Clay-rich soils and sediments are key components of near-surface systems, influencing water retention, ion exchange, and structural stability. Their complex dielectric behavior under moist conditions arises from electrostatic interactions between charged mineral surfaces and exchangeable cations, forming diffuse double layers that govern transport and retention processes. This study investigates the broadband dielectric relaxation of four water-saturated clay minerals (kaolin, illite, and two sodium-activated bentonites) in the 1 MHz to 5 GHz frequency range using coaxial probe measurements. The dielectric spectra were parameterized using two phenomenological models - the Generalized Dielectric Relaxation Model (GDR) and the Combined Permittivity and Conductivity Model (CPCM) - alongside two theoretical mixture models: the Augmented Broadband Complex Dielectric Mixture Model (ABC-M) and the Complex Refractive Index Model (CRIM). These approaches were evaluated for their ability to link dielectric relaxation behavior to petrophysical parameters such as cation exchange capacity (CEC), volumetric water content (VWC), and porosity. The results show distinct spectral signatures correlating with clay mineralogy, particularly in the low-frequency range. Relaxation parameters, including relaxation strength and apparent DC conductivity, exhibit strong relationships with CEC, emphasizing the influence of clay-specific surface properties. Expansive clays like bentonites showed enhanced relaxation due to ion exchange dynamics, while deviations in a soda-activated bentonite highlighted the impact of chemical treatments on dielectric behavior. This study provides a framework for linking clay mineral physics with electromagnetic methods, with implications for soil characterization, hydrological modeling, geotechnical assessment, and environmental monitoring.

In order to numerically solve high-dimensional nonlinear PDEs and alleviate the curse of dimensionality, a stochastic particle method (SPM) has been proposed to capture the relevant feature of the solution through the adaptive evolution of particles [J. Comput. Phys. 527 (2025) 113818]. In this paper, we introduce an active birth-death dynamics of particles to improve the efficiency of SPM. The resulting method, dubbed SPM-birth-death, sample new particles according to the nonlinear term and execute the annihilation strategy when the number of particles exceeds a given threshold. A rigorous error estimation for SPM-birth-death is established, elucidating the first-order convergence in time and space, as well as half-order accuracy in the initial sample size with explicit variance estimates. We also extend the analysis framework to SPM and provide theoretical justification for the existing numerical convergence study. Our theoretical results reveal that the introduced active birth-death dynamics of particles results into less frequent resampling and SPM-birth-death is thus able to achieve higher efficiency than SPM. Validating benchmarks are provided. In particular, preliminary numerical experiments on the Allen-Cahn equation demonstrate that SPM-birth-death can achieve smaller errors at the same computational cost compared with the original SPM.

Rates for particle interaction processes and decays will be modified in a Lorentz-violating quantum field theory, because of changes to the particle kinematics$-$particularly through the modified dispersion relations affecting the outgoing particle phase space. We outline here these changes to the rates for three-particle decays. Considering a process with a constant scattering amplitude (not directly modified by the Lorentz violation), we calculate leading order corrections to the kinematics for a decay into three identical spinless particles whose propagation is affected by a $c_{μν}$-type symmetric tensor background. We examine the angular distribution of the daughter particles and describe the shape of the corresponding Dalitz plot outlining the kinematically allowed region, according to two toy models for the $c_{μν}$ textures. Precision measurements of the boundaries of this region could be used to constrain Lorentz violation coefficients for the particles involved in processes such as $η\rightarrow 3π^{0}$.

Spin splitting in emerging altermagnets is non-relativistic and momentum-dependent, yet energy-independent, and localized in momentum space, posing challenges for practical applications. Here, we propose an intercalation-driven paradigm for altermagnets to attain ameliorative electronic structures, multiferroic characteristics, and anomalous and spin transport functionalities. As a representative system, we investigate electrochemistry- and self-intercalated V2Se2O bilayers, building on the recently reported room-temperature K- and Rb-intercalated V2Se2O family [Nat. Phys. 2025, 21, 754; Nat. Phys. 2025, 21, 760], utilizing density functional theory, Wannier function analyses, Monte Carlo simulations, and non-equilibrium Green function methods. Intercalation induces room-temperature intralayer ferrimagnetic and interlayer ferromagnetic order (358 K for Li-intercalation and 773 K for V-intercalation), ferroelasticity (~1 % signal intensity), in-plane uniaxial magnetic anisotropy, and metallization, while also modifying the anomalous Hall effect. Notably, Li- and V-intercalated V2Se2O bilayers exhibit enhanced spin splitting and half-metallic behavior, respectively, yielding near-perfect spin filtering efficiency. Intercalation substantially enhances spin transport in V2Se2O-based devices, enabling giant magnetoresistance (877 %), ultra-high thermal tunneling magnetoresistance (~12000 %), and observable spin Seebeck and temperature negative differential resistance effects. This intercalation-driven paradigm expands altermagnetic functionalities through multifunctional integration, offering promising avenues for advanced, miniaturized, room-temperature exploitation of anomalous, electron, and spin transport properties.

We calculate the valence parton distribution function~(PDF) of pion, within the theoretical approach based on spectral function representation of valence quarks in the pion. In this approach we assume that the soft partonic structure of pion is defined by the valence $q\bar q$ cluster, consisting of current quarks, embedded in the residual field of the pion. The valence PDF is calculated using phenomenological Light-Front wave functions for the $q\bar q$ cluster and the residual field. Our result indicates that the peak position of the x weighted valence PDF~(xPDF) depends on parameters characterizing the virtuality of the cluster and the mass of the residual system. Magnitudes of these parameters are obtained by fitting to the height and the peak position of empirical pion xPDF evaluated at starting $Q_0$. They indicate that unlike the nucleon case, very little residual mass is needed to describe existing pion PDFs at intermediate x. They also indicate that the $q\bar q$-cluster is highly virtual and on average the interacting quark carries almost all of the momentum of the $q\bar q$ cluster. This picture is consistent with the dominance of the Feynman mechanism leaving little room for the hard component in the PDF, and practically describing $\sim (1-x)$ behavior observed recently at $x\to 1$ limit. Our non-trivial observation is that the height and the peak position of xPDF define the analytic behavior of valence PDF at $x\to 1$ limit.

We investigate weak-type $(1, 1)$ boundedness of sparse operators with respect to Lebesgue measure. Specifically, we find the Bellman function maximizing level sets of sparse operators (localized to an interval) and use this to find the exact weak-$(1,1)$ norm of these sparse operators.

Understanding how higher-order interactions shape the energy landscape of coupled oscillator networks is crucial for characterizing complex synchronization phenomena. Here, we investigate a generalized Kuramoto model with triadic interactions, combining deterministic basin analysis, noise-induced transitions, and quantum annealing methods. We uncover a dual effect of higher-order interactions: they simultaneously expand basins for non-twisted states while contracting those of twisted states, yet modify potential well depths for both. As triadic coupling strengthens, higher-winding-number states and non-twisted states gain stability relative to synchronized states. The system exhibits remarkable stability asymmetry, where states with small basins can possess deep potential wells, making them highly resistant to noise-induced transitions once formed. These findings extend quasipotential theory to high-dimensional networked systems and offer new insights for controlling synchronization in complex systems.

We compare the total capital efficiency of secure restaking and Proof-of-Stake (PoS) protocols. First, we consider the sufficient condition for the restaking graph to be secure. The condition implies that it is always possible to transform such a restaking graph into separate secure PoS protocols. Next, we derive two main results: upper and lower bounds on the required extra stakes to add to the validators of the secure restaking graph to be able to transform it into secure PoS protocols. In particular, we show that the restaking savings compared to PoS protocols can be very large and can asymptotically grow as a square root of the number of validators. We also study a complementary question of aggregating secure PoS protocols into a secure restaking graph and provide matching lower and upper bounds on the PoS savings.

We explore the reconstruction of bulk local states in three-dimensional flat spacetime (Flat$_3$) using states from two-dimensional Carrollian conformal field theories (CCFT$_2$), proposed as dual field theories in one lower dimension. For massive scalar-type bulk excitations, reconstruction is achieved through states in the induced representation. This method successfully reproduces the bulk massive scalar spectrum and the bulk-to-bulk propagator. Additionally, we identify a new flat limit from AdS$_3$ and dS$_3$ spacetimes, further validating our proposal for scalar reconstruction in Flat$_3$/CCFT$_2$.

Immersed boundary methods (IBMs) facilitate the simulation of flows around stationary, moving, and deforming bodies on Cartesian grids. However, extending these simulations to the large grid sizes required for realistic flow problems remains a significant computational challenge. In this work, we present the implementation and acceleration of ViCar3D, a sharp-interface immersed boundary solver, on graphical processing units (GPUs). We utilize OpenACC, CUDA Fortran and MPI to reprogram \emph{ViCar3D}, a sharp-interface immersed boundary solver, on multi-GPU architectures. Verification and scalability studies are performed for two benchmark cases: two-dimensional flow past a circular cylinder and direct numerical simulation (DNS) of flow past a finite rectangular wing. For the latter, we observe an approximately 20X speedup (node-to-node comparison) relative to the CPU-based implementation. The GPU-accelerated solver is capable of simulating complex 3D flows with up to 200 million mesh points on a single node equipped with four GPUs. Strong and weak scaling tests demonstrate maximum scaling efficiencies of 92\% and 93\%, respectively, on multi-GPU systems. We further test the code to simulate fluid flow past complex-shaped single-body and multi-body cases.

We address the algorithmic problem of determining the reversible Markov chain $\tilde X$ that is closest to a given Markov chain $X$, with an identical stationary distribution. More specifically, $\tilde X$ is the reversible Markov chain with the closest transition matrix, in the Frobenius norm, to the transition matrix of $X$. To compute the transition matrix of $\tilde X$, we propose a novel approach based on Riemannian optimization. Our method introduces a modified multinomial manifold endowed with a prescribed stationary vector, while also satisfying the detailed balance conditions, all within the framework of the Fisher metric. We evaluate the performance of the proposed approach in comparison with an existing quadratic programming method and demonstrate its effectiveness through a series of synthetic experiments, as well as in the construction of a reversible Markov chain from transition count data obtained via direct estimation from a stochastic differential equation.

We developed a new model for the production and propagation of spectrally resolved primary and secondary Cosmic Ray (CR) nuclei elements within the framework of the Cosmic Ray Energy Spectrum (CRESP) module of the PIERNIK MHD code. We extend the algorithm to several CR nuclei and demonstrate our code's capability to model primary and secondary CR species simultaneously. Primary C, N, and O are accelerated in supernova (SN) remnants. The spallation collisions of the primary nuclei against the thermal ISM protons lead to secondary Li, Be, and B products. All the CR species evolve according to the momentum-dependent Fokker-Planck equations that are dynamically coupled to the MHD system of equations governing the evolution of the ISM. We demonstrate the operation of this system in the gravity stratified box reproducing the Milky Way conditions in the Sun's local environment. We perform a parameter study by investigating the impact of the SN rate, the CR parallel diffusion coefficient $D_\parallel$, and the rigidity-dependent diffusion coefficient power index $δ$. A novel result of our investigation is that the secondary-to-primary flux ratio \BtoC increases with increasing diffusion coefficient, due to the weaker vertical magnetic field resulting from CR buoyancy effects. Moreover, a higher SN rate leads to lower values of \BtoC because of stronger winds and the shorter residence time of primary CR particles in dense disk regions.

Stellar-mass binary black holes (BBH) may form, evolve, and merge within the dense environments of active galactic nuclei (AGN) disks, thereby contributing to the BBH population detected by gravitational wave (GW) observatories. Mergers occurring in AGN disks may be gravitationally lensed by the supermassive black hole at the AGN centre. The probability of such lensing events has been approximately estimated by using the Einstein criterion in previous work. However, a more reasonable approach to calculating the lensing probability should be based on whether the detector can distinguish the lensed GW waveform from the unlensed one. In this work, we calculate the lensing probability of LIGO sources embedded in AGN disk by relating threshold mismatch to the signal-to-noise ratio of the observed events. For the sensitivity of LIGO-Virgo-KAGRA O3 observation runs, our results indicate that the lensing probability is several times higher than previous estimates. If AGNs are indeed the primary formation channel for BBHs, we could quantify the probability of detecting the lensed GW events in such a scenario. The non-detections, on the other hand, will place stricter constraints on the fraction of AGN disk BBHs and even the birthplaces of BBH mergers.

We investigate the properties of the mean Poynting vector in global thermal equilibrium, which can be non-zero in non-reciprocal electromagnetic systems. Using dyadic Green's functions and the fluctuation-dissipation theorem, we provide a general proof that the mean Poynting vector is divergence-free under equilibrium conditions. Relying on this proof, we explicitly demonstrate that for systems where a normal mode expansion of the Green's function is applicable, the divergence of the equilibrium mean Poynting vector vanishes. As concrete examples, we also examine the equilibrium mean Poynting vector near a planar non-reciprocal substrate and in configurations involving an arbitrary number of dipolar non-reciprocal objects in free space. Finally, we argue that the so-called persistent heat current, while present in equilibrium, cannot be detected through out-of-equilibrium heat transfer measurements.

There has been much recent interest in the chiral motion of achiral particles that sediment in a viscous fluid in a regime where inertial effects can be neglected. This occurs in a broad range of applications such as those involving biological objects like algae, ultra-thin graphene flakes, or colloidal suspensions. It is known that articles with two planes of symmetry can be categorised as ``settlers'', ``drifters'' or ``flutterers'', where the latter sediment along chiral trajectories despite their achiral shapes. Here we analyse the sedimentation of circular disks bent into a U-shape (``flutterers'') and show how their behaviour changes when we break one of their symmetries by pinching the disks along their axis. The ``fluttering'' behaviour is found to be robust to such shape changes, with the trajectories now evolving towards helical paths. However, the behaviour changes when the degree of pinching becomes too strong, at which point the particles become ``drifters'' which sediment steadily without rotation. We establish criteria for the transition between the two types of behaviour and confirm our predictions in experiments. Finally, we discuss the implications of our observations for the dispersion of dilute suspensions made of such particles.

We develop a tensor categorical duality in the sprit of the Tannaka-Krein duality for the C*-algebras admitting the Yetter-Drinfeld module structure over a compact quantum group. Under this duality, given a reduced compact quantum group G, the Yetter-Drinfeld G-C*-algebras correspond to the bimodule categories over the representation category Rep(G), satisfying a certain centrality condition.

With the increasing importance of data privacy, Local Differential Privacy (LDP) has recently become a strong measure of privacy for protecting each user's privacy from data analysts without relying on a trusted third party. In this paper, we consider the problem of high-utility differentially private release. Given a domain of items and a distance-defined utility function, our goal is to design a differentially private mechanism that releases an item with the global expected error as small as possible. The most common LDP mechanism for this task is the Generalized Randomized Response (GRR) mechanism that treats all candidate items equally except for the true item. In this paper, we introduce Bipartite Randomized Response mechanism (BRR), which adaptively divides all candidate items into two parts by utility rankings. In the local search phase, we confirm how many high-utility candidates to be assigned with high release probability, which gives the locally optimal bipartite classification of all candidates. For preserving LDP, the global search phase uniformly selects the smallest number of dynamic high-utility candidates obtained locally. In particular, we give explicit formulas on the uniform number of dynamic high-utility candidates. The global expected error of our BRR can theoretically deliver a decrease with an asymptotically exact ratio, and when the privacy budget is set to $3$ the expected error can be reduced by $66.4\%$. Extensive experiments demonstrate that BRR outperforms the state-of-the-art methods across the standard metrics and datasets.

In this paper we investigate the validity of Hopf's Lemma for a (possibly sign-changing) function $u \in H^s_0(Ω)$ satisfying \[ (-Δ)^s u(x) \geq c(x)u(x) \quad \text{in }Ω,\] where $Ω\subset \mathbb{R}^N$ is an open, bounded domain, $c \in L^\infty(Ω)$, and $(-Δ)^s u$ is the fractional Laplacian of $u$. We show that, under suitable assumptions, the validity of Hopf's Lemma for $u$ at a point $x_0 \in \partial Ω$ is essentially equivalent to the validity of Hopf's Lemma for the Caffarelli-Silvestre extension of $u$ at the point $(x_0,0) \in \mathbb{R}^N \times \mathbb{R}^+$. We also provide a slightly more precise characterization of a dichotomy result stated in a recent paper by Dipierro, Soave and Valdinoci.

Studying heavy-quark hadrons is crucial due to the nonperturbative nature of low-energy QCD, with Heavy-Quark Symmetry (HQS) serving as a key framework for understanding their spin and flavor symmetries. However, a key issue is that the theoretically expected $B^{(*)}\bar{K}$ molecular states have not yet been observed, although they are considered the bottom-quark counterparts of the observed $\bar{D}^{(*)}\bar{K}$ molecular states (corresponding to $D_{s0}(2317/2460)^{-}$), which challenges the universality of HQS. The main goal of this work is to search for the theoretically predicted $B\bar{K}$ and $B^{*}\bar{K}$ molecular states, namely $B_{s0}^{*}(5725)$ and $B_{s1}^{*}(5778)$, via the reactions $K^{-}p \to Λ_{b}^{0} B_{s0}^{*}$ and $K^{-}p \to Λ_{b}^{0} B_{s1}^{*}$. Within an effective Lagrangian framework, we compute the relevant cross sections, considering $t$-channel $B^{(*)}$ exchanges and $K^{-}p$ initial-state interactions (ISI). The results show that the production cross sections of $B_{s0}^{*}(5725)$ and $B_{s1}^{*}(5778)$ can reach the order of 0.01~nb, and we suggest that experiments searching for $B_{s0}^{*}(5725)$ are best performed at $P_{K^{-}} = 12.18~\mathrm{GeV}$, while higher energies are most favorable for producing $B_{s1}^{*}(5778)$. The ISI play a crucial role, as they not only significantly enhance the production cross sections of $B_{s0}^{*}(5725)$ and $B_{s1}^{*}(5778)$ (by roughly one order of magnitude) but also markedly affect the angular distributions of the produced particles. We also calculated the production cross sections of the conventional quark-antiquark states $B_{s0}^*(5700)$ and $B_{s1}^*(5720)$, which are found to be nearly the same as those of $B_{s0}^{*}(5725)$ and $B_{s1}^{*}(5778)$. Although their internal structures remain ambiguous, these results can inform future experimental searches at CERN and J-PARC.

We propose a characterization of a $p$-Laplace higher eigenvalue based on the inverse iteration method with balancing the Rayleigh quotients of the positive and negative parts of solutions to consecutive $p$-Poisson equations. The approach relies on the second eigenvalue's minimax properties, but the actual limiting eigenvalue depends on the choice of initial function. The well-posedness and convergence of the iterative scheme are proved. Moreover, we provide the corresponding numerical computations. As auxiliary results, which also have an independent interest, we provide several properties of certain $p$-Poisson problems.

Electron-electron and electron-phonon interactions are responsible for the formation of spin, charge, and superconducting correlations in layered quantum materials. A paradigmatic model for such materials that captures both kinds of interactions is the two-dimensional Hubbard-Holstein model with a dispersionless Einstein phonon. In this work, we provide a detailed analysis of the magnetic, density, and superconducting fluctuations at and away from half-filling. To that end, we employ the functional renormalization group using the recently introduced extension of the single-boson exchange formulation. More precisely, we go beyond previous approaches to the model by resolving the full frequency dependence of the two-particle vertex and taking into account the feedback from the electronic self-energy. We perform broad parameter scans in the space of Hubbard repulsion, electron-phonon coupling strength, and phonon frequency to explore the leading magnetic, density, and superconducting susceptibilities from the adiabatic to the anti-adiabatic regime. Our numerical data reveal that self-energy effects lead to an enhancement of the $d$-wave superconducting susceptibility towards larger phonon frequencies, in contrast to earlier isotope-effect studies. At small phonon frequencies, large density contributions to the $s$-wave superconducting susceptibility change sign and eventually lead to a reduction of $s$-wave superconductivity with increasing electron-phonon coupling, signaling the breakdown of Migdal-Eliashberg theory. We analyze our findings systematically, employing detailed diagnostics of the intertwined fluctuations and pinning down the various positive and negative isotope effects of the physical susceptibilities.

We uncover a class of inter-species topological phases in a one-dimensional lattice, loaded with two species of non-identical particles interacting via a dynamical gauge field (DGF). Two types of topological states are found to emerge from different inter-species topology activated by the DGF. Specifically, edge confined states with co-localization of both species arise from an extrinsic inter-species topology, which can be decomposed into the single-particle topology for each species. On the other hand, bulk bound states with extended distribution emerge from an intrinsic inter-species topology that cannot be understood from single-particle ones. The two classes of inter-species topology are found to be independent of each other, characterized by different sets of inter-species topological invariants. Thus, their topological states can coexist in certain parameter regimes and compete with each other, leading to distinguished dynamical signatures. We further propose a feasible cold-atom realization of our model to demonstrate experimental accessibility of inter-species topological phases. Our work establishes inter-species topology as a new organizing principle of topological matter, revealing how correlations between distinct particle species can generate topological phenomena beyond single-particle paradigms.

Graph alignment - identifying node correspondences between two graphs - is a fundamental problem with applications in network analysis, biology, and privacy research. While substantial progress has been made in aligning correlated Erdős-Rényi graphs under symmetric settings, real-world networks often exhibit asymmetry in both node numbers and edge densities. In this work, we introduce a novel framework for asymmetric correlated Erdős-Rényi graphs, generalizing existing models to account for these asymmetries. We conduct a rigorous theoretical analysis of graph alignment in the sparse regime, where local neighborhoods exhibit tree-like structures. Our approach leverages tree correlation testing as the central tool in our polynomial-time algorithm, MPAlign, which achieves one-sided partial alignment under certain conditions. A key contribution of our work is characterizing these conditions under which asymmetric tree correlation testing is feasible: If two correlated graphs $G$ and $G'$ have average degrees $λs$ and $λs'$ respectively, where $λ$ is their common density and $s,s'$ are marginal correlation parameters, their tree neighborhoods can be aligned if $ss' > α$, where $α$ denotes Otter's constant and $λ$ is supposed large enough. The feasibility of this tree comparison problem undergoes a sharp phase transition since $ss' \leq α$ implies its impossibility. These new results on tree correlation testing allow us to solve a class of random subgraph isomorphism problems, resolving an open problem in the field.

Teleportation, a novel scheme, initially posited by Bennett \textit{et.al}, has been studied here in the context of sending a single qutrit from Alice to Bob using two qutrit entangled channels as resources. In this paper we have considered two special two qutrit entangled states, which belong to $SU(3)$ group, as useful resources for teleportation. For the successful teleportation, these entangled states have been chosen as quantum channels shared between Alice and Bob. Another entangled basis of two qutrit states have been used as auxiliary states, which would help Alice to manipulate with her channel so that the single qutrit she holds can be successfully teleported to Bob. Bob's choices of measurement operators influence the retrieval of Alice's single qutrit.

We investigate medium-induced modifications to jet substructure observables that characterize hard components in central Pb-Pb collisions at $\sqrt{s_{NN}}=5.02$~TeV. Using a multistage Monte Carlo simulation of in-medium jet shower evolution, we explore flavor-dependent medium effects through simulations of inclusive and $γ$-tagged jets. The results show that quark jets undergo a nonmonotonic modification compared with gluon jets in observables such as the Pb-Pb to $p$-$p$ ratio of the soft drop prong angle $r_g$, the relative prong transverse momentum $k_{T,g}$, and the groomed mass $m_g$ distributions. Due to this nonmonotonic modification, $γ$-tagged jets, enriched in quark jets, provide surprisingly clear signals of medium-induced structural modifications, distinct from effects dominated by selection bias. Further systematic studies demonstrate that these effects are dominated by recoil medium response. This work highlights the potential of hard substructures in $γ$-tagged jets as powerful tools for probing the jet-medium interactions in high-energy heavy-ion collisions, in particular by enabling detailed investigations of jet-medium parton scatterings via their associated medium response. All simulations for $γ$-tagged jet analyses carried out in this paper used triggered events containing at least one hard photon, which highlights the utility of these observables for future Bayesian analysis.

This paper investigates the optimal control of an epidemic governed by a SEIR model with an operational delay in vaccination. We address the mathematical challenge of imposing hard healthcare capacity constraints (e.g., ICU limits) over an infinite time horizon. To rigorously bridge the gap between theoretical constraints and numerical tractability, we employ a variational framework based on Moreau--Yosida regularization and establish the connection between finite- and infinite-horizon solutions via $Γ$-convergence. The necessary conditions for optimality are derived using the Pontryagin Maximum Principle, allowing for the characterization of boundary-maintenance arcs where the optimal strategy maintains the infection level precisely at the capacity boundary. Numerical simulations illustrate these theoretical findings, quantifying the shadow prices of infection and costs associated with intervention delays.

Let $G$ be a locally elliptic group, $(Φ,Ψ)$ a complementary pair of Young functions, and $ω: G \rightarrow [1,\infty)$ a weight function on $G$ such that the weighted Orlicz space $L^Φ(G,ω)$ is a Banach $*$-algebra when equipped with the convolution product and involution $f^*(x):=\overline{f(x^{-1})}$ ($f \in L^Φ(G,ω)$). Such a weight always exists on $G$ and we call it an $L^Φ$-weight. We assume that $1/ω\in L^Ψ(G)$ so that $L^Φ(G,ω) \subseteq L^1(G)$. This paper studies the spectral theory and primitive ideal structure of $L^Φ(G,ω)$. In particular, we focus on studying the Hermitian, Wiener and $*$-regularity properties on this algebra, along with some related questions on spectral synthesis. It is shown that $L^Φ(G,ω)$ is always quasi-Hermitian, weakly-Wiener and $*$-regular. Thus, if $L^Φ(G,ω)$ is Hermitian, then it is also Wiener. Although, in general, $L^Φ(G,ω)$ is not always Hermitian, it is known that Hermitianness of $L^1(G)$ implies Hermitianness of $L^Φ(G,ω)$ if $ω$ is sub-additive. We give numerous examples of locally elliptic groups $G$ for which $L^1(G)$ is Hermitian and sub-additive $L^Φ$-weights on these groups. In the weighted $L^1$ case, even stronger Hermitianness results are formulated.

Antiferromagnets on the Shastry-Sutherland lattice have attracted lots of research interest due to the possible existence of deconfined criticality. In the present work, we study the $J_1$-$J_2$-$J_r$ model using Variational Monte Carlo (VMC) method, where $J_1$, $J_2$, $J_r$ stand for the nearest-neighbor, next nearest neighbor and ring exchange interactions respectively. An empty plaquette (EP) phase with spontaneous mirror symmetry breaking is reproduced. However, the EP phase in the VMC approach is $Z_2$ deconfined and have Majorana-type gapless spinon excitations, which is qualitatively different from the EP phase in literature. The central observation of the present study is the gapless $Z_2$ Quantum spin liquid phase resulting from the competition between the EP phase, the full plaquette (FP) phase and the antiferromagnetic Néel phase. While the phase transition from the $Z_2$ QSL phase to the EP phase is likely of Landau-Ginzburg type, the continuous transitions to the confined FP and Néel phases are exotic and need to be further explored.

Distributed networks are prone to errors so verifying their output is critical. Hence, we develop LOCAL certification protocols for graph properties in which nodes are given certificates that allow them to check whether their network as a whole satisfies some fixed property while only communicating with their local network. Most known LOCAL certification protocols are specifically tailored to the problem they work on and cannot be translated more generally. Thus we target general protocols that can certify any property expressible within a certain logical framework. We consider Monadic Second Order Logic (MSO$_2$), a powerful framework that can express properties such as non-$k$-colorability, Hamiltonicity, and $H$-minor-freeness. Unfortunately, in general, there are MSO$_2$-expressible properties that cannot be certified without huge certificates. For instance, non-3-colorability requires certificates of size $Ω(n^2/\log n)$ on general $n$-vertex graphs (Göös, Suomela 2016). Hence, we impose additional structural restrictions on the graph. We provide a LOCAL certification protocol for certifying any MSO$_2$-expressible property on graphs of bounded treewidth and, consequently, a LOCAL certification protocol for certifying bounded treewidth. That is for each integer $k$ and each MSO$_2$-expressible property $Π$ we give a LOCAL Certification protocol to certify that a graph satisfies $Π$ and has treewidth at most $k$ using certificates of size $\mathcal{O}(\log n)$ (which is asymptotically optimal). Our LOCAL certification protocol requires only one round of distributed communication, hence it is also proof-labeling scheme. Our result improves upon work by Fraigniaud, Montealegre, Rapaport, and Todinca (Algorithmica 2024), Bousquet, Feuilloley, Pierron (PODC 2022), and the very recent work of Baterisna and Chang.

The NP-hardness of the closest vector problem (CVP) is an important basis for quantum-secure cryptography, in much the same way that integer factorisation's conjectured hardness is at the foundation of cryptosystems like RSA. Recent work with heuristic quantum algorithms (arXiv:2212.12372) indicates the possibility to find close approximations to (constrained) CVP instances that could be incorporated within fast sieving approaches for factorisation. This work explores both the practicality and scalability of the proposed heuristic approach to explore the potential for a quantum advantage for approximate CVP, without regard for the subsequent factoring claims. We also extend the proposal to include an antecedent "pre-training" scheme to find and fix a set of parameters that generalise well to increasingly large lattices, which both optimises the scalability of the algorithm, and permits direct numerical analyses. Our results further indicate a noteworthy quantum speed-up for lattice problems obeying a certain `prime' structure, approaching fifth order advantage for QAOA of fixed depth p=10 compared to classical brute-force, motivating renewed discussions about the necessary lattice dimensions for quantum-secure cryptosystems in the near-term.