Papers

Neural autoregressive solvers for the Multi-Attribute Vehicle Routing Problem (MAVRP) reach competitive cost but offer no per-step justification, a problem when dispatchers must validate, accept, or compare them. We open two complementary black boxes in one protocol. On the encoder side, linear probes, spontaneous-organization metrics, rank-based richness measures, and discovered-direction analyses with intervention validation characterize how the latent represents constraint families at the graph, node, and edge level. On the decoder side, three attribution methods (gradient, integrated gradients, DeepLIFT) feed three reading angles: abductive, contrastive against the best feasible alternative, and counterfactual (smallest input change that switches the action or restores feasibility). Explanations are scored on fidelity, concentration, stability, sanity, and actionability. Across six variants combining three encoders (Attention baseline, Unimp, UnimpMoe) with two decoders (Hard-Mask, Recourse), we find that graph inductive bias improves both representational predictability and decoder sanity, that the Mixture-of-Experts encoder represents constraints in a distributed rather than axis-aligned way, and that the Recourse training regime, not merely its softer mask, produces policies that represent infeasibility usefully, exposing make-feasible counterfactuals that Hard-Mask policies fail to produce even when fed infeasible alternatives externally.

In this paper, we study the category $\mathcal{O}$ of representations of shifted Yangians associated to a simply-laced simple Lie algebra $\mathfrak{g}$ over $\mathbb{C}$. In particular, we prove that the (complexified) Grothendieck ring of this category is isomorphic to the Cox ring of the open bi-infinite Bott-Samelson variety, which is a pro-variety we construct from Bott-Samelson varieties for alternating heaps. Using work of Francone-Leclerc, we prove a conjecture of Hernandez-Zhang by identifying the above Grothendieck ring with a cluster algebra defined by Geiss-Hernandez-Leclerc. Our methods also yield an action of the Langlands dual group $G^{\vee}$ on this Grothendieck ring, and show that the shifted coproducts defined in work of the first and fifth authors with collaborators give rise to coproducts for truncated shifted Yangians. This machinery then allows us to prove further conjectures of Frenkel-Hernandez and Geiss-Hernandez-Leclerc on extended $QQ$-systems, and to obtain a generalization of a duality defined by Hernandez-Leclerc.

This paper concerns a conformally invariant elliptic problem on $\mathbb R^n$ driven by $(-\Delta)^{n/2}$ that has a nonlocal exponential nonlinearity of Choquard type. The problem under consideration is a nonlocal generalization of the constant $Q$-curvature problem on $\mathbb R^n$. We classify the asymptotic behavior at infinity of all solutions that satisfy a suitable integrability assumption. Under a growth restriction at infinity and a lower bound on the energy we provide an explicit classification for solutions. The classification is heuristically consistent with the classification of the corresponding local problem.

We study an equivariant version of grid homology for strongly invertible knots. We consider symmetric grid diagrams encoding the strong inversion, prove that every strongly invertible knot admits such a representation, and establish the corresponding analogue of Cromwell's theorem. A symmetric grid induces an involution on the associated grid complex. We study the mapping cone of the sum of the identity and this involution, proving that its homology is an invariant of strongly invertible knots. Finally, we define equivariant analogues of the tau invariant and of the Legendrian grid invariants, and show that the former provide lower bounds for equivariant unknotting numbers and for the genus of simple equivariant cobordisms.

We establish an explicit symmetric spectral reciprocity formula for twisted fourth moments of $\mathrm{GL}_{2}$ $L$-functions over an arbitrary number field. A Rankin–Selberg kernel gives a new analytic regularization of the formal Petersson identity for products of Eisenstein series, based on the rank decomposition $4=2+2$. The formula preserves the nonnegativity of the amplified fourth-moment spectral side at general levels. We further identify a new square-level concentration phenomenon: after amplification, the dominant off-diagonal contribution comes from representations whose conductor is the square of a product of two distinct primes. Using square-level local test functions in the relative trace formula, we prove new hybrid subconvexity bounds for twisted $L$-functions. Combining these bounds with symmetric spectral reciprocity and amplification, we obtain the uniform estimate \begin{align*} L(1/2,\pi)\ll_{F,\varepsilon} C(\pi)^{\frac{1}{4}-\frac{1}{120}+\varepsilon} \end{align*} for every unitary cuspidal automorphic representation $\pi$ of $\mathrm{GL}_{2}/F$ with arbitrary central character. We also obtain stronger bounds for fixed central characters, a prime-level fifth-moment estimate, and applications to Rankin–Selberg and ring class $L$-functions.

Let $M$ be a von Neumann algebra which acts in a standard way on the Hilbert space $H$, let $G$ be a subgroup of the group of all $*$-automorphisms of $M$. We prove that $M$ is $G$-finite (in the sense of I. Kovács and Szücs, i.e. the set of all normal, $G$-invariant states on $M$ is separating) if and only if, for every $x\in M$ and all $\xi,\eta\in H$, the coefficient function $g\mapsto \langle g(x)\xi|\eta\rangle$ is weakly almost periodic.

Large Language Models (LLMs) offer a natural interface for translating human objectives into reward signals for cooperative multi-agent reinforcement learning (MARL), yet the training-time dynamics of this integration remain poorly understood. We show that dynamically updating LLM-generated reward weights during off-policy MARL violates the stationarity assumption of Potential-Based Reward Shaping (PBRS) and contaminates the experience replay buffer, whose stored transitions carry reward labels computed under stale shaping weights. We characterise the result as a regime-dependent failure whose severity depends on how competent the unshaped baseline already is. To control it we propose two stabilisation strategies: a Phase-Based Freeze Schedule that enforces strict stationarity within training phases, and Exponential Moving Average (EMA) smoothing that bounds per-episode weight drift. We evaluate across three cooperative environments and five random seeds with QMIX, complemented by an exploratory VDN extension, yielding a three-regime taxonomy. In the augmentative regime (Simple Spread), where the baseline is functional (74.4 %), EMA significantly improves success to 86.7 % ($+12.3$ pp, $p<0.01$) while naive dynamic updates collapse it to 15.2 %. In the essential regime (Level-Based Foraging), where the baseline is broken (0.1 %), any shaping unlocks the task (95.9 % under EMA). In the supplementary regime (SMAC 3m), where the baseline is near-saturated (98.8 %), stabilised shaping preserves performance (99.9 %) while unstabilised shaping adds variance without gain. These findings establish reward-signal stationarity as a necessary design constraint and indicate that regime placement is a practical predictor of whether dynamic LLM shaping helps or harms.

Several positivity questions in stochastic processes, dependence modeling, fractional analysis, and renewal theory reduce to a common recognition task: after normalization, identify the object as a Laplace transform, a potential density, an inverse-flow coefficient, or a finite kernel average, and then read the sign pattern from that representation. We develop this recognition calculus for completely monotone functions, Bernstein functions, special Bernstein functions, and probabilistic realizations through subordinators and mixing measures. The main affirmative results settle three narrowly stated source questions in the conventions used by their source papers. Möhle's Problem 6.3 on the block-counting process of exchangeable coalescents with residual singleton mass (dust) is proved by a finite-simplex ordered-pair kernel certificate. For the Pearse–Bondell power-divergence copula generators, we prove complete monotonicity of the inverse throughout the remaining strict negative range $\lambda\le-1$ identified in their Section 3.8. Together with the special cases already verified in the source paper, this yields Archimedean copulas in every dimension for $\lambda\le-1$. The Bendikov–Cygan monotonicity question for discrete renewal sequences attached to special Bernstein functions is answered by representing the potential kernel as a Gamma average of a nonincreasing density. Supporting representation and boundary results cover Sibisi's Prabhakar–Pollard $Q$-measure, the Mecke–Nagel–Weiss atom at zero, and the cubic branch criterion $a^2\ge3b$.

We study two recently discovered "dimension-free" Monte Carlo sampling algorithms, the multiproposal and multiple-try preconditioned Crank-Nicolson methods (mpCN and MTpCN). These methods were designed to address certain non-parametric (i.e. infinite-dimensional) sampling problems, defined relative to a Gaussian reference measure, by combining proposal and acceptance mechanisms that take non-trivial advantage of parallel computing architectures. We provide the first rigorous analysis of both algorithms, establishing exponential convergence to the target measure through the weak Harris framework, both for a finite number of proposals and in the infinite-proposal limit. The resulting mixing rates are independent of the dimension and uniform in the number of proposals, and apply to targets with bounded, Lipschitz log-likelihoods, without requiring convexity. At the center of the analysis are two new coupling constructions, together with analytical tools of independent interest, yielding Wasserstein contraction estimates, $L^2$ spectral gaps, and associated statistical guarantees (laws of large numbers, central limit theorems, and non-asymptotic concentration bounds) for the corresponding Monte Carlo estimators. These theoretical results are complemented by a numerical study on benchmark problems with complex posterior geometries and high-dimensional structure, comparing mpCN and MTpCN against standard pCN and independent parallel-chain implementations. The experiments indicate that the multiproposal methods can offer a shorter warm-up phase and greater robustness to the choice of tuning parameters as the number of proposals grows.

Quantum low-density parity-check (LDPC) codes are a promising route to fault-tolerant quantum computation, but their use requires efficient preparation of encoded states. Standard encoder constructions generate circuits through fixed algebraic procedures, yet the resulting circuit can contain substantial redundancy. We formulate LDPC encoder preparation as a circuit-resynthesis problem: given the linear-reversible matrix implemented by the encoder's CNOT block, we seek a lower-cost equivalent circuit that can be routed efficiently on the target hardware and which mitigates noise. We propose a novel optimization approach referred as two-sided Hamming descent and a noise-aware optimization pipeline for this task. Across several families of Calderbank-Shor-Steane (CSS) LDPC encoders, including Bivariate Bicycle, hypergraph-product, and entanglement-assisted codes, the proposed pipeline produces substantially smaller and shallower encoder circuits than the standard constructions and the synthesis baselines considered, cutting gate counts by 53.8% in aggregate across the benchmark and by up to 68% on the Bivariate Bicycle family. The gains remain visible after routing, where the two-qubit depth is reduced by up to 71% and translate into higher-fidelity state preparation under circuit-level noise. On the Bivariate Bicycle family, live-range scheduling further reduces routed preparation failure by up to 13.7% without adding two-qubit gates to the selected circuit. These results indicate that encoder-matrix resynthesis, combined with hardware-calibrated selection and scheduling, is an effective compiler-level tool for preparing quantum LDPC code states.

We extend the distributional framework developed in the companion paper [Zolla, arXiv:2605.15716] to the nonlinear case, focusing on the second-order ($\chi^{(2)}$) response responsible for second-harmonic generation (SHG). Starting from the most general tensorial nonlocal second-order constitutive relation and combining a spatial moment expansion with a distributional thin-layer limit, we show that the full complexity of the nonlinear interfacial response condenses, at leading order, into two scalars, the nonlinear surface susceptibilities $\chi^{(2),s}_\parallel$ and $\chi^{(2),s}_\perp$, associated with the tangential and normal components of the electric field, respectively. A key structural result is established: via a marginal integration over one field argument, the nonlinear surface problem reduces recursively to an effective linear one, whose surface susceptibility is determined by the bulk nonlinear kernel alone. Generalized nonlinear Maxwell boundary conditions are derived explicitly for planar and spherical interfaces, and curvature corrections are obtained systematically. The formalism is illustrated on Gaussian, Yukawa, and tensorial Lorentz kernels.

We show convergence of a continuous data assimilation method for the Oberbeck-Boussinesq system in the dimension $d=2,3$. Our working hypothesis is boundedness of the reference solution, while the synchronized solution satisfies the equations in a weak sense. The main tool is the relative energy inequality for stochastic problems.

We propose and analyse a mathematical model of evolutionary adaptation for non-degenerate (permanent) replicator systems, in which the fitness landscape matrix evolves on a slow timescale – the evolutionary time – while the species dynamics unfold on a fast timescale. Under a two-timescale separation justified by Tikhonov's theorem, the adaptation problem reduces to maximising the mean fitness at steady state over a convex admissible set of fitness landscape matrices. We derive a fitness variation formula and establish necessary and sufficient conditions for a fitness maximum, showing that the optimisation reduces at each step to a linear programming problem. The algorithm is applied to four canonical replicator systems: the hypercycle, the bi-hypercycle, the anthill system, and the RNA molecule network. In all cases the evolutionary process follows a universal three-phase pattern: an initial phase of fitness growth without equilibrium shift, during which purely altruistic replication gives way to mixed altruistic-selfish behaviour; a second phase of dominant species emergence; and a stabilisation phase analogous to the error catastrophe threshold in quasispecies models. A key consequence is that all evolved systems acquire resistance to parasitic species. We further prove that without non-degeneracy constraints the process leads to sequential species annihilation, with a provable spectral lower bound on fitness increase by dimension reduction.

The assessment of planktonic standing stocks and microorganism structures is critical for understanding upper ocean biological processes. Currently, autonomous underwater vehicles (AUVs) equipped with in-situ optical imaging and artificial intelligence (AI) methods offer a promising solution for persistent surveillance, mapping and monitoring of planktonic life. However, current AI methods often lack robustness in dynamic, unstructured environments, where environmental noise and non-biological artifacts lead to frequent misclassifications. Standard convolutional neural network (CNN) classifiers often struggle with such conditions, leading to misclassifications that require time-consuming manual validation by marine biologists. To address this issue, we propose a novel robustness verification framework for in-situ plankton classifiers based on reachability analysis. We also introduce a continuous-time neural ordinary differential equation (neural ODE) classification model leveraging the high-resolution imaging capabilities of the SilCam particle imager. In this paper, we demonstrate the effectiveness of the proposed framework by formally verifying the robustness of the neural ODE model against environmental perturbations. We demonstrate that our verification framework acts as an automated filter providing formal guarantees of model stability against ambiguous data, thereby improving the reliability of autonomous sampling and reducing the post-processing workload.

In this work, minimum-time navigation on Riemannian manifolds is analyzed by means of Finsler geometry. The introduced notion of a superwind being anisotropic, reducible, and unknown a priori encompasses a wind in the spirit of Zermelo's navigation, and a gravitational wind in the sense of a slippery slope model, including Matsumoto's navigation on a mountain slope as the particular cases. The considered generalization also deploys a non-uniform slope, where the counterbalance to the transverse impact of superwind causing a lateral drift varies over space. This requires in particular an extension of the existing theory of general $(\alpha, \beta)$-metrics, which is presented in our study. The solution of the navigation problem is given by a new Finsler metric and the time-minimizing geodesics, which are determined. Moreover, the thorough analysis establishes the necessary and sufficient conditions for strong convexity under which the resultant velocity defines a Finsler metric. For completeness, a variety of comprehensive and comparative examples are explored in dimension two.

Non-classical knot theory refers to a family of extensions of classical knot theory in which one or more of the basic ingredients of the classical framework are modified. These modifications may affect the local structure of crossings, the allowed Reidemeister-type moves, the ambient space, the global form of diagrams, or the additional data carried by diagrams. Examples include skein modules of three-manifolds, pseudo knots, singular knots, stuck knots, bonded knots, tied links, virtual knots, welded knots, knotoids, and related braid-type structures. In this survey, we present an overview of several non-classical knot theories from a comparative geometric and algebraic perspective. We examine how changes in ambient topology, new crossing types, rigidity constraints, auxiliary relational data, modified isotopy relations, and open or virtual diagrammatic settings lead to generalized knot theories with distinct topological and combinatorial features. Particular emphasis is placed on braid-theoretic formulations, skein-theoretic methods, trace constructions, and extensions of classical polynomial invariants. We conclude by outlining open problems concerning generalized algebraic structures, hybrid diagrammatic theories, and the relationships among different non-classical knot theories.

The Vlasov-Hartree system is a mean-field model for a mixture of infinitely many interacting bosons and fermions where the bosons are described quantum mechanically and the fermions are described classically. This paper studies the well-posedness and dispersive properties of the Vlasov-Hartree system with initial data of arbitrary size. We prove that the Vlasov-Hartree system is globally well-posed in a low-regularity functional framework where the particle trajectories are meaningfully defined, but which includes discontinuous fermion densities. Moreover, when the interaction between the bosons and fermions is repulsive, we prove that the system exhibits dispersion in the form of time decay estimates for the particle densities and fields. When the interaction is attractive, we show that, at worst, the fields exhibit very mild growth in time.

Let S be a compact topological surface with finitely many genus and finitely many holes and let D be a polygonal decomposition of S. In this paper, we consider the generalized hyperbolic conical circle packings associated with D. We first show that the boundary value problem has a unique solution k by prescribing total geodesic curvatures of generalized hyperbolic conical circles centered at interior vertices and geodesic curvatures of generalized hyperbolic conical circles centered at boundary vertices. Then we show that such a solution k can be obtained by taking a limit of the packings inductively modified by Thurston's algorithm via an arbitrarily chosen initial generalized hyperbolic conical circle packing associated with D and with given boundary values. Thirdly, we develop the so-called discrete Schwarz-Pick lemma for the solution packing k on D.

We study a nonlocal continuity equation on $\mathbb{R}^d$ in which a probability density is driven by the competition between attraction toward a prescribed background measure $\omega$ and self-repulsion among particles, governed respectively by the power-law kernels $\psi_a(x) = |x|^{1+a}$ and $\psi_r(x) = |x|^{1+r}$ with exponents $a, r \in [0,1)$. We establish global Lagrangian well-posedness via a squared-radius regularization, obtaining {finite-time $L^\infty$ and moment bounds}, $W^{n,\infty}$ regularity, and uniqueness in the Lagrangian class. When the initial data is compactly supported and attraction dominates ($a > r$, or $a = r$ with $\omega(\mathbb{R}^d) > 1$), we prove that the support remains uniformly bounded at all time; a counterexample shows this fails for $a = r > 1$. For the attractive-dominant nonquadratic range $0 \leq r \leq a < 1$, we characterize zero-flux stationary states via a free-boundary problem involving a fractional Laplacian operator, reducing the stationarity condition to a fractional exterior Dirichlet problem. This characterization allows us to exhibit explicit examples of stationary measures in dimensions $d \in \{1,2,3\}$. Numerical particle simulations {are consistent} with the theoretical stationary profiles. Finally, for $0 \le r \le a < 1$, we prove that every global solution whose energy is bounded from below and whose moments are bounded uniformly in time converges, along a sequence of times, to a zero-flux stationary state. Full narrow convergence follows whenever the omega-limit set contains a single stationary state.

In this paper, we refine Borel's 1903 result in Diophantine approximation by providing sharper bounds for the minimum of three consecutive approximation coefficients $\Theta_n(x)$, defined for any real number $x$ with regular continued fraction (RCF) expansion $x=[0;a_1,a_2,\dots]$ as $\Theta_n = q_n^2\left| x-\frac{p_n}{q_n}\right|$. Here $\frac{p_n}{q_n}$ is the $n$th RCF convergent of $x$. Borel's result states that for all (irrational) $x$ and all $n\in\mathbb{N}$, $$ \min \left\{ \Theta_{n-1}(x),\Theta_n(x),\Theta_{n+1}(x)\right\} \leq \frac{1}{\sqrt{5}}. $$ We focus on the situation where $a_{n+1}=1$, since otherwise a result by F.~Bagemihl and J.R.~McLaughlin from 1966 implies that the Borel-bound $1/\sqrt{5}$ can already be improver to $1/\sqrt{8}$.

The Cauchy dual subnormality problem asks whether the Cauchy dual of a $2$-isometry is subnormal. This problem has attracted considerable attention due to its connections with moment theory, operator theory, and complex function theory. In particular, it has been closely related to operator-theoretic models on function spaces such as de Branges–Rovnyak spaces and Dirichlet-type spaces. In this paper, we investigate the Cauchy dual subnormality problem in the multivariable setting of toral $2$-isometries. We show that if $\mathbf{T} = (T_1, T_2)$ is an analytic left-inverse commuting pair of toral $2$-isometries satisfying the joint kernel condition, then it is unitarily equivalent to an operator-valued weighted shift with invertible weights $\{W_{I}^{(j)}:j=1,2,I\in \mathbb{Z}_{+}^2\},$ where the initial weights $W_{0,0}^{(1)}$ and $W_{0,0}^{(2)}$ are positive operators. Moreover, if these initial weights commute, then the Cauchy dual $\mathbf{T}' := (T_1', T_2')$ is subnormal. We also construct an example in which the initial weights do not commute, and the corresponding Cauchy dual tuple $\mathbf{T'}=(T_1',T_2')$ fails to be subnormal. Finally, we establish a complete set of unitary invariants for pure toral $2$-isometries.

Let $\lambda_j(K)$ be the $j$th Dirichlet eigenvalue of a convex body $K$. It is well known that $\lambda_1$ satisfies a Brunn–Minkowski inequality: $K \mapsto \lambda_1(K)^{-1/2}$ is concave on the family of convex bodies. We show that no analogous statement holds for higher eigenvalues. More precisely, for any $j \geq 2$ and $N \geq 2$, if $K \mapsto (f \circ \lambda_j)(K)$ is concave on the family of convex bodies in $\mathbb{R}^N$ for some function $f: (0, \infty) \to \mathbb{R}$, then $f$ must be constant.

For a map of Tambara functors $R \rightarrow T$ we define when $T$ is separable over $R$ and if $T$ carries an action by some finite group $H$ that fixes $R$ we also define when $R \rightarrow T$ is an $H$-Galois extension of Tambara functors. We show that flat and separable extensions are (formally) étale in the sense of Hill and that Galois extensions are separable. We prove that the Burnside Tambara functor is separably closed and we express Nullstellensatzian Tambara functors as filtered colimits of Galois extensions.

Cyclic Lotka-Volterra systems in form of ODEs and PDEs are solved by linearly implicit Kahan's method, that preserves the quadratic Poisson bracket, phase space volume, Hamiltonian ad Casimirs. The solutions are predicted applying dynamic mode decomposition (DMD). Numerical results show that the extended DMD with quadratic dictionaries and Hankel DMD with the delay embeddings, predict the periodic solutions with high accuracy, whereas the standard DMD fails. The Hamiltonians and Casimirs are also preserved accurately by the extend DMD and Hankel DMD.

In static risk measurement, law invariance expresses the principle that the risk of a position should depend only on its distribution, and not on the particular probability space on which it is represented. In a dynamic setting, the same principle leads naturally to adapted law invariance: the risk assessment should depend only on the probabilistic structure of the financial position together with the way information about it is revealed over time. We show that, for time-consistent risk measures, adapted law invariance is equivalent to a recursive one-step conditional-law representation. More precisely, assuming Fatou regularity, the one-step risk evaluations are exactly conditional lifts of static law-invariant risk measures, and the full dynamic risk measure is obtained by backward composition of these one-step maps. Convexity and coherence of the dynamic risk measure are characterized by the corresponding properties of the static one-step risk measures. This identifies adapted law invariance as the dynamic counterpart of ordinary law invariance. It also clarifies the strength of terminal-law invariance, as it appears in the rigidity theorem of Kupper and Schachermayer: it does not distinguish risks with the same distribution but different times of resolution. We further obtain an adapted Kusuoka representation in the coherent case and establish an extension of the Kupper–Schachermayer theorem.

The Dvoretzky–Kiefer–Wolfowitz–Massart inequality gives an upper bound on the probability that the empirical distribution function of a finite sequence of independent random variables deviates from its theoretical value. It is widely used in statistics in tests and in the production of confidence intervals. The original proof by Massart, later reformulated by Dudley, is very long and technical. Recently, Reeve slightly extends Dvoretzky–Kiefer–Wolfowitz–Massart inequality and presents an alternative, much shorter proof, that uses a continuous time martingale. We present a simpler and less technical proof of the original Dvoretzky–Kiefer–Wolfowitz–Massart inequality, based on a discrete-time martingale and Sion's minimax theorem.

In this paper, our aim is to establish several sharp and improved Landau-type theorems for $K$-quasiregular harmonic mappings $f=h+\overline{g}$ in the unit disk $\Bbb{D} = \{z\in\Bbb{C}: |z|<1\}$. Under various boundedness assumptions on the analytic part $h$ or its derivative, we obtain explicit univalence radii and corresponding schlicht disk radii that significantly improve upon existing estimates in the literature. We also establish new Landau-type theorems under novel hypotheses. We provide examples to illustrate our results, and comprehensive numerical tables present quantitative values of the radii for various parameter choices, demonstrating the effectiveness of our results.

We answer affirmatively a question of Fulger and Langer on the interplay between slope semistability and positivity for projective bundles with vanishing discriminant. The main result of this paper was obtained by Chatgpt 5.5 pro, and the Danus system based on the Rethlas system.

We solve two open problems about spatial (or geometric) quantiles and depth. First we show that in infinite dimension, the spatial distribution function and the associated spatial quantiles characterize the underlying distribution, which has been established in Koltchinski (1997) in finite dimension but remained unknown in infinite dimension. Second, and more surprisingly, we show that the spatial depth also fully characterizes probability measures, which has been an open problem even in finite dimension since the introduction of these concepts in Chaudhuri (1996) and Vardi & Zhang (2000). Our results provide theoretical foundations for nonparametric depth-based statistical inference and introduce novel proof techniques to investigate these questions for other depth and quantile concepts.

Let $f_r(k)$ be the smallest $n$ such that every $r$-coloring of $\{1,2,\ldots,n\}$ has a monochromatic solution to the equation \[\frac{1}{x_1}+\frac{1}{x_2}+\cdots+\frac{1}{x_k}=\frac{1}{x_{k+1}}, \] where $x_1,x_2,\ldots,x_k$ are not necessarily distinct. In this paper, we prove that $f_r(2)\geq 4^r/2$ for all $r\geq1$, and $f_r(k)\geq(2^r-1)k^r$ for all $k\geq3$ and $r\geq1$. When $r=2$, we show that, if $k=3\cdot2^m$ for some positive integer $m$, then $f_2(k)=3k^2$; and if $k=p^m$ for some odd prime number $p$ and positive integer $m$, then $f_2(k)\geq3k^2+1$. We also provide new computational results for $f_2(k)$ and $f_3(k)$, as well as a generalization of our lower bounds for $f_2(k)$ to equations with general coefficients.