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Model counting is a fundamental problem in many practical applications, including query evaluation in probabilistic databases and failure-probability estimation of networks. In this work, we focus on a variant of this problem where the underlying formula is expressed in the Disjunctive Normal Form (DNF), also known as #DNF. This problem has been shown to be #P-complete, making it often intractable to solve exactly. Much research has therefore focused on obtaining approximate solutions, particularly in the form of $(\varepsilon, \delta)$ approximations. The primary contribution of this paper is a new approach, called pepin, an approximate #DNF counter that significantly outperforms prior state-of-the-art approaches. Our work is based on the recent breakthrough in the context of the union of sets in the streaming model. We demonstrate the effectiveness of our approach through extensive experiments and show that it provides an affirmative answer to the challenge of efficiently computing #DNF., Comment: 13 pages, 7 Figures

Coronal jets are one of the most common eruptive activities in the solar atmosphere. They are related to rich physics processes, including but not limited to magnetic reconnection, flaring, instabilities, and plasma heating. Automated identification of off-limb coronal jets has been difficult due to their abundant nature, complex appearance, and relatively small size compared to other features in the corona. In this paper, we present an automated coronal jet identification algorithm (AJIA) that utilizes true and fake jets previously detected by a laborious semi-automated jet detection algorithm (SAJIA, Liu et al. 2023) as the input of an image segmentation neural network U-NET. It is found that AJIA could achieve a much higher (0.81) detecting precision than SAJIA (0.34), meanwhile giving the possibility of whether each pixel in an input image belongs to a jet. We demonstrate that with the aid of artificial neural networks, AJIA could enable fast, accurate, and real-time coronal jet identification from SDO/AIA 304 \AA observations, which are essential in studying the collective and long-term behavior of coronal jets and their relation with the solar activity cycles., Comment: 13 pages, 7 figures

Approximate model counting is the task of approximating the number of solutions to an input Boolean formula. The state-of-the-art approximate model counter for formulas in conjunctive normal form (CNF), ApproxMC, provides a scalable means of obtaining model counts with probably approximately correct (PAC)-style guarantees. Nevertheless, the validity of ApproxMC's approximation relies on a careful theoretical analysis of its randomized algorithm and the correctness of its highly optimized implementation, especially the latter's stateful interactions with an incremental CNF satisfiability solver capable of natively handling parity (XOR) constraints. We present the first certification framework for approximate model counting with formally verified guarantees on the quality of its output approximation. Our approach combines: (i) a static, once-off, formal proof of the algorithm's PAC guarantee in the Isabelle/HOL proof assistant; and (ii) dynamic, per-run, verification of ApproxMC's calls to an external CNF-XOR solver using proof certificates. We detail our general approach to establish a rigorous connection between these two parts of the verification, including our blueprint for turning the formalized, randomized algorithm into a verified proof checker, and our design of proof certificates for both ApproxMC and its internal CNF-XOR solving steps. Experimentally, we show that certificate generation adds little overhead to an approximate counter implementation, and that our certificate checker is able to fully certify $84.7\%$ of instances with generated certificates when given the same time and memory limits as the counter., Comment: The extended version, including the appendix, of the paper to be published in CAV24. The associated artifact is available at https://doi.org/10.5281/zenodo.10948839

The frustrated ladder with alternate ferromagnetic(F) exchange $-J_F$ and AF exchange $J_A$ to first neighbors and F exchange $-J_L$ to second neighbors is studied by exact diagonalization (ED) and density matrix renormalization group (DMRG) calculations in systems of $2N$ spins-1/2 with periodic boundary conditions. The ground state is a singlet $(S = 0)$ and the singlet-triplet gap $\varepsilon_T$ is finite for the exchanges considered. Spin-1/2 string correlation functions $g_1(N)$ and $g_2(N)$ are defined for an even number $N$ of consecutive spins in systems with two spins per unit cell; the ladder has string order $g_2(\infty)> 0$ and $g_1(\infty) = 0$. The minimum $N^*$ of $g_2(N)$ is related to the range of ground-state spin correlations. Convergence to $g_2(\infty)$ is from below, and $g_1(N)$ decreases exponentially for $N \geq N^*$. Singlet valence bond (VB) diagrams account for the size dependencies. The frustrated ladder at special values of $J_F$, $J_L$ and $J_A$ reduces to well-known models such as the spin-1 Heisenberg antiferromagnet and the $J_1-J_2$ model, among others. Numerical analysis of ladders matches previous results for spin-1 gaps or string correlation functions and extends them to spin-1/2 systems. The nondegenerate singlet ground state of ladder is a bond-order wave, a Kekul\'e VB diagrams at $J_L = J_F/2 \leq J_A$, that is reversed on interchanging $-J_F$ and $J_A$. Inversion symmetry is spontaneously broken in the dimer phase of the $J_1-J_2$ model where the Kekul\'e diagrams are the doubly degenerate ground states at $J_2/J_1 = 1/2$., Comment: 12 pages, 11 figures

In this brief review, we introduce a new spin ladder system called skewed spin ladders and discuss the exotic quantum phases of this system. The spin ladders studied are the 5/7, 3/4 and 3/5 systems corresponding to alternately fused 5 and 7 membered rings; 3 and 4 membered rings; and 3 and 5 membered rings. These ladders show completely different behaviour as the Hamiltonian model parameter is changed. When the Hamiltonian parameter is increased the 5/7 ladder switches from an initial singlet ground state to progressively higher spin ground state and then to a reentrant singlet state before finally settling to the highest spin ground state whose spin equals the number of unit cells in the system. The 3/4 ladder goes from a singlet ground state to a high spin ground state with each unit cell contributing spin 1 to the state, as the model parameter is increased. The 3/5 ladder shows a singlet ground state for small parameters and high spin ground state for intermediate values of the parameter and for still higher parameters, a reentrant singlet ground state. They can also show interesting magnetization plateaus as illustrated by studies on a specific spin ladder.

Photomultiplier tubes (PMTs) are traditionally an integral part of large underground experiments as they measure the light emission from particle interactions within the enclosed detection media. The BUTTON experiment will utilise around 100 PMTs to measure the response of different media suitable for rare event searches. A subset of low-radioactivity 10-inch Hamamatsu R7081 PMTs were tested, characterised, and compared to manufacture certification. This manuscript describes the laboratory tests and analysis of gain, peak-to-valley ratio and dark rate of the PMTs to give an understanding of the charge response, signal-to-noise ratio and dark noise background as an acceptance test of the suitability of these PMTs for water-based detectors. Following the evaluation of these tests, the PMT performance agreed with the manufacturer specifications. These results are imperative for modeling the PMT response in detector simulations and providing confidence in the performance of the devices once installed in the detector underground., Comment: 11 pages, 11 figures

The past three decades have witnessed notable success in designing efficient SAT solvers, with modern solvers capable of solving industrial benchmarks containing millions of variables in just a few seconds. The success of modern SAT solvers owes to the widely-used CDCL algorithm, which lacks comprehensive theoretical investigation. Furthermore, it has been observed that CDCL solvers still struggle to deal with specific classes of benchmarks comprising only hundreds of variables, which contrasts with their widespread use in real-world applications. Consequently, there is an urgent need to uncover the inner workings of these seemingly weak yet powerful black boxes. In this paper, we present a first step towards this goal by introducing an approach called CausalSAT, which employs causal reasoning to gain insights into the functioning of modern SAT solvers. CausalSAT initially generates observational data from the execution of SAT solvers and learns a structured graph representing the causal relationships between the components of a SAT solver. Subsequently, given a query such as whether a clause with low literals blocks distance (LBD) has a higher clause utility, CausalSAT calculates the causal effect of LBD on clause utility and provides an answer to the question. We use CausalSAT to quantitatively verify hypotheses previously regarded as "rules of thumb" or empirical findings such as the query above. Moreover, CausalSAT can address previously unexplored questions, like which branching heuristic leads to greater clause utility in order to study the relationship between branching and clause management. Experimental evaluations using practical benchmarks demonstrate that CausalSAT effectively fits the data, verifies four "rules of thumb", and provides answers to three questions closely related to implementing modern solvers., Comment: 17 pages, 3 figures, to be published in SAT23

Dedicated treatment of symmetries in satisfiability problems (SAT) is indispensable for solving various classes of instances arising in practice. However, the exploitation of symmetries usually takes a black box approach. Typically, off-the-shelf external, general-purpose symmetry detection tools are invoked to compute symmetry groups of a formula. The groups thus generated are a set of permutations passed to a separate tool to perform further analyzes to understand the structure of the groups. The result of this second computation is in turn used for tasks such as static symmetry breaking or dynamic pruning of the search space. Within this pipeline of tools, the detection and analysis of symmetries typically incurs the majority of the time overhead for symmetry exploitation. In this paper we advocate for a more holistic view of what we call the SAT-symmetry interface. We formulate a computational setting, centered around a new concept of joint graph/group pairs, to analyze and improve the detection and analysis of symmetries. Using our methods, no information is lost performing computational tasks lying on the SAT-symmetry interface. Having access to the entire input allows for simpler, yet efficient algorithms. Specifically, we devise algorithms and heuristics for computing finest direct disjoint decompositions, finding equivalent orbits, and finding natural symmetric group actions. Our algorithms run in what we call instance-quasi-linear time, i.e., almost linear time in terms of the input size of the original formula and the description length of the symmetry group returned by symmetry detection tools. Our algorithms improve over both heuristics used in state-of-the-art symmetry exploitation tools, as well as theoretical general-purpose algorithms., Comment: appeared at SAT 2023; second version corrects some minor inaccuracies

The magnetization $M(h)$ is used to identify three singlet quantum phases of the ladder with isotropic exchange interactions. The Dimer phase with frustrated F exchanges in rungs and legs has a first-order $M(h)$ transition at $0$ K from singlet to ferromagnetic at the saturation field $h_s$. The Haldane-DAF phase with strong F exchange in rungs and net AF exchange between rungs has continuous $M(h)$ and is adiabatically connected to the $S = 1$ Heisenberg AF chain. The AF phase with strong F exchange in legs and net AF exchange between legs has continuous $M(h)$ and is adiabatically connected to the spin-1/2 $J_1-J_2$ model with $J_1 > 0$ and $J_2 < 0$. All three singlet phases have finite gaps to the lowest triplet state., Comment: 6 pages, 4 figues

Traditional Boolean satisfiability (SAT) solvers based on the conflict-driven clause-learning (CDCL) framework fare poorly on formulas involving large numbers of parity constraints. The CryptoMiniSat solver augments CDCL with Gauss-Jordan elimination to greatly improve performance on these formulas. Integrating the TBUDDY proof-generating BDD library into CryptoMiniSat enables it to generate unsatisfiability proofs when using Gauss-Jordan elimination. These proofs are compatible with standard, clausal proof frameworks., Comment: Presented at 2022 Workshop on the Pragmatics of SAT

Power-law distributions have been studied as a significant characteristic of non-linear dissipative systems. Since discovering the power-law distribution of solar flares that was later extended to nano-flares and stellar flares, it has been widely accepted that different scales of flares share the same physical process. Here, we present the newly developed Semi-Automated Jet Identification Algorithm (SAJIA) and its application for detecting more than 1200 off-limb solar jets during Solar Cycle 24. Power-law distributions have been revealed between the intensity/energy and frequency of these events, with indices found to be analogous to those for flares and coronal mass ejections (CMEs). These jets are also found to be spatially and temporally modulated by the solar cycle forming a butterfly diagram in their latitudinal-temporal evolution, experiencing quasi-annual oscillations in their analysed properties, and very likely gathering in certain active longitudinal belts. Our results show that coronal jets display the same nonlinear behaviour as that observed in flares and CMEs, in solar and stellar atmospheres, strongly suggesting that they result from the same nonlinear statistics of scale-free processes as their counterparts in different scales of eruptive events. Although these jets, like flares and other large-scale dynamic phenomena, are found to be significantly modulated by the solar cycle, their corresponding power-law indices still remain similar., Comment: 10 figures, 1 table, to be published in ApJ Supplement

We investigate the origin in the early Solar System of the short-lived radionuclide 244Pu (with a half life of 80 Myr) produced by the rapid (r) neutron-capture process. We consider two large sets of r-process nucleosynthesis models and analyse if the origin of 244Pu in the ESS is consistent with that of the other r and slow (s) neutron-capture process radioactive nuclei. Uncertainties on the r-process models come from both the nuclear physics input and the astrophysical site. The former strongly affects the ratios of isotopes of close mass (129I/127I, 244Pu/238U, and 247Pu/235U). The 129I/247Cm ratio, instead, which involves isotopes of a very different mass, is much more variable than those listed above and is more affected by the physics of the astrophysical site. We consider possible scenarios for the evolution of the abundances of these radioactive nuclei in the galactic interstellar medium and verify under which scenarios and conditions solutions can be found for the origin of 244Pu that are consistent with the origin of the other isotopes. Solutions are generally found for all the possible different regimes controlled by the interval ($\delta$) between additions from the source to the parcel of interstellar medium gas that ended up in the Solar System, relative to decay timescales. If r-process ejecta in interstellar medium are mixed within a relatively small area (leading to a long $\delta$), we derive that the last event that explains the 129I and 247Cm abundances in the early Solar System can also account for the abundance of 244Pu. Due to its longer half life, however, 244Pu may have originated from a few events instead of one only. If r-process ejecta in interstellar medium are mixed within a relatively large area (leading to a short $\delta$), we derive that the time elapsed from the formation of the molecular cloud to the formation of the Sun was 9-16 Myr., Comment: 17 pages, 3 figures, published on Universe as part of the Special Issue "Nuclear Astrophysics in the Era of High Precision Astronomy"

The spin$-1/2$ chain with ferromagnetic exchange $J_1 < 0$ between first neighbors and antiferromagnetic $J_2 > 0$ between second neighbors supports two spin-Peierls (SP) instabilities depending on the frustration $\alpha = J_2/\vert J_1\vert$. Instead of chain dimerization with two spins per unit cell, $J_1-J_2$ models with $\alpha > 0.65$ and linear spin-phonon coupling are unconditionally unstable to sublattice dimerization with four spins per unit cell. Unequal $J_1$ to neighbors to the right and left extends the model to gapped ($\gamma > 0$) chains with conditional SP transitions at $T_{SP}$ to dimerized sublattices and a weaker specific heat $C(T)$ anomaly. The spin susceptibility $\chi(T)$ and $C(T)$ are obtained in the thermodynamic limit by a combination of exact diagonalization of small systems with $\alpha > 0.65$ and density matrix renormalization group (DMRG) calculations of systems up to $N \sim 100$ spins. Both $J_1-J_2$ and $\gamma > 0$ models account quantitatively for $\chi(T)$ and $C(T)$ in the paramagnetic phase of $\beta$-TeVO$_{4}$ for $T > 8$ K, but lower $T$ indicates a gapped chain instead of a $J_1-J_2$ model as previously thought. The same parameters and $T_{SP} = 4.6$ K generate a $C(T)/T$ anomaly that reproduces the anomaly at the $4.6$ K transition of $\beta$-TeVO$_{4}$, but not the weak $\chi(T)$ signature., Comment: 12 pages, 13 figures

Radioactive nuclei are the key to understanding the circumstances of the birth of our Sun because meteoritic analysis has proven that many of them were present at that time. Their origin, however, has been so far elusive. The ERC-CoG-2016 RADIOSTAR project is dedicated to investigating the production of radioactive nuclei by nuclear reactions inside stars, their evolution in the Milky Way Galaxy, and their presence in molecular clouds. So far, we have discovered that: (i) radioactive nuclei produced by slow ($^{107}$Pd and $^{182}$Hf) and rapid ($^{129}$I and $^{247}$Cm) neutron captures originated from stellar sources - asymptotic giant branch (AGB) stars and compact binary mergers, respectively - within the galactic environment that predated the formation of the molecular cloud where the Sun was born; (ii) the time that elapsed from the birth of the cloud to the birth of the Sun was of the order of 10$^7$ years, and (iii) the abundances of the very short-lived nuclei $^{26}$Al, $^{36}$Cl, and $^{41}$Ca can be explained by massive star winds in single or binary systems, if these winds directly polluted the early Solar System. Our current and future work, as required to finalise the picture of the origin of radioactive nuclei in the Solar System, involves studying the possible origin of radioactive nuclei in the early Solar System from core-collapse supernovae, investigating the production of $^{107}$Pd in massive star winds, modelling the transport and mixing of radioactive nuclei in the galactic and molecular cloud medium, and calculating the galactic chemical evolution of $^{53}$Mn and $^{60}$Fe and of the p-process isotopes $^{92}$Nb and $^{146}$Sm., Comment: accepted for publication on Universe Special Issue AGB Stars - In Honor of Professor Maurizio Busso on the Occasion of His 70th Birthday

Barium (Ba) stars are polluted by material enriched in the slow neutron capture (s-process) elements synthesised in the interior of their former asymptotic giant branch (AGB) companion star, which is now a white dwarf. We compare individual Ba star abundance patterns to AGB nucleosynthesis models to verify if the AGB model mass is compatible with independently derived AGB mass. We selected a sample of 28 Ba stars for which both self-consistent spectroscopic observation and analysis are available and stellar mass determinations, via positioning the star on the HR diagram and comparing with evolutionary tracks. For this sample stars we considered both previously and recently derived elemental abundances. Then, we performed a detailed comparison of these s-process elemental abundances to different AGB nucleosynthesis models from the Monash and the FRUITY theoretical data sets. We simplified the binary mass transfer by calculating dilution factors to match the [Ce/Fe] value of each star when using different AGB models, and we then compared the diluted model abundances to the complete Ba-star abundance pattern. Our comparison confirms that low mass, non-rotating AGB stellar models with 13C as the main neutron source are the polluters of the vast majority of the considered Ba stars. Out of the 28 stars, in 21 cases the models are in good agreement with both the determined abundances and the independently derived AGB mass, although in 16 cases higher observed abundances of Nb, Ru, Mo and/or Nd, Sm than predicted. For 3 stars we obtain a match to the abundances only by considering models with masses lower than those independently determined. Finally, 4 stars show much higher first s-process peak abundance values than the model predictions, which may represent the signature of a physical and/or nucleosynthetic process that is not represented in the set of models considered here., Comment: accepted for publication in A&A

Observational pre-cursors of large solar flares provide a basis for future operational systems for forecasting. Here, we study the evolution of the normalized emergence (EM), shearing (SH) and total (T) magnetic helicity flux components for 14 flaring with at least one X-class flare) and 14 non-flaring ($<$ M5-class flares) active regions (ARs) using the Spaceweather Helioseismic Magnetic Imager Active Region Patches vector magnetic field data. Each of the selected ARs contain a $\delta$-type spot. The three helicity components of these ARs were analyzed using wavelet analysis. Localised peaks of the wavelet power spectrum (WPS) were identified and statistically investigated. We find that: i) the probability density function of the identified WPS peaks for all the EM, SH and T profiles can be fitted with a set of Gaussian functions centered at distinct periods between $\sim$ 3 to 20 hours. ii) There is a noticeable difference in the distribution of periods found in the EM profiles between the flaring and non-flaring ARs, while no significant difference is found in the SH and T profiles. iii) In flaring ARs, the distributions of the shorter EM/SH/T periods ($<$ 10 hrs) split up into two groups after flares, while the longer periods ($>$ 10 hrs) do not change. iv) When the EM periodicity does not contain harmonics, the ARs do not host a large energetic flare. Finally, v) significant power at long periods ($\sim$ 20 hour) in the T and EM components may serve as pre-cursor for large energetic flares.

Analysis of inclusions in primitive meteorites reveals that several short-lived radionuclides (SLRs) with half-lives $0.1-100$ Myr existed in the early Solar System (ESS). We investigate the ESS origin of $^{107}$Pd, $^{135}$Cs, and $^{182}$Hf, which are produced by $slow$ neutron captures (the $s$-process) in asymptotic giant branch (AGB) stars. We modelled the galactic abundances of these SLRs using the OMEGA+ galactic chemical evolution (GCE) code and two sets of mass- and metallicity-dependent AGB nucleosynthesis yields (Monash and FRUITY). Depending on the ratio of the mean life $\tau$ of the SLR to the average length of time between the formation of AGB progenitor $\gamma$, we calculate timescales relevant for the birth of the Sun. If $\tau/\gamma\gtrsim2$, we predict self-consistent isolation times between 9 and 26 Myr by decaying the GCE predicted $^{107}$Pd/$^{108}$Pd, $^{135}$Cs/$^{133}$Cs, and $^{182}$Hf/$^{180}$Hf ratios to their respective ESS ratios. The predicted $^{107}$Pd/$^{182}$Hf ratio indicates that our GCE models are missing $9-73\%$ of $^{107}$Pd and $^{108}$Pd in the ESS. This missing component may have come from AGB stars of higher metallicity than those that contributed to the ESS in our GCE code. If $\tau/\gamma\lesssim0.3$, we calculate instead the time ($T_{\rm LE}$) from the last nucleosynthesis event that added the SLRs into the presolar matter to the formation of the oldest solids in the ESS. For the 2 M$_\odot$, $Z=0.01$ Monash model we find a self-consistent solution of $T_{\rm LE}=25.5$ Myr.

Given a Boolean formula $\varphi$ over the set of variables $X$ and a projection set $\mathcal{P} \subseteq X$, a subset of variables $\mathcal{I}$ is independent support of $\mathcal{P}$ if two solutions agree on $\mathcal{I}$, then they also agree on $\mathcal{P}$. The notion of independent support is related to the classical notion of definability dating back to 1901, and have been studied over the decades. Recently, the computational problem of determining independent support for a given formula has attained importance owing to the crucial importance of independent support for hashing-based counting and sampling techniques. In this paper, we design an efficient and scalable independent support computation technique that can handle formulas arising from real-world benchmarks. Our algorithmic framework, called Arjun, employs implicit and explicit definability notions, and is based on a tight integration of gate-identification techniques and assumption-based framework. We demonstrate that augmenting the state of the art model counter ApproxMC4 and sampler UniGen3 with Arjun leads to significant performance improvements. In particular, ApproxMC4 augmented with Arjun counts 387 more benchmarks out of 1896 while UniGen3 augmented with Arjun samples 319 more benchmarks within the same time limit.

The antiferromagnetic $J_1-J_2$ model is a spin-1/2 chain with isotropic exchange $J_1 > 0$ between first neighbors and $J_2 = \alpha J_1$ between second neighbors. The model supports both gapless quantum phases with nondegenerate ground states and gapped phases with $\Delta(\alpha) > 0$ and doubly degenerate ground states. Exact thermodynamics is limited to $\alpha = 0$, the linear Heisenberg antiferromagnet (HAF). Exact diagonalization of small systems at frustration $\alpha$ followed by density matrix renormalization group (DMRG) calculations returns the entropy density $S(T,\alpha,N)$ and magnetic susceptibility $\chi(T,\alpha,N)$ of progressively larger systems up to $N = 96$ or 152 spins. Convergence to the thermodynamics limit, $S(T,\alpha)$ or $\chi(T,\alpha)$, is demonstrated down to $T/J \sim 0.01$ in the sectors $\alpha < 1$ and $\alpha > 1$. $S(T,\alpha)$ yields the critical points between gapless phases with $S^\prime(0,\alpha) > 0$ and gapped phases with $S^\prime(0,\alpha) = 0$. The $S^\prime(T,\alpha)$ maximum at $T^*(\alpha)$ is obtained directly in chains with large $\Delta(\alpha)$ and by extrapolation for small gaps. A phenomenological approximation for $S(T,\alpha)$ down to $T = 0$ indicates power-law deviations $T^{-\gamma(\alpha)}$ from $\exp(-\Delta(\alpha)/T)$ with exponent $\gamma(\alpha)$ that increases with $\alpha$. The $\chi(T,\alpha)$ analysis also yields power-law deviations, but with exponent $\eta(\alpha)$ that decreases with $\alpha$. $S(T,\alpha)$ and the spin density $\rho(T,\alpha) = 4T\chi(T,\alpha)$ probe the thermal and magnetic fluctuations, respectively, of strongly correlated spin states. Gapless chains have constant $S(T,\alpha)/\rho(T,\alpha)$ for $T < 0.10$. Remarkably, the ratio decreases (increases) with $T$ in chains with large (small) $\Delta(\alpha)$., Comment: 12 pages, 14 figures

The low temperature thermodynamics of correlated 1D fermionic models with spin and charge degrees of freedom is obtained by exact diagonalization (ED) of small systems and followed by density matrix renormalization group (DMRG) calculations that target the lowest hundreds of states $\{E(N)\}$ at system size $N$ instead of the ground state. Progressively larger $N$ reaches $T < 0.05t$ in correlated models with electron transfer $t$ between first neighbors and bandwidth $4t$. The size dependence of the many-fermion basis is explicitly included for arbitrary interactions by scaling the partition function. The remaining size dependence is then entirely due to the energy spectrum $\{E(N)\}$ of the model. The ED/DMRG method is applied to Hubbard and extended Hubbard models, both gapped and gapless, with $N_e = N$ or $N/2$ electrons and is validated against exact results for the magnetic susceptibility $\chi(T)$ and entropy $S(T)$ per site. Some limitations of the method are noted. Special attention is given to the bond-order-wave phase of the extended Hubbard model with competing interactions and low $T$ thermodynamics sensitive to small gaps., Comment: 14 figures