We are concerned with a semilinear elliptic equation in the half-space, subject to a nonlinear dynamic boundary condition. We establish the global well-posedness of solutions in a new setting for the problem, namely the framework of Morrey spaces. These are strictly larger than $L^{p}$ and weak-$L^{p}$ spaces, accommodating a broader class of rough initial data, including homogeneous and nondecaying (at infinity) profiles. In our analysis, we consider functional spaces invariant under the natural scaling of the problem, which enables the construction of self-similar solutions. To achieve this, we need to derive key estimates in Morrey spaces for certain interior and boundary operators that arise from the corresponding integral formulation. Furthermore, we obtain some qualitative properties of the solutions, such as positivity, symmetry, and asymptotic stability. Leveraging this last property, we show the existence of self-similar attractor basins and construct a class of solutions that are asymptotically self-similar.

A designer relies on an experimenter to provide information to a decision maker, but the experimenter has incentives to persuade rather than merely transmit information. Anticipating this motive, the designer can restrict the set of admissible experiments, but cannot prevent the experimenter from garbling any admissible experiment. We model this situation as delegation over experiments. The optimal delegation set can be obtained by comparing maximally informative experiments among those the experimenter has no incentive to garble. When the experimenter's preferences are $S$-shaped, we fully characterize such experiments as double censorship. Relative to the full delegation outcome, upper censorship, double censorship features an intermediate pooling region, inducing a smaller pooling region for the highest states. We show that the designer strictly benefits from imposing a nontrivial delegation set to constrain the experimenter's ability to persuade while retaining valuable information provision.

Data analyses are often constructed in an imperative manner, where commands representing actions taken on the data are issued sequentially. The publication of these commands, along with the data, is essential to the reproducibility of the analysis by others. However, simply presenting the code and the results of running the code can hide important details about the data analyst's premises, expectations, and assumptions about the data. Understanding this analysis reasoning can be critical to evaluating the quality of an analysis and for suggesting possible improvements. We argue that a formal representation of a data analysis that externalizes its logical construction offers more useful information for statically illustrating an analyst's reasoning. Such a formal representation would allow for the evaluation of some aspects of a data analysis without the need for the data, the visualization of the logical connections leading to a conclusion, and the ability to assess the sensitivity of an analyst's assumptions to unexpected features in the data. In this paper we describe an implementation of this formal representation and how it might be applied to some common data analysis tasks.

Since the prediction of a time-reversal symmetry breaking (TRSB) $d+id^\prime$ state in twisted bilayer cuprate superconductors by Can et al. in 2021, several experiments have attempted to detect this state, yielding conflicting results. At present, it is not clear which differences in samples or experimental conditions might explain these discrepancies. In this work, we perform a tight-binding lattice model calculation with phenomenological interlayer tunneling, examining the order parameter as a function of twist angle, interlayer tunneling, doping, and temperature. We observe the TRSB state to be correlated to the position of the Van Hove singularity in the normal state which changes not only as a function of doping but also the tunneling strength. Two such phases are identified as nominally consistent with in-plane $d+id'$ and $d+is$ order, but with unexpected transformation properties under bilayer symmetry operations. We calculate the Josephson critical current, in particular examining the angle dependence for various tunneling strengths. Finally, we discuss the existing experiments in the context of our results.

Communication is a crucial social factor in the success of software projects, as positively or negatively perceived statements can influence how recipients feel and affect team collaboration through emotional contagion. Whether a developer perceives a written message as positive, negative, or neutral is likely shaped by multiple factors. In this paper, we investigate how mood traits and states, life circumstances, project phases, and group dynamics relate to the perception of text-based messages in software development. We conducted a four-round survey study with 81 students in team-based software projects. Across rounds, participants reported these factors and labeled 30 decontextualized statements for sentiment, including meta-data on labeling rationale and uncertainty. Our results show: (1) Sentiment perception is only moderately stable within individuals, and label changes concentrate on ambiguity-prone statements; (2) Correlation-level signals are small and do not survive global multiple-testing correction; (3) In statement-level repeated-measures models (GEE), higher mood trait and reactivity are associated with more positive (and less neutral) labeling, while predictors of negative labeling are weaker and at most trend-level (e.g., task conflict); (4) We find no clear evidence of systematic project-phase effects. Overall, sentiment perception varies within persons and is strongly statement-dependent. Although our study was conducted in an academic setting, the observed variability and ambiguity effects suggest caution when interpreting sentiment analysis outputs and motivate future work with contextualized, in-project communication.

This project is aimed at studying the first-order phase transitions, that is presumed to have ensued in the early universe, and its consequences on the primordial gravitational waves. The effects of bubble nucleation, growth, and coalescence are reviewed. The resulting first-order phase transition is taken as the source of the gravitational waves that were produced, in order to determine the energy density, amplitude, and frequency spectra of the relic gravitational wave background. This is accomplished by modelling the first-order phase transition as a turbulent fluid and employing relativistic hydrodynamic equations to estimate the required physical quantities. Two models are majorly studied for all the analysis done in this project. Both models compute the necessary gravitational wave spectra using the exponential Kraichnan function as the temporal decorrelation function. Also, both models contemplate the turbulence in the flow of plasma to be stationary, obeying the conditions dictated by the Kolmogorov turbulence. However, the first model uses a de-coherence function that depends on the wavenumber and time, while, the second model uses the top hat correlation, to compute the anisotropic stress. The new model introduced here adheres to the freely decaying turbulence model, but employs the time dependent de-coherence function in its computations.

Quantum batteries are quantum systems that store energy and deliver it on demand, and their practical value hinges on how fast they can be charged. While collective charging protocols and global control are known to enhance charging power, it remains unclear how the battery's internal interaction architecture itself constrains performance. Here we study interacting fermionic batteries whose internal couplings are encoded by a graph adjacency matrix, charged via a simple interaction with an external fermionic device. We prove that the star topology maximises the early time charging power, which proxies the maximal average power - a widely used quantum battery quality metric. We substantiate the result numerically by an exhaustive sweep over all graphs with $N\leq 7$ vertices and by benchmarks against random graph ensembles at larger $N$. Our findings shed light on architecture as a controllable knob for fast charging and motivate hub-and-spoke designs in scalable quantum-battery platforms.

The information flows in serverless platforms are complex and non-conservative. This is a direct result of how independently deployed functions interact under the platform coarse-grained control mechanisms. To manage this complexity, we introduce a topological model for serverless services. Using Hodge decomposition, we can separate observed operational flows into two distinct categories. They include components that can be corrected locally and harmonic modes that persist at any scale. Our analysis reveals that these harmonic flows emerge naturally from different types of inter-function interactions. They should be understood as structural properties of serverless systems, not as configuration errors. Building on this insight, we present an iterative method for analyzing inter-function flows. This method helps deriving practical remediation strategies. One such strategy is the introduction of "dumping effects" to contain harmonic inefficiencies, offering an alternative to completely restructuring the service's topological model. Our experimental results confirm that this approach can uncover latent architectural structures.

Efficient entanglement distillation is a central task in quantum information science and future quantum networks. At the core of distillation protocols are the quantum error correction and detection schemes which enhance the fidelity of entangled pairs. Conventional protocols focus on digital systems, which typically require complicated compiled circuits, high-fidelity multi-qubit operations and delicate pulse-level control that impose high demands on near-term hardware. Crucially, the leading physical platforms for quantum networks, trapped ions and neutral atoms, are governed by native many-body Hamiltonians inherently suited for analog, continuous-time evolution. Adopting these natural dynamics is simpler than engineering digital logic via delicate pulse-level control. Motivated by this experimental reality, we seek to leverage the intrinsic analog capabilities for efficient entanglement distillation. In this work, we introduce the Hamiltonian entanglement distillation protocol, which exploits the intrinsic information scrambling generated by random time evolution under native Hamiltonians. We establish a quantitative connection between output fidelity and Out-of-Time-Order Correlators, showing that efficient scrambling directly implies good distillation performance. Since generic Hamiltonians are naturally efficient scramblers, the capability for distillation is ubiquitous: almost all Hamiltonians in the Hilbert space suffice for high-fidelity distillation. Numerical simulations of representative Rydberg-atom and trapped-ion systems further confirm that robust performance could be achieved using only short-range interactions and evolution times feasible in current experiments. By avoiding the complexity of digital circuit control, our approach substantially relaxes experimental requirements, providing a scalable route to entanglement engineering on current analog quantum platforms.

The temporal resolution of Low-Gain Avalanche Detectors (LGADs), also known as Ultra-Fast Silicon Detectors (UFSDs), is governed by two contributions: jitter, arising from electronic noise and signal slew rate, and the Landau noise term, arising from the non-uniform energy deposition of minimum ionizing particles (MIPs). We show that a correct simulation of the initial ionization alone significantly overestimates the measured Landau noise. Two additional physical mechanisms are necessary to reproduce the data: space charge effects during electron/hole drift, which smooth the granularity of the initial charge distribution, and gain saturation during multiplication, which preferentially suppresses large-amplitude fluctuations. All steps of the model have been implemented in the fast simulation program Weightfield2 (WF2). The model is validated against several independent experimental observations: the evolution of the measured charge distribution with gain, the temporal resolution of events in the Landau tail, and the thickness dependence of timing performance. We also discuss a data-driven gain measurement method based on gain saturation, and implications for gain layer design.

Vulnerabilities emanating from DRAM errors pose a vexing problem that remains, as of yet, unsolved and elusive but cannot be ignored. Prior defenses focused on specific details of early RowHammer attacks and fail to generalize with the generalizations of recent RowHammer attacks. Even worse, it is presently not clear that techniques from prior defenses will be able to cope with these generalizations or if an entirely new approach is required. Although still work-in-progress, we have identified a new approach that combines memory allocation with principles underlying software diversity and shows promising early results. At first glance, software diversity seems to be an unlikely contender, since it faces seemingly insurmountable obstacles, primarily the lack of sufficient entropy in memory subsystems. Our system - called MAD, short for memory allocation diversity - leverages two novel, complementary spatial diversification techniques to overcome this entropy obstacle. Entropy aside, MAD offers ease-of-implementation, negligible performance impact, and is both hardware and software agnostic. From a security perspective, MAD's goal is to deter RowHammer attacks by delaying them to the maximum extent possible. Such a delay opens the door for a variety of additional responses, e.g., proactive rebooting, or complementary in-depth analysis of ongoing attacks that would be too slow for an always-on defense.

Understanding ion-matter interactions at the atomistic level is key to advancing materials for the semiconductor industry, space systems, and nuclear fusion technologies. However, most atomistic frameworks still rely on simplified descriptions of how ions transfer energy to the electronic subsystem, overlooking the sensitivity of this process to the actual ion path. Existing electron-ion interaction models, such as the tensorial unified two-temperature model, were developed to study self-irradiation scenarios, but their suitability for light-ion irradiation remains unexplored. Here, we propose that for light projectiles, stepping back from the tensorial formulation toward a simpler, local model of electronic stopping provides a more efficient and physically transparent trajectory-dependent description. We parameterize and validate both models for hydrogen and helium in tungsten using ab initio electronic stopping data and large-scale ion range simulations, benchmarked against existing experimental data. This provides a consistent framework for including nonadiabatic electronic stopping in atomistic simulations of light-ion energy dissipation.

A combinatorial code $\mathcal{C}$ is a collection of subsets of $[n]$, or equivalently a set of points in $\{0,1\}^n$. A morphism of codes is a map from one combinatorial code to another such that the coordinates of points in the image can be expressed as products of coordinates in the domain. By representing morphisms of codes as binary matrices, we show that any morphism of codes is part of a Galois connection where its adjoint is boolean multiplication by the representative matrix. We use this to characterize those morphisms of codes which allow to factor a boolean matrix, with applications to estimating boolean matrix rank. Morphisms also induce a partial order on (isomorphism classes of) codes. We determine the covering relations in this partial order for which the two adjoint maps are mutual inverses in terms of \emph{free} neurons, a combinatorial condition on the index corresponding to the covering maps. We introduce the \emph{defect} of a code as a new tool to study this poset and show that defect decreases by exactly 0 or 1 under a covering map.

Heavy-ion experiments at the CERN SPS began in the mid-1980s to study nuclear matter at extreme temperatures and densities. The program started with light ions, such as oxygen and sulphur, at energies of 60A GeV and 200A GeV, later advancing to lead ions at 158A GeV. A series of experiments, employing novel detector technologies, explored various signatures of quark-gluon plasma (QGP) formation. In February 2000, these results led CERN to announce evidence for the QGP formation. Subsequently, an energy scan was conducted with lead ions from 20A GeV to 158A GeV, to locate the threshold of QGP creation.

We introduce extensions of Khovanov homology and the Lee and Bar-Natan spectral sequences for links in $ \mathbb{RP}^3 $. These extensions are distinct to those previously defined by Asaeda-Przytycki-Sikora (and Gabrovšek's generalization), Chen, and Manolescu-Willis. The new Lee and Bar-Natan theories each yield Rasmussen invariants (that are distinct to one another). The invariant extracted from the new Lee homology is distinct to that defined by Manolescu-Willis; it is unclear if the same is true for the new Bar-Natan homology and that defined by Chen.

A reduced local field model derived from full-wave electromagnetic simulations shows that photonic nanojet formation corresponds to an emergent mesoscopic funnel of propagating power flux sustained by an effective free-space transverse mode structure. This interpretation moves beyond purely geometric-optics or interference-based explanations by identifying a self-consistent redistribution of phase gradients and effective longitudinal wavenumber near the nanojet waist. The model quantitatively captures characteristic nanojet morphology, including the formation and local structure of the jet waist. It also yields a geometry-independent lower bound on the nanojet waist, linking transverse confinement to the effective axial wavenumber through an explicit trade-off. The model establishes a direct connection between full-wave Maxwell fields and a reduced free-space oscillator description, yielding new physical insight into nanojet confinement and suggesting design principles for nanojet-assisted imaging, lithography, and subwavelength field localization.

With the advancement of precision medicine there is an increasing need for design and analysis methods in clinical trials with the objective of investigating effect heterogeneity and estimating subgroup effects. As this requires precise estimation of interaction effects, borrowing information from external data sources including retrospective studies and early phase clinical trials to enrich the trial in sparse subgroups is pertinent. Motivated by a trial in gastric cancer we consider a practical design and analysis framework for borrowing from external data sources that only partially inform the inference. As the analysis model we propose an individually weighted model where the external data are weighted based on their fit with the target population based on the distribution of a set of covariates. In a simulation study we assess the performance of the model under various scenarios and make comparisons to dynamic borrowing. In addition, we provide a Bayesian design framework where design priors are extracted from the external data to determine decision boundaries and sample sizes. The design procedure is demonstrated within the context of our motivating example.

MnBi$_2$Te$_4$ and MnBi$_4$Te$_7$ are antiferromagnetic topological insulators belonging to the MnBi$_{2n}$Te$_{3n+1}$ series, where structural layering provides a natural route to tune magnetic interaction in van der Waals magnets. Despite extensive interest in their topological properties, how the insertion of Bi$_2$Te$_3$ quintuple layers modifies magnetic critical fluctuations near the antiferromagnetic transition remains unresolved. Here, we combine scanning tunneling microscopy (STM), critical scaling analysis, and magnetocaloric measurements to directly correlate real-space structures with magnetic criticality. STM reveals atomically flat septuple-layer terraces in MnBi$_2$Te$_4$ whereas MnBi$_4$Te$_7$ displays coexisting septuple and quintuple layer terminations reflecting its alternating stacking sequence. MnBi$_2$Te$_4$ exhibits robust three-dimensional Ising-like critical behavior together with a distinct low-temperature first-order transition. In contrast, MnBi$_4$Te$_7$ displays crossover-dominated criticality arising from weakened interlayer exchange and competing magnetic phases. Correspondingly, the magnetocaloric response differs significantly between the two compounds. MnBi$_2$Te$_4$ shows dual-type magnetocaloric behavior with a sharp field-induced sign reversal of the isothermal magnetic entropy change ($-ΔS_M$). It exhibits both inverse ($-ΔS_M < 0$) and conventional ($-ΔS_M > 0$) magnetocaloric effects. In contrast, MnBi$_4$Te$_7$ shows only conventional magnetocaloric response with a broad positive entropy peak. These results establish structural layering as a key parameter governing magnetic critical fluctuations and magnetocaloric behavior in MnBi$_{2n}$Te$_{3n+1}$ topological magnets.

We introduce the notion of strong regularity for subanalytic sheaves and establish estimates for the supports and microsupports of their multi-microlocalizations. As applications, we study subanalytic sheaves of Whit- ney and temperate holomorphic solutions of regular D-modules along an involutive subbundle. In this setting we prove initial value theorems for multi-microlocal objects with growth conditions and division theorems for temperate and Whitney multi-microfunctions. As a consequence, we obtain a multi-microlocal version of Bochner's tube theorem for solution sheaves of strongly asymptotically developable functions.

The original binary-encounter Bethe model of Kim and Eugene Rudd (1994 Phys. Rev. A 50 3954-67) has proven to be an accurate analytical representation of total impact ionisation cross sections of electrons colliding with atoms and molecules. It is based on a decomposition into partial ionisation cross sections from electrons in bound orbitals. Despite the model's accuracy for total ionisation, its individual partial cross sections for ionisation rely on thresholds calculated theoretically which systematically overestimate the experimental orbital binding energies. Here, we examine the BEB model's performance when based on experimental ionisation thresholds. The resulting partial cross sections of the various final (excited) ionic states produced could help to prefigure subsequent optical radiations and non-radiative transitions in models of plasma physics.

Partitions with distinct even parts have long been the subject of extensive research. In this paper, We present some new perspectives on such partitions from a combinatorial viewpoint, and connect them with signed partitions and bicolored partitions, thereby obtaining several partition identities. We construct bijective proofs for each of our results. Furthermore, these bijections will partially answer the combinatorial problems posed by Andrews-El Bachraoui and K$\imath$l$\imath$ç-Kurşungöz. respectively.

In interacting quantum many-body systems, relaxation toward equilibrium reflects a competition between internal chaotic dynamics and environmental dissipation. While conventional Markovian baths typically produce exponential decay, non-Markovian dissipation can give rise to more intricate behavior, including algebraic relaxation. We study an open Sachdev-Ye-Kitaev (SYK) model coupled to a pseudogapped fermionic bath, using the Keldysh formalism to compute steady-state correlations in the large-$N$ limit. Our results uncover a rich dynamical phase diagram, with regimes of bath-driven power-law relaxation, chaos-driven exponential decay, and an intermediate pre-relaxation phase where exponential decay crosses over to algebraic decay. These findings demonstrate that non-Markovian environments can qualitatively reshape relaxation mechanisms in strongly correlated quantum many-body systems.

Most data-driven analysis and control methods rely on centralized access to system measurements. In contrast, we consider a setting in which the measurements are distributed across multiple agents and raw data are not shared. Each agent has access only to locally held samples, possibly as little as a single measurement, and agents exchange only locally computed signals. Consequently, no individual agent possesses sufficient information to identify the entire system or synthesize a controller independently. To address this limitation, we develop distributed dynamical algorithms that enable the agents to collectively compute global system certificates from local data. Two problems are addressed. First, for stable linear time-invariant (LTI) systems, the agents compute a Lyapunov certificate by solving the Lyapunov equation in a fully distributed manner. Second, for general LTI systems, they compute the stabilizing solution of the algebraic Riccati equation and hence the optimal linear-quadratic regulator (LQR). An initially proposed scheme guarantees practical convergence, while a subsequent augmented PI-type algorithm achieves exact convergence to the desired solution. We further establish robustness of the resulting LQR controller to uncertainty and measurement noise. The approach is illustrated through distributed Lyapunov certification of a quadruple-tank process and distributed LQR design for helicopter dynamics.

Let R be a d-dimensional Cohen-Macaulay complete local ring with infinite residue field k. The dominant index $\operatorname{dx}(R)$ is by definition the least number of extensions necessary to build k in the singularity category $\operatorname{D^{sg}}$ out of each nonzero object, up to finite direct sums, direct summands and shifts. The local ring R is called uniformly dominant if $\operatorname{dx}(R)$ is finite. In this paper, we prove that R is uniformly dominant with $\operatorname{dx}(R)\le6d+5$ if R has codimension 2 and is not a complete intersection. Also, we show that R is uniformly dominant with $\operatorname{dx}(R)\le d+1$ if R is Burch, and with $\operatorname{dx}(R)\le d$ if R is either a quasi-fiber product ring, or has multiplicity at most 5 and is not Gorenstein. A result on hypersurfaces by Ballard, Favero and Katzarkov is recovered, and results on Burch rings and quasi-fiber product rings by Takahashi are refined.

Earth science datasets are growing rapidly in both volume and structural complexity. They increasingly contain richly labelled data with heterogeneous metadata and complex internal constraints that impose dependencies between variables and dimensions. Datacubes have become a common abstraction for organising such datasets, but traditional dense and orthogonal datacube models struggle to represent irregular, sparse or branching data spaces efficiently. In this paper, we introduce a generalised data hypercube representation based on compressed tree structures, which enables an accurate and compact description of complex data spaces. We describe the design of this representation and analyse its ability to capture sparsity and conditional relationships while remaining efficient to traverse. Using a concrete implementation, we study the performance characteristics of compressed tree data hypercubes and demonstrate their effectiveness as fast, cache-like indices over large backend data stores. Building on this representation, we present an integrated feature extraction system that operates directly on tree-based data hypercubes within the Polytope framework. By embedding data access strategies into the data hypercube abstraction itself, the system enables precise, sub-field data extraction and supports flexible, user-driven access patterns. We evaluate the performance of the integrated system and show how it enables new ways of interacting with complex datasets that are difficult to support using traditional access models. This work bridges the gap between expressive data hypercube models and efficient data access methods. In particular, it provides a unified framework that combines tree-based data representations with feature extraction capabilities. The proposed approach therefore offers a foundation for scalable and user-centric access to large heterogeneous Earth science datasets.

We present a scheme for implementing a high-fidelity non-linear phase shift on a photonic state. The scheme is based on repeated scattering off a two-level quantum emitter embedded in a chiral or one-sided waveguide. The waveguide is equipped with elements inducing second-order dispersion and temporal phase shifts, which effectively form a harmonic trap and confine the photon pulses to a Gaussian shape. The same quantum emitter can be used for each scattering, and thus, only one quantum emitter is needed in this scheme. To illustrate the application of our scheme for photonic quantum computing and quantum communication, we analyze the implementation of a control-Z gate and a deterministic Bell-state analyzer for photonic qubits. Through numerical optimization, we show that we can reach a control-Z gate fidelity of $\mathcal{F} \sim 99.2\%$ ($\mathcal{F} \sim 96\%$) and a success probability of $P_s \sim 99.6 \%$ ($P_s\sim 98 \%$) for a Bell-state measurement with $N=17$ ($N=5$) scatterings.

This article covers polyhomogeneous mapping properties of the Radon transform $R$ of smooth functions on the open unit ball $Ω\subset\mathbb{R}^n$ and the back-projection operator $R^*$ on $Z=(-1,1)\times S^{n-1}\subset\mathbb{R}\times S^{n-1}$. We construct a double $b$-fibration which desingularizes the point-hyperplane relation of $\overlineΩ$ as the total space of a fibration over $\overline{Z}$. We provide formulas for $R$ and $R^*$ in operations generated by the associated $b$-fibrations and sharper estimates on the polyhomogeneous mapping properties of $R$ and $R^*$ compared to classic estimates using classic Mellin functional techniques. We include a discussion of a one (complex) parameter family of normal operators associated to $R$ mapping $C^{\infty}(\overlineΩ)$ to itself.

An integral domain $D$ is called a \emph{prime-divisor-finite domain} (PDF-domain) if every nonzero element has only finitely many nonassociate prime divisors. A domain $D$ is said to be a \emph{tightly prime-divisor-finite domain} (TPDF-domain) if it is a PDF-domain and every nonzero nonunit element admits at least one prime divisor. In this paper, we study TPDF-domains. We investigate some basic properties of these domains and examine the behavior of the TPDF property under standard constructions such as localization, $D+M$ constructions, and polynomial rings.

Accurate prediction of physical fields is critical in various engineering applications, including thermal management in electronic systems, airfoil shape optimization in aerospace, and flow field control in hypersonic vehicles. This study employs the Denoising Diffusion Probabilistic Models (DDPMs) for predicting the temperature field caused by the thermal diffusion, and the flow fields spanning from incompressible to hypersonic regimes. A conditional DDPM framework is first validated with a steady-state thermal diffusion problem by predicting the temperature distribution around a plate with holes. Strong agreement with ground truth data is shown with an average error of approximately 0.013 for plates with a central circular hole. The model also delivers high accuracy in critical regions, such as near the inner circular or square holes. Its performance is further evaluated on incompressible flow around an airfoil and hypersonic flow over a compression ramp, confirming robust predictive capability across diverse flow conditions. Additionally, a latent-space implementation of DDPM is introduced, which employs an Autoencoder (AE) for dimensionality reduction and reconstruction of the physical data. The resulting Latent Diffusion Model (LDM) maintains reconstruction quality comparable to the standard DDPM while substantially reducing the computational cost of the diffusion training process. When applied to hypersonic flow over a compression ramp in the original parameter space, LDM predictions align well with ground truth, achieving a deviation of only 4.28% in separation length estimation. This work confirms the high predictive accuracy of the DDPM framework and highlights the efficiency gains from performing diffusion in a learned latent space. The findings establish an efficient framework for high fidelity generative modeling of complex thermal/flow fields.

In this paper, we study quadratic growth solutions $u$ of fully nonlinear elliptic equations of the form $F(D^2u,x)=f$ in $\mathbb{R}^n$, where $f$ is periodic and $F$ has the periodicity in $x$. Under the assumption that the oscillation of $F(M,x)$ in $x$ is ``small", we establish the existence and Liouville type results for quadratic growth solutions, which can be expressed into the sum of a quadratic polynomial and a periodic function. Consequently, these results are generalization of the existing results for linear elliptic equations $a_{ij}D_{ij}u=0$ and fully nonlinear elliptic equations $F(D^2u)=f$ with the periodic data.