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The online ordered Ramsey game is played between two players, Builder and Painter, on an infinite sequence of vertices with ordered graphs $(G_1,G_2)$, which have linear orderings on their vertices. On each turn, Builder first selects an edge before Painter colors it red or blue. Builder's objective is to construct either an ordered red copy of $G_1$ or an ordered blue copy of $G_2$, while Painter's objective is to delay this for as many turns as possible. The online ordered Ramsey number $r_o(G_1,G_2)$ is the number of turns Builder takes to win in the case that both players play optimally. Few lower bounds are known for this quantity. In this paper, we introduce a succinct proof of a new lower bound based on the maximum left- and right-degrees in the ordered graphs. We also upper bound $r_o(G_1,G_2)$ in two cases: when $G_1$ is a cycle and $G_2$ a complete bipartite graph, and when $G_1$ is a tree and $G_2$ a clique., Comment: 8 pages

De novo molecule design has become a highly active research area, advanced significantly through the use of state-of-the-art generative models. Despite these advances, several fundamental questions remain unanswered as the field increasingly focuses on more complex generative models and sophisticated molecular representations as an answer to the challenges of drug design. In this paper, we return to the simplest representation of molecules, and investigate overlooked limitations of classical generative approaches, particularly Variational Autoencoders (VAEs) and auto-regressive models. We propose a hybrid model in the form of a novel regularizer that leverages the strengths of both to improve validity, conditional generation, and style transfer of molecular sequences. Additionally, we provide an in depth discussion of overlooked assumptions of these models' behaviour.

We present the design and observations of low resolution JWST/NIRSpec PRISM spectroscopy from the Ultradeep NIRSpec and NIRCam ObserVations before the Epoch of Reionization (UNCOVER) Cycle 1 JWST Treasury program. Targets are selected using JWST/NIRCam photometry from UNCOVER and other programs, and cover a wide range of categories and redshifts to ensure the legacy value of the survey. These categories include the first galaxies at $z\gtrsim10$, faint galaxies during the Epoch of Reionization ($z\sim6-8$), high redshift AGN ($z\gtrsim6$), Population III star candidates, distant quiescent and dusty galaxies ($1\lesssim z \lesssim 6$), and filler galaxies sampling redshift--color--magnitude space from $z\sim 0.1-13$. Seven NIRSpec MSA masks across the extended Abell 2744 cluster were observed, along with NIRCam parallel imaging in 8 filters (F090W, F115W, F150W, F200W, F277W, F356W, F410M, F444W, F480M) over a total area of ~26 arcmin$^2$, overlapping existing HST coverage from programs including the Hubble Frontier Fields and BUFFALO. We successfully observed 553 objects down to $m_{\mathrm{F444W}}\sim30\mathrm{AB}$, and by leveraging mask overlaps, we reach total on-target exposure times ranging from 2.4-16.7h. We demonstrate the success rate and distribution of confirmed redshifts, and also highlight the rich information revealed by these ultradeep spectra for a subset of our targets. An updated lens model of Abell 2744 is also presented, including 14 additional spectroscopic redshifts and finding a total cluster mass of $M_{\mathrm{SL}}=(2.1\pm0.3)\times10^{15}\,\mathrm{M}_{\odot}$. We publicly release reduced 1D and 2D spectra for all objects observed in Summer 2023 along with a spectroscopic redshift catalog and the updated lens model of the cluster (https://jwst-uncover.github.io/DR4.html)., Comment: 19 pages, 9 figures, 4 tables, submitted to ApJ, comments welcome! Data available at: https://jwst-uncover.github.io/DR4.html (v2: figure format correction)

We study linear scaling relations in electron-phonon superconductors. By combining numerical and analytical techniques, we find linear Homes scaling relations between the zero-temperature superfluid density and the normal-state DC conductivity. This is due to Galilean invariance being broken, either via a large impurity scattering rate or inelastic scattering of electrons and Einstein phonons at large electron-phonon coupling. Our work thus shows that Homes scaling is more universal than either cuprate or BCS-like physics, and is instead a fundamental result in a wide class of superconductors., Comment: Main article: 7 pages, 3 figures. Supplemental material: 32 pages, 14 figures

Using deep imaging from the CANDELS and HFF surveys, we present bulge+disc decompositions with GalfitM for $\sim$17,000 galaxies over $0.2 \leq z\leq 1.5$. We use various model parameters to select reliable samples of discs and bulges, and derive their stellar masses using an empirically calibrated relation between mass-to-light ratio and colour. Across our entire redshift range, we show that discs follow stellar mass-size relations that are consistent with those of star-forming galaxies, suggesting that discs primarily evolve via star formation. In contrast, the stellar mass-size relations of bulges are mass-independent. Our novel dataset further enables us to separate components into star-forming and quiescent based on their specific star formation rates. We find that both star-forming discs and star-forming bulges lie on stellar mass-size relations that are similar to those of star-forming galaxies, while quiescent discs are typically smaller than star-forming discs and lie on steeper relations, implying distinct evolutionary mechanisms. Similar to quiescent galaxies, quiescent bulges show a flattening in the stellar mass-size relation at $\sim$10$^{10}$M$_\odot$, below which they show little mass dependence. However, their best-fitting relations have lower normalisations, indicating that at a given mass, bulges are smaller than quiescent galaxies. Finally, we obtain rest-frame colours for individual components, showing that bulges typically have redder colours than discs, as expected. We visually derive UVJ criteria to separate star-forming and quiescent components and show that this separation agrees well with component colour. HFF bulge+disc decomposition catalogues used for these analyses are publicly released with this paper., Comment: 30 pages, 18 figures, and 6 tables. Resubmitted to MNRAS after addressing a thorough and constructive referee report

We report the first detection of coherent elastic neutrino-nucleus scattering (CEvNS) on germanium, measured at the Spallation Neutron Source at Oak Ridge National Laboratory. The Ge-Mini detector of the COHERENT collaboration employs large-mass, low-noise, high-purity germanium spectrometers, enabling excellent energy resolution, and an analysis threshold of 1.5 keV electron-equivalent ionization energy. We observe a on-beam excess of 20.6$_{+7.1}^{-6.3}$ counts with a total exposure of 10.22 GWhkg and we reject the no-CEvNS hypothesis with 3.9 sigma significance. The result agrees with the predicted standard model of particle physics signal rate within 2 sigma., Comment: 7 pages, 5 figures

Sensor-aided beamforming reduces the overheads associated with beam training in millimeter-wave (mmWave) multi-input-multi-output (MIMO) communication systems. Most prior work, though, neglects the challenges associated with establishing multi-user (MU) communication links in mmWave MIMO systems. In this paper, we propose a new framework for sensor-aided beam training in MU mmWave MIMO system. We leverage the beamspace representation of the channel that contains only the angles-of-departure (AoDs) of the channel's significant multipath components. We show that a deep neural network (DNN)-based multimodal sensor fusion framework can estimate the beamspace representation of the channel using sensor data. To aid the DNN training, we introduce a novel supervised soft-contrastive loss (SSCL) function that leverages the inherent similarity between channels to extract similar features from the sensor data for similar channels. Finally, we design an MU beamforming strategy that uses the estimated beamspaces of the channels to select analog precoders for all users in a way that prevents transmission to multiple users over the same directions. Compared to the baseline, our approach achieves more than 4$\times$ improvement in the median sum-spectral efficiency (SE) at 42 dBm equivalent isotropic radiated power (EIRP) with 4 active users. This demonstrates that sensor data can provide more channel information than previously explored, with significant implications for machine learning (ML)-based communication and sensing systems.

By classical results of Malcev, cancellative monoids need not be group-embeddable. In this paper, we describe and give presentations for and study an infinite family $\mathcal{M}_n$ of cancellative monoids which are not group-embeddable, originating from Malcev's original work. We show that $\mathcal{M}_n$ is right LCM for $n \geq 2$, owing to applications in the study of $\mathrm{C}^*$-algebras by Brix, Bruce and Dor-On. We finish by showing that $\mathcal{M}_1$ is not right LCM, but is $2$-aligned., Comment: 10 pages

In various biomedical studies, the focus of analysis centers on the magnitudes of data, particularly when algebraic signs are irrelevant or lost. To analyze the magnitude outcomes in repeated measures studies, using models with random effects is essential. This is because random effects can account for individual heterogeneity, enhancing parameter estimation precision. However, there are currently no established regression methods that incorporate random effects and are specifically designed for magnitude outcomes. This article bridges this gap by introducing Bayesian regression modeling approaches for analyzing magnitude data, with a key focus on the incorporation of random effects. Additionally, the proposed method is extended to address multiple causes of informative dropout, commonly encountered in repeated measures studies. To tackle the missing data challenge arising from dropout, a joint modeling strategy is developed, building upon the previously introduced regression techniques. Two numerical simulation studies are conducted to assess the validity of our method. The chosen simulation scenarios aim to resemble the conditions of our motivating study. The results demonstrate that the proposed method for magnitude data exhibits good performance in terms of both estimation accuracy and precision, and the joint models effectively mitigate bias due to missing data. Finally, we apply proposed models to analyze the magnitude data from the motivating study, investigating if sex impacts the magnitude change in diaphragm thickness over time for ICU patients.

Given a $k$-uniform hypergraph $G$ and a set of $k$-uniform hypergraphs $\mathcal{H}$, the generalized Ramsey number $f(G,\mathcal{H},q)$ is the minimum number of colors needed to edge-color $G$ so that every copy of every hypergraph $H\in \mathcal{H}$ in $G$ receives at least $q$ different colors. In this note we obtain bounds, some asymptotically sharp, on several generalized Ramsey numbers, when $G=K_n$ or $G=K_{n,n}$ and $\mathcal{H}$ is a set of cycles or paths, and when $G=K_n^k$ and $\mathcal{H}$ contains a clique on $k+2$ vertices or a tight cycle.

The ongoing fifth-generation (5G) standardization is exploring the use of deep learning (DL) methods to enhance the new radio (NR) interface. Both in academia and industry, researchers are investigating the performance and complexity of multiple DL architecture candidates for specific one-sided and two-sided use cases such as channel state estimation (CSI) feedback, CSI prediction, beam management, and positioning. In this paper, we set focus on the CSI prediction task and study the performance and generalization of the two main DL layers that are being extensively benchmarked within the DL community, namely, multi-head self-attention (MSA) and state-space model (SSM). We train and evaluate MSA and SSM layers to predict the next slot for uplink and downlink communication scenarios over urban microcell (UMi) and urban macrocell (UMa) OFDM 5G channel models. Our numerical results demonstrate that SSMs exhibit better prediction and generalization capabilities than MSAs only for SISO cases. For MIMO scenarios, however, the MSA layer outperforms the SSM one. While both layers represent potential DL architectures for future DL-enabled 5G use cases, the overall investigation of this paper favors MSAs over SSMs.

We introduce an interesting class of left adequate monoids which we call pretzel monoids. These, on the one hand, are monoids of birooted graphs with respect to a natural `glue-and-fold' operation, and on the other hand, are shown to be defined in the category of left adequate monoids by a natural class of presentations. They are also shown to be the free idempotent-pure expansions of right cancellative monoids, making them, in some sense, the left adequate analogues of Margolis-Meakin expansions for inverse monoids. The construction recovers the second author's geometric model of free left adequate monoids when the right cancellative monoid is free., Comment: 34 pages, 10 figures

In this paper, we describe the "Medium Bands, Mega Science" JWST Cycle 2 survey (JWST-GO-4111) and demonstrate the power of these data to reveal both the spatially-integrated and spatially-resolved properties of galaxies from the local universe to the era of cosmic dawn. Executed in November 2023, MegaScience obtained ~30 arcmin^2 of deep multiband NIRCam imaging centered on the z~0.3 Abell 2744 cluster, including eleven medium-band filters and the two shortest-wavelength broad-band filters, F070W and F090W. Together, MegaScience and the UNCOVER Cycle 1 treasury program provide a complete set of deep (~28-30 mag) images in all NIRCam medium- and broad-band filters. This unique dataset allows us to precisely constrain photometric redshifts, map stellar populations and dust attenuation for large samples of distant galaxies, and examine the connection between galaxy structures and formation histories. MegaScience also includes ~17 arcmin^2 of NIRISS parallel imaging in two broad-band and four medium-band filters from 0.9-4.8um, expanding the footprint where robust spectral energy distribution (SED) fitting is possible. We provide example SEDs and multi-band cutouts at a variety of redshifts, and use a catalog of JWST spectroscopic redshifts to show that MegaScience improves both the scatter and catastrophic outlier rate of photometric redshifts by factors of 2-3. Additionally, we demonstrate the spatially-resolved science enabled by MegaScience by presenting maps of the [OIII] line emission and continuum emission in three spectroscopically-confirmed z>6 galaxies. We show that line emission in reionization-era galaxies can be clumpy, extended, and spatially offset from continuum emission, implying that galaxy assembly histories are complex even at these early epochs. We publicly release fully reduced mosaics and photometric catalogs for both the NIRCam primary and NIRISS parallel fields., Comment: 21 pages, 10 figures. Fully reduced imaging, photometric catalogs, and photometric redshift fits publicly available at https://jwst-uncover.github.io/megascience/

Codebook-based beam selection is one approach for configuring millimeter wave communication links. The overhead required to reconfigure the transmit and receive beam pair, though, increases in highly dynamic vehicular communication systems. Location information coupled with machine learning (ML) beam recommendation is one way to reduce the overhead of beam pair selection. In this paper, we develop ML-based location-aided approaches to decouple the beam selection between the user equipment (UE) and the base station (BS). We quantify the performance gaps due to decoupling beam selection and also disaggregating the UE's location information from the BS. Our simulation results show that decoupling beam selection with available location information at the BS performs comparable to joint beam pair selection at the BS. Moreover, decoupled beam selection without location closely approaches the performance of beam pair selection at the BS when sufficient beam pairs are swept., Comment: Submitted to IEEE BlackSeaCom 2024, 6 pages, 5 figures

Additive manufacturing is an expanding multidisciplinary field encompassing applications including medical devices, aerospace components, microfabrication strategies, and artificial organs. Among additive manufacturing approaches, light-based printing technologies, including two-photon polymerization, projection micro stereolithography, and volumetric printing, have garnered significant attention due to their speed, resolution and/or potential applications for biofabrication. In this study, we introduce dynamic interface printing (DIP), a new 3D printing approach that leverages an acoustically modulated, constrained air-liquid boundary to rapidly generate cm-scale three-dimensional structures within tens of seconds. Distinct from volumetric approaches, this process eliminates the need for intricate feedback systems, specialized chemistry, or complex optics while maintaining rapid printing speeds. We demonstrate the versatility of this technique across a broad array of materials and intricate geometries, including those that would be impossible to print via conventional layer-by-layer methods. In doing so, we demonstrate the rapid fabrication of complex structures in-situ, overprinting, structural parallelisation, and biofabrication utility. Moreover, we showcase that the formation of surface waves at this boundary enables enhanced mass transport, material flexibility, and permits three-dimensional particle patterning. We therefore anticipate that this approach will be invaluable for applications where high resolution, scalable throughput, and biocompatible printing is required., Comment: 63 Pages, 4 Figures, 18 Supplementary Figures

Multipliers can be used to guarantee both the Lyapunov stability and input-output stability of Lurye systems with time-invariant memoryless slope-restricted nonlinearities. If a dynamic multiplier is used there is no guarantee the closed-loop system has finite incremental gain. It has been suggested in the literature that without this guarantee such a system may be critically sensitive to time-varying exogenous signals including noise. We show that multipliers guarantee the power gain of the system to be bounded and quantifiable. Furthermore power may be measured about an appropriate steady state bias term, provided the multiplier does not require the nonlinearity to be odd. Hence dynamic multipliers can be used to guarantee Lurye systems have low sensitivity to noise, provided other exogenous systems have constant steady state. We illustrate the analysis with an example where the exogenous signal is a power signal with non-zero mean.

Obtaining accurate and timely channel state information (CSI) is a fundamental challenge for large antenna systems. Mobile systems like 5G use a beam management framework that joins the initial access, beamforming, CSI acquisition, and data transmission. The design of codebooks for these stages, however, is challenging due to their interrelationships, varying array sizes, and site-specific channel and user distributions. Furthermore, beam management is often focused on single-sector operations while ignoring the overarching network- and system-level optimization. In this paper, we proposed an end-to-end learned codebook design algorithm, network beamspace learning (NBL), that captures and optimizes codebooks to mitigate interference while maximizing the achievable performance with extremely large hybrid arrays. The proposed algorithm requires limited shared information yet designs codebooks that outperform traditional codebooks by over 10dB in beam alignment and achieve more than 25% improvements in network spectral efficiency., Comment: To be submitted to IEEE Transactions on Wireless Communications

The use of one-bit analog-to-digital converter (ADC) has been considered as a viable alternative to high resolution counterparts in realizing and commercializing massive multiple-input multiple-output (MIMO) systems. However, the issue of discarding the amplitude information by one-bit quantizers has to be compensated. Thus, carefully tailored methods need to be developed for one-bit channel estimation and data detection as the conventional ones cannot be used. To address these issues, the problems of one-bit channel estimation and data detection for MIMO orthogonal frequency division multiplexing (OFDM) system that operates over uncorrelated frequency selective channels are investigated here. We first develop channel estimators that exploit Gaussian discriminant analysis (GDA) classifier and approximated versions of it as the so-called weak classifiers in an adaptive boosting (AdaBoost) approach. Particularly, the combination of the approximated GDA classifiers with AdaBoost offers the benefit of scalability with the linear order of computations, which is critical in massive MIMO-OFDM systems. We then take advantage of the same idea for proposing the data detectors. Numerical results validate the efficiency of the proposed channel estimators and data detectors compared to other methods. They show comparable/better performance to that of the state-of-the-art methods, but require dramatically lower computational complexities and run times.

We report the spectroscopic confirmation of a massive ($\log(M_\star/M_\odot)=10.34 \pm_{0.07}^{0.06}$), HST-dark ($m_\mathrm{F150W} - m_\mathrm{F444W} = 3.6$) quiescent galaxy at $z_{spec}=3.97$ in the UNCOVER survey. NIRSpec/PRISM spectroscopy and a non-detection in deep ALMA imaging surprisingly reveals that the galaxy is consistent with a low ($<$10 $M_\odot \ \mathrm{yr^{-1}}$) star formation rate despite evidence for moderate dust attenuation. The F444W image is well modeled with a two component \sersic fit that favors a compact, $r_e\sim200$ pc, $n\sim2.9$ component and a more extended, $r_e\sim1.6$ kpc, $n\sim1.7$ component. The galaxy exhibits strong color gradients: the inner regions are significantly redder than the outskirts. Spectral energy distribution models that reproduce both the red colors and low star formation rate in the center of UNCOVER 18407 require both significant ($A_v\sim1.4$ mag) dust attenuation and a stellar mass-weighted age of 900 Myr, implying 50\% of the stars in the core already formed by $z=7.5$. Using spatially resolved annular mass-to-light measurements enabled by the galaxy's moderate magnification ($\mu=2.12\pm_{0.01}^{0.05}$) to reconstruct a radial mass profile from the best-fitting two-component \sersic model, we infer a total mass-weighted $r_\mathrm{eff} = 0.72 \pm_{0.11}^{0.15}$ kpc and log$(\Sigma_\mathrm{1 kpc} \ [\mathrm{M_\odot/kpc^2}]) = 9.61 \pm_{0.10}^{0.08}$. The early formation of a dense, low star formation rate, and dusty core embedded in a less attenuated stellar envelope suggests an evolutionary link between the earliest-forming massive galaxies and their elliptical descendants. Furthermore, the disparity between the global, integrated dust properties and the spatially resolved gradients highlights the importance of accounting for radially varying stellar populations when characterizing the early growth of galaxy structure., Comment: 17 pages, 9 figures, 2 tables. Resubmitted to ApJ after response to referee and update to include new medium band imaging from the JWST MEGASCIENCE program. Comments welcome!

Fluid antenna systems (FASs) can reconfigure their locations freely within a spatially continuous space. To keep favorable antenna positions, the channel state information (CSI) acquisition for FASs is essential. While some techniques have been proposed, most existing FAS channel estimators require several channel assumptions, such as slow variation and angular-domain sparsity. When these assumptions are not reasonable, the model mismatch may lead to unpredictable performance loss. In this paper, we propose the successive Bayesian reconstructor (S-BAR) as a general solution to estimate FAS channels. Unlike model-based estimators, the proposed S-BAR is prior-aided, which builds the experiential kernel for CSI acquisition. Inspired by Bayesian regression, the key idea of S-BAR is to model the FAS channels as a stochastic process, whose uncertainty can be successively eliminated by kernel-based sampling and regression. In this way, the predictive mean of the regressed stochastic process can be viewed as the maximum a posterior (MAP) estimator of FAS channels. Simulation results verify that, in both model-mismatched and model-matched cases, the proposed S-BAR can achieve higher estimation accuracy than the existing schemes., Comment: Accepted by IEEE WCNC 2024. This paper proposes S-BAR as a general solution to estimate FAS channels. More insights can be found in the journal version of this paper: arXiv:2312.06551. arXiv admin note: substantial text overlap with arXiv:2312.06551

The paper focuses on real-time facial expression recognition (FER) systems as an important component in various real-world applications such as social robotics. We investigate two hardware options for the deployment of FER machine learning (ML) models at the edge: neuromorphic hardware versus edge AI accelerators. Our study includes exhaustive experiments providing comparative analyses between the Intel Loihi neuromorphic processor and four distinct edge platforms: Raspberry Pi-4, Intel Neural Compute Stick (NSC), Jetson Nano, and Coral TPU. The results obtained show that Loihi can achieve approximately two orders of magnitude reduction in power dissipation and one order of magnitude energy savings compared to Coral TPU which happens to be the least power-intensive and energy-consuming edge AI accelerator. These reductions in power and energy are achieved while the neuromorphic solution maintains a comparable level of accuracy with the edge accelerators, all within the real-time latency requirements.

Background. The Expected Value of Sample Information (EVSI) measures the expected benefits that could be obtained by collecting additional data. Estimating EVSI using the traditional nested Monte Carlo method is computationally expensive but the recently developed Gaussian approximation (GA) approach can efficiently estimate EVSI across different sample sizes. However, the conventional GA may result in biased EVSI estimates if the decision models are highly nonlinear. This bias may lead to suboptimal study designs when GA is used to optimize the value of different studies. Therefore, we extend the conventional GA approach to improve its performance for nonlinear decision models. Methods. Our method provides accurate EVSI estimates by approximating the conditional benefit based on two steps. First, a Taylor series approximation is applied to estimate the conditional benefit as a function of the conditional moments of the parameters of interest using a spline, which is fitted to the samples of the parameters and the corresponding benefits. Next, the conditional moments of parameters are approximated by the conventional GA and Fisher information. The proposed approach is applied to several data collection exercises involving non-Gaussian parameters and nonlinear decision models. Its performance is compared with the nested Monte Carlo method, the conventional GA approach, and the nonparametric regression-based method for EVSI calculation. Results. The proposed approach provides accurate EVSI estimates across different sample sizes when the parameters of interest are non-Gaussian and the decision models are nonlinear. The computational cost of the proposed method is similar to other novel methods. Conclusions. The proposed approach can estimate EVSI across sample sizes accurately and efficiently, which may support researchers in determining an economically optimal study design using EVSI., Comment: 11 pages, 2 figures, presented at 44th Medical Decision Making Annual North American Meeting

Configuring the hybrid precoders and combiners in a millimeter wave (mmWave) multiuser (MU) multiple-input multiple-output (MIMO) system is challenging in frequency selective channels. In this paper, we develop a system that uses compressive estimation on the uplink to configure precoders and combiners for the downlink (DL). In the first step, the base station (BS) simultaneously estimates the channels from all the mobile stations (MSs) on each subcarrier. To reduce the number of measurements required, compressed sensing techniques are developed that exploit common support on the different subcarriers. In the second step, exploiting reciprocity and the channel estimates, the base station designs hybrid precoders and combiners. Two algorithms are developed for this purpose, with different performance and complexity tradeoffs: 1) a factorization of the purely digital solution, and 2) an iterative hybrid design. Extensive numerical experiments evaluate the proposed solutions comparing to state-of-the-art strategies, and illustrating design tradeoffs in overhead, complexity, and performance.

The effective sample size (ESS) measures the informational value of a probability distribution in terms of an equivalent number of study participants. The ESS plays a crucial role in estimating the Expected Value of Sample Information (EVSI) through the Gaussian approximation approach. Despite the significance of ESS, existing ESS estimation methods within the Gaussian approximation framework are either computationally expensive or potentially inaccurate. To address these limitations, we propose a novel approach that estimates the ESS using the summary statistics of generated datasets and nonparametric regression methods. The simulation results suggest that the proposed method provides accurate ESS estimates at a low computational cost, making it an efficient and practical way to quantify the information contained in the probability distribution of a parameter. Overall, determining the ESS can help analysts understand the uncertainty levels in complex prior distributions in the probability analyses of decision models and perform efficient EVSI calculations., Comment: 5 pages

We introduce a new tool useful for greedy coloring, which we call the forb-flex method, and apply it to odd coloring and proper conflict-free coloring of planar graphs. The odd chromatic number, denoted $\chi_{\mathsf{o}}(G)$, is the smallest number of colors needed to properly color $G$ such that every non-isolated vertex of $G$ has a color appearing an odd number of times in its neighborhood. The proper conflict-free chromatic number, denoted $\chi_{\mathsf{PCF}}(G)$, is the smallest number of colors needed to properly color $G$ such that every non-isolated vertex of $G$ has a color appearing uniquely in its neighborhood. Our new tool works by carefully counting the structures in the neighborhood of a vertex and determining if a neighbor of a vertex can be recolored at the end of a greedy coloring process to avoid conflicts. Combining this with the discharging method allows us to prove $\chi_{\mathsf{PCF}}(G) \leq 4$ for planar graphs of girth at least 11, and $\chi_{\mathsf{o}}(G) \leq 4$ for planar graphs of girth at least 10. These results improve upon the recent works of Cho, Choi, Kwon, and Park., Comment: 32 pages, 11 figures

In this paper, we investigate finite solvable tidy groups. We classify the tidy $\{ p, q \}$-groups. Combining this with a previous result, we are able to characterize the finite tidy solvable groups. Using this characterization, we bound the Fitting height of finite tidy solvable groups and we prove that the quotients of finite tidy solvable groups are tidy.