutions exhibit localized monotonic transitions separating regions of nearly uniform velocity and are interpreted as analytical representations of streamwise streaks. The model predicts characteristic streak properties, including spacing, thickness, intensity, and streamwise extent, which are shown to be consistent in order of magnitude with experimental observations of near-wall streaks. The results provide a unified analytical description of mean velocity profiles, secondary flows, and streak formation in wall-bounded turbulence, and suggest a mechanism linking transverse velocity fluctuations to the emergence of coherent streamwise structures.

With the rapid evolution of wireless technologies, Wi-Fi has expanded beyond its original role in data transmission to support various emerging applications, particularly in physical-layer security, including device authentication, user authentication, and secret key generation. Despite extensive research on Wi-Fi Channel State Information (CSI)-based physical-layer security, its vulnerabilities remain largely unexplored. In this work, we propose BFIAttack, a novel attack that exploits Beamforming Feedback Information (BFI) to reconstruct the CSI of a legitimate user or device, thereby compromising Wi-Fi-based physical-layer security. We realize the attack by leveraging a closed-form CSI reconstruction method for the single-antenna station scenario and a maximum likelihood estimation-based CSI reconstruction for the multi-antenna station scenario. Moreover, we exploit spatial similarities among antenna pairs to refine the reconstructed CSI and enhance attack effectiveness. Experimental results show that BFIAttack achieves an average attack success rate of $73\%$ in multi-antenna station scenarios with no more than five attack attempts, and over $93\%$ in single-antenna station scenarios with only a single attempt. BFIAttack reveals critical vulnerabilities in existing Wi-Fi-based physical-layer security.

The observation of the Chiral Magnetic Effect (CME) in heavy-ion collisions remains challenging because of large flow-induced backgrounds and experimental constraints. We demonstrate that the forward-backward multiplicity asymmetry (FBMA) provides a robust and experimentally accessible control parameter to separate the flow background from CME signal in the collisions of deformed nuclei, such as prolate uranium where FBMA is naturally enhanced and correlated with the initial-state geometry. Monte Carlo Glauber simulations indicate that varying FBMA within a fixed centrality class modulates ellipticity largely independently of the magnetic-field correlator, establishing FBMA as a practical tool for disentangling CME signals from flow driven background.

The dependence of mechanical properties on microscopic interactions remains a central problem in the physics of disordered solids near the jamming transition. We numerically and theoretically investigate the mechanical response of jammed cohesive granular materials using discrete element simulations and effective medium theory (EMT). We find that the shear modulus exhibits pronounced hysteresis under compression and decompression, even though the interparticle force law itself is strictly history-independent. While such hysteresis disappears for purely repulsive particles when mechanical properties are characterized in terms of pressure, it persists in cohesive packings, indicating that pressure is not a unique state variable for cohesive particles. Extending EMT to cohesive interactions, we show that the functional form of the shear modulus remains the same for both repulsive and cohesive particles, but that attractive interactions violate marginal stability. The resulting deviation from marginal stability generates excess rigidity, as predicted by a scaling relation. This prediction is quantitatively verified by numerical simulations and explains the persistent hysteresis in cohesive packings.

We study the three-dimensional incompressible Navier-Stokes system on $\mathbb{R}^3$ with an additional dissipative nonlocal term \[ \partial_t u + (u\cdot\nabla)u + \nabla p = νΔu + Lu, \qquad {\rm div}\, u = 0, \] where $L$ is a self-adjoint Fourier multiplier whose symbol is comparable to $-|ξ|^{2α}$ for some $α>1$. We first identify a sharp Fourier-symbol criterion distinguishing lower-order convolution perturbations from genuinely regularizing nonlocal corrections. In the resulting hyperdissipative class we prove the exact $L^2$ energy identity, global weak solvability for every $α>1$, and local strong well-posedness in $H^s(\mathbb{R}^3)$ for $s>\frac52$. We then show that the Lions exponent $α=\frac54$ remains the critical energy-growth threshold in this nonlocal setting: if $α\ge \frac54$, every $H^s$ solution is global, while for every $α>1$ one has global strong solvability for sufficiently small $H^s$ data. Finally, for the vanishing-hyperdissipation approximation of the classical three-dimensional Navier-Stokes equations, we prove a near-singular divergence principle: if the classical flow blows up at a first singular time $T_*$ in a continuation norm $X$, then the corresponding regularized family cannot remain uniformly bounded in $X$ on any interval approaching $T_*$. This identifies the precise point at which the fixed-parameter global theory degenerates in the Navier-Stokes limit.

Shaping the reachable set of a dynamical system is a fundamental challenge in control design, with direct implications for both performance and safety. This paper considers the problem of selecting the optimal input matrix for a linear system that maximizes warping of the reachable set along a direction of interest. The main result establishes that under certain assumptions on the dynamics, the problem reduces to a finite number of linear optimization problems. When these assumptions are relaxed, we show heuristically that the same approach yields good results. The results are validated on two systems: a linearized ADMIRE fighter jet model and a damped oscillator with complex eigenvalues. The paper concludes with a discussion of future directions for reachable set warping research.

Mutually Unbiased Bases (MUBs) constitute a fundamental geometric structure in quantum theory, known for providing an optimal measurement scheme for quantum state tomography. In prime and prime-power dimensions, analytical constructions of maximal sets of MUBs are well-known and standard construction relies on the Weyl-Heisenberg (WH) group and finite fields. In non-prime-power dimensions, on the other hand, the existence of such maximal sets remains an open question. We present a generalized numerical method of constructing MUBs without any reliance on a priori group structure or specific algebraic frameworks. Formulating the problem at the level of Gram matrix, we reduce the search for complete sets of $d+1$ MUBs in dimension $d$ to a phase space optimisation problem. We use the fact that the MUB Gram matrix is a projection matrix, and the third- and fourth-order trace constraints are necessary and sufficient conditions for a valid projection matrix. We further develop a classification framework based on third-order Bargmann invariants and automorphism groups, allowing us to probe the underlying algebraic and geometric structure of the resulting configurations. Numerical applications of this method in dimensions $3$, $4$, and $5$ demonstrate that all numerically constructed solutions are mutually isomorphic, are isolated points in phase space, and possess automorphism groups that coincide exactly with the Clifford group, the normalizer of the WH group. Though the scope of the search was limited, in dimension $d = 6$ our numerical search yielded no MUBs within explored parameter space.

The enumeration of lozenge tilings of hexagons with holes has been studied intensively in recent years. Researchers tried to find shapes and positions of holes in hexagonal regions so that the number of lozenge tilings of the resulting regions is given by a simple product formula. In the present work, we consider new regions that are hybrids of regions studied by the first author (hexagons with intrusions) and Ciucu (F-cored hexagons). Then, we show that the tiling generating functions of these new regions under a certain weight are given by simple product formulas. To give a proof, we present shuffling theorems for lozenge tilings of hexagons with intrusions, which give simple relations between the tiling generating functions of two related hexagonal regions with intrusions.

In the classic online graph balancing problem, edges arrive sequentially and must be oriented immediately upon arrival, to minimize the maximum in-degree. For adversarial arrivals, the natural greedy algorithm is $O(\log n)$-competitive, and this bound is the best possible for any algorithm, even with randomization. We study this problem in the i.i.d. model where a base graph $G$ is known in advance and each arrival is an independent uniformly random edge of $G$. This model generalizes the standard power-of-two choices setting, corresponding to $G = K_n$, where the greedy algorithm achieves an $O(\log\!\log n)$ guarantee. We ask whether a similar bound is possible for arbitrary base graphs. While the greedy algorithm is optimal for adversarial arrivals and also for i.i.d. arrivals from regular base graphs (such as $G = K_n$), we show that it can perform poorly in general: there exist mildly irregular graphs $G$ for which greedy is $\widetildeΩ(\log n)$-competitive under i.i.d. arrivals. In sharp contrast, our main result is an $O(\log\!\log n)$-competitive online algorithm for every base graph $G$; this is optimal up to constant factors, since an $Ω(\log\!\log n)$ lower bound already holds even for the complete graph $G = K_n$. The key new idea is a notion of log-skewness for graphs, which captures the irregular substructures in $G$ that force the offline optimum to be large. Moreover, we show that any base graph can be decomposed into ``skew-biregular'' pieces at only $O(\log\!\log n)$ scales of log-skewness, and use this to design a decomposition-based variant of greedy that is $O(\log\!\log n)$-competitive.

Cross-correlation functions (CCFs) are classical tools for studying lead-lag relationships between paired time series, but they are most often used descriptively rather than inferentially. Motivated by mouse experiments on gut-brain interactions in reward learning, we carry out a two-sample hypothesis test for formal statistical inference on collections of subject-specific CCF curves. In our application, each experimental session yields two related CCFs describing the temporal association of dopamine activity with locomotor velocity and acceleration, which leads naturally to a multivariate functional data formulation. We treat each empirical CCF as a functional observation indexed by lag and test equality of mean multivariate CCF functions across groups using integrated and maximum-type global statistics, \(F_{\mathrm{int}}\) and \(F_{\max}\), constructed from pointwise Hotelling \(T^2\) statistics. The integrated test targets broad differences across the lag domain, whereas the maximum test is sensitive to local differences. Applied to free-feeding and intragastric infusion datasets, the proposed methods detect substantial differences in dopamine-locomotion coupling across brain region and biological sex in the free-feeding experiment, with more selective effects in the infusion setting. The proposed framework provides a flexible and rigorous FDA-based approach for comparing dynamic dependence structures across experimental conditions.

We report a comprehensive study of the temperature-dependent structural, magnetic, vibrational, and dielectric properties of Al-substituted M-type hexaferrites SrFe$_{12-x}$Al$_x$O$_{19}$. Neutron powder diffraction and Mössbauer spectrometry show that Al$^{3+}$ preferentially replaces Fe$^{3+}$ at spin-up octahedral sites (2a, 12k), disrupting the exchange coupling with the spin-down 4f tetrahedral sites and leading to a progressive reduction of site-specific magnetic moments and a systematic decrease in the Curie temperature, supported by temperature dependent susceptibility measurements. Raman spectroscopy reveals pronounced phonon anomalies near $T_C$, particularly in modes associated with bipyramidal Fe-O vibrations, reflecting the weakening of both 4e-12k and 4e-4f exchange pathways. However, the coercive field exhibits a dramatic increase, reaching $μ_0H_C$ $\sim$ 1.2 T for SrFe$_{9.6}$Al$_{2.4}$O$_{19}$, among the largest values reported for this class. Susceptibility measurements suggest that Al substitution, while weakening the superexchange network, contributes to the stabilization of single-domain behavior.

Understanding the factors that control superconductivity is essential for discovering new superconducting materials using high-throughput elemental substitution. Focusing on the recently predicted ambient-pressure superconducting X2MH6 family, we disentangle the phononic and electronic contributions to Tc to determine how isoelectronic substitution alters superconductivity. While substitution affects both phononic and electronic properties, the electronic contribution plays the dominant role in determining Tc in the X2MH6 family. We show that the electronic contribution is affected by three key factors: the X-H bond distance, the electron localization function networking value of hydrogen, and the hydrogen-projected density of states at the Fermi level. A combined figure of merit derived from these parameters exhibits a robust correlation with Tc across the family. We further show that pressure produces competing effects on superconductivity: it enhances the electronic contribution by shortening X-H bonds, but simultaneously weaken the phononic contribution by increasing phonon frequencies. The net pressure dependence of Tc therefore results from the balance between these opposing tendencies. By disentangling and analyzing the electronic and phononic mechanisms, this work provides comprehensive insight into superconductivity in X2MH6 hydrides and offers practical guidance for designing new high-Tc hydride superconductors.

Accurate modeling of magnetic hysteresis is essential for high-fidelity power electronics device simulations. The transient hysteresis phenomena such as the ringing effect and the minor loops are the bottleneck for the accurate hysteresis modeling and the core losses estimation. To capture the hysteresis loops with both the macro structure and the micro transient details, in this paper, we propose the multi-scale ResNet augmented Fourier Neural Operator (Res-FNO). The framework employs a hybrid input structure that combines sequential time-series data with scalar material labels through specialized feature engineering. Specifically, the time derivative of magnetic flux density ($\frac{dB}{dt}$) is incorporated as a critical physical feature to enhance the model sensitivity to high-frequency oscillations and minor loop triggers. The proposed architecture synergizes global spectral modeling with localized refinement by integrating a multi-scale ResNet path in parallel with the FNO blocks. This design allows the global operator path to capture the underlying physical evolution while the local refinement path, compensates for spectral bias and reconstructs fine-grained temporal details. Extensive experimental validation across diverse magnetic materials from 79 to Material 3C90 demonstrates the strong generalization capability of the proposed Res-FNO, proving its robust ability to model complex ringing effects and minor loops in realistic power electronic applications.

Sixth-generation (6G) networks are expected to provide ubiquitous connectivity across terrestrial and non-terrestrial domains. This will be possible by integrating non-terrestrial networks (NTNs) to extend coverage to underserved areas. Antennas are central to this vision, with multiple-input multiple-output (MIMO) technologies receiving the most attention due to their ability to exploit spatial multiplexing to improve link capacity and reliability. However, conventional MIMO can consume significant energy, as each antenna element typically requires an independent RF chain. This limitation is particularly critical in non-terrestrial systems, where onboard energy resources are limited. Holographic MIMO (HMIMO) has emerged as a promising alternative in this context. These systems are based on theoretically continuous apertures, where radiation is generated through controlled modulation of surface impedance. This enables beamforming mechanisms with significantly fewer RF chains, reducing power consumption. In this work, we make the case for HMIMO as a suitable candidate for NTN integration within IMT-2030 systems. We discuss its advantages over conventional MIMO and present a case study of HMIMO integration in LEO-based multi-user communication.

We study a logarithmic fractional Schrödinger--Poisson system in \(\R^{3}\): \begin{equation*} \begin{cases} \varepsilon^{2α}(-Δ)^αu+V(x)u+ϕu=u\log u^{2}+|u|^{p-2}u, & \text{in }\R^{3},\\ \varepsilon^{2α}(-Δ)^αϕ=u^{2}, & \text{in }\R^{3}. \end{cases} \end{equation*} Here \(α\in\bigl(\frac34,1\bigr)\), \(4<p<2_α^{*}=\frac{6}{3-2α}\), and \(V\) satisfies a global potential condition. Using a suitable Orlicz-type Banach space, we establish a \(C^{1}\) variational framework for the problem and combine the Nehari manifold method with Lusternik--Schnirelmann category theory. We then prove that, for every fixed \(δ>0\) and all sufficiently small \(\varepsilon>0\), the system admits at least \(\operatorname{cat}_{M_δ}(M)\) distinct positive solutions. Moreover, the maximum points of these solutions concentrate near the global minimum set of \(V\) as \(\varepsilon\to0\).

This letter presents a framework for space-to-ground wireless energy transfer (WET) for wirelessly chargeable devices (WCD) located in remote areas or disaster situations. We consider a grid of multi-antenna satellites that charge a WCD within line-of-sight. Closed-form expressions for harvested energy are derived considering maximum ratio transmission (MRT) ensuring that the WCD meets its circuit charging threshold $P_{th}$. Simulations elucidate that milli-joule-level energy can be harvested during satellite grid visibility, with charging efficiency influenced by the number of satellites, their altitude, charging frequency, and grid inclination.

P. Di Francesco first introduced the "Aztec triangle" in his study of the relationship between the twenty-vertex model and domino tilings. He conjectured an exact formula for the number of tilings of the Aztec triangle, and it has since been proved by several authors. In an attempt to prove the conjecture, M. Ciucu showed that the tiling number of the Aztec triangle divides the tiling number of a new region called the "cruciform region," a superposition of two Aztec rectangles. Ciucu proved that the number of domino tilings of a cruciform region is given by a simple product formula. In this paper, we generalize Ciucu's tiling formula by providing a generating-function formula for the cruciform region.

The development of quantum networks (QNs) relies on efficient mechanisms for distributing entanglement among multiple quantum users (QUs) under practical system constraints. This paper investigates the problem of entanglement rate maximization in a dual-connectivity (DC) wireless quantum network comprising multiple quantum base stations (QBSs). Under the DC architecture, each QU can associate with up to two QBSs, thereby enhancing resource utilization compared to conventional single-connectivity (SC) schemes. The joint QBS-QU association and entanglement generation rate allocation problem is formulated as a mixed-integer nonlinear programming problem that incorporates practical constraints, including limited QBS entanglement generation capacity as well as heterogeneous minimum entanglement rate demands and fidelity requirements for QUs. To efficiently solve this challenging problem, an alternating optimization (AO) algorithm is developed, which decomposes the original formulation into entanglement rate allocation and association subproblems. Simulation results demonstrate that the proposed DC architecture significantly outperforms SC schemes, while the AO algorithm achieves near-optimal performance with substantially reduced computational complexity.

AI in education is commonly delivered through web-based systems such as online forms and institutional platforms. However, these approaches can exclude teachers in low-resource contexts, where everyday mobile platforms like WhatsApp serve as primary digital infrastructure. To address this gap, we present a field pilot in Cameroon that deploys a WhatsApp-based chatbot with LLM-supported content for teacher professional development (TPD), compared with an online form baseline. The system was evaluated through a mixed-methods study with 47 primary school teachers, integrating quantitative measures with qualitative insights from interviews and participant feedback. Results show that the chatbot was rated higher in perceived usability and overall experience, while learnability remained comparable. These improvements were driven by platform familiarity, low interaction overhead, and the modular structure of LLM-supported content, but were constrained by connectivity limitations, prepaid data costs, and multilingual needs (English/French). Building on these findings, we outline design directions for multilingual, culturally grounded interaction and for supporting prompting and reflection in AI use. More broadly, this work points to Thoughtful AI that supports reflection, relevance, and sustained professional growth.

We find that reinforcement exponentially reduces computation time of the quantum search problem from $\sqrt{D}$ to $\ln D$ in a $D$-dimensional system. Therefor, a reinforced quantum search is expected to exhibit an exponentially larger noise threshold compared to a standard search algorithm in a noisy environment. We use numerical simulations to characterize the level of noise tolerance via reinforcement in the presence of both coherent and incoherent noise, considering a system of $N$ qubits and a single $D$-level (qudit) system. Our results show that reinforcement significantly enhances the algorithm's success probability and improves the scaling of its computation time with system size. These findings indicate that reinforcement offers a promising strategy for error mitigation, especially when a precise noise model is unavailable.

Movable Antenna (MA) technology is emerging as a promising advancement with the potential to significantly enhance the performance of future wireless communication and sensing systems. In this paper, we address two-dimensional (2D) direction of arrival (DOA) estimation via joint shape-position optimization. Specifically, we formulate an optimization problem aimed at minimizing the Cramér-Rao Bound (CRB) based on a 2D DOA estimation model for MA systems. To tackle the highly non-convex nature of this CRB minimization, we investigate the spatial utilization of the movable region (MR) under minimum antenna spacing constraints. By demonstrating that an equilateral triangle yields the minimum overlap area, we strategically design an equilateral triangular MR. This specific geometric configuration enables the exploitation of structural symmetry to simplify the geometric constraints, which effectively reduces the complexity of solving the optimization problem. Subsequently, we derive the optimal MA positions by selecting the candidate locations farthest from the centroid of MR. The results demonstrate that the proposed joint shape-position optimization substantially enhances 2D DOA estimation performance.

We formulate Lagrangian descriptors (LDs) in the path integral framework. Averaging the classical LD over fluctuations about extremal trajectories defines a quantum LD that incorporates quantum effects. Invariant manifolds, which sharply organize classical transport, become finite-width phase space structures under quantum fluctuations, and their overlap provides a geometric mechanism consistent with tunneling as fluctuation-induced delocalization of transport barriers. We demonstrate this approach for the Hamiltonian saddle, where path integral sampling reveals manifold broadening and barrier penetration. This establishes a geometric framework for studying phase space transport and tunneling beyond the classical regime, while also providing a natural route toward the application of LDs to field theory.

Let $p$ be a prime, let $q=p^n$, and let $D\subseteq \mathbb{F}_q^\ast$. A celebrated result of Carlitz and McConnel states that if $D$ is a proper subgroup of $\mathbb{F}_q^\ast$, and $f:\mathbb{F}_q\to\mathbb{F}_q$ is a function such that $(f(x)-f(y))/(x-y)\in D$ for all $x\neq y$, then $f$ must be of the form $f(x)=ax^{p^j}+b$. In this paper, we extend their result to the setting where $D$ is a union of cosets of a fixed subgroup of $\mathbb{F}_q^\ast$, under a mild assumption. In a similar spirit, we also investigate maximum cliques in related Cayley graphs over finite fields, strengthening several results of Blokhuis, Sziklai, and Asgarli and Yip.

Integrating distributed energy resources (DERs) is a critical step toward addressing the global climate crisis. This transformation has driven the transition from traditional consumers to prosumers and given rise to new energy sharing business models. Existing works have extensively studied prosumer energy sharing mechanisms, yet little attention has been paid to privacy protection, particularly when communication constraints are taken into account. In this paper, we study an energy sharing mechanism where information is exchanged over wireless channels via over-the-air (OTA) multiple-input multiple-output (MIMO) aggregation to exploit spectral efficiency for scalable prosumer coordination. To characterize the privacy leakage risk during data transmission process, we introduce an adversarial attack model and demonstrate that, under certain conditions, the platform can extract and recover prosumers' private parameters from the base station observations. To safeguard the energy sharing mechanism against such attacks, we propose a differentially private equilibrium-seeking algorithm, analyze the achievable privacy level, and establish convergence guarantees that quantify the impact of privacy on the convergence accuracy. Numerical experiments demonstrate that our approach effectively protects prosumers' privacy while converging to near-optimal solutions.