A graph $G$ is $D$-distinguishable if there is a labeling of its vertices with $D$ labels such that the only automorphism of $G$ which preserves the labeling is the identity. The distinguishing number of $G$ is the minimum value $D$ for which $G$ is $D$-distinguishable. The fixing number of $G$ is the minimum cardinality of a subset of the vertices of $G$ which is fixed pointwise only by the trivial automorphism. We prove that the fixing number of any $2$-distinguishable tree of order $n \geq 3$ is at most $4n/11$, or at most $(D-1)n / (D+1)$ for a $D$-distinguishable tree ($D \geq 3$). For every $D$ and $r$ at least $2$, we characterize the $D$-distinguishable trees with radius $r$ by constructing a universal tree $T_r^D$ which has the property that a tree $T$ of radius $r$ is $D$-distinguishable if and only if $T$ is a union of branches of $T_r^D$. We obtain a similar collection of universal trees for the property of having a constant paint cost spectrum, i.e., the minimum size of the complement of a color class in a distinguishing $D$-coloring of $T$ is equal to the fixing number. Finally, we prove bounds on the distinguishing and fixing numbers of a tree in terms of the eccentr
A finite non-abelian group $H$ is hamiltonian if all of its subgroups are normal. We compute the minimal orders of graphs having a hamiltonian group as their automorphism group. The fixing number of a graph $Γ$ is the minimum cardinality of a subset $S$ of $V(Γ)$ such that the stabilizer of $S$ is trivial. For a given finite group $G$, the fixing set is defined as the set comprising all possible fixing numbers of graphs having group $G$ as their automorphism groups. We determine the fixing sets corresponding to finite hamiltonian groups.
Gauge fixing is an essential step in lattice QCD calculations, particularly for studying gauge-dependent observables. Traditional iterative algorithms are computationally expensive and often suffer from critical slowing down and scaling bottlenecks on large lattices. We present a novel machine learning framework for lattice gauge fixing, where Wilson lines are utilized to construct gauge transformation matrices within a convolutional neural network. The model parameters are optimized via backpropagation, and we introduce a hybrid strategy that combines a neural-network-based transformation with subsequent iterative methods. Preliminary tests on SU(3) gauge theory ensembles for Coulomb gauge demonstrate the potential of this approach to improve the efficiency of lattice gauge fixing. Furthermore, we show that the model exhibits lattice size transferability, where parameters optimized on smaller lattices remain effective for larger volumes without additional training. This framework provides a scalable path toward mitigating critical slowing down in high-precision gauge fixing.
Despite the operational importance of hot fixes, large-scale evidence on how they reshape routine maintenance workflows, particularly in the era of autonomous coding agents, remains limited. We analyse hot fixes present in over 61,000 GitHub repositories from the Hao-Li/AIDev dataset and find consistent patterns of urgency: reduced collaboration (typically a single contributor), smaller and more targeted changes (median 2-3 commits and files, with <10 line modifications), limited review (often fewer than two reviewers), and substantially fewer test file modifications than regular bug fixes, consistent with their urgency-driven character. Leveraging the same urgency contexts, we examine differences between human- and AI-agent-authored hot fixes, revealing over 10 distinct repair behaviours, thus offering insights into future human-automation collaboration for hot fixing. Our study is the first to empirically analyse hot fix code changes at scale using a repository-level operationalisation of urgency. The comparison of human and agentbehaviours delineates their distinct characteristics, providing a foundation for understanding hot fixing in real-world practice
Lattice QCD provides a first-principles framework for solving Quantum Chromodynamics (QCD). However, its application to off-shell partons has been largely restricted to the Landau gauge, as achieving high-precision $ξ$-gauge fixing on the lattice poses significant challenges. Motivated by a universal power-law dependence of off-shell parton matrix elements on gauge-fixing precision in the Landau gauge, we propose an empirical precision extrapolation method to approximate high-precision $ξ$-gauge fixing. By properly defining the bare gauge coupling and then the effective $ξ$, we validate our $ξ$-gauge fixing procedure by successfully reproducing the $ξ$-dependent RI/MOM renormalization constants for local quark bilinear operators at 0.3\% level, up to $ξ\sim 1$.
Regression bugs refer to situations in which something that worked previously no longer works currently. Such bugs have been pronounced in the Linux kernel. The paper focuses on regression bug tracking in the kernel by considering the time required to fix regression bugs. The dataset examined is based on the regzbot automation framework for tracking regressions in the Linux kernel. According to the results, (i) regression bug fixing times have been faster than previously reported; between 2021 and 2024, on average, it has taken less than a month to fix regression bugs. It is further evident that (ii) device drivers constitute the most prone subsystem for regression bugs, and also the fixing times vary across the kernel's subsystems. Although (iii) most commits fixing regression bugs have been reviewed, tested, or both, the kernel's code reviewing and manual testing practices do not explain the fixing times. Likewise, (iv) there is only a weak signal that code churn might contribute to explaining the fixing times statistically. Finally, (v) some subsystems exhibit strong effects for explaining the bug fixing times statistically, although overall statistical performance is modest but
Quantum annealing can efficiently obtain solutions to combinatorial optimization problems. Size-reduction methods are used to treat large-scale combinatorial optimization problems that cannot be input directly into a quantum annealer because of its size limitation. Various size-reduction methods using fixing spins have been proposed as quantum-classical hybrid methods to obtain solutions. However, the high performance of these hybrid methods is yet to be clearly elucidated. In this study, we adopted a parameterized fixing spins method to verify the effects of fixing spins. The results revealed that setting the appropriate number of spins of the subproblem is crucial for obtaining a satisfactory solution, and the energy gap expansion is confirmed after fixing spins.
An automorphism group of a graph $G$ is the set of all permutations of the vertex set of $G$ that preserve adjacency and non adjacency of vertices in a graph. A fixing set of a graph $G$ is a subset of vertices of $G$ such that only the trivial automorphism fixes every vertex in $S$. Minimum cardinality of a fixing set of $G$ is called the fixing number of $G$. In this article, we define a fractional version of the fixing number of a graph. We formulate the problem of finding the fixing number of a graph as an integer programming problem. It is shown that a relaxation of this problem leads to a linear programming problem and hence to a fractional version of the fixing number of a graph. We also characterize the graphs $G$ with the fractional fixing number $\frac{|V(G)|}{2}$ and the fractional fixing number of some families of graphs is also obtained.
We analyze how gauge fixing, which is required by any practical continuum approach to gauge systems, can interfere with the physical symmetries of such systems. In principle, the gauge fixing procedure, which deals with the (unphysical) gauge symmetry, should not interfere with the other (physical) symmetries. In practice, however, there can be an interference which takes two different forms. First, depending on the considered gauge, it might not always be simple or possible to devise approximation schemes that preserve the physical symmetry constraints on (gauge-independent) observables. Second, even at an exact level of discussion, the (gauge-dependent) effective action for the gauge field, and thus the related vertex functions, may not reflect the physical symmetries of the problem. We illustrate these difficulties using a very general class of gauge fixings that contains the usual gauge fixings as particular cases. Using background field techniques, we then propose specific gauge choices that allow one to keep the physical symmetries explicit, both at the level of the observables and at the level of the effective action for the gauge field. Our analysis is based on the notion o
To support software developers in finding and fixing software bugs, several automated program repair techniques have been introduced. Given a test suite, standard methods usually either synthesize a repair, or navigate a search space of software edits to find test-suite passing variants. Recent program repair methods are based on deep learning approaches. One of these novel methods, which is not primarily intended for automated program repair, but is still suitable for it, is ChatGPT. The bug fixing performance of ChatGPT, however, is so far unclear. Therefore, in this paper we evaluate ChatGPT on the standard bug fixing benchmark set, QuixBugs, and compare the performance with the results of several other approaches reported in the literature. We find that ChatGPT's bug fixing performance is competitive to the common deep learning approaches CoCoNut and Codex and notably better than the results reported for the standard program repair approaches. In contrast to previous approaches, ChatGPT offers a dialogue system through which further information, e.g., the expected output for a certain input or an observed error message, can be entered. By providing such hints to ChatGPT, its su
The fixing number of a graph $G$ is the smallest cardinality of a set of vertices $S$ such that only the trivial automorphism of $G$ fixes every vertex in $S$. The fixing set of a group $Γ$ is the set of all fixing numbers of finite graphs with automorphism group $Γ$. Several authors have studied the distinguishing number of a graph, the smallest number of labels needed to label $G$ so that the automorphism group of the labeled graph is trivial. The fixing number can be thought of as a variation of the distinguishing number in which every label may be used only once, and not every vertex need be labeled. We characterize the fixing sets of finite abelian groups, and investigate the fixing sets of symmetric groups.
In this paper we derive a gauge fixing identity by varying the covariant gauge fixing term in $Z[A,J,η, {\bar η}]$ in the background field method of QCD in pure gauge. Using this gauge fixing identity we establish a relation between $Z[J,η,{\bar η}]$ in QCD and $Z[A,J,η, {\bar η}]$ in background field method of QCD in pure gauge. We show the validity of this gauge fixing identity in general non-covariant and general Coulomb gauge fixings respectively. This gauge fixing identity is used to prove factorization theorem in QCD at high energy colliders and in non-equilibrium QCD at high energy heavy-ion colliders.
Regardless of the long history of gauge theories, it is not well recognized under which condition gauge fixing at the action level is legitimate. We address this issue from the Lagrangian point of view, and prove the following theorem on the relation between gauge fixing and Euler-Lagrange equations: In any gauge theory, if a gauge fixing is complete, i.e., the gauge functions are determined uniquely by the gauge conditions, the Euler-Lagrange equations derived from the gauge-fixed action are equivalent to those derived from the original action supplemented with the gauge conditions. Otherwise, it is not appropriate to impose the gauge conditions before deriving Euler-Lagrange equations as it may in general lead to inconsistent results. The criterion to check whether a gauge fixing is complete or not is further investigated. We also provide applications of the theorem to scalar-tensor theories and make comments on recent relevant papers on theories of modified gravity, in which there are confusions on gauge fixing and counting physical degrees of freedom.
For classical gravitational systems the lapse function and the shift vector are usually determined by imposing appropriate gauge fixing conditions and then demanding their preservation with respect to the dynamics generated by a canonical Hamiltonian. Effective descriptions encoding quantum geometric effects motivated by loop quantum gravity for symmetry reduced models are often captured by polymerization of connection (or related) variables in gauge fixing conditions as well as constraints. Usually, one chooses the same form of polymerization in both cases. A pertinent question is if the dynamical stability of the effective gauge fixing conditions under the effective dynamics generated by the polymerized canonical Hamiltonian is provided by the lapse function and the shift vector obtained from the polymerization of their classical counterparts. If this is the case, then we say that gauge fixing and polymerization commute. In this manuscript we investigate these issues and obtain consistency conditions for the commutativity of gauge fixing and polymerization. Our analysis shows that such a commutativity occurs in rather special situations and reveals pitfalls in making seemingly we
We apply Dirac's gauge fixing procedure to (2+1)-gravity with vanishing cosmological constant. For general gauge fixing conditions based on two point particles, this yields explicit expressions for the Dirac bracket. We explain how gauge fixing is related to the introduction of an observer into the theory and show that the Dirac bracket is determined by a classical dynamical r-matrix. Its two dynamical variables correspond to the mass and spin of a cone that describes the residual degrees of freedom of the spacetime. We show that different gauge fixing conditions and different choices of observers are related by dynamical Poincaré transformations. This allows us to locally classify all Dirac brackets resulting from the gauge fixing and to relate them to a set of particularly simple solutions associated with the centre-of-mass frame of the spacetime.
We consider projections of SU(2) lattice link variables onto Z(2) center and U(1) subgroups, with and without gauge-fixing. It is shown that in the absence of gauge-fixing, and up to an additive constant, the static quark potential extracted from projected variables agrees exactly with the static quark potential taken from the full link variables; this is an extension of recent arguments by Ambjorn and Greensite, and by Ogilvie. Abelian and center dominance is essentially trivial in this case, and seems of no physical relevance. The situation changes drastically upon gauge fixing. In the case of center projection, there are a series of tests one can carry out, to check if vortices identified in the projected configurations are physical objects. All these criteria are satisfied in maximal center gauge, and we show here that they all fail in the absence of gauge fixing. The non-triviality of center projection is due entirely to the maximal center gauge-fixing, which pumps information about the location of extended physical objects into local Z(2) observables.
The gauge fixing procedure for N=1 supersymmetric Yang-Mills theory (SYM) is proposed in the context of the stochastic quantization method (SQM). The stochastic gauge fixing, which was formulated by Zwanziger for Yang-Mills theory, is extended to SYM_4 in the superfield formalism by introducing a chiral and an anti-chiral superfield as the gauge fixing functions. It is shown that SQM with the stochastic gauge fixing reproduces the probability distribution of SYM_4, defined by the Faddeev-Popov prescription, in the equilibrium limit with an appropriate choice of the stochastic gauge fixing functions. We also show that the BRST symmetry of the corresponding stochastic action and the power counting argument in the superfield formalism ensure the renormalizability of SYM_4 in this context.
LaTeX is a widely-used document preparation system. Its powerful ability in mathematical equation editing is perhaps the main reason for its popularity in academia. Sometimes, however, even an expert user may spend much time fixing an erroneous equation. In this paper, we present EqFix, a synthesis-based repairing system for LaTeX equations. It employs a set of fixing rules and can suggest possible repairs for common errors in LaTeX equations. A domain-specific language is proposed for formally expressing the fixing rules. The fixing rules can be automatically synthesized from a set of input-output examples. An extension of relaxers is also introduced to enhance the practicality of EqFix. We evaluate EqFix on real-world examples and find that it can synthesize rules with high generalization ability. Compared with a state-of-the-art string transformation synthesizer, EqFix solved 37% more cases and spent less than half of their synthesis time.
A global supersymmetry (SUSY) in supersymmetric gauge theory is generally broken by gauge fixing. A prescription to extract physical information from such SUSY algebra broken by gauge fixing is analyzed in path integral framework. If $δ_{SUSY}δ_{BRST}Ψ= δ_{BRST}δ_{SUSY}Ψ$ for a gauge fixing ``fermion'' $Ψ$, the SUSY charge density is written as a sum of the piece which is naively expected without gauge fixing and a BRST exact piece. If $δ_{SUSY}δ_{SUSY}δ_{BRST}Ψ= δ_{BRST}δ_{SUSY}δ_{SUSY}Ψ$, the equal-time anti-commutator of SUSY charge is written as a sum of a physical piece and a BRST exact piece. We illustrate these properties for N=1 and N=2 supersymmetric Yang-Mills theories and for a D=10 massive superparticle (or ``D-particle'') where the $κ$-symmetry provides extra complications.
It has been suggested that using a gauge fixing Lagrangian that is not quadratic in a gauge fixing condition is most appropriate for gauge theories formulated on a hypersphere. We reexamine the appropriate ghost action that is to be associated with gauge fixing, applying a technique that has been used for ensuring that the propagator for a massless spin-two field is transverse and traceless. It is shown that this non-quadratic gauge fixing Lagrangian leads to two pair of complex Fermionic ghosts and two Bosonic real ghosts.