For many years it was routine to use equal model prior probabilities in Bayesian model uncertainty analysis. At least twenty years ago it became clear that this was problematic, leading to support of much too large models in the increasingly huge model spaces being considered in genomics and other fields. A popular replacement was to adopt a suggestion of Harold Jeffreys for the variable selection problem in which a total of $k$ possible variables are being considered for inclusion in the model: give the collection of all models containing $d$ variables ($d = 0, . . . , k$) prior probability $1/(k + 1)$ and then divide this prior probability equally among the models in the collection. Many other choices of model prior probabilities that impose severe parsimony have also been introduced. We begin by reviewing the problems with using equal model prior probabilities and then discuss some serious problems with the Jeffreys choice. Finally, we introduce and study a number of objective alternative choices of model prior probabilities, from both numerical and theoretical perspectives.

This thesis is devoted to the first-quantized approach to quantum field theory, commonly known as the 'Worldline Formalism'. It collects most of the works completed by the author during the PhD, illustrating the versatility and efficiency of this formalism across a broad range of physical contexts. The applications discussed fall into two broad categories: perturbative and non-perturbative analyses. In particular, the thesis investigates how quantum particles interact with strong background fields, how scattering amplitudes can be efficiently computed, and how perturbative expansions of the heat kernel can be systematically performed. These studies highlight recent advances in extending the worldline approach to increasingly complex situations. Different field theories are examined from this first-quantized perspective and are organized according to the spin of the particle under consideration. The discussion begins with the seemingly simple scalar case, progresses through spin 1, both in the abelian and the non-abelian case, and concludes with spin-2 particles, both in the massless and in the massive realizations. Through this sequence of examples, the thesis aims to demonstrate the flexibility as well as the computational power of the Worldline Formalism in addressing fundamental open problems in theoretical physics.

For a metric space $(X, d)$, a family $\mathcal{H}$ of locality sensitive hash functions is called $(r, cr, p_1, p_2)$ sensitive if a randomly chosen function $h\in \mathcal{H}$ has probability at least $p_1$ (at most $p_2$) to map any $a, b\in X$ in the same hash bucket if $d(a, b)\leq r$ (or $d(a, b)\geq cr$). Locality Sensitive Hashing (LSH) is one of the most popular techniques for approximate nearest-neighbor search in high-dimensional spaces, and has been studied extensively for Hamming, Euclidean, and spherical geometries. An $(r, cr, p_1, p_2)$-sensitive hash function enables approximate nearest neighbor search (i.e., returning a point within distance $cr$ from a query $q$ if there exists a point within distance $r$ from $q$) with space $O(n^{1+ρ})$ and query time $O(n^ρ)$ where $ρ=\frac{\log 1/p_1}{\log 1/p_2}$. But LSH for hyperbolic spaces $\mathbb{H}^d$ remains largely unexplored. In this work, we present the first LSH construction native to hyperbolic space. For the hyperbolic plane $(d=2)$, we show a construction achieving $ρ\leq 1/c$, based on the hyperplane rounding scheme. For general hyperbolic spaces $(d \geq 3)$, we use dimension reduction from $\mathbb{H}^d$ to $\mathbb{H}^2$ and the 2D hyperbolic LSH to get $ρ\leq 1.59/c$. On the lower bound side, we show that the lower bound on $ρ$ of Euclidean LSH extends to the hyperbolic setting via local isometry, therefore giving $ρ\geq 1/c^2$.

We present a constitutive model for the passive and active response of skeletal muscles. At variance with more classical approaches, the model is developed exploiting adapted Voigt representations of strain and stress tensors within the context of nonlinear Cauchy elasticity. This framework allows us to identify non-trivial stress-strain relations in a rather direct way from experimental data, enhancing the mechanical interpretability of the material functions that describe the tissue response and obtaining additional insight on the distinct role of the contractile fibres and of the surrounding extracellular matrix. We propose a two-material model, with an additive splitting of the stress contributions, in which only one component depends on an activation parameter. The constitutive model for the passive behaviour satisfactorily predicts the nonlinear stress response to elongation at different relative orientations with respect to the fibre direction and highlights the dominant role of the extracellular matrix. The activation model, essentially determined by the mechanics of the contractile fibres, captures well the isometric stress response through the prescription of an elasto-plastic evolution of the along-fibre active strain.

Multi-TeV gamma-ray emission around eHWC J1850+001 (a source from the first HAWC catalog of gamma-ray sources emitting above 56 TeV) is spatially coincident with the pulsar wind nebula (PWN) G32.64+0.53, powered by PSR J1849-0001. The absence of counterparts in radio, optical, and GeV energy ranges, contrasted with clear detections in X-rays and very-high-energy (VHE) gamma-rays, is indicative of a non-thermal leptonic origin for the nebula. We apply a systematic analysis pipeline, including a sophisticated model for the Galactic diffuse emission, to 2860 days of data from the HAWC Observatory. Our detailed analysis confirms that the ultra-high-energy (UHE) emission originates from G32.64+0.53, and we measure its spectrum up to 270 TeV with significant emission well beyond 100 TeV. We fit the multi-wavelength observations with a time-dependent leptonic model powered by the pulsar's rotational energy, and the results establish the nebula as a leptonic PeV accelerator, capable of accelerating electrons to a maximum energy of $E_{\mathrm{cut}}=1.5_{-0.6}^{+1.7}~\mathrm{PeV}$. The model also constrains the true age of the system to $26.8~\mathrm{kyr}$ and the nebular magnetic field to a low value of $2.5 ~\mathrm{μG}$, supporting a leptonic PWN origin for the observed UHE emission.

The rapid emergence of flying qubits in graphene and other low-dimensional conductors is pushing quantum electronics into an ultrafast regime where conventional transport theories no longer apply. In these systems, single-electron wave packets propagate coherently over micrometer scales while interacting with collective charge excitations on comparable time scales. Yet existing theoretical frameworks describe either fermionic single-particle dynamics or bosonic plasmonic modes, without reconciling the two. Here we introduce a unified theory of dynamical quantum transport that bridges this long-standing divide. Starting from a gauge-invariant scattering approach, we show how a time-dependent single-electron excitation self-consistently generates a propagating internal potential that behaves as a collective plasmonic mode. Electrons propagate at the Fermi velocity while simultaneously 'surfing' on this self-induced plasmon wave, whose velocity is renormalized by Coulomb interactions and screening. This dynamical mean-field framework captures photon-assisted transport, charge relaxation, and edge magnetoplasmon dynamics within a single description and remains valid far beyond the low-frequency limit. By unifying single-electron and plasmonic pictures, our results provide a timely foundation for the interpretation and control of flying-qubit experiments in graphene at gigahertz/terahertz frequencies.

We study the problem of optimally hedging the price exposure of liquidity positions in constant-product automated market makers (AMMs) when the hedge is funded by collateralized borrowing. A liquidity provider (LP) who borrows tokens to construct a delta-neutral position faces a trade-off: higher hedge ratios reduce price exposure but increase liquidation risk through tighter collateral utilization. We model token prices as correlated geometric Brownian motions and derive the hedge ratio h that maximizes risk-adjusted return subject to a liquidation-probability constraint expressed via a first-passage-time bound. The unconstrained optimum h* admits a closed-form expression, but at h* the liquidation probability is prohibitively high. The practical optimum h** = min(h*, h_bar(alpha)) is determined by the binding liquidation constraint h_bar(alpha), which we evaluate analytically via the first-passage-time formula and confirm with Monte Carlo simulation. Simulations calibrated to on-chain data validate the analytical results, demonstrate robustness across realistic parameter ranges, and show that the optimal hedge ratio lies between 50% and 70% for typical DeFi lending conditions. Practical guidelines for rebalancing frequency and position sizing are also provided.

Longlshort-term memory (LSTM) is a deep learning model that can capture long-term dependencies of wireless channel models and is highly adaptable to short-term changes in a wireless environment. This paper proposes a simple LSTM model to predict the channel transfer function (CTF) for a given transmitter-receiver location inside a bus for the 60 GHz millimetre wave band. The average error of the derived power delay profile (PDP) taps, obtained from the predicted CTFs, was less than 10% compared to the ground truth.

Power Delay Profile (PDP) plays a crucial role in wireless communications, providing information on multipath propagation and signal strength variations over time. Accurate detection of peaks within PDP is essential to identify dominant signal paths, which are critical for tasks such as channel estimation, localization, and interference management. Traditional approaches to PDP analysis often struggle with noise, low resolution, and the inherent complexity of wireless environments. In this paper, we evaluate the application of traditional and modern deep learning neural networks to reconstruction-based anomaly detection to detect multipath components within the PDP. To further refine detection and robustness, a framework is proposed that combines autoencoders and Density-Based Spatial Clustering of Applications with Noise (DBSCAN) clustering. To compare the performance of individual models, a relaxed F1 score strategy is defined. The experimental results show that the proposed framework with transformer-based autoencoder shows superior performance both in terms of reconstruction and anomaly detection.

We study the fundamental communication limits of information-theoretic secure aggregation in a hierarchical network consisting of a server, multiple relays, and multiple users per relay. Communication proceeds over two rounds and two hops, and the system is subject to arbitrary user and relay dropouts. Up to $T$ users may collude with either the server or any single relay. The server aims to recover the sum of the inputs of all users that survive the first round, while learning no additional information beyond the aggregate sum and the inputs of the colluding users. Each relay, however, must learn nothing about the users' inputs except for the information revealed by the inputs of the colluding users under the same collusion model. We introduce a four-dimensional rate tuple that captures the communication cost across rounds and hops. Under a delayed message availability model, we establish necessary and sufficient conditions for feasibility and fully characterize the optimal first-round communication rates. For the second round, we characterize the optimal user-to-relay rate and derive lower and upper bounds on the relay-to-server rate. While these bounds do not coincide in general, they are tight in certain regimes of interest. Our results reveal a sharp threshold phenomenon: secure aggregation is feasible if and only if the total number of surviving users across surviving relays exceeds the collusion threshold. Achievability is established via a vector linear coding scheme with carefully structured correlated randomness exhibiting MDS-like properties, ensuring correctness and information-theoretic security under all possible dropout patterns. Entropic converse bounds are also derived.

We study the Couette Taylor instabilities for an incompressible viscous fluid between two coaxial cylinders of nearly equal radii, allowing counter-rotation with the ratio of rotation rate $μ\in [-1,1]$. Working in a rotating frame and in a small-gap and small-viscosity regime, we derive the corresponding limiting Navier Stokes system and analyze the linear stability of the Couette flow. In particular, we numerically compute the critical Taylor number for general perturbations and identify a transition near $μ_c \approx -0.8$: for $μ> μ_c$ the most unstable mode is axisymmetric, whereas for $μ< μ_c$ the most unstable mode is non-axisymmetric. Near criticality, slowly varying traveling waves are governed by a time-independent Ginzburg Landau equation. The nonlinear coefficient changes sign near $\hatμ_c \approx -0.65$, yielding a supercritical regime for $μ> \hatμ_c$ and a subcritical regime for $μ_c < μ< \hatμ_c$. In the subcritical range, we classify small-amplitude steady states, including Taylor vortex flows, wavy vortices, a two-parameter family of quasi-periodic flows, and a localized traveling perturbation of the Couette flow.

Designing effective reduced-order models (ROMs) for parametrized transport-dominated problems remains challenging because of the well-known Kolmogorov barrier. Autoencoder-based nonlinear ROMs have been developed to improve the compression ability for such systems. However, despite their stronger compression ability, autoencoder-based ROMs constructed in the Eulerian frame may fail to accurately predict future solutions, due to the poor coherence between historical and future solutions in the Eulerian frame. In contrast, we show that representing transport-dominated dynamics in the Lagrangian frame can lead to a significantly faster decay of the Kolmogorov n-width and improve coherence between historical and future solutions. Building on these insights, we develop two non-intrusive ROMs leveraging Lagrangian data: a Lagrangian autoencoder-based ROM and a Lagrangian parametric dynamic mode decomposition. Numerical experiments demonstrate that these Lagrangian ROMs achieve more accurate and stable future predictions than their Eulerian counterparts.

We study an application of fair division theory to school redistricting. Procaccia, Robinson, and Tucker-Foltz (SODA 2024) recently proposed a mathematical model to generate redistricting plans that provide theoretically guaranteed fairness among demographic groups of students. They showed that an almost proportional allocation can be found by adding $O(g \log g)$ extra seats in total, where $g$ is the number of groups. In contrast, for three or more groups, adding $o(n)$ extra seats is not sufficient to obtain an almost envy-free allocation in general, where $n$ is the total number of students. In this paper, we focus on the case of two groups. We introduce a relevant relaxation of envy-freeness, termed 1-relaxed envy-freeness, which limits the capacity violation not in total but at each school to at most one. We show that there always exists a 1-relaxed envy-free allocation, which can be found in polynomial time.

Vocal training is difficult because the muscles that control pitch, resonance, and phonation are internal and invisible to learners. This paper investigates how Electromyography (EMG) and ultrasonic imaging (UI) can make these muscles observable for training purposes. We report three studies. First, we analyze the EMG and UI data from 16 singers (beginners, experienced & professionals), revealing differences among three vocal groups of the muscle control proficiency. Second, we use the collected data to create a system that visualizes an expert's muscle activity as reference. This system is tested in a user study with 12 novices, showing that EMG highlighted muscle activation nuances, while UI provided insights into vocal cord length and dynamics. Third, to compare our approach to traditional methods (audio analysis and coach instructions), we conducted a focus group study with 15 experienced singers. Our results suggest that EMG is promising for improving vocal skill development and enhancing feedback systems. We conclude the paper with a detailed comparison of the analyzed modalities (EMG, UI and traditional methods), resulting in recommendations to improve vocal muscle training systems.

We consider degenerate nonautonomous energies $$ \int_Ωf(x, Dv)\, dx, $$ for vector-valued functions $v \in W^{1,1}(Ω, \mathbb{R}^N)$, where the integrand $f(x,P)$ satisfies growth and weak uniform quasiconvexity assumption associated with the double phase function $H(x,t)=t^p + a(x)t^q$. We establish partial Hölder regularity for the gradients of minimizers under suitable, and possibly minimal, regularity assumptions on $H$ and $f$. Our approach relies on two approximation results: $\mathcal{A}$-harmonic approximation and a variational version of the $ϕ$-harmonic approximation.

We investigate how anisotropic strain reorganizes commensurate moiré superlattices and electronic structure in twisted bilayer graphene across a finite range of twist angles. Motivated by experiments demonstrating robust magic angle phenomenology under angular disorder and heterostrain, we construct exact commensurate moiré supercells generated by general anisotropic strain applied to the top graphene layer. At fixed effective moiré deformation, with particular focus on the $\pm 6.008^\circ$ twist angle, we uncover two distinct and symmetry inequivalent commensurate geometries: tilted two dimensional moiré superlattices and quasi one dimensional stripe like patterns. Anisotropic strain qualitatively reshapes the low energy band structure by reducing the number of Dirac points in the moiré Brillouin zone, leading to sharply different electronic and magnetic field responses in these regimes. Strikingly, tilted two dimensional structures near pristine angles preserve bandwidths and AA region localization comparable to the unstrained case, providing a natural explanation for the persistence of magic angle physics over a finite distortion window, whereas quasi one dimensional moiré patterns exhibit dimensional reduction and immediate Hofstadter butterfly splitting at infinitesimal magnetic fields. Our results identify anisotropic strain as a unifying geometric mechanism controlling commensurate moiré physics beyond the pristine twist angle limit.

In this paper, we introduce the notion of a $w$-Hilbert domain and investigate its basic properties. More precisely, we explore its relationship with Hilbert domains, strong Mori domains, and UMT domains by providing various examples using $D+M$ constructions. Furthermore, we establish necessary and sufficient conditions for the polynomial ring and the Anderson ring to be $w$-Hilbert domains, and compare the $w$-dimension of the polynomial ring with that of its base ring.

A partition is $t$-regular if none of its parts is divisible by $t$. Let $p(N,t)$ be the number of $(t+1)$-regular partitions of a positive integer $N$. In 1971, Hagis proved an asymptotic formula for $p(N,t)$ using the circle method, when $t$ fixed. In this article, we use the saddle point method and extend the result of Hagis in different ranges of $t$, obtaining explicit bounds. We also discuss an application of our result to estimate zeros in the character table of the symmetric group.

Hippocampal neurons exhibit precise phase locking to network oscillations, but the computational principle governing this temporal precision is still unclear. Neural information is conveyed jointly by firing rates and spike timing, but existing models treat these dimensions separately, limiting mechanistic interpretation of spike-field coupling and its reported association with spectral features such as the aperiodic slope. Here we show that hippocampal phase locking emerges from a fundamental dynamical mechanism referred to as forced phase integration that separates neural information into orthogonal magnitude (what) and phase (when) coordinates. To formalize this principle, the unified complex-valued neuron (UCN) has been developed, a biologically grounded generative framework in which spike timing arises from phase accumulation while spike magnitude encodes instantaneous signal strength. This framework reproduces biological spike-theta synchronization and enables mechanistic re-evaluation of slope-locking associations, demonstrating that previously reported effects arise from oscillatory contamination rather than causal modulation. These findings establish a unified phase-native principle of neural timing and coding.

In this paper, we give an elementary new method for determining the rational points on algebraic curves using torsion packets. We also provide examples of curves for which all rational points can be completely determined by our method.

In magnetic heterostructures with broken inversion symmetry, the Rashba effect gives rise to a gradient-free interaction between magnons and phonons, which we term interfacial spin-lattice coupling. Here, we investigate the dynamic consequences of this interfacial coupling in ferromagnetic heterostructures. By expressing the interaction in terms of circular variables for magnetization and lattice displacement, we reveal a direct interface-induced helicity-helicity coupling hat does not rely on lattice deformation gradients. Consequently, it leads to helicity-dependent phonon absorption, enabling in-plane acoustic waves to exert a spin torque on the magnetization, which becomes dominant in thin magnetic films. Our findings highlight the crucial, yet overlooked, role of inversion-asymmetric interfaces in angular-momentum conversion between spin and lattice, opening up possibilities for efficient phonon-driven magnetic devices that are enabled by interface engineering.

The study of dilepton emission from hot hadronic matter provides a unique probe of the medium properties in heavy-ion collisions. In strong magnetic fields expected in non-central collisions, the emission spectrum can develop anisotropic features. While the impact of a magnetic field on the dilepton emission rate has been explored, higher-order azimuthal anisotropy--characterized by flow coefficients beyond elliptic flow--remains an open question, particularly in the low invariant-mass region where medium effects are most pronounced. We investigate higher-order anisotropy in the dilepton emission rate from a magnetized hot hadronic medium. Our results reveal a continuous dilepton spectrum with strong Landau-cut contributions at low invariant masses due to the background magnetic field. The emission rate exhibits significant azimuthal-angle dependence in this region, characterized by flow coefficients $v_n$. Odd coefficients vanish by symmetry, while the elliptic flow $v_2$ is positive and oscillatory at low invariant masses--driven by Landau-level quantization of pions--but negligible at higher masses. Higher-order coefficients $v_4,v_6$ show similar trends, with notable structures at low masses and negligible values at higher masses. The transverse-momentum dependence increases with magnetic field strength, while temperature effects are minimal. Our findings demonstrate that the magnetic field induces significant higher-order azimuthal anisotropy in dilepton emission at low invariant masses, with $v_2,v_4,v_6$ exhibiting distinct oscillatory behavior linked to pion Landau quantization. This resolves the gap in understanding the anisotropic response of dilepton emission in magnetized media. The results highlight dileptons as sensitive probes of magnetic-field effects in heavy-ion collisions, offering new avenues to constrain field strengths and hadronic dynamics in the QGP.

In this paper, we study the following Patlak-Keller-Segel model with $p$-Laplacian diffusion \begin{align*} \left\{ \begin{aligned} &ρ_t=\nabla \cdot \left( \left| \nabla ρ\right|^{p-2}\nabla ρ\right) -χ\nabla \cdot \left( ρ\nabla c \right), &0=\varDelta c+ρ^m, \end{aligned}\right. \end{align*} and the exponent $m>0$ is chosen as $$ m = \frac{(p-2)N + p}{N}. $$ This relation ensures the scale invariance of the system and is conjectured to be the critical exponent that separates global boundedness from finite-time blow-up. We prove that, at the critical threshold $m=\frac{(p-2)N + p}{N}$, the system indeed admits finite-time blow-up solutions. More precisely, in the slow diffusion regime $p>2$, there exist backward self-similar blow-up solutions that are radially decreasing, compactly supported, and concentrate into a Dirac $δ$-measure at the blow-up time $T$; and their supports shrink toward the origin at the rate $(T-t)^{\frac1{mN}}$. For the fast diffusion case $1<p\le 2$, we show that there are no backward self-similar blow-up solutions with finite-mass. Additionally, we also explore forward self-similar solutions in both the slow diffusion and fast diffusion cases. These solutions also carry finite mass and exhibit a Dirac $δ$-singularity at the initial moment. Specifically, in the slow diffusion case, the support expands at the rate $t^{\frac1{mN}}$, whereas in the fast diffusion case, the solution becomes strictly positive for all positive times. Our work provides the first blow up analysis for the $p$-Laplacian Keller-Segel system when $p\ne 2$, and it confirms that the exponent $m$ given above is indeed the sharp threshold between global existence and finite time singularity formation.

The interaction of oxygen with cobalt and cobalt-based alloys has been a very important topic in the field of spintronics as it leads to enhanced orbital anisotropy and interfacial Dzyaloshinskii-Moriya interaction (DMI), which are crucial in the context of applications such as magnetic tunnel junctions (MTJs) based data storage and domain wall (DW) motion. To understand the complex and interesting relationship between oxygen and ferromagnetic (FM)/heavy metal (HM) interfaces, we studied controlled oxygen ion implantation in a cobalt layer located in a Pt/Co 1.2 /Pt (nm) multilayer with a specific structure. At high implantation fluence, the perpendicular anisotropy was lost, as verified by in-plane hysteresis measurements. Under low magnetic field conditions, the DW dynamics of Co/Pt multilayers were analyzed, highlighting key parameters such as DW velocity, roughness amplitude, and roughness exponent. After O+-ion implantation, the DW velocity increased by more than 50 times, rising from 5 um/s to 300 um/s compared with the as-deposited multilayer. The fundamental cause of this improvement is the structural and magnetic changes brought by the implantation, which successfully lower the energy barriers preventing DW movements. The results show how oxygen implantation can be used to precisely tailor the ferromagnetic interfaces, leading to promised improvements in the functionality of next-generation spintronic devices.

The interplay between magnetic order and electronic topology in van der Waals materials enables extreme responses to external stimuli. The nodal-line semiconductor Mn3Si2Te6 exemplifies this, exhibiting colossal angular magnetoresistance (CAMR) where resistivity changes by orders of magnitude upon rotating the magnetic field. While this phenomenon implies a profound coupling between spin orientation and charge transport, the microscopic magnetic potentials driving spin orientations remain elusive. Here, we combine thermodynamic torque magnetometry and electron spin resonance spectroscopy to reconstruct the magnetic anisotropy energy that controls magnetization rotation in Mn3Si2Te6. We show that low-temperature ground state is a coherent canted ferrimagnet stabilized by competing second- (K1) and fourth-order (K2) magnetic anisotropy. Crucially, torque requires a substantial symmetry-allowed sixth-order term (K3), which provides near-plane stiffness and sustains canting at high fields. Using the resulting anisotropy parameters, we compute the non-linear relation between field angle θ_H and magnetization angle θ_M and reparameterize CAMR in terms of θ_M, providing a concrete magnetic basis for how sharp angular transport features can emerge near the in-plane configuration.

Graph pattern counting serves as a cornerstone of network analysis with extensive real-world applications. Its integration with local differential privacy (LDP) has gained growing attention for protecting sensitive graph information in decentralized settings. However, existing LDP frameworks are largely ad hoc, offering solutions only for specific patterns such as triangles and stars. A general mechanism for counting arbitrary graph patterns, even for the subclass of acyclic patterns, has remained an open problem. To fill this gap, we present the first general solution for counting arbitrary acyclic patterns under LDP. We identify and tackle two fundamental challenges: generalizing pattern construction from distributed data and eliminating node duplication during the construction. To address the first challenge, we propose an LDP-tailored recursive subpattern counting framework that incrementally builds patterns across multiple communication rounds. For the second challenge, we apply a random marking technique that restricts each node to a unique position in the pattern during computation. Our mechanism achieves strong utility guarantees: for any acyclic graph pattern with $k$ edges, we achieve an additive error of $\tilde{O}(\sqrt{N}d(G)^k)$, where $N$ is the number of nodes and $d(G)$ is the maximum degree of the input graph $G$. Experiments on real-world graph datasets across multiple types of acyclic patterns demonstrate that our mechanisms achieve up to $46$-$2600\times$ improvement in utility and $300$-$650\times$ reduction in communication cost compared to the baseline methods.

Given a finite group $G$, the \emph{Prime Order Element (POE) Graph} $Γ(G)$ consists of the group elements as the vertices, and two vertices $x$ and $y$ are adjacent if and only if $o(xy)$ is prime. This paper presents a thorough structural and spectral analysis of the POE graphs associated with the finite Abelian groups of different types. The order of a finite Abelian group may be a prime or a product of primes, which influences the structure of POE graphs. The POE graph is connected when the order of the Abelian group is a square-free integer. The POE graphs of the other Abelian groups have multiple connected components. Some of these components are isomorphic to the POE graph of a lower-order group. We study various graph-theoretic properties of the components, including regularity and bipartiteness. Arranging the elements of the group in a number of particular orders, we observe the block structure in the adjacency matrix of POE graphs. It assists us in investigating the spectral properties of POE graphs. We explicitly derive the characteristic polynomials governing both integral and irrational eigenvalues, and compute the eigenvalues with multiplicity in terms of the structure of the graphs.

The undirected zero divisor graph of a commutative ring with unity \( R \), denoted by \( Γ(R) = (V(Γ(R)), E(Γ(R))) \). The vertex set \( V(Γ(R)) \) consists of all the non-zero zero-divisors of \( R \). The edge set \( E(Γ(R)) \) is defined by the set \( \{ e = a_1 a_2 \mid a_1 \cdot a_2 = 0 \text{ and } a_1, a_2 \in V(Γ(R)) \} \). The barycentric subdivision of $Γ$ is the process of subdividing each edge by inserting new vertex in the graph $Γ$. In this article, we have focused on the fault-tolerant metric dimension of the barycentric subdivision of zero divisor graph of the group of integers modulo \( n \), represented by \( fdim(BS(Γ(\mathbb{Z}_n )\), where \( n = pq \); \( p \) and \( q \) are distinct odd primes with \( q > p \). We also demonstrate that \( fdim(BS(Γ(\mathbb{Z}_n) \geq q - 1 \) for every \( n = pq \), where \( p \) and \( q \) are any distinct odd primes with \( q > p \).

Faceted search acts as a critical bridge for navigating massive ecommerce catalogs, yet traditional systems rely on static rule-based extraction or statistical ranking, struggling with emerging vocabulary, semantic gaps, and a disconnect between facet selection and underlying retrieval. In this paper, we introduce GenFacet, an industrial-grade, end-to-end generative framework deployed at JD.com. GenFacet reframes faceted search as two coupled generative tasks within a unified Large Language Model: Context-Aware Facet Generation, which dynamically synthesizes trend-responsive navigation options, and Intent-Driven Query Rewriting, which translates user interactions into precise search queries to close the retrieval loop. To bridge the gap between generative capabilities and search utility, we propose a novel multi-task training pipeline combining teacher-student distillation with GRPO. This aligns the model with complex user preferences by directly optimizing for downstream search satisfaction. Validated on China's largest selfoperated e-commerce platform via rigorous offline evaluations and online A/B tests, GenFacet demonstrated substantial improvements. Specifically, online results reveal a relative increase of 42.0% in facet Click-Through Rate (CTR) and 2.0% in User Conversion Rate (UCVR). These outcomes provide strong evidence of the benefits of generative methods for improving query understanding and user engagement in large-scale information retrieval systems.

In this paper, we concentrate on investigating the self-similar singular solutions of Keller-Segel model with signal consumption ($-uv^α$) and singular sensitivity. We perform a detailed exploration into the existence and decay rate of self-similar solutions, particularly, the permissibility of arbitrary mass for these solutions across all possible cases. Based on these findings, we can delve deeper into verifying that these self-similar solutions $(u, v)$ exhibit varying degrees of singularity depending on the value of $α$ and the spatial dimension. Our analysis reveals that the component $u$ (with arbitrary mass) of the solution consistently behaves analogous to heat kernel, that is, $u$ exhibiting a Dirac $δ$ initial singularity identical to that of the fundamental solution, and converges to $0$ in the sense of the $L^p$-norm ($p>1$) as time approaches infinity. However, the initial behavior of the other component $v$ varies significantly based on the value of $α$ and the spatial dimension, exhibiting regularity (not singular), less singularity, or strong singularity (more singular than fundamental solution). Moreover, both $u$ and $v$ undergo instantaneous smoothing, becoming smooth immediately after $t>0$. This phenomenon reveals the adaptive strategies of cells in high-density aggregation environments to prevent resource depletion, reflecting an optimization process of self-organizing behavior.