We show that the affine cactus group is a CAT$(0)$ group for all degrees. Furthermore, we show that the affine cactus group $AJ_3$ of degree three is a hyperbolic group.

We analyse the rôle of approximate axionic shift symmetries in gravity-decoupling limits arising in type II Calabi-Yau compactifications. Associated with each shift symmetry there is an axionic string whose tension constrains the gradient of the Kähler potential, as expected in regimes where gravity becomes weakly coupled. The gradients of these tensions define vector fields on moduli space, analogous to those associated with BPS particle masses. Together, they characterise the evolution of different effective field theory sectors along asymptotic limits, encoding both their coupling to gravity and their kinetic mixing. By deriving upper bounds on the inner products of these vector fields, we show that they split into mutually orthogonal subsets, one of which decouples from gravity. Finally, we relate the Laplacian of certain axionic string tensions to a divergent moduli space curvature.

Building on the $f$-information model of Bloedel et al. (2025), this paper introduces a one-parameter family of information acquisition models and characterizes optimal information acquisition. This family extends the mutual information model (Matějka and McKay, 2015) while preserving its analytical tractability. The information cost is derived from the $α$-divergence, which nests the KL-divergence ($α=-1$), the reverse KL-divergence ($α=1$), and the squared Hellinger distance ($α=0$), and is represented in closed form via the $α$-integration of Amari (2007). The optimal choice probabilities belong to the $q$-exponential family, which appears in nonextensive statistical mechanics (Tsallis, 1988) and in the $q$-logit model of traffic route choice (Nakayama, 2013). This family reduces to the modified logit in the mutual information case (Matějka and McKay, 2015). We further show that the relationship between payoffs and the set of actions chosen with positive probability in each state changes qualitatively across ranges of $α$.

Studies of Fermi-Large Area Telescope (LAT) data coincident with dwarf spheroidal satellite galaxies (dSphs) of the Milky Way (MW) have put the most stringent constraints on models of annihilating dark matter (DM) with candidate masses in the GeV-TeV range. Recent results found the presence of small, local significance excesses from these targets, at the 2-3 sigma level. However, these excesses disagree on the predicted properties of the DM candidate, and their significance vanishes when considering the correction factors for the number of trials. In this work, we apply key improvements to the analysis of the dSphs. We use stricter cuts on the data, implement a method to adaptively model the background, and assume an updated framework for DM annihilation. We find that our improved background modeling leads to a better agreement between the model and the data. This produces an increase in the local and global significance of the dSphs excess compared to previous studies. Finally, we find that the DM properties obtained in this work are less dependent on the sample of dSphs being considered compared to previous studies, while remaining in agreement with the predictions from the Galactic center excess observed by the Fermi-LAT and the antiproton excess observed by the Alpha Magnetic Spectrometer (AMS-02). Considering our improvements, a future significant increase in the number of dwarfs may lead to a definitive confirmation or exclusion of the DM interpretation of the Galactic center excess.

Orbital inclination is crucial in determining the binary mass. The astrometric excess noise contains the orbital motion information, which can be used to constrain the inclination. We aim to constrain the orbital inclination of a binary system by combining radial velocity measurements with the astrometric excess noise from the Gaia DR3 solution. The astrometric excess noise is directly related to the orbital parameters. For a binary system with a radial velocity solution, it can be treated as a function of the orbital inclination. Using the Gaia nominal scanning law and the estimated centroid uncertainties, we simulate Gaia astrometric epoch observations to reproduce the expected excess noise. By sampling different inclinations and comparing the resulting simulated excess noise with the value reported in Gaia DR3, we can constrain the inclination to a specific interval. We have developed a method to constrain the orbital inclination within a specific range, enabling a more accurate determination of the binary mass, particularly for spectroscopic binaries. Internal and external validations demonstrate the robustness of the method, although certain limitations remain. It is most reliable for systems exhibiting a strong astrometric signal of binary motion, while caution is required when applying it to binaries with weak astrometric wobbles or poorly sampled orbits.

This work studies a nonlocal extension of the Klausmeier vegetation model in $\mathbb{R}^N$ $(N \ge 1)$ that incorporates both local and nonlocal diffusion. The biomass dynamics are driven by a nonlocal convolution operator, representing anomalous and faster dispersal than the standard Laplacian acting on the water component. Using semigroup theory combined with a duality argument, we establish global well-posedness and uniform boundedness of classical solutions. Numerical simulations based on the Finite Difference Method with Forward Euler integration illustrate the qualitative effects of nonlocal diffusion and kernel size on vegetation patterns. The results demonstrate that nonlocal interactions significantly influence the spatial organization of vegetation, producing richer and more coherent structures than those arising in the classical local model.

We report an example of a many-body system, derived from the double kicked top (DKT), with non-chaotic yet mean-ergodic dynamics that displays \textit{strong} eigenstate thermalization hypothesis (ETH) in the quantum regime. The analysis addresses a key open question: whether \textit{strong} ETH is a quantum analog of ergodicity (or mean-ergodicity). Despite non-chaotic dynamics, the fluctuations of the diagonal matrix elements of an observable scale as $D^{-1/2}$, where $D$ denotes the Hilbert space dimension. Furthermore, the off-diagonal matrix elements show parameter-independent distribution, together with a smooth function $f_O(\bar{E}, ω)$ that becomes nearly uniform in the large-$k_θ$ domain. Our findings show that even mean-ergodic and non-chaotic systems can exhibit \textit{strong} ETH.

Effective community governance hinges on understanding what specific residents think and need. Recent work has used large language models (LLMs) to simulate human respondents, offering a scalable, reproducible way to study human attitudes and behaviors at low cost. However, these studies typically prompt the model with just a few demographic variables (age, gender, income), simulating only general role types. This is insufficient for community governance, where decisions depend on the views of specific residents. We bridge this gap with an integrated research framework covering dataset, benchmark, algorithm, and system. The dataset comprises approximately 1.2 million characters of first-person narrative collected through two-hour semi-structured interviews with each of 92 residents in an urban community, organized around nine community-governance domains. The benchmark probes 18 mainstream LLMs across four prompting strategies and shows that adding rich life-history profiles meaningfully raises fidelity above the no-profile baseline, but this gain comes with more input tokens per call from the longer prompts they require. The algorithm, curriculum-LoRA, is a parameter-efficient personalization framework that, by closing this fidelity-cost gap, matches the strongest baseline's fidelity at roughly 10x lower per-call cost and Pareto-dominates every configuration tested. The system integrates curriculum-LoRA into a closed-loop policy-evaluation pipeline. Together, these results bring individual-level LLM-based resident simulation within reach of resource-constrained local administrations, enabling community-governance decisions to be systematically pre-evaluated in silico before real-world deployment.

Decision-focused learning (DFL) is attractive for portfolio optimization because it trains predictors according to downstream decision quality rather than prediction accuracy alone. However, SPO(Smart, Predict then Optimize surrogate)-based DFL may produce inflated return signals and unstable portfolio reallocations. This study provides a KKT-based interpretation showing that portfolio decisions can be viewed as ranking over risk- and transaction-cost-adjusted marginal scores. Empirically, we examine prediction inflation and excessive turnover in SPO-trained portfolios, and evaluate clipping, min-max rescaling, and partial portfolio adjustment as practical stabilization mechanisms. The results suggest that realistic output constraints and portfolio-level turnover control improve the implementability of SPO-based portfolio strategies.

Predicting diffusion in chemically complex alloys remains challenging due to the strong dependence of migration barriers on local atomic environments. Migration barriers computed using density functional theory and nudged elastic band calculations are represented via a tensorial cluster expansion including transition states and deployed in on-lattice kinetic Monte Carlo simulations. Applied to the Ta-W system, the framework captures nontrivial composition-dependent diffusion behavior arising from a crossover between solute trapping and percolated low-barrier transport pathways, yielding a maximum in the apparent activation energy near intermediate compositions. This approach establishes a general and scalable route for integrating first-principles transition-state energetics into mesoscale kinetic simulations, enabling predictive multiscale modeling of diffusion in chemically complex materials and providing a pathway for uncovering emergent transport phenomena.

We study missing-link inference in partially observed networks by systematically comparing dyadic link prediction (LP) with hyperlink prediction (HP) and an estimation-based ERGM comparator. LP serves as the primary baseline, using classical heuristics computed on the observed graph. HP extends this framework by scoring candidate higher-order structures (cliques) via lifted dyadic scores and via the CHEbyshev Spectral HyperlInk pREdictor (CHESHIRE). All methods are evaluated under a common masking protocol that removes dyadic evidence induced by held-out hyperlinks to ensure comparability. Across public email and covert-network datasets, LP remains strong for dyadic recovery, while HP -- particularly CHESHIRE -- provides gains when the inferential target is higher-order group structure. ERGMs offer an interpretable dependence-based complement through conditional tie probabilities. The contribution is a comparative, reproducible evaluation clarifying when LP, HP, and ERGM estimation are most appropriate under network missingness.

We study a fully connected Hopfield-type associative memory network with online activity-dependent synaptic plasticity, where neural states and synaptic couplings coevolve during retrieval. Using the generating-functional formalism, we derive a dynamical mean-field theory (DMFT) in the large-system limit with extensively many stored random patterns, and show that the many-body dynamics reduces to an effective single-site stochastic process with colored Gaussian crosstalk noise and delayed feedback terms. Numerical solutions of the DMFT equations agree well with direct simulations. We find that moderate plasticity enlarges the basin of attraction and increases the maximum retrievable memory load by generating a positive delayed feedback that stabilizes retrieval against crosstalk noise. However, excessively strong plasticity causes the network to imprint the imperfect initial cue itself, leading to spurious attractors and degraded retrieval performance. Consequently, an optimal plasticity strength emerges from the trade-off between memory stabilization and premature cue imprinting. These results extend the DMFT description of associative memory to networks with coevolving neural and synaptic dynamics.

Block codes are considered for improving the reliability of messages stored in a computer memory with both stuck-at defects and random errors. It is assumed that the side information about the state of the defects is available to the encoder, but not to the decoder. A novel recursive construction of a set of masks is developed such that it can satisfy any $s$ stuck-at errors in a $2^m$ binary sequence, when $s \leq m$. We prove that the masks generated in this way are codewords in a Reed-Muller $RM(s-1, m)$ code. The constructed set contains no more than $2^s m^{s-1}$ masks. We provide the lower and the upper bound on the size of the stuck-at redundancy, a fixed subset of mask bits that uniquely represents each mask in the set. The stuck-at code constructed in this way is a non-linear code. It is also a subcode of an $RM(r,m)$ code, with $ r \geq s-1$, that can be used for additional random error correction. The encoding requires no mask search and is straightforward based on the description of the recursive construction. The decoding is done in a single attempt and requires almost no additional complexity or latency.

In this article, we design an original solver based on Quantized Tensor Trains (QTT) for linear elliptic equations with heterogeneous coefficient field, that allows for extremely fine meshes. It can achieve full-field simulations in dimensions $d=2$ and $d=3$ with a number of Degrees of Freedom (DoFs) up to $20$ orders of magnitude beyond the classical solvers, recovering accurately the solution as well as its gradient in the $\LL^2$ norm. For treating such an enormous amount of data, the solver crucially relies on the exponential compression properties of QTTs. This significantly improves upon the existing literature. The main ingredient of the proposed solver consists in the introduction of a penalization term involving the Helmholtz--Leray projector in the equation governing the gradient unknown. For practical reasons related to the expression of the Helmholtz--Leray projector, the penalized equation is solved in Fourier space. The primal solution is then obtained from the gradient via the Green operator. A core property of the solver is that it is unconditionally stable with respect to the mesh size. Based on numerical evidence supported by mathematical analysis, we show that reliable gradients and solutions can be obtained, and guaranteed by the proposed a posteriori error estimator. As an illustration, we successfully solve an elliptic equation in a microstructured material with up to $10^{37}$ virtual degrees of freedom in dimension $d=3$.

This paper takes an ecological approach toward large-scale models of hybrid human-AI intelligence. Emerging models of human-AI interaction predominantly advance the complementarity thesis variously dubbed human-AI collaboration and human-AI hybrid intelligence. However, this constitutes an over-simplification of the modalities of human-AI interaction and possibility-space for both individual and collective action that human-AI interaction potentiates. To fill these gaps, this paper develops a decision and game-theoretic approach to the human-AI delegation-verification dilemma. First, we map out canonical decision-theoretic strategies that account for adaptive user trajectories, modeling how agents transition between strategies based on interaction feedback to reach stable equilibria. Second, we scale individually stable strategies to collective equilibria using three extrapolation principles: (a) non-communicative aggregation (b) local social signaling and (c) institutional norms setting. The analysis identifies the emergence of sociotechnical lock-in, a macro-behavioral state where individually adaptive delegation, in the absence of communicative and institutional safeguards, aggregates into a systemic collective action problem modeled as a prisoner's dilemma that degrades shared epistemic standards. We argue that adoption under higher communicative standards and institutional norms can mitigate suboptimal collective equilibria by imposing social commitments on individual users.

Valley polarized twisted bilayer dice lattice hosts topologically nontrivial flat bands far from charge neutrality due to broken time reversal symmetry, whereas the ones in the vicinity of it remain topologically trivial. However, when both valleys are taken into consideration, the time reversal symmetry is preserved, which poses a serious hindrance to enumerate the valley specific topological phases that rely on the detection of the Berry curvature. In this work, we demonstrate that such a twisted structure with an applied uniaxial strain exhibits a nonlinear Hall effect far from charge neutrality. We ascertain that the nonlinear anomalous Hall signals can serve as a probe for topological phase transitions associated with a specific energy state that is constrained to reside at the lower edge of the middle subband and controlled via a staggered mass. Specifically, we show that the nonlinear anomalous Hall response undergoes a sign reversal across the topological phase boundaries. By tuning the carrier density, we compute the nonlinear Hall response obtained from the Berry curvature dipole, both in the chiral limit, and also when the chiral symmetry is broken. It is further seen that the nonlinear Hall effect is significantly enhanced in the broken chiral symmetry regime.

Neural operators are increasingly used as accelerators inside classical numerical methods, but it is rarely clear which architectural ingredients matter for which application. We answer this question for one important use case: the coarse-space correction inside a two-level preconditioner for discretised linear partial differential equations. We systematically vary four DeepONet-like architectures along two design axes: input discretisation (sampling versus integration against a basis) and source-term linearity. In doing this, we show that the favourable corner of this 2$\times$2 design is occupied by a single architecture, the Neural Green's Operator (NGO), and that moving away from it produces predictable failure modes: structurally non-symmetric preconditioned spectra, breakdown of preconditioned conjugate gradients on self-adjoint problems, and stagnation on non-self-adjoint ones. Used as a coarse-space correction, the NGO matches the iteration count of an exact coarse solve on diffusion and advection-diffusion problems. The principle generalises: integrating inputs against the basis used for the output is what allows a neural operator to serve as a Galerkin-type coarse-space correction.

In recent years, Deep-Learning Earth System Models (DL-ESMs) have emerged as promising, computationally efficient complements to traditional Earth system models. Here, we present an evaluation framework for testing DL-ESMs from an Earth system model-development perspective using standardized diagnostics from the PCMDI Metrics Package (PMP). This framework allows DL-ESMs, including Ai2's ACE2 and Google's NeuralGCM, to be assessed with metrics that quantify their ability to reproduce climatology, major modes of variability, monsoon behavior, and precipitation variability relative to observational reference datasets and CMIP-class benchmarks. By evaluating DL-ESMs with tools commonly used for traditional models, we extend their assessment beyond short-range forecast skill and toward longer Earth System-relevant applications. The results identify encouraging strengths in several large-scale fields and modes of variability, while also highlighting persistent challenges in precipitation, tropical variability, and long-run stability for some model versions. This evaluation is a critical step toward building trust in DL-ESMs, guiding future model development, and clarifying their fit-for-purpose for Earth system science applications.

We study a distributed consensus problem on a complete communication network of $n$ vertices, each holding one of two opinions. The vertices communicate in rounds, possibly in the presence of adversarial noise, and exchange information until they all agree on a single opinion. We consider a general class of protocols, where the vertices randomly sample neighbors and update their own opinion according to an update function $f$ depending on the sampled opinions. A prominent example is the $k$-maj protocol, where every vertex adopts the majority opinion of $k$ randomly sampled neighbors, breaking ties uniformly. We consider the runtime $R_n$ that is the number of rounds until all vertices agree on the same opinion, which we call the dominating opinion $D_n$. In our main result we describe the limiting distributions of these two key quantities for a large class of update functions $f$, for arbitrary initial configurations and under the presence of an adversary who may alter the opinions of up to $o(\sqrt{n})$ vertices in each round. We show that there are $f$-specific constants $γ, m > 0$ such that $R_n$ centers around $μ_n = \frac{1}{2}\log_γn + \log_m\ln n$, and we describe the asymptotic distribution of $R_n - μ_n$. In particular, we show that it does not converge, and that it becomes asymptotically periodic both in the $\log n$ as well as the $\log\log n$ scale. Applied to $k$-maj, our results show, among other things, that $γ_{k\text{-maj}} = \binom{k-1}{\lfloor k/2 \rfloor}2^{1-k}k \sim ({2k}/π)^{1/2}$.

Energy demands from data centers have surged and stressed the grid in recent years. Electric grids require balancing supply and demand every second, motivating demand response (reduction) from large loads, including data centers. This can be achieved by rescheduling jobs on a physical machine. Its real-time implementation is uncertain due to fluctuating resource utilization, and rescheduling incurs quality-of-service (QoS) losses that providers are unwilling to disclose. We propose a restless multi-armed bandit (RMAB) framework, in which the grid operator requests load reductions without access to detailed job-rescheduling procedures. Using open-source virtual machine (VM) datasets, we model job arrivals and rescheduling at each data center as a restless arm in a Markov decision process (MDP) and derive Whittle-index-based policies using the learned transition function via Thompson sampling. To overcome the weakness of an increasingly long learning process due to an enlarged state space, we use a mixed strategy that includes a global upper confidence bound (UCB) and encodes trust indices to enhance robustness and accelerate learning. Results show that the proposed mixed-strategy algorithm remains robust across varying state-space sizes and consistently outperforms the pure Thompson-Whittle (TW) algorithm, especially when contextual information is noisy. It also demonstrates superior performance compared to the state-of-the-art EXP4 framework. We provided open-source code to ensure reproducibility.

We compute the logarithmic one-loop matter contribution to the thermodynamics and homogeneous twistless torsional Newton--Cartan scale response of a four-dimensional Lifshitz black brane. The background is the neutral planar member of the analytic Einstein--Maxwell--dilaton Lifshitz black-brane family, while the quantum fields are treated as probes: a real scalar with arbitrary mass and nonminimal curvature coupling, a four-component Dirac spinor, and an Abelian Maxwell field with its Faddeev--Popov ghosts. For each spin sector, the logarithmic coefficient separates into a smooth radial heat-kernel contribution, governed by $\calC_1$, and a horizon-localized conical contribution, governed by $\calW_h$. This separation identifies which part of the matter one-loop logarithm is visible in the boundary Lifshitz/TTNC Ward identity and which part instead belongs to horizon replica entropy. The smooth coefficient $\calC_1$ controls the source-induced homogeneous projection of the TTNC Weyl Ward identity, namely the smooth boundary-source contribution to the Ward combination $zε-2p$, whereas the thermodynamic logarithmic entropy is controlled by $\calC_{\rm therm}=\calC_1+zL^2\calW_h$. We give closed expressions for $\calC_1$, $\calW_h$, and $\calC_{\rm therm}$ for the scalar, Dirac, and Maxwell probe sectors, identify the gauge-field contact contribution in the conical entropy, and verify that the smooth source response vanishes in the relativistic planar $z=1$ limit while the standard horizon heat-kernel entropy coefficient remains.

Let $\mathcal{A}$ be a (not necessarily rational) conformal net. We show that the braided $\mathrm{W}^*$-tensor category $\text{Rep}(\mathcal{A})$ of representations of $\mathcal{A}$ is canonically a balanced $\mathrm{W}^*$-tensor category. The balance is given by the action of $e^{-2πi L_0}$, where $L_0$ denotes the generator of rotations on $S^1$. In arXiv:2606.03623, we generalize this result to the larger context of a group acting on $\mathcal{A}$. We provide here a more accessible proof for the case where no group is present.

Let $k$ be an algebraically closed field of prime characteristic $p$. Let $G$ be a finite group. We investigate the Brauer indecomposability of Scott $kG$-modules in relation to the kernel of modules. We generalize a criterion for Brauer indecomposability. We also prove that, in certain cases, Brauer indecomposability of a Scott $kG$-module can be lifted from that of a Scott module over a $p$-local subgroup.

We study the classification problem for anomaly-free 6D $\mathcal N=(1,0)$ supergravities with a gauged abelian R-symmetry and one tensor multiplet. We present eleven new models with gauge group $G_{\mathrm{non-Abelian}}\times U(1)_R$ that satisfy the local Green--Schwarz factorization condition, together with several recently proposed global consistency conditions. In particular, the low-rank models we found are precisely where some of the recent enumeration literature is least directly applicable. These examples suggest that the landscape of anomaly-free gauged $U(1)_R$ supergravities may be richer than previously recognized while still remaining highly constrained. We analyze the arithmetic structure of the anomaly coefficients, including their integral pairings, embeddability into rank-two unimodular charge lattices, the characteristic-vector condition and ghost-free gauge-field conditions. We show that $n_V \equiv 8 \pmod{12}$ is necessary and sufficient for the unimodular embeddability in the rank-two case, when the gauge group does not contain $SU(2)$, $SU(3)$ and $G_2$. For the characteristic-vector condition we verify sufficiency for the branches realized by our examples and identify a remaining branch requiring additional exclusion. We also present a detailed discussion of the contribution to the anomaly polynomial when the $D_4$ Lie algebra is present. These results sharpen the boundary between anomaly-free 6D spectra, global-consistency constraints, and possible UV realization in string theory or F-theory.

In this paper, we establish a new lower bound for the number of low-lying zeros of Dirichlet $L$-functions $L(s, χ)$ on the critical line within extremely short intervals. Specifically, for a sufficiently large prime $P$ and real number $T \in [a_1/\sqrt{\log P}, 1]$, we prove that the sum of the number of zeros on the critical line $N_0(T, χ)$ over characters $χ\bmod P$ satisfies $$ \sum_{χ\bmod P} N_0(T, χ) \gg T^2 P\sqrt{\log P} .$$ Traditional approaches encounter significant technical barriers in this short-interval regime. The Levinson method fails due to its own inherent limitations in handling such restricted intervals , while standard applications of the Selberg mollifier are hindered by the emergence of complex, inseparable cross-terms that are difficult to evaluate. To overcome these obstacles, we introduce a novel analytic framework utilizing high-dimensional Mellin transforms. This approach systematically manages the multi-variable series generated by the mollifier calculations. By explicitly resolving these cross-term obstructions, we extract the localized lower bound, providing a robust method that circumvents the short-interval bottleneck and offers potential applicability to the zero statistics of higher-rank $L$-function families.

Hilbert's sixth problem calls for the axiomatization of physics, particularly the derivation of macroscopic statistical laws from microscopic mechanical principles. A conceptual difficulty arises in classical probability theory: in continuous spaces every individual microstate has probability zero. In this paper, we introduce a probabilistic framework based on Soft Logic and Soft Numbers in which point events possess infinitesimal Soft probabilities rather than the classical zero. We show that Soft probability can be interpreted as an infinitesimal refinement of classical probability and discuss its implications for statistical mechanics and Hilbert's sixth problem. In addition, we show rigorously how to construct a Mobius strip, based on the soft numbers, and we discuss how this Mobius strip representation with soft numbers allows for a deeper understanding of the nature and character of Hilbert's sixth problem. Inspired by the collapsing of that classical probability to zero, we suggest adding an axiom for an Infinitesimal Probability into the list of Kolmogorov's five Probability axioms. Furthermore, we suggest a probabilistic framework based on Soft Numbers for assigning values to probabilities of impossible events of a discrete random variable with realizations outside its support (which, in the ordinary probability, collapse to zero). This assignment of Soft Number values is based on an extension of the Pascal triangle to have soft zeros outside of the regular Pascal triangle (with real values) based on factorials of negative numbers.

Let $0<α<2$, $β>0$ and $α/2<|s|\leq 1$. In a previous work, we obtained all possible values of the Lebesgue exponent $p=p(γ)$ for which the Fourier transform of $ E_{α,β}(e^{\dot{\imath}πs} |\cdot|^γ )$ is an $L^{p}(\mathbb{R}^d)$ function, when $γ>(d-1)/2$. We recover the more interesting lower regularity case $0<γ\leq (d-1)/2$, using tools from the Littlewood-Paley theory. This question arises in the analysis of certain space-time fractional diffusion and Schrödinger problems and has been solved for the particular cases $α\in (0,1)$, $β=α,1$, and $s=-1/2,1$ via asymptotic analysis of Fox $H$-functions. The Littlewood-Paley theory provides a simpler proof that allows considering all values of $β,γ>0$ and $s\in (-1,1]\setminus [-α/2,α/2]$. This enabled us to prove various key estimates for a general class of nonlocal space-time problems.

Coincident Peak (CP) pricing is widely used in U.S. electricity markets to allocate capacity and transmission costs. This paper develops a behavioral game-theoretic framework for CP-driven load shifting that couples a nonlinear cost-allocation model with day-ahead (one-shot) and real-time (sequential-learning) decision processes. We examine two update rules, namely best-response dynamics (BRD) and fictitious-play dynamics (FPD), across continuous and finite action spaces to quantify how flexibility, action resolution, and participation influence peak outcomes. Using ERCOT peak-day data, we find that FPD reliably reduces system peaks, whereas BRD is more variable and can increase peaks under tight-capacity conditions. Finer action resolution improves peak shaving, while the number of participants is largely neutral when aggregate flexibility is fixed. Meanwhile, information-provider signals can induce herding, whereas response-aware or diverse signals improve peak shaving. These results highlight both the potential and limits of CP pricing: smoothing information and enabling granular control are as important as the amount of available flexibility. The framework offers practical guidance for system operators and consumers: For ISOs, broadcasting smoothed CP signals and setting minimum controllable-capacity thresholds enhance coordination. For consumers, greater flexibility and finer control resolution improve both cost savings and peak-shaving performance.

We present the results of a multi-methodological study aimed at investigating the interaction between graphene and Tritium during its $β$-decay to Helium, under different levels of loading and geometrical configurations. We combine Density Functional Theory (DFT), to evaluate the interaction potentials, with calculations of the decay rate, in order to study the consequences that the presence of the substrate has on the $β$-decay spectrum of Tritium. We determine the shape of the event rate, accounting for the effects of (part of) the corresponding condensed matter degrees of freedom. In the context of future neutrino experiments, our results provide important information aimed at the optimization of hosting material, as well as the determination of the physics reach. Furthermore, our work outlines a novel theoretical and computational scheme to address a question at the boundary between high and low energy physics. This requires non-conventional declinations of DFT combined with full quantum treatments of the nuclear configuration involved in the decay process.

This paper presents the first method for constructing bases for polynomial spline spaces over an arbitrary T-meshes (PT-splines for short). We construct spline basis functions for an arbitrary T-mesh by first converting the T-mesh into a diagonalizable one via edge extension, ensuring a stable dimension of the spline space. Basis functions over the diagoalizable T-mesh are constructed according to the three components in the dimension formula corresponding to cross-cuts, rays, and T $l$-edges in the diagonalizable T-mesh, and each component is assigned some local tensor product B-splines as the basis functions. We prove this set of functions constitutes a basis for the diagonalizable T-mesh. To remove redundant edges from extension, we introduce a technique, termed Extended Edge Elimination (EEE) to construct a basis for an arbitrary T-mesh while reducing structural constraints and unnecessary refinements. The resulting PT-spline basis ensures linear independence and completeness, supported by a dedicated construction algorithm. A comparison with LR B-splines, which may lack linear independence and are limited to LR-meshes, highlights the PT-spline's versatility across any T-mesh. Examples are also provided to demonstrate that dimensional instability in spline spaces is related with basis function degradation and that PT-splines are advantageous over HB-splines for certain hierarchical T-meshes.