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We study a pickup-and-delivery problem that arises when customers randomly submit requests over the course of a day from a choice of vendors on a collaborative e-commerce portal. Based on the attributes of a customer request, a dispatcher dynamically schedules the delivery service on either a dedicated vehicle or a crowdshipper, both of whom experience time dependent travel times. While dedicated vehicles are available throughout the day, the availability of crowdshippers is unknown a priori and they appear randomly for only portions of the day. With an objective of minimizing the sum of routing costs, piece-rate crowdshipper payments, and lateness charges, we model the uncertainty in request arrivals and crowdshipper appearances as a Markov decision process. To determine an action at each decision epoch, we employ a heuristic that partially destroys the existing routes and repairs them guided by a parameterized cost function approximation that accounts for the remaining temporal capacity of delivery vehicles. Through a set of computational experiments, we demonstrate the improvement of our approach over a myopic approach in several key performance metrics. In addition, we conduct computational experiments demonstrate the impact of inserting wait time in the route scheduling and the benefit of explicitly modeling time-dependent travel times. Through our computational testing, we also investigate the effect of demand management mechanisms that facilitate many-to-one request bundles or one-to-many request bundles on reducing the cost to service requests.

This article is the continuation of \cite{ohlmann2021illposedness} where we exhibited the ill-posedness of a quasi-linear wave equation in dimension $2$ for initial data in $H^{7/4}(\ln H^{7/4})$. Here, we look at modifications of the equation and show that the blow-up phenomenon still occurs. First, we study another equation with the same characteristics but a different underlying ODE. Later, we study an equation where a $x_2$ dependency is introduced. The latter case constitutes the main contribution of this paper, as we are able to show that the solution still behaves pathologically without having an explicit formula for either the characteristics or the values of the solution. Finally, we study the case where a perturbation of the initial data is introduced.

Stellar mergers are one important path to highly magnetised stars. Mergers of two low-mass white dwarfs may create up to every third hot subdwarf star. The merging process is usually assumed to dramatically amplify magnetic fields. However, so far only four highly magnetised hot subdwarf stars have been found, suggesting a fraction of less than $1\%$. We present two high-resolution magnetohydrodynamical (MHD) simulations of the merger of two helium white dwarfs in a binary system with the same total mass of $0.6\,M_\odot$. We analyse one equal-mass merger with two $0.3\,M_\odot$ white dwarfs, and one unequal-mass merger with a $0.25\,M_\odot$ white dwarf and a $0.35\,M_\odot$ white dwarf. We simulate the inspiral, merger, and further evolution of the merger remnant for about $50$ rotations. We find efficient magnetic field amplification in both mergers via a small-scale dynamo, reproducing previous results of stellar merger simulations. The magnetic field saturates at similar strength for both simulations. We then identify a second phase of magnetic field amplification in both merger remnants that happens on a timescale of several tens of rotational periods of the merger remnant. This phase generates a large-scale ordered azimuthal field. We identify it as a large-scale dynamo driven by the magneto-rotational instability (MRI). Finally, we suggest that in the unequal-mass merger remnant, helium burning will eventually start in a shell around a cold core. The convection zone this generates will coincide with the region that contains most of the magnetic energy, probably erasing the strong, ordered field. The equal-mass merger remnant instead will probably ignite burning in the center, retaining its ordered field. Therefore, the mass ratio of the initial merger could be the selecting factor that decides if a merger remnant will stay highly magnetised long after the merger., Comment: 12 pages, 9 figures, submitted to A&A, comments welcome

In the context of two-player games over graphs, a language $L$ is called positional if, in all games using $L$ as winning objective, the protagonist can play optimally using positional strategies, that is, strategies that do not depend on the history of the play. In this work, we describe the class of parity automata recognising positional languages, providing a complete characterisation of positionality for $\omega$-regular languages. As corollaries, we establish decidability of positionality in polynomial time, finite-to-infinite and 1-to-2-players lifts, and show the closure under union of prefix-independent positional objectives, answering a conjecture by Kopczy\'nski in the $\omega$-regular case., Comment: Full version of a paper accepted at LICS 2024

We propose to study transformations on graphs, and more generally structures, by looking at how the cut-rank (as introduced by Oum) of subsets is affected when going from the input structure to the output structure. We consider transformations in which the underlying sets are the same for both the input and output, and so the cut-ranks of subsets can be easily compared. The purpose of this paper is to give a characterisation of logically defined transductions that is expressed in purely structural terms, without referring to logic: transformations which decrease the cut-rank, in the asymptotic sense, are exactly those that can be defined in monadic second-order logic. This characterisation assumes that the transduction has inputs of bounded treewidth; we also show that the characterisation fails in the absence of any assumptions., Comment: 22 pages, 13 figures

In this article, we study the well-posedness of the energy-critical half-wave maps equation (HWM) in dimension $1$. The half-wave maps equation emerges from the continuum limit of the Haldane Shastry spin chains and has been shown to arise as the continuum limit of Calogero-Moser classical spin systems. In higher dimension $d\geq 5$, it has been shown that (HWM) is well-posed by Krieger and Sire. This result has been improved by Krieger and Kiesenhofer to $d = 4$ but the Strichartz estimate on which the argument is built no longer holds for smaller dimensions. A Lax-pair structure has been revealed for this equation by Lenzmann and G\'erard, indicating complete integrability and the fact that rational solutions stay rational for all time. The well-posedness of the (HWM) equation in lower dimensions remains an open problem. Here, we show the well-posedness of (HWM) in the rational case for finite times with separated poles, and for large and infinite times with distinct speeds of propagation.

The merger of a white dwarf (WD) and a neutron star (NS) is a relatively common event that will produce an observable electromagnetic signal. Furthermore, the compactness of these stellar objects makes them an interesting candidate for gravitational wave (GW) astronomy, potentially being in the frequency range of LISA and other missions. To date, three-dimensional simulations of these mergers have not fully modelled the WD disruption, or have used lower resolutions and have not included magnetic fields even though they potentially shape the evolution of the merger remnant. In this work, we simulate the merger of a 1.4$M_\odot$ NS with a 1$M_\odot$ carbon oxygen WD in the magnetohydrodynamic moving mesh code \AREPO. We find that the disruption of the WD forms an accretion disk around the NS, and the subsequent accretion by the NS powers the launch of strongly magnetized, mildly relativistic jets perpendicular to the orbital plane. Although the exact properties of the jets could be altered by unresolved physics around the NS, the event could result in a transient with a larger luminosity than kilonovae. We discuss possible connections to fast blue optical transients (FBOTs) and long-duration gamma-ray bursts. We find that the frequency of GWs released during the merger is too high to be detectable by the LISA mission, but suitable for deci-hertz observatories such as LGWA, BBO or DECIGO., Comment: Accepted for publication in A&A. 13 pages, 11 figures

We study the existence of positional strategies for the protagonist in infinite duration games over arbitrary game graphs. We prove that prefix-independent objectives in {\Sigma}_0^2 which are positional and admit a (strongly) neutral letter are exactly those that are recognised by history-deterministic monotone co-B\"uchi automata over countable ordinals. This generalises a criterion proposed by [Kopczy\'nski, ICALP 2006] and gives an alternative proof of closure under union for these objectives, which was known from [Ohlmann, TheoretiCS 2023]. We then give two applications of our result. First, we prove that the mean-payoff objective is positional over arbitrary game graphs. Second, we establish the following completeness result: for any objective W which is prefix-independent, admits a (weakly) neutral letter, and is positional over finite game graphs, there is an objective W' which is equivalent to W over finite game graphs and positional over arbitrary game graphs.

We propose an algorithm that aims at minimizing the inter-node communication volume for distributed and memory-efficient tensor contraction schemes on modern multi-core compute nodes. The key idea is to define processor grids that optimize intra-/inter-node communication volume in the employed contraction algorithms. We present an implementation of the proposed node-aware communication algorithm into the Cyclops Tensor Framework (CTF). We demonstrate that this implementation achieves a significantly improved performance for matrix-matrix-multiplication and tensor-contractions on up to several hundreds modern compute nodes compared to conventional implementations without using node-aware processor grids. Our implementation shows good performance when compared with existing state-of-the-art parallel matrix multiplication libraries (COSMA and ScaLAPACK). In addition to the discussion of the performance for matrix-matrix-multiplication, we also investigate the performance of our node-aware communication algorithm for tensor contractions as they occur in quantum chemical coupled-cluster methods. To this end we employ a modified version of CTF in combination with a coupled-cluster code (Cc4s). Our findings show that the node-aware communication algorithm is also able to improve the performance of coupled-cluster theory calculations for real-world problems running on tens to hundreds of compute nodes., Comment: 15 pages, 4 figures

In stellar-dense environments, stars can collide with each other. For collisions close to a supermassive black hole (SMBH), the collisional kinetic energy can be so large that the colliding stars can be completely destroyed, potentially releasing an amount of energy comparable to that of a supernova. Such violent collisions, which we call BH-driven disruptive collisions (BDCs), have been examined mostly analytically, with the non-linear hydrodynamical effects being left largely unstudied. Using the moving-mesh hydrodynamics code {\small AREPO}, we investigate high-velocity ($>10^{3}$ km/s) collisions between 1M$_{\odot}$ giants with varying radii, impact parameters, and initial approaching velocities, and estimate their observables. Very strong shocks across the collision surface efficiently convert $\gtrsim10\%$ of the initial kinetic energy into radiation energy. The outcome is a gas cloud expanding supersonically, homologously, and quasi-spherically, generating a flare with a peak luminosity $\simeq 10^{41}-10^{44}$ erg/s in the extreme UV band ($\simeq 10$ eV). The luminosity decreases approximately following a power-law $t^{-0.7}$ initially, then $t^{-0.4}$ after $t\simeq$10 days at which point it would be bright in the optical band ($\lesssim 1$eV). Subsequent, and possibly even brighter, emission would be generated due to the accretion of the gas cloud onto the nearby SMBH, possibly lasting up to multi-year timescales. This inevitable BH-collision product interaction can contribute to the growth of BHs at all mass scales, in particular, seed BHs at high redshifts. Furthermore, the proximity of the events to the central BH makes them a potential tool for probing the existence of dormant BHs, even very massive ones which cannot be probed by tidal disruption events., Comment: 16 pages, 14 figures, 2 tables, accepted for publication in MNRAS, comments welcome, movies here: https://www.youtube.com/playlist?list=PLxLK3qI02cQd9lyIo6DIqm1tQnx_-G3Ut

This short note establishes positionality of mean-payoff games over infinite game graphs by constructing a well-founded monotone universal graph., Comment: 4 pages, 2 figures

We are witnessing the dawn of gravitational wave (GW) astronomy. With currently available detectors, observations are restricted to GW frequencies in the range between ${\sim} 10\,\mathrm{Hz}$ and $10\,\mathrm{kHz}$, which covers the signals from mergers of compact objects. The launch of the space observatory LISA will open up a new frequency band for the detection of stellar interactions at lower frequencies. In this work, we predict the shape and strength of the GW signals associated with common-envelope interaction and merger events in binary stars, and we discuss their detectability. Previous studies estimated these characteristics based on semi-analytical models. In contrast, we used detailed three-dimensional magnetohydrodynamic simulations to compute the GW signals. We show that for the studied models, the dynamical phase of common-envelope events and mergers between main-sequence stars lies outside of the detectability band of the LISA mission. We find, however, that the final stages of common-envelope interactions leading to mergers of the stellar cores fall into the frequency band in which the sensitivity of LISA peaks, making them promising candidates for detection. These detections can constrain the enigmatic common-envelope dynamics. Furthermore, future decihertz observatories such as DECIGO or BBO would also be able to observe this final stage and the post-merger signal, through which we might be able to detect the formation of Thorne-\.Zytkow objects., Comment: Accepted for publication in A&A, 12 pages, 8 figures

We use model-theoretic tools originating from stability theory to derive a result we call the Finitary Substitute Lemma, which intuitively says the following. Suppose we work in a stable graph class C, and using a first-order formula {\phi} with parameters we are able to define, in every graph G in C, a relation R that satisfies some hereditary first-order assertion {\psi}. Then we are able to find a first-order formula {\phi}' that has the same property, but additionally is finitary: there is finite bound k such that in every graph G in C, different choices of parameters give only at most k different relations R that can be defined using {\phi}'. We use the Finitary Substitute Lemma to derive two corollaries about the existence of certain canonical decompositions in classes of well-structured graphs. - We prove that in the Splitter game, which characterizes nowhere dense graph classes, and in the Flipper game, which characterizes monadically stable graph classes, there is a winning strategy for Splitter, respectively Flipper, that can be defined in first-order logic from the game history. Thus, the strategy is canonical. - We show that for any fixed graph class C of bounded shrubdepth, there is an O(n^2)-time algorithm that given an n-vertex graph G in C, computes in an isomorphism-invariant way a structure H of bounded treedepth in which G can be interpreted. A corollary of this result is an O(n^2)-time isomorphism test and canonization algorithm for any fixed class of bounded shrubdepth., Comment: 30 pages, 2 figures

A class of graphs $\mathscr{C}$ is monadically stable if for any unary expansion $\widehat{\mathscr{C}}$ of $\mathscr{C}$, one cannot interpret, in first-order logic, arbitrarily long linear orders in graphs from $\widehat{\mathscr{C}}$. It is known that nowhere dense graph classes are monadically stable; these encompass most of the studied concepts of sparsity in graphs, including graph classes that exclude a fixed topological minor. On the other hand, monadic stability is a property expressed in purely model-theoretic terms and hence it is also suited for capturing structure in dense graphs. For several years, it has been suspected that one can create a structure theory for monadically stable graph classes that mirrors the theory of nowhere dense graph classes in the dense setting. In this work we provide a step in this direction by giving a characterization of monadic stability through the Flipper game: a game on a graph played by Flipper, who in each round can complement the edge relation between any pair of vertex subsets, and Connector, who in each round localizes the game to a ball of bounded radius. This is an analog of the Splitter game, which characterizes nowhere dense classes of graphs (Grohe, Kreutzer, and Siebertz, J.ACM'17). We give two different proofs of our main result. The first proof uses tools from model theory, and it exposes an additional property of monadically stable graph classes that is close in spirit to definability of types. Also, as a byproduct, we give an alternative proof of the recent result of Braunfeld and Laskowski (arXiv 2209.05120) that monadic stability for graph classes coincides with existential monadic stability. The second proof relies on the recently introduced notion of flip-wideness (Dreier, M\"ahlmann, Siebertz, and Toru\'nczyk, arXiv 2206.13765) and provides an efficient algorithm to compute Flipper's moves in a winning strategy.

This paper is concerned with games of infinite duration played over potentially infinite graphs. Recently, Ohlmann (LICS 2022) presented a characterisation of objectives admitting optimal positional strategies, by means of universal graphs: an objective is positional if and only if it admits well-ordered monotone universal graphs. We extend Ohlmann's characterisation to encompass (finite or infinite) memory upper bounds. We prove that objectives admitting optimal strategies with $\varepsilon$-memory less than $m$ (a memory that cannot be updated when reading an $\varepsilon$-edge) are exactly those which admit well-founded monotone universal graphs whose antichains have size bounded by $m$. We also give a characterisation of chromatic memory by means of appropriate universal structures. Our results apply to finite as well as infinite memory bounds (for instance, to objectives with finite but unbounded memory, or with countable memory strategies). We illustrate the applicability of our framework by carrying out a few case studies, we provide examples witnessing limitations of our approach, and we discuss general closure properties which follow from our results., Comment: 52 pages, 21 figures

We develop a spatially realistic model of mutualistic metacommunities that exploits the joint structure of spatial and interaction networks. This model exhibits a sharp transition between a stable non-null equilibrium state and a global extinction state. This behaviour allows defining a threshold on colonisation/extinction parameters for the long-term metacommunity persistence. This threshold, the 'metacommunity capacity', extends the metapopulation capacity concept and can be calculated from the spatial and interaction networks without simulating the whole dynamics. In several applications we illustrate how the joint structure of the spatial and the interaction networks affects metacommunity capacity. It results that a weakly modular spatial network and a power-law degree distribution of the interaction network provide the most favourable configuration for the long-term persistence of a mutualistic metacommunity. Our model that encodes several explicit ecological assumptions should pave the way for a larger exploration of spatially realistic metacommunity models involving multiple interaction types.

Objectives: To compare artificial intelligence (AI) as a second reader in detecting lung nodules on chest X-rays (CXR) versus radiologists of two binational institutions, and to evaluate AI performance when using two different modes: automated versus assisted (additional remote radiologist review). Methods: The CXR public database (n = 247) of the Japanese Society of Radiological Technology with various types and sizes of lung nodules was analyzed. Eight radiologists evaluated the CXR images with regard to the presence of lung nodules and nodule conspicuity. After radiologist review, the AI software processed and flagged the CXR with the highest probability of missed nodules. The calculated accuracy metrics were the area under the curve (AUC), sensitivity, specificity, F1 score, false negative case number (FN), and the effect of different AI modes (automated/assisted) on the accuracy of nodule detection. Results: For radiologists, the average AUC value was 0.77 $\pm$ 0.07, while the average FN was 52.63 $\pm$ 17.53 (all studies) and 32 $\pm$ 11.59 (studies containing a nodule of malignant etiology = 32% rate of missed malignant nodules). Both AI modes -- automated and assisted -- produced an average increase in sensitivity (by 14% and 12%) and of F1-score (5% and 6%) and a decrease in specificity (by 10% and 3%, respectively). Conclusions: Both AI modes flagged the pulmonary nodules missed by radiologists in a significant number of cases. AI as a second reader has a high potential to improve diagnostic accuracy and radiology workflow. AI might detect certain pulmonary nodules earlier than radiologists, with a potentially significant impact on patient outcomes., Comment: This paper is in submission process to the European Radiology journal

We study turn-based quantitative games of infinite duration opposing two antagonistic players and played over graphs. This model is widely accepted as providing the adequate framework for formalizing the synthesis question for reactive systems. This important application motivates the question of strategy complexity: which valuations (or payoff functions) admit optimal positional strategies (without memory)? Valuations for which both players have optimal positional strategies have been characterized by Gimbert and Zielonka for finite graphs and by Colcombet and Niwi\'nski for infinite graphs. However, for reactive synthesis, existence of optimal positional strategies for the opponent (which models an antagonistic environment) is irrelevant. Despite this fact, not much is known about valuations for which the protagonist admits optimal positional strategies, regardless of the opponent. In this work, we characterize valuations which admit such strategies over infinite game graphs. Our characterization uses the vocabulary of universal graphs, which has also proved useful in understanding recent breakthrough results regarding the complexity of parity games. More precisely, we show that a valuation admitting universal graphs which are monotone and well-ordered is positional over all game graphs, and -- more surprisingly -- that the converse is also true for valuations admitting neutral colors. We prove the applicability and elegance of the framework by unifying a number of known positionality results, proving new ones, and establishing closure under lexicographical products. Finally, we discuss a class of prefix-independent positional objectives which is closed under countable unions., Comment: 51 pages, 20 figures

Modeling the evolution of progenitors of gravitational-wave merger events in binary stars faces two major uncertainties: the common-envelope phase and supernova kicks. These two processes are critical for the final orbital configuration of double compact-object systems with neutron stars and black holes. Predictive one-dimensional models of common-envelope interaction are lacking and multidimensional simulations are challenged by the vast range of relevant spatial and temporal scales. Here, we present three-dimensional hydrodynamic simulations of the common-envelope interaction of an initially $10\,M_{\odot}$ red supergiant primary star with a black-hole and a neutron-star companion. We show that the high-mass regime is accessible to full ab-initio simulations. Nearly complete envelope ejection is reached assuming that all recombination energy still available at the end of our simulation continues to help unbinding the envelope. In contrast to previous assumptions, we find that the dynamical plunge-in of both companions terminates at orbital separations too wide for gravitational waves to merge the systems in a Hubble time. We discuss the further evolution of the system based on analytical estimates. A subsequent mass-transfer episode from the remaining $3\,M_{\odot}$ core of the supergiant to the compact companion does not shrink the orbit sufficiently either. A neutron-star--neutron-star and neutron-star--black-hole merger is still expected for a fraction of the systems if the supernova kick aligns favorably with the orbital motion. For double neutron star (neutron-star--black-hole) systems we estimate mergers in about $9 \%$ ($1 \%$) of cases while about $77 \%$ ($94 \%$) of binaries are disrupted, i.e., supernova kicks actually enable gravitational-wave mergers in our cases; however, we expect a reduction in predicted gravitational-wave merger events. (abbr.), Comment: 16 pages, 11 figures, accepted by A&A

Asymmetric shapes and evidence for binary central stars suggest a common-envelope origin for many bipolar planetary nebulae. The bipolar components of the nebulae are observed to expand faster than the rest and the more slowly expanding material has been associated with the bulk of the envelope ejected during the common-envelope phase of a stellar binary system. Common-envelope evolution in general remains one of the biggest uncertainties in binary star evolution and the origin of the fast outflow has not been explained satisfactorily. We perform three-dimensional magnetohydrodynamic simulations of common-envelope interaction with the moving-mesh code AREPO. Starting from the plunge-in of the companion into the envelope of an asymptotic giant branch star and covering hundreds of orbits of the binary star system, we are able to follow the evolution to complete envelope ejection. We find that magnetic fields are strongly amplified in two consecutive episodes. First, when the companion spirals in the envelope and, second, when it forms a contact binary with the core of the former giant star. In the second episode, a magnetically-driven, high-velocity outflow of gas is launched self-consistently in our simulations. The outflow is bipolar and the gas is additionally collimated by the ejected common envelope. The resulting structure reproduces typical morphologies and velocities observed in young planetary nebulae. We propose that the magnetic driving mechanism is a universal consequence of common envelope interaction responsible for a substantial fraction of observed planetary nebulae. Such a mechanism likely also exists in the common-envelope phase of other binary stars that lead to the formation of Type Ia supernovae, X-ray binaries and gravitational-wave merger events., Comment: 9 pages, 11 figures; accepted for publication by A&A

We consider the problem of automatically constructing computer programs from input-output examples. We investigate how to augment probabilistic and neural program synthesis methods with new search algorithms, proposing a framework called distribution-based search. Within this framework, we introduce two new search algorithms: Heap Search, an enumerative method, and SQRT Sampling, a probabilistic method. We prove certain optimality guarantees for both methods, show how they integrate with probabilistic and neural techniques, and demonstrate how they can operate at scale across parallel compute environments. Collectively these findings offer theoretical and applied studies of search algorithms for program synthesis that integrate with recent developments in machine-learned program synthesizers., Comment: Attached repository: https://github.com/nathanael-fijalkow/DeepSynth/

We propose a variant of an algorithm introduced by Schewe and also studied by Luttenberger for solving parity or mean-payoff games. We present it over energy games and call our variant the ESL algorithm (for Energy-Schewe-Luttenberger). We find that using potential reductions as introduced by Gurvich, Karzanov and Khachiyan allows for a simple and elegant presentation of the algorithm, which repeatedly applies a natural generalisation of Dijkstra's algorithm to the two-player setting due to Khachiyan, Gurvich and Zhao., Comment: 14 pages, 5 figures. arXiv admin note: substantial text overlap with arXiv:2110.04533

We study the algorithm of Gurvich, Khachyian and Karzanov (GKK algorithm) when it is ran over mean-payoff games with no simple cycle of weight zero. We propose a new symmetric analysis, lowering the $O(n^2 N)$ upper-bound of Pisaruk on the number of iterations down to $N + Ep + Em$, which is smaller than $nN$, where $n$ is the number of vertices, $N$ is the largest absolute value of a weight, and $Ep$ and $Em$ are respectively the largest finite energy and dual-energy values of the game. Since each iteration is computed in $O(m)$, this improves on the state of the art pseudopolynomial $O(mnN)$ runtime bound of Brim, Chaloupka, Doyen, Gentilini and Raskin, by taking into account the structure of the game graph. We complement our result by showing that the analysis of Dorfman, Kaplan and Zwick also applies to the GKK algorithm, which is thus also subject to the state of the art combinatorial runtime bound of $O(m 2^{n/2})$., Comment: 12 pages, 7 figures

The notion of separating automata was introduced by Bojanczyk and Czerwinski for understanding the first quasipolynomial time algorithm for parity games. In this paper we show that separating automata is a powerful tool for constructing algorithms solving games with combinations of objectives. We construct two new algorithms: the first for disjunctions of parity and mean payoff objectives, matching the best known complexity, and the second for disjunctions of mean payoff objectives, improving on the state of the art. In both cases the algorithms are obtained through the construction of small separating automata, using as black boxes the existing constructions for parity objectives and for mean payoff objectives., Comment: In Proceedings GandALF 2021, arXiv:2109.07798. arXiv admin note: substantial text overlap with arXiv:2104.05262

Due to the increasing number of observations Type Ia supernovae are nowadays regarded as a heterogeneous class of objects consisting of several subclasses. One of the largest of these is the class of Type Iax supernovae (SNe Iax) which have been suggested to originate from pure deflagrations in CO Chandrasekhar-mass white dwarfs (WDs). Although a few deflagration studies have been carried out, the full diversity of the class is not captured yet. We therefore present a parameter study of single-spot ignited deflagrations with varying ignition locations, central densities, metallicities and compositions. We also explore a rigidly rotating progenitor and carry out 3D hydrodynamic simulations, nuclear network calculations and radiative transfer. The new models extend the range in brightness covered by previous studies to the lower end. Our explosions produce $^{56}$Ni masses from $5.8 \times 10^{-3}$ to $9.2 \times 10^{-2}\,M_\odot$. In spite of the wide exploration of the parameter space the main characteristics of the models are primarily driven by the mass of $^{56}$Ni. Secondary parameters have too little impact to explain the observed trend among faint SNe~Iax. We report kick velocities of the bound explosion remnants from $6.9$ to $369.8\,$km$\,s^{-1}$. The wide exploration of the parameter space and viewing-angle effects in the radiative transfer lead to a significant spread in the synthetic observables. The trends towards the faint end of the class are, however, not reproduced. This motivates a quantification of the systematic uncertainties in the modeling procedure and the influence of the $^{56}$Ni-rich bound remnant. While the pure deflagration scenario remains a favorable explanation for bright and intermediate luminosity SNe~Iax, the possibility that SNe~Iax do not consist of a single explosion scenario needs to be considered., Comment: 27 pages, 17 figures, 3 tables, accepted for publication in A&A