In this paper, we investigate the existence and limit behaviours of travelling solitary waves of the form $ψ(t,x)=e^{iλt}φ\left(x-vt\right)$ to the nonlinear pseudo-relativistic Schrödinger equation \[ i\partial_t ψ=(\sqrt{-Δ+m^2})ψ- |ψ|^{\frac{2}{N}}ψ-μ|ψ|^{q}ψ~~\text{ on }\mathbb{R}^N, \] for $m\ge 0$ and $|v|<1$. To this end, we introduce and analyse an associated constrained variational problem, whose minimizers are termed boosted ground states and the parameter $λ$ is obtained as a Lagrangian multiplier. We first provide a complete classification for the existence and nonexistence of such boosted ground states. Based on this classification, we then study several limiting profiles, for which the exact blow-up rate is also established.

In this paper we give a complete description of all possible automorphism groups of real $\mathbb{R}$-rational del Pezzo surfaces $X$ of degree $4$, using the description of $X$ as the blow-up of some smooth real quadric surface $Q$ in $\mathbb{P}^{3}_{\mathbb{R}}$. We examine all possible ways to blow up $4$ geometric points on $Q$, illustrate in each case the $\operatorname{Gal}(\mathbb{C}/\mathbb{R})$-action on the conic bundle structures on $X_{\mathbb{C}}$, and use it to give a geometric description of the real automorphism group $\operatorname{Aut}_{\mathbb{R}}(X)$ by generators in terms of automorphisms and birational automorphisms of $Q$. As a consequence, we get which finite subgroups of $\operatorname{Bir}_{\mathbb{C}}(\mathbb{P}^{2})$ can act faithfully by automorphisms on real $\mathbb{R}$-rational del Pezzo surfaces of degree $4$.

Inspired by the recent discovery of metallic spin supersolidity and its giant magnetocaloric effect in the rare-earth alloy EuCo$_2$Al$_9$ [Nature 651, 61 (2026)], we perform a combined study through electronic structure analysis, effective spin model, and Monte Carlo simulations on a stacked triangular lattice, and reveal a novel mechanism for the emergence of 3D spin supersolid in a metallic antiferromagnet. From first-principles inputs, we derive a minimal spin model on a stacked triangular lattice (STL), which arises from the interplay between Ruderman-Kittel-Kasuya-Yosida (RKKY) and dipolar interactions and accurately reproduces the experimental thermodynamics. Based on the STL model, we identify a ground state that simultaneously breaks discrete lattice translational symmetry and continuous spin-rotational symmetry -- the hallmark of a spin supersolid. Furthermore, we present the field-temperature phase diagram of the 3D STL model and discuss the various magnetic phases and associated phase transitions. Under zero field, the spin supersolid Y order establishes in two steps: an upper transition at $T_{N1}$, where an emergent U(1) symmetry appears and the system enters a fluctuating collinear regime, followed by a lower transition at $T_{N2}$ into the spin supersolid Y phase. In contrast, the supersolid V phase undergoes a single phase transition at $T_N^V$. Our results not only provide a comprehensive theoretical understanding of the metallic spin supersolid reported for EuCo$_2$Al$_9$ but also pave the way for further experimental investigations into its supersolid transitions and universality class.

The discovery of spin supersolid and its giant magnetocaloric effect has opened a new arena in frustrated quantum magnets and cutting-edge cryogenics. The intermetallic EuCo2Al9 (ECA), for the first time, extends this intriguing phase from Mott insulators to a highly conductive metal [1]. In this work, we systematically study the electrical transport properties of ECA, where itinerant electrons serve as a sensitive probe for the spin supersolid states. We observe anomalies both in the temperature-dependent resistivity and field-dependent magnetoresistance and Hall signals, which are attributed to response of electrons to the Eu2+ spins and their fluctuations. Moreover, Shubnikov-de Haas quantum oscillations at high magnetic field reveal pronounced band splitting in the spin polarized state. Our results reveal an intimate correspondence between electrical transport and magnetic transitions in ECA, deepening the understanding of this metallic spin supersolid.

We introduce a model of quantum walkers interacting with a magnetic impurity localized at the origin. First, we study a model of a single quantum walker interacting with a localized magnetic impurity. For a simple case of parameter values, we analytically obtain the eigenvalues and the eigenvectors of bound states, in which the quantum walker is bound to the magnetic impurity. Second, we study a model with two quantum walkers and one magnetic impurity, in which the two quantum walkers indirectly interact with each other via the magnetic impurity, as in the Kondo model. We numerically simulate the collision dynamics when the spin-spin interaction at the origin is of the XX type and the SU(2) Heisenberg type. In the case of the XX interaction, we calculate the entanglement negativity to quantify how much the two quantum walkers are entangled with each other, and find that the negativity increases drastically upon the collision of the two walkers. We compare the time dependence for different statistics, namely, fermionic, bosonic, and distinguishable walkers. In the case of the SU(2) interaction, we simulate the dynamics starting from the initial state in which one fermionic walker is in a bound eigenstate around the origin and the other fermionic walker is a delta function colliding with the first walker. We find that a bound eigenstate closest to the singlet state of the first walker and the magnetic impurity is least perturbed by the collision of the second walker. We speculate that this is a manifestation of Kondo physics at the lowest level of the real-space renormalization-group procedure.

The study addressed the problem of Anonymous Multi-Agent Path-finding (AMAPF). Unlike the classical formulation, where the assignment of agents to goals is fixed, in the anonymous MAPF setting it is irrelevant which agent reaches specific goal, provided that all goals are occupied. Most existing multi-agent pathfinding algorithms rely on a discrete representation of the environment (e.g., square grids) and do not account for the sizes of agents. This limits their applicability in real-world scenarios, such as trajectory planning for mobile robots in warehouses. Conversely, methods operating in continuous space typically impose substantial restrictions on the input data, such as constraints on the distances between initial and goal positions or between start/goal positions and obstacles. In this work, we considered one of the AMAPF algorithms designed for continuous space, where agents are modeled as disks of equal size. The algorithm requires a strict minimum separation of $4$ agent radii between any start/goal positions. Proposed a modification aimed at relaxing the constraints and reduce this limit from $4$ to $2\sqrt{3}$. We theoretically demonstrated that the proposed enhancements preserve original theoretical properties, including the guarantee that all agents will eventually achieve their goals safely and without collisions.

We develop a new benchmarking scheme for the Decoded Quantum Interferometry (DQI) algorithm quantifying the number of quantum gates required to obtain an optimal solution to a problem amenable to DQI. We apply the benchmarking scheme to the Binary Paint Shop Problem (BPSP) in order to benchmark the performance of DQI against a state of the art classical solver. To do so, we provide an explicit construction of a quantum circuit implementation of a greedy decoder for low-density parity check codes arising from max-2-XORSAT problems.

Distributionally balanced sampling designs are low-discrepancy probability designs obtained by minimizing the expected discrepancy between the auxiliary-variable distribution of a random sample and the target population distribution. Existing constructions rely on circular population sequences, which restrict the design space by forcing samples to be contiguous blocks of a sequence. We propose a new construction based on minimum tactical configurations that removes this topological constraint. The resulting designs are fixed-size, have equal inclusion probabilities, and belong to the class with minimum feasible configuration size. We develop both a simple initialization valid for arbitrary population and sample sizes and a spatial initialization that yields a lower initial expected discrepancy, together with a simulated annealing algorithm for optimization within this class. In simulations and empirical examples, the proposed method outperforms state-of-the-art alternatives in terms of distributional fit, balance, and spatial spread.

The dynamics of glassy materials slows down upon cooling, typically showing either Arrhenius or super-Arrhenius behavior. However, it was recently shown that 2D cell-based models for biological tissues can be continuously tuned between Arrhenius and sub-Arrhenius dynamics. In previous work, using the 2D Voronoi model, we proposed that such atypical dynamical behavior could be a generic feature of the broad class of mechanically under-constrained materials. Our earlier study had left two important points open: (1) many 2D systems are affected by long-wavelength fluctuations and the 2D melting scenario, and (2) the 2D Voronoi model sits exactly at the isostatic point, making it a marginal case rather than a strictly under-constrained one. Both points complicate the interpretation of our 2D Voronoi model results and their generalization to other systems; to remedy this, here we use large-scale simulations to study the glassy behavior of the 3D extension of the Voronoi model. We first show that the structural relaxation time $τ_α$ of the 3D Voronoi model can be tuned between sub-Arrhenius and Arrhenius behavior, like the 2D Voronoi model. We then establish that the four-point susceptibility, the structure factor, and the model's mechanical properties all display trends consistent with the 2D Voronoi model. These results provide strong evidence that sub-Arrhenius glassy dynamics are a generic feature of under-constrained materials across dimensions. Our work thus broadens the class of disordered materials known to have highly unusual glassy phenomenology.

We report a search for the rare decay $B^{+} \rightarrow K^{+} τ^{+} τ^{-}$ using $1.2 \times 10^9$ $Υ(4S)$ mesons produced near threshold in electron-positron collisions and collected by the Belle and Belle~II experiments. We fully reconstruct the hadronic decay of one $B$ meson produced in the $Υ(4S)\rightarrow B^{+} B^{-}$ decay, and search for $B^{\pm}\rightarrow K^{\pm} τ^{+}τ^{-}$ candidates among the remaining collision products, reconstructing a charged kaon and leptonic decays of the $τ$ leptons. We optimize the selection for best sensitivity and look for an excess over background at low values of the residual energy detected in the calorimeter after full event reconstruction. We observe no significant excess and set the limit $\mathcal{B}(B^{+}\rightarrow K^{+}τ^{+}τ^{-})< 0.56\times 10^{-3}$ at the 90% confidence level, improving on the only previous result by a factor of four.

Distinguishing active galaxies from star-forming galaxies is essential for understanding galaxy evolution. Diagnostic methods like the BPT (Baldwin, Phillips, and Terlevich) diagram use optical emission-line ratios to separate galaxies. However, with growing availability of large surveys and high-resolution instruments, manually identifying galaxy types has become increasingly challenging. In this study, we investigate machine learning to classify active and star-forming galaxies using properties like stellar mass, stellar velocity dispersion, colour, redshift, and [O III] luminosity. These new approaches enable faster AGN/star-forming galaxy classification than the BPT diagram and provide a flexible, scalable alternative that can complement traditional diagnostics, particularly for large surveys or low-quality data. We employ four classification algorithms -- Decision Tree, Random Forest, Support Vector Classifier (SVC), and k-Nearest Neighbours (KNN) -- using the Galaxy Zoo 1 dataset derived from the SDSS sample. The dataset contains 47,675 galaxies within the redshift range 0.02--0.05, including 17,002 pure star-forming and 2,254 active galaxies, labeled using the BPT diagram. These labels train and evaluate our models through confusion matrices, learning curves, and receiver operating characteristic (ROC) curves. Among the four algorithms, the SVC and Random Forest models achieve the highest accuracy of approximately 93\%, while KNN shows the lowest at 88\%. Furthermore, we estimate supermassive black hole masses using stellar velocity dispersion ($σ$) and the $M_{\rm BH}-σ$ relation. We apply four regression models -- Random Forest Regressor, Support Vector Regressor (SVR), KNN Regressor, and Polynomial Regression. All four models produce similar results, with $R^2$ values from 0.75 to 0.77, indicating consistent performance.

We consider a far generalization of the well-known stable roommates and non-bipartite stable allocation problems. In its setting, one is given a finite non-bipartite graph $G=(V,E)$ with nonnegative integer edge capacities $b(e)\in{\mathbb Z}_+$, $e\in E$, in which for each vertex (``agent'') $v\in V$, the preferences on the set $E_v$ of its incident edges are given via a choice function $C_v$ acting on the vectors in ${\mathbb Z}_+^{E_v}$ bounded by the capacities and obeying the standard axioms of substitutability and size monotonicity. We refer to the related stability problem as the stable partnership problem with integer choice functions, or SPPIC for short. Extending well-known results for particular cases, we give a solvability criterion for SPPIC and develop an algorithm of finding a stable solution, called a stable partnership, or establishing that there is none. Moreover, in general the algorithm constructs a pair $(x,{\cal K})$ such that $x\in {\mathbb Z}_+^E$ and ${\cal K}$ is a set of pairwise edge-disjoint odd cycles in $G$ satisfying the following properties: if ${\cal K}=\emptyset$, then $x$ is a stable partnership, whereas if ${\cal K}$ is nonempty, then a stable partnership does not exist, and in this case, the set ${\cal K}$ is determined canonically. Our constructions essentially use earlier author's results on the corresponding bipartite counterpart of SPPIC. Keywords: stable marriage problem, stable roommates problem, stable partition, stable allocation, choice function

One of the most intriguing proposals for wavefunction collapse is the Diosi Penrose model, in which collapse is driven by stochastic fluctuations of the Newtonian potential. We argue that a closely related effective structure can emerge in string theory if, as recently suggested, the present cosmic acceleration is sourced by instant folded strings and their decay products. A key difference, however, is that in this stringy setting the noise is naturally colored in time rather than white. As a result, the scenario is significantly less constrained by existing experiments than the standard Diosi Penrose model.

The advent of quantum computing will pose great challenges to the current communication systems, requiring essential changes in the establishment of security associations in traditional architectures. In this context, the multi-technological and heterogeneous nature of 5G networks makes it a challenging scenario for the introduction of quantum communications. Specifically, 5G networks support the unification of non-3GPP access technologies (i.e. Wi-Fi), which are secured through the IPsec protocol suite and the Non-3GPP Interworking Function (N3IWF) entity. These mechanisms leverage traditional public key cryptography and Diffie-Hellman key exchange mechanisms, which should be updated to quantum-safe standards. Therefore, in this paper we present the design and development of a Quantum Key Distribution (QKD) based non-3GPP access mechanism for 5G networks, integrating QKD keys with IPsec tunnel establishment. Besides, we also demonstrate the feasibility of the system by experimental validation in a testbed with commercial QKD equipment and an open-source 5G core implementation. Results show that the time required to complete the authentication and IPsec security association establishment is 4.62% faster than traditional cryptography PSK-based systems and 5.17% faster than the certificate-based system, while ensuring Information-Theoretic Security (ITS) of the QKD systems.

Folded sampling replaces clipping in analog-to-digital converters by reducing samples modulo a threshold, thereby avoiding saturation artifacts. We study the reconstruction of bandlimited functions from folded samples and show that, for equispaced sampling patterns, the recovery problem is inherently unstable. We then prove that imposing any a priori energy bound restores stability, and that this regularization effect extends to non-uniform sampling geometries. Our analysis recasts folded-sampling stability as an infinite-dimensional lattice shortest-vector problem, which we resolve via harmonic-analytic tools (the spectral profile of Fourier concentration matrices) and, alternatively, via bounds for integer Tschebyschev polynomials. Our work brings context to recent results on injectivity and encoding guarantees for folded sampling and further supports the empirical success of folded sampling under natural energy constraints.

We analyze quasinormal modes, grey-body factors, and absorption cross-sections of a massive scalar field in four-dimensional Einstein--Gauss--Bonnet black-hole spacetimes within a stability-constrained coupling window. High-order WKB-Padé spectra show that increasing field mass typically reduces damping and drives the system toward long-lived, quasi-resonant behavior. The scattering sector follows the same potential-barrier physics: larger effective barriers suppress transmission and low-frequency absorption, while the Gauss--Bonnet coupling has a comparatively mild impact over the stable range. These results provide a compact baseline for massive-field spectroscopy in higher-curvature black-hole backgrounds.

Sublinearly Morse directions in proper geodesic spaces are defined by sublinearly Morse stability. In this paper we offer an alternative characterization for sublinearly Morse geodesic lines via middle recurrence. We then study first passage percolation (FPP) on proper geodesic graphs of bounded degree. We associate an i.i.d. collection of random passage times to each edge. Under suitable conditions on the passage time distribution, we prove that sublinearly Morse boundaries are invariant under first passage percolation.

A recurring debate in the philosophy of statistics concerns what, exactly, should count as a measure of evidence for or against a given hypothesis. P-values, likelihood ratios, and Bayes factors all have their defenders. In this paper we add two additional candidates to this list: the e-value and its sequential analogue, the e-process. E-values enjoy several desirable properties as measures of evidence: they combine naturally across studies, handle composite hypotheses, provide long-run error rates, and admit a useful interpretation as the wealth accrued by a bettor in a game against the null distribution. E-processes additionally handle optional stopping and optional continuation. This work examines the extent to which e-values and e-processes satisfy the evidential desiderata of different statistical traditions, concluding that they combine attractive features of p-values, likelihood ratios, and Bayes factors, and merit serious consideration as interpretable and intuitive measures of statistical evidence.

Computing derivatives of observables with respect to parameters of the theory is a powerful tool in lattice QCD, as it allows the study of physical effects not directly accessible in the original Monte Carlo simulation. Prominent examples of this include the impact of the up-down quark mass difference and electromagnetic corrections. In this work, we present a new approach based on automatic differentiation to evaluate such derivatives to arbitrarily high orders, where particular emphasis will be placed on strong isospin-breaking effects and on the propagation of derivatives through the conjugate gradient algorithm in the computation of correlation functions.

The rapid deployment of distributed energy resources (DERs) is one of the essential efforts to mitigate global climate change. However, a vast number of small-scale DERs are difficult to manage individually, motivating the introduction of virtual power plants (VPPs). A VPP operator coordinates a group of DERs by setting suitable prices, and aggregates them for interaction with the power grid. In this context, optimal pricing plays a critical role in VPP operation. This paper proposes a robust optimal operation model for VPPs that considers uncertainty in the price elasticity of demand. Specifically, the demand elasticity is found to be influenced by the pricing decision, giving rise to decision-dependent uncertainty (DDU). An improved column-and-constraint (C&CG) algorithm, together with tailored transformation and reformulation techniques, is developed to solve the robust model with DDU efficiently. Case studies based on actual electricity consumption data of London households demonstrate the effectiveness of the proposed model and algorithm.

We identify a geometric principle governing the location of Hopf and Bogdanov--Takens bifurcations in planar predator--prey systems. The prey coordinate of any coexistence equilibrium undergoing such a bifurcation lies between consecutive critical points of the prey nullcline. The mechanism is algebraic. At critical points of the nullcline, the vanishing of its derivative induces constraints on the Jacobian that prevent the spectral conditions required for bifurcation from being satisfied. We refer to this phenomenon as \emph{spectral rigidity}. The principle is established for three model families and one discrete counterpart with qualitatively different nullcline geometries: a quadratic case (Bazykin model), a cubic case (Holling type~IV with harvesting), and a rational case (Crowley--Martin functional response). In each case, the localization follows from explicit parametric characterizations and symbolic reduction. The analysis extends to discrete-time systems. For a map obtained by forward Euler discretization of the Crowley--Martin model, the Neimark--Sacker bifurcation occurs on the descending branch of the nullcline, providing a continuous--discrete duality governed by the same mechanism. We conjecture that this localization holds for general smooth prey nullclines, with critical points acting as spectral barriers that organise the bifurcation structure.

Ion irradiation is a versatile tool for nanostructuring surfaces, yet the roles of energy deposition and dissipation at the surface and in ultrathin materials remain poorly understood. In this study, we investigate nanopore formation in monolayer MoS$_2$ on different substrates under irradiation of highly charged ions (HCIs) and swift heavy ions (SHIs): two types of ions that, despite having vastly different kinetic energies, interact primarily with the electronic system of the target. Using scanning transmission electron microscopy, we quantify pore radii and pore formation efficiencies for suspended MoS$_2$, MoS$_2$ on SiO$_2$, bilayer MoS$_2$ and MoS$_2$ on gold. Both pore size and pore formation efficiency exhibit a pronounced dependence on the type of substrate. Pores are largest and most frequent in MoS$_2$ on SiO$_2$, while the gold substrate massively quenches pore formation. The results indicate that the observed pore dimensions under both HCI and SHI irradiation are consistent with a central role of substrate and interface-dependent electronic dissipation pathways.

Accurate modeling of ultrasound wave propagation is essential for high-fidelity simulation and imaging in ultrasonic testing. A primary challenge lies in characterizing the excitation source, particularly for transducers with large apertures relative to the acoustic wavelengths. In such cases, non-uniform excitation and spatial interference significantly affect the resulting radiation patterns. This paper proposes a distributed source inversion strategy to reconstruct an effective spatio-temporal transducer model that reproduces experimentally measured wavefields. The reconstructed source model captures aperture-dependent phase and amplitude variations without the need for detailed knowledge of the transducer structure. The approach is validated using directivity measurements on an aluminum half-cylinder, where simulations incorporating the reconstructed source model show close agreement with experimental directivity patterns and waveform shapes. Finally, synthetic studies on reverse time migration and full-waveform inversion demonstrate that accurate transducer modeling is critical for the success of simulation-based imaging and inversion workflows and significantly improves reconstruction quality.

We investigate topological phase transitions for the Haldane and Kane-Mele model in a lattice with $p6$ symmetry, which consists of triangles and hexagons arranged in a two-dimensional geometry. For the Haldane model, which breaks time-reversal symmetry, we calculate the Chern number using a multi-band non-Abelian Wilson loop formalism. By varying the hopping parameters in the triangles and hexagons independently, a large variety of topological phases emerge. In the presence of a next-next-nearest neighbor hopping, the phase diagram becomes even richer, with regions exhibiting high Chern numbers. Then, we consider the Kane-Mele model, for which time-reversal symmetry is preserved, and calculate the number of $π$-crossings in the Concentric Wilson Loop Spectrum (CWLS). This method is appropriate to determine the topological invariant for systems hosting time-reversal and rotational symmetry, but lacking all other symmetries. According to a classification based on $K$-theory, the CWLS invariant reveals topological properties even when more conventional invariants fail to detect them. The formalism was previously successfully applied to systems with 3- and 4-fold symmetry. Here, we surprisingly find that for the 6-fold-symmetry model investigated, the topology identified by this invariant is fragile, therefore questioning the claim that this should be the strong invariant missing in a complete classification of topological insulators.

We investigate a class of locally complicated self-affine functions defined via the $Q_s$-representation of real numbers. In particular, we compute local Hölder exponents at points with given asymptotic frequencies of digits in their $Q_s$-representation. Furthermore, we establish conditions under which these functions possess continuum level sets. Finally, for self-affine functions satisfying additional conditions, we describe the geometric structure of the set of maximum points and show that this set can be fractal.

This article provides a gentle, visual introduction to the basic concepts of differential geometry appropriate for students familiar with special relativity. Visual methods are used to explain basics of differential geometry and build intuition for all types of Riemannian and Lorentzian manifolds of constant curvature. A visual derivation of the Thomas precession is given, showcasing the utility of differential geometry while also pointing a spotlight at certain intricacies of Minkowski space crucial from a pedagogical perspective. In addition, a straightforward method to generate some Carter-Penrose diagrams -- suitable for students with no differential geometry knowledge -- is presented, and a new method of indicating distortion on spacetime diagrams is shown.

Recent laser altimeters are able to not only measure the ranging distance between the spacecraft and the surface but also the full time-of-flight of the photons or pulse shape. This new capabilities allows to measure the intra-footprint properties: surface slope distribution and surface microtexture. Here we simulate and discuss for the first time the effect of surface microtexture, especially for ice covered surface with longer penetration depth. Using the WARPE simulation software, two kind of microtextures are simulated: compact slab and granular. Laser pulse shape for an ideal instrument is simulated using physical properties such as the grain size, material composition, thickness, compacity (filling factor, porosity) rather than radiative properties. The effects of these parameters on the pulse shape are discussed as well in the range that could be possibly be observed with actual BELA measurement. Finally, examples of WARPE's simulated pulse shapes are used as input in the precise simulation chain of the BELA measurement output, to further assess the capability to detect variation in surface microtexture.

Given a polynomial $\sum_νa_νX^ν$ of degree $<d$, bounded by one on the unit disk, how large can $\lvert a_0+a_1+\ldots+a_n \rvert$ ($n<d$) get? This question dates back at least to the 1952 thesis work of H. S. Shapiro. In 1978, D. J. Newman gave an exact answer for $d=2(n+1)$, but there does not seem to have been further progress on the question since. We study variations on this theme, obtaining exact answers for some related coefficient sums, and answer the original question in an asymptotic sense, provided that $n$ is not too large in terms of $d$. The latter is achieved via a quantitative Eneström--Kakeya theorem, while the former is based on certain identities for carefully selected Lagrange interpolators. From the interpolation approach we also obtain a general inequality for coefficient sums $\lvert t_0 a_0 + \ldots + t_{d-1} a_{d-1} \rvert$ for arbitrary complex numbers $t_0,\ldots,t_{d-1}$. This inequality fails to be sharp in general, yet it is in some cases and also yields non-trivial bounds for Shapiro's problem for some choices of $n$ and $d$.

Let $\mathbb{K}$ be an algebraically closed field, and $A \subset \mathbb{K}[x_{1}, \ldots, x_n]$ be a subalgebra of finite codimension. It is known that there exists a (not necessarily unique) finite filtration of $\mathbb{K}$-algebras \[ A = A_{0} \subset A_{1} \subset \ldots \subset A_m = \mathbb{K}[x_{1}, \ldots, x_n], \] where each $A_i$ can be written as the kernel of some linear functional $L_{i + 1} : A_{i + 1} \to \mathbb{K}$, and each $L_i$ is either a derivation or of the form $L_i : f \to c(f(\mathbfα) - f(\mathbfβ))$ for some $\mathbfα, \mathbfβ \in \mathbb{K}^{n}$ and $c \in \mathbb{K}$. We investigate the structure of these filtrations and linear functionals. Our main result shows that each such $L_i$ which is a derivation may be written as a linear combination of partial derivatives evaluated at points of $\mathbb{K}^{n}$.

qcombo is a Python package for the symbolic evaluation of commutators between general quantum many-body operators expressed in normal-ordered form using the generalized Wick theorem. The package provides an automated and systematic framework for generating the corresponding algebraic expressions, significantly reducing the risk of human error in lengthy and complex analytical derivations. It is designed to assist the development and implementation of modern many-body methods in nuclear physics, quantum chemistry, and related fields. The functionality and workflow of the package are demonstrated through an application to the in-medium similarity renormalization group (IMSRG) method, which has been widely used for nuclear ab initio calculations. As a representative example, qcombo is employed to automatically generate the complete set of multi-reference IMSRG flow equations with operators truncated at the normal-ordered three-body level.