Papers

The Squier complex $\mathcal{D}(\cP)$ has been defined in \cite{LDHM1} as a combinatorial 2-complex. Also several extensions of $\mathcal{D}(\cP)$ are defined and used extensively in \cite{LDHM2}, \cite{SES} and \cite{EP}. Since the results obtained in those papers rely on the assumption that $\mathcal{D}(\cP)$ and its extensions in \cite{LDHM2} and \cite{SES} are CW-complexes of dimensions 2 and 3, respectively, and no proofs are given to support these preconditions, we provide the respective proofs in the present paper.

For noncompact smooth Kähler-Ricci shrinkers, we prove a Matsushima theorem identifying the centralizer of the soliton field with the complexification of its Killing centralizer, uniqueness up to biholomorphism on a fixed complex manifold, and K-polystability of the associated polarized Fano fibration.

The computational complexity of simplicial homology has been studied since the question was raised by Kaibel and Pfetsch, and clique homology was later shown to be $\mathsf{QMA}_1$-hard by Crichigno and Kohler. King and Kohler subsequently introduced the weighted, spectrally gapped version considered here and developed a reduction intended to establish the same hardness result in this setting. We give a complete proof of the hard direction. The earlier King-Kohler argument contains a gap in the analysis of the nontrivially folded local gadgets, so the required topological and spectral properties have to be re-established on the clique complexes actually produced by the reduction. This is not a local repair of the published proof: the argument requires a new analysis of the local topology, the filtered cochain complexes, and the interaction of many overlapping gadgets. Our proof uses exact finite certificates for the local combinatorial data and relative homology to determine the effect of each gadget on the encoded qubit space. We then prove the finite-dimensional filtered-Hodge statement needed to recover the corresponding rank-one low-energy penalty and derive a quantitative global estimate that transfers the source-Hamiltonian promise gap to the final weighted Hodge Laplacian. Consequently, for fixed efficiently computable $k(n)$ and inverse-polynomial $\gamma(n)$, weighted gapped clique homology is $\mathsf{QMA}_1(\mathcal G)$-hard. The previously known $\mathsf{QMA}$-containment argument is logically independent of the gap and remains unaffected.

We initiate the study of weak Schur sampling (WSS) under noisy inputs. Many optimal quantum learning algorithms rely on this measurement, for instance in quantum spectrum estimation (QSE) and quantum state tomography (QST). The standard analysis of WSS, QSE and QST assumes that the $n$ input copies of the target state $\rho$ are identical, i.e. $\rho^{\otimes n}$. We study what happens when they are not, a more realistic noisy scenario that breaks the very permutation symmetry on which WSS is built. In the simplest model the input is a noisy product state: the copies are independent but not necessarily identical and each is $\epsilon$-close in trace distance to the $d$-dimensional target state $\rho$. A naive argument via the data-processing inequality allows the per-copy errors to accumulate to $n\epsilon$ on the measurement outcome. We show that for QSE via WSS this does not happen: the expected estimation error in total variation is at most $\epsilon+\eta(n,d)$, where $\eta(n,d)=O(d/\sqrt n)$ is the noiseless error, and the additive noise term $\epsilon$ is optimal. The output error does not grow with $n$ even though the input error does; in this sense WSS is robust. Beyond noisy product states, for arbitrary input $\omega$ we show that the error is at most $\eta(n,d)+\frac{1}{n}\|\omega-\rho^{\otimes n}\|_{W_1}$, where $W_1$ is the quantum Wasserstein distance of De Palma et al. In particular, our results imply, for the first time, noise-robustness of Keyl and Werner's seminal algorithm for QSE, more than two decades after its introduction. Finally, we show that the noise cannot be too bad: within our model of robustness the promise on the input cannot be weakened to closeness of single-copy marginals or to a global trace-distance budget alone, as in both cases some inputs defeat every measurement and estimator.

For classical–quantum wiretap channels with a transitive covariant input action, both output ensembles admit a description as unitary group orbits of fixed seed states. We use this structure to construct finite-block preprocessing schemes whose private information is an entropy difference evaluated at two distributions related by group convolution. Applying the construction to product groups allows correlations across any fixed number of channel uses. We give a sufficient condition for the associated uniform-input bound to equal the single-use private information. Constant-composition channel codes and universal$_2$ hashing achieve the resulting rates, with trace-distance leakage controlled by sandwiched Rényi information. For uniform outer randomization, the representation structure reduces the comparison-state optimization to Eve's invariant states. For the hybrid binary channel considered by Tikku, Berta and Renes, numerical optimization at three and four uses gives rates above our numerical estimate of the optimized noisy-repetition benchmark wherever that estimate is positive on the tested grid. The improvement comes from noise distributions beyond the family of independent physical flips and a symmetric logical flip. We also prove nonadditivity at an explicit channel parameter using a two-use encoder, an analytic single-use converse, and certified entropy bounds. Finally, we relate the hybrid model to binary phase-shift keying with a fixed receiver measurement.

We study the average sizes of ideals and antichains in several families of posets arising from rooted unlabeled trees. For ideals, we encode order ideals by coloring vertices red, with the condition that every descendant of a red vertex is also red. Using generating functions and singularity analysis, we show that for every family governed by a quadratic root decomposition, the average number of red vertices in a tree of size $n$ is asymptotic to $n/3$. This includes binary plane trees, plane 1-2 trees, plane 2-trees, and 0-1-trees, while unrestricted rooted plane trees instead have average ideal size asymptotic to $2n/5$. For antichains, represented by pairwise incomparable blue vertices, the corresponding asymptotic constants depend on the particular branching function; for example, binary plane trees have average antichain size asymptotic to $n/6$. We also consider plane trees in which every vertex has at most $k$ children and prove that the asymptotic proportion of red vertices increases strictly with $k$, from $1/3$ when $k=2$ to the limiting value $2/5$ for unrestricted rooted plane trees. The asymptotic proportion of blue vertices also increases strictly with $k$, tending to $3/10$. The proofs are based on bivariate generating functions together with the analytic implicit-function and smooth implicit-function methods of analytic combinatorics.

We study a physics-informed neural network (PINN) approximation of a coupled PDE system that models the interaction between fluid and elastic dynamics. The canonical heat-wave model considered here couples a parabolic and a hyperbolic equation through transmission conditions on a fixed boundary interface, and serves as a prototype for the Stokes-elastic systems that arises in real world applications. This coupling makes the error analysis more delicate than that for a single PDE. We bound the error between the exact and PINN solutions by the associated PINN residuals, and we moreover prove that the generalization error is controlled by the training error, the number of quadrature points, and the size of the network. {To the best of our knowledge}, these are the first error estimates for a PINN approximation of a coupled fluid-structure interaction PDE system. We also show that numerical experiments with respect to a given test problem.

A Hardy space approach to the Nyman-Beurling and Báez-Duarte criterion for the Riemann Hypothesis (RH) was introduced in [7]. It states that the RH holds if and only if the span of a particular sequence of functions $(h_k)_{k\geq2}$, denoted $\mathcal{N}$, is dense in the Hardy space $H^2$. In this note we construct a bounded operator $T$ on $H^{2}$ such that if the closure of $\mathcal{N}$ is $T$-invariant, then either the RH holds or there exists an infinite progression of $\zeta$-zeros. In particular, this yields an operator-theoretic criterion for the Riemann hypothesis as an invariant subspace question.

This is a very first draft of a manuscript about existence of solutions of differential inclusions involving chattering.

Why are some dynamic lot-sizing problems polynomial? We address this question by introducing Grid Theory, a structural framework based on cumulative production and the additive structure of production bounds. For a general single-item dynamic lot-sizing model with lower and upper production bounds, there exists an optimal extreme solution in which, within each regeneration interval, all but at most one production quantity lie on a boundary value. This induces additive grids, and the Main Grid Theorem establishes that an optimal cumulative production trajectory can be restricted to these discrete sets. Although the resulting grids may be exponentially large, we introduce the notion of additive dimension to capture production-bound profiles whose boundary sums admit a low-dimensional representation. We show that bounded additive dimension yields a polynomially constructible grid envelope and a polynomial time grid-based dynamic programming algorithm. The framework extends to separable concave costs and establishes polynomial solvability of several families, including constant capacities, minimum order quantities, a fixed number of capacity levels, fixed-degree polynomial capacities, periodic capacities, and piecewise polynomial capacities. In particular, polynomiality may hold even when the number of distinct capacity values grows with the planning horizon. Grid Theory thus identifies additive structure, rather than the number of distinct resource values, as a sufficient mechanism for polynomial solvability.

Let $\mathcal{C}_\varepsilon$ denote the time it takes a Brownian motion to come within distance $\varepsilon$ of every point in the two dimensional torus $\mathbb{T}^2$ of unit area. We prove that for an explicit non-random centering $m_\varepsilon$, the variables $\{\sqrt{\mathcal{C}_{\varepsilon}} - m_\varepsilon \}$ converge in distribution as $\varepsilon \to 0$, to a randomly shifted Gumbel law.

We refute the Strong Roberson Conjecture, which asserts that adding any graph outside a class closed under minors and disjoint unions strictly increases the distinguishing power of homomorphism counts from that class. More precisely, we construct connected graphs $H$ for which counts from graphs excluding $H$ as a minor determine the number of homomorphisms from $H$ to any target graph. We also refute the analogous conjecture with immersions in place of minors. We give explicit infinite families of excluded graphs, including cubic bipartite graphs that yield counterexamples for both relations. The proof introduces a method for deriving exact homomorphism count dependence from modular equivalences. We obtain these equivalences for infinitely many primes using prime-order automorphisms of graphs that exclude their orbit quotients as minors or immersions.

We introduce a global frequency function for harmonic functions on rotationally symmetric Cartan-Hadamard manifolds. In real hyperbolic space, we also identify a different frequency function associated with the normalized spherical $L^2$-mass of a hyperbolic harmonic function and prove its monotonicity. As a consequence, we obtain a logarithmic convexity theorem for the normalized spherical $L^2$-mass and a hyperbolic analog of the classical three-sphere theorem.

For a compact Riemann surface $X$, Simpson associated to each stable graded Higgs bundle $(E,\theta)$ a locus $W_{[(E,\theta)]}^1 \subset M_{\mathrm{dR}}(X,n)$, consisting of flat bundles that admit a Simpson filtration whose associated graded Higgs bundle is $(E,\theta)$. He conjectured that $W_{[(E,\theta)]}^1$ is Zariski closed in $M_{\mathrm{dR}}(X,n)$. We prove this conjecture in arbitrary rank. The proof is based on the Quillen geometry of the determinant-of-cohomology line bundle over the moduli of holomorphic bundles. To any holomorphic family of flat bundles, we associate a determinant-of-cohomology line bundle equipped with a canonical holomorphic determinant connection, and derive explicit formulas for its curvature. For a family of flat bundles in $W_{[(E,\theta)]}^1$, we then prove that the determinant connection form is exact. This exactness forces the associated determinant frame to extend as a nowhere-vanishing frame across any one-parameter degeneration, which controls the limiting filtration and yields the desired closedness.

We study an infinite-dimensional Skorokhod problem on the positive cone of $L_1([0,1])$, where a regulator process constrains a free càdlàg path to remain nonnegative through an oblique reflection mechanism determined by a positive linear integral operator $\boldsymbol{F}$ on $L_1([0,1])$ with spectral radius less than $1$. This constrained process $Z$ remains in the positive cone, and the regulator $Y$ is nondecreasing with respect to the cone order and satisfies a pointwise complementarity condition, increasing only on those space-time regions where the corresponding component of $Z$ vanishes. This formulation extends the classical multidimensional Skorokhod problem on the nonnegative orthant to an infinite-dimensional setting. Under suitable assumptions on the operator $\boldsymbol{F}$ that guarantee a dominant positive eigenfunction, we proceed through a diagonalization argument to give an alternative representation of the system in which the $L_1$ norm of the linear operator is less than 1. From this characterization, we establish existence and uniqueness of solutions for arbitrary input paths $X\in D([0,T],L_1([0,1]))$. We further show that the associated Skorokhod map is well defined and Lipschitz continuous with respect to both the uniform and Skorokhod $J_1$ topologies. As a consequence, solutions may be obtained as limits of multidimensional Skorokhod problems, providing a constructive characterization of the infinite-dimensional reflection mechanism. These results provide a framework for the study of constrained stochastic systems with infinitely many interacting components and provide a foundation for reflected stochastic processes evolving in function spaces.

In this article, we give a global construction of the Virasoro conformal blocks on Teichmüller space of Riemann surfaces of genus $g$ with $m$ marked points, when the central charge is $c>25$. We prove that they are global holomorphic sections of a holomorphic line bundle and they satisfy the Ward identity, which encodes their conformal invariance. For the sphere with $3$ points, the space of conformal blocks is one-dimensional. For a given marked Riemann surface and a marked pair of pants decomposition, it is in general an infinite dimensional Hilbert space, isomorphic to $L^2(\R_+^{3g-3+m})$ with respect to some measure involving the DOZZ structure constants. We show that it carries a projective unitary representation of the mapping class group. More generally, there are unitary isomorphisms for each Moore-Seiberg move, changing a marked pair of pants decomposition into another one.

This paper is the second part of the proof of the conformal bootstrap for Liouville conformal field theory on compact surfaces with boundary. It is devoted to the spectral theory of the half-annulus semigroup and to the resulting bootstrap formula. Building on the Segal axioms and gluing properties established in Part~1~\cite{GRW1}, we identify the generator of the half-annulus semigroup with the boundary Hamiltonian and establish its spectral decomposition by scattering methods. This yields a direct-integral decomposition of the boundary state space into irreducible Virasoro representations, governed by the boundary spectral measure. Combining this decomposition with Ward identities for boundary deformations, we express general correlation functions as integrals over spectral parameters attached to cutting curves, with integrands given by products of bulk and boundary structure constants and the corresponding conformal blocks. These results also underpin the analyticity and symmetry properties of conformal blocks~\cite{GRSSbloc}, and contribute to the construction of mapping class group representations on spaces of conformal blocks~\cite{PapierBlocs}.

The generalization of the Kähler metric in the complex Finsler setting admits four natural versions. Among these, the strongly Kähler–Finsler condition is known to coincide with one of the intermediate conditions. In this paper, we establish the equivalence between the weakly Kähler–Finsler condition and the remaining intermediate condition. Moreover, we obtain new necessary and sufficient conditions for a weakly Kähler–Finsler metric to be strongly Kähler–Finsler. Using these criteria, we construct, for the first time, smooth examples of weakly Kähler–Finsler metrics which are not Kähler–Finsler, via the Kobayashi metrics on smooth bounded strongly convex domains.

We extend to the Willmore flow the phase-field neural operator approach proposed by Bretin, Denis, Masnou, and Terii (2022) for mean curvature motion in two and three dimensions. The resulting lightweight, single-time-step architecture is motivated by a Bence-Merriman-Osher-type expansion of the Allen-Cahn semigroup, and is trained using trajectories generated by a hybrid minimizing movement scheme. In our experiments with non-oriented interfaces, unconstrained convolution kernels produce anisotropic dynamics, whereas radial kernels enhance the evolution process. We also construct three-dimensional kernels from trained two-dimensional kernels via their radial Fourier profiles, which provides an effective initialization for the non-oriented 3D model. Finally, we show applications to curve and surface reconstruction from unoriented point clouds, including reconstruction using the transferred model without the need for additional 3D training.

Given a Finsler manifold $(M,F)$ and suitable nested regions $B_1\subset B_2\subset B_3$, with $M\setminus B_1$ connected and non-contractible, as in the model case of concentric balls in $\mathbb R^n$, $n\geq2$, we consider conformal deformations $\varphi_\lambda F$ which can be chosen so as to leave $F$ unchanged outside $B_3$ and grow on $B_2\setminus B_1$. This growth creates an energy barrier, diverging with $\lambda$, that low-energy curves cannot cross. The minimax families can therefore be chosen in the path space of $M\setminus B_1$, which contains compact subsets of arbitrarily large category. Consequently, for every $m\in\mathbb N$ and all sufficiently large $\lambda$, there exist at least $m$ geodesics joining two prescribed points, all avoiding $B_1$. This extends to the Finsler setting an idea of Capozzi, Fortunato and Greco. The same scheme applies to Randers metrics $\alpha+\beta$, with $\alpha$ deformed and $\beta$ unchanged, as well as to fixed-energy trajectories of non-relativistic electromagnetic systems and relativistic Lorentz-force trajectories, and light rays in spacetimes endowed with a causal Killing field. In the non-relativistic case only the scalar potential is perturbed, while the magnetic field is preserved. When the Killing field is not everywhere timelike, the Fermat principle yields a possibly singular Randers–Kropina metric on a spacelike hypersurface. Under a natural admissible category condition, combining localized barriers with Randers approximation gives $m$ Randers–Kropina geodesics with pairwise distinct lengths, equivalently $m$ future-pointing light rays with pairwise distinct arrival times.

Let $X$ be the moduli space of stable trace-free rank $r$ Higgs bundles over a smooth projective curve $C$ with fixed determinant $\Lambda\in\Pic(C)$ and $\gcd(r,\deg\Lambda)=1$. We prove that the $\cc^*$-equivariant quantum product of a divisor on $X$ is given by the Steinberg correspondence of the Lagrangian Steinberg cycle in $X\times X$. The Steinberg cycle is the image of the reduced virtual fundamental cycle of the $2$-pointed Kontsevich stable map space to $X$ under evaluation map. For $\Gamma=\Pic^0(C)[r]$, we give a character decomposition for $\Gamma$-invariant curve classes or finite orbit sums. In rank two and genus two we explicitly determine its fifteen endoscopic lines and the reduced spectral discriminant. %The resulting endoscopic support statement is cohomological, not a determination of the entire Chow cycle. For $\widehat X=[X/\Gamma]$, which is the moduli space of stable $\PGL_r$-Higgs bundles over $C$. We consider the orbifold quantum cohomology theory from a gerbe-twisted theory. The complex K-theory $\KU^*(X)$ and twisted K-theory $\KU^*([X/\Gamma],\alpha)$ by the lifting $\mu_r$-gerbe $\alpha$ of the universal projective bundle is an isomorphism, and is given by the Fourier-Mukai transformation. We prove that the Fourier-Mukai transformation is compatible with the quantum product by the divisors.

We present several geometric constructions of graphs with rapidly growing chromatic numbers, most of which are triangle-free or have large girth.

Decision making under uncertainty requires balancing the value of additional information against the cost of delaying the decision until that information becomes available. This challenge is especially prevalent for embodied decisions such as those faced by animals moving between food sources and robots selecting navigation targets, where movement through space shapes both the quality of information and the cost of acting. We consider a version of this problem, where the decision is on choosing among rewards of uncertain value, and those rewards can only be attained through changing the decision maker's state sufficiently. Our focus is on interactions between reward learning and state evolution, restated as a deterministic optimal control problem with two terminal alternatives, one with a known reward and the other learned gradually through noisy observations. We demonstrate that delayed reward attainment is often optimal in a variety of systems, especially so when the alternatives are initially difficult to distinguish and the cost of waiting is relatively low. For a subset of problems, we also derive sufficient conditions that allow shortening the planning horizon a priori, guaranteeing that further observations would not affect the value function. Finally, when observation quality depends on the state, we show that it is often optimal for a controller to evolve the state strategically toward more informative sensing locations before making a decision, in line with the behavior broadly observed in biological and engineered systems.

We give two alternate formulations for the second moment formula of the Siegel transform over $\mathrm{Sp}(2n, \mathbb{Z}) \backslash \mathrm{Sp}(2n,\mathbb{R})$, originally established by Kelmer and Yu. We also provide several applications that demonstrate their usefulness.

In this paper we study a discrete homotopy theory for undirected graphs. More precisely, we consider $A$-homotopy groups, first defined in the work of Barcelo, Kramer, Laubenbacher and Weaver. For every degree $n$ and every group $G$ (Abelian if $n\geq 2$), we construct a graph whose $n$-th $A$-homotopy group is isomorphic to $G$ and whose other $A$-homotopy groups all vanish. Using these, we show that for every infinite sequence of groups $G_1,G_2,\ldots$, with $G_i$ Abelian for $i\geq 2$, there is a graph $X$ such that the $i$-th $A$-homotopy group of $X$ is isomorphic to $G_i$ for all $i$. Along the way, we identify relative $A$-homotopy groups with relative classical homotopy groups, and prove a discrete version of the Blakers-Massey theorem. We then construct partial Postnikov systems and (full) Whitehead towers.

Let $M^n$, $n\ge3$, be the smooth, connected, complete, noncompact boundary of a closed convex set with nonempty interior in $\mathbb{R}^{n+1}$. We prove that its normalized scalar-curvature integral at infinity is at most $4\pi\omega_{n-2}(1-\operatorname{AVR}(M))$, except when, up to an ambient rigid motion, $M$ is the product of a smooth closed convex surface in $\mathbb{R}^3$ and $\mathbb{R}^{n-2}$. These exceptional products have zero asymptotic volume ratio and limit $8\pi\omega_{n-2}$. We also characterize the equality case for the $4\pi\omega_{n-2}(1-\operatorname{AVR}(M))$ bound. The proof uses exterior tube volumes together with the cone Steiner and Gauss–Bonnet formulas. An elementary comparison of intrinsic and Euclidean radial distances proves existence of the intrinsic limit and identifies its exact value from the cone coefficients. The same conclusions extend to orientable convex submanifolds of arbitrary codimension

The paper contains three results. First, we give a global quotient presentation of the supermoduli space of genus $g>1$ super Riemann surfaces. Second, we prove that the moduli stack of marked super Riemann surfaces, which we call super Teichmüller space, is a complex analytic Deligne Mumford superstack. Third, we construct an étale atlas for super Teichmüller space whose underlying topological space is a disjoint union of $2^{2g}$ contractible unit balls.

We study gradient-type dynamics for weak stochastic control through the laws of controlled diffusions on path space. Although the objective is generally nonconvex in the drift, a uniformly convex running cost induces a strictly convex functional of the corresponding path law and hence a natural Bregman geometry. We construct the associated implicit proximal scheme and prove its well-posedness and convergence to a continuous-time mirror flow. When the mirror cost coincides with the running cost and the law-dependent potential is linear, the flow becomes affine in centered entropy coordinates. This yields a variational construction of the resulting Newton flow, uniform stability estimates, and exponential convergence to the optimizer. For a general uniformly convex mirror cost and a nonlinear convex potential of the law, we construct the natural-gradient flow globally in learning time using BMO estimates for quadratic BSDEs, and establish quantitative convergence and a first-order discretization error. We further analyze interacting finite-particle approximations, proving exponential convergence to the finite-particle optimum and an O(1/N) normalized discrepancy from the mean-field optimum. In the Markovian setting, reflection coupling yields uniform spatial regularity and exponential convergence of the feedback controls on Rd, including for nonquadratic state-dependent Hamiltonians.

Inverse game theory seeks to learn agents' unknown objectives from observed equilibrium behavior. Existing residual-based approaches can lead to non-convex problems and need not ensure strong monotonicity of the learned game, limiting reliable equilibrium prediction. We develop a tractable convex framework for learning static and dynamic non-cooperative games using first-order and Nikaido-Isoda (NI) loss. Specialized operator parameterizations, including a Helmholtz–Hodge decomposition in a reproducing kernel Hilbert space (RKHS), enforce strong monotonicity for quadratic, nonparametric, non-quadratic, and linear-quadratic dynamic games. We establish finite-sample out-of-sample prediction guarantees, using Rademacher complexity to match existing bounds for the non-quadratic setting. For noisy sequential data, we develop a robust receding-horizon learning scheme whose adaptive regularization admits a maximum a posteriori (MAP) interpretation and is computed using randomized trace estimation. Numerical experiments, including a stylized autonomous-vehicle collision-avoidance application, demonstrate predictive accuracy and robustness to measurement noise.

Given a graph $G$ and a positive integer $k$, a packing $k$-domatic coloring of $G$ is a function $f: V(G) \to \{1,\ldots,t\}$ such that (1) for any color $j \in \{1,\ldots, t\}$ and any two distinct vertices $u,v\in V(G)$ with $f(u)=f(v)=j$ we have $d_G(u,v)>j$, and (2) $V(G)$ admits a partition into $k$ sets $A_1,\ldots,A_k$ such that for any $v\in V(G)$ and any $i\in\{1,\ldots,k\}$ there exists a vertex $w\in A_i$ such that $d_G(v,w)\le f(w)$. The minimum integer $t$ such that $G$ admits a packing $k$-domatic coloring of $G$ using colors in $\{1,\ldots,t\}$ is denoted by $\chi_{\rho,k}(G)$. The condition (1) in the definition of a packing $k$-domatic coloring implies that $G$ is a packing coloring, hence $\chi_{\rho,k}(G)\ge \chi_\rho(G)$ holds for any graph $G$, where $\chi_\rho(G)$ is the packing chromatic number of $G$. On the other hand, the new concept also leads to a generalization of the domatic number of a graph due to which one can easily see that $\chi_{\rho,2}(G)=\chi_\rho(G)$ holds for any graph with no isolated vertices. One of the main result in this paper is that $\chi_{\rho,3}(G)=\chi_\rho(G)$ holds in any connected graph $G$ with minimum degree at least $2$, and the bound is best possible in two different senses. In addition, we provide several exact values and bounds on the new invariant in paths and cycles. We prove that $\chi_{\rho,k}(P_n)=k$ for any $k\in\{3,4,5\}$ as soon as $n\ge 8$, and, in contrast, $\chi_{\rho,k}(P_n)>k$ for any $n\ge k\ge 12$. We also prove the exact values of $\chi_{\rho,k}(P_\infty)$ when $k\in \{3,4,5\}$ for the two-way infinite path $P_\infty$, and exact values of $\chi_{\rho,k}(C_n)$ when $k\in\{3,4\}$ for all cycles $C_n$. We also consider a general framework of $S$-packing $k$-domatic colorings, where $S$ is an arbitrary sequence of non-negative integers, and present some basic results in this context.