Papers

Real-time services, such as VoIP and large-scale neural network training, require strict transmission delay guarantees. While routing under hop constraints is tractable, real-world delays increase sharply with equipment load, typically modeled using the M/M/1 queuing function where delay is inversely proportional to available bandwidth. We investigate the resulting Delay-Constrained Maximum Concurrent Flow (DCMCF) problem, which seeks to maximize the minimum throughput across all commodities. The problem's complexity stems from the conditional and non-linear nature of the delay constraints, which are active only along the specific paths used by the flow. We prove that DCMCF is strongly NP-hard, even for single-source/single-destination instances. To address the inherent non-convexity of the problem, we introduce a new convex relaxation expressed through second-order cone constraints, obtained from the convex envelope of a function representing the conditional delay associated with a single arc of a given path. The relaxation is shown to outperform existing formulations based on disjunctive programming. Leveraging this result, we develop a polynomial-time approximation algorithm with a provable performance guarantee and present numerical experiments demonstrating the effectiveness of the proposed approach.

We study how the response of a Bayesian posterior statistic to future observations changes as information accumulates. For a non-decreasing function $T$, define $\Pi_n^T=\E[T(\Theta)\vert \mathcal F_n]$, where $\Theta$ has an arbitrary prior and the observations come from a one-parameter exponential family. Conditioning on the same current value of $\Pi^T$, we show that the posterior statistic after additional observations is larger in convex order when the current posterior is based on fewer observations. We also prove preservation of convexity: the expected value of a convex function of the future posterior statistic is convex in the current posterior statistic. Together, these two properties provide structural tools for establishing time-monotonicity results in dynamic Bayesian decision and optimal stopping problems. If the exponential family contains an infinitely divisible distribution, the results extend to a continuous-time observation model through a family of Lévy processes.

We introduce a nonlocal regularization of optimal transport that bridges Kantorovich and Monge formulations. The regularization penalizes oscillations through an interaction kernel singular near the diagonal. We show that, for sufficiently strong singularities, every finite-energy transport plan is induced by a map, yielding equivalence between the regularized Kantorovich and Monge problems. For fractional kernels, the regularizer reduces to a fractional Sobolev seminorm that can allow jump discontinuities, leading to existence results for broad classes of source and target measures. We derive first-order optimality conditions and analyze the vanishing-regularization limit. The regularized functionals $\Gamma$-converge to the classical Kantorovich problem, while the first-order $\Gamma$-limit selects, whenever they exist, finite-energy Monge minimizers with minimal nonlocal energy. When no such minimizer exists, we identify a different asymptotic regime and establish a sharp scaling law in a prototypical mass-splitting example. Finally, after lifting the problem to the product of transport plans, we derive a dual formulation using the cone of copositive functions and propose a fixed-point numerical scheme illustrating the regularization.

The Prouhet–Tarry–Escott (PTE) problem has many generalizations and has been studied in various algebraic domains. In this paper, we prove that finite subsets $S$ of integral domains with small additive doubling constant (but still a power of $|S|$) always contain solutions to Wright's generalization of the PTE problem: there are small subsets $A$ and $B$ of the same size such that $\sum_{a\in A} a^j=\sum_{b\in B} b^j$ for $1\le j\le k$, but not for $j=k+1$. More generally, our method gives simultaneous solutions for $m$ systems, with pairwise distinct $(k+1)$-th power sums. In contrast with the classical case $S\subseteq [N]$, where the problem has been studied by Wooley and others using Vinogradov's mean value theorem, our approach is based on polynomial identities and additive properties of $S$. We also discuss barriers to extending these results to broader settings.

Maxwell observed that the graph of any rigid generic framework in $\mathbb{R}^d$ on $n$ vertices has at least $dn-\binom{d+1}{2}$ edges. In this article we prove that graphs whose complement has maximum degree at most two and no component isomorphic to a triangle or a square are rigid in the maximum dimension allowed by this observation. In particular, this determines the precise maximum dimension in which the graph obtained from a complete graph $K_{2m}$ by deleting a perfect matching is rigid, resolving a recent conjecture of Lew. We also deduce bounds on the rigidity of complements of bounded-degree graphs more generally, which significantly improve existing degree-based bounds.

Let $F$ be a non-archimedean local field of characteristic zero. Let $M$ be a Levi subgroup of $G = \mathrm{SL}(n,F)$. Consider the action on the discrete series of $M$ of the group of unitary unramified characters of $M$ (where a character acts by twisting). Given an orbit $\mathcal{O}$ for that action, let $W_\mathcal{O}$ be the global stabilizer of $\mathcal{O}$ in the Weyl group of $M$. We construct families of orbits $\mathcal{O}$ for which the action of $W_\mathcal{O}$ has no fixed point. More precisely, let $n \geq 8$ be an integer that is not prime and not equal to $9$. We attach families of orbits without fixed points to any divisor $m \geq 2$ of $n$ such that $m \;| \; q-1$ where $q$ is the order of the residue field of $F$. The Levi subgroup $M \subset G$ for the corresponding families has $m$ blocks of size $k\geq 2$ and $2$ blocks of size $m$. The core of the paper is the construction of superculpidal representations of $\mathrm{GL}(m,F)$ and $\mathrm{GL}(k,F)$ satisfying some conditions that we require to construct the families of orbits $\mathcal{O}$ that we expects. In addition, we study the fixed point problem for the dimensions $n$ that we excluded above, i.e., prime or small dimensions. We prove that there is always a fixed point under the action of $W_\mathcal{O}$ in those cases.

Motivated by the Casas-Loday-Pirashvili maps concerning representations of $n$-Leibniz algebras, we give an intrinsic characterization of the cohomology of $n$-Lie algebras with coefficient in a representation space $V$. We investigated the cohomology theory of $n$-Lie algebras in three different ways: cohomology theory of Leibniz algebras, infinitesimal deformation and abelian extension. In the main part of this paper, we solve the extending problems for $n$-Lie algebras. A necessary and sufficient unified-product criterion is obtained. The case of crossed products, sparse non-abelian extensions, matched pairs are studied as special cases. We also investigate the factorizations, deformation maps, and complements problem for $n$-Lie algebras. An appendix removes the general non-reduced unified product with all intermediate mixed components.

This note develops three multiplication-closed families of stochastic matrices for consensus under arbitrary switching. The first is a support-dominance enlargement of an earlier power-generated construction, in which exact support-pattern matching is relaxed to a dominance-type condition without losing closure under multiplication. The second is a fixed-core class with an exact uniform scrambling horizon and corresponding convergence-rate guarantees for switching products. The third and main contribution is the almost Sarymsakov class, obtained by allowing variable mixing-core dimensions under suitable structural conditions. This new semigroup class properly contains the classical Sarymsakov class and admits a dimension-only scrambling-horizon bound. Examples illustrate the necessity of the proposed structural assumptions and clarify the sharpness or limitations of the derived bounds.

The Kolmogorov's 0-1 law is so powerful and fundamental in Probability theory that the result of the law of large numbers is its corollary. The latter is the theoretical bedrock of modern day statistical machine learning. Expected resurgence in the interest of the law's proof is challenged by its measure theoretic nature. By observing 0-1 law as False-Truth law like in logic, we show here that the proof of the law is simplified. This simplicity comes from the fact that our approach hardly needs any measure theory beyond defining an obvious probability space. A novel contribution via this article is the use of logic within the proof of a fundamental result in the theory of probability.

Let $I$ be a monomial ideal in a standard graded polynomial ring and let $\overline{I}$ denote its integral closure. We study the relationship between $\mathrm{v}(I)$ and $\mathrm{v}(\overline{I})$. We prove that $\mathrm{v}(\overline{I}) \leq \mathrm{v}(I)$ for monomial ideals in two variables, for equigenerated monomial ideals in three variables and for several special classes of monomial ideals, while providing examples showing that this inequality does not hold in general. For the edge ideal $I(G)$ of a connected graph $G$, we show that $\mathrm{v}(I(G)^k)=\mathrm{v}(\overline{I(G)^k}) = 2k-1$ for all $k \geq 1+|E(G)|$. Moreover, when $G$ is disconnected, we prove that $\mathrm{v}(\overline{I(G)^k})\leq\mathrm{v}({I(G)^k})$ for all sufficiently large $k$.

We investigate the $L_2$ Turán problems for various small directed graphs, specifically focusing on self-converse tournaments and stability versions. First, we determine the exact maximum $L_2$ norm squared of the out-degree sequence for digraphs avoiding the transitive tournament $TT_4$ and the strongly connected tournament $R_4$, answering open questions from recent paper. We prove that the complete directed 3-partite Turán graph $T_3(m)$ exactly maximizes the $L_2$ norm squared for $TT_4$-free digraphs. For $R_4$-free digraphs, the maximum is achieved by $T_3(m)$ except when $m \equiv 1 \pmod 3$, where peeling off a terminal sink vertex to form $T_3(m-1) \to v$ strictly increases the objective. We complement these results with exact values and a conjecture for the regular tournament $Reg_5$. Furthermore, we prove a stability version for $\vec{C}_3$-free digraphs: any sequence of digraphs asymptotically achieving the maximum $L_2$ density must have an edit distance of $O(\delta^{1/2})m^2$ to the extremal ordered digon-chain $\vec{F}_{m,2}$.

We study regression under bounded responses in terms of excess mean squared error. When the comparator class is finite, this setting is known as model selection aggregation, and achieving minimax excess risk requires improper learning algorithms. Contrary to this, in the universal learning framework no improperness is needed, as simple empirical risk minimization achieves the best-possible exponential learning rate. Hence, the two frameworks suggest different optimal algorithmic principles. This poses the question of best-of-both-worlds guarantees: Are minimax and universal exponential rates achievable by the same algorithm? For finite hypothesis classes, we answer this question in the affirmative by showing that the $Q$-aggregation estimator - which is known to achieve minimax optimal tails - achieves exponential universal rates. A wide range of other estimators and algorithmic principles (ERM, sequential averaging, pruning, and star estimation) do not achieve both. For countably infinite hypothesis classes, we answer the question in the negative by showing that there is an inherent trade-off between achieving exponential universal and minimax uniform rates. This trade-off is exactly traced by combining optimal algorithms from each world using $Q$-aggregation. Besides these results, we prove several additional structural results about universal rates in learning with squared loss.

We consider density estimation under the relaxed local differential privacy condition that the privatized distributions are $\alpha$-close in total variation distance. We show that adding independent noise with a convenient symmetrized Gamma distribution to each sensitive observation attains the $\alpha$-TV-LDP. We prove that the deconvolution estimator of $r$-Sobolev smooth functions attains the pointwise rate $(n\alpha)^{-\frac{2r-1}{2r}}$ up to log factors which is faster than $(n\alpha^2)^{-\frac{2r-1}{2r+1}}$ under the classical $\alpha$-LDP and closer to the nonprivate minimax rate $n^{-\frac{2r-1}{2r}}$. Next, we use a Goldenshluger-Lepski procedure to build a free of the smoothness adaptive procedure and show optimality of our rates in the convolution model of our privatisation scheme. We illustrate the benefits of this simple privacy mechanism by implementing a neural network estimator which does not need to add more noise in the optimization steps. Numerical results show significant improvement of the estimation rate over the Laplace and the private-SGD mechanisms.

We study the distribution of the positive sojourn time for a one-dimensional two state quantum walk, conditioned on return to the origin. Konno showed that, for the Hadamard walk, this conditional distribution is exactly uniform at times divisible by $4$. In this paper, we investigate whether this finite time exact uniformity characterizes the Hadamard coin within the family of rotation coins. For a fixed initial state, we prove that the following three conditions are equivalent for rotation coins: the conditional distribution is exactly uniform at time $8$; the conditional distribution is exactly uniform at every time $4m$ with $m\ge2$; and the coin is the Hadamard coin. Thus, the uniformity phenomenon found by Konno is characterized as a rigidity phenomenon of the Hadamard coin within the rotation coin family. The proof uses a matrix-valued generating function for paths returning to the origin. We analyze the algebraic structure arising from an absorbing process on the half line. Finally, by comparing low degree coefficients at time $8$, we show that exact uniformity forces the rotation coin to be the Hadamard coin.

We give a presentation for the Kauffman bracket skein algebra of a closed or one-holed oriented surface of genus $2$. This is the first such result for genus greater than $1$.

We study the growth of the number of invisible lattice points along powers of quadratic polynomials. Let $f(x)=Ax^2+Bx+C\in\mathbb{Z}[x]$ have a positive leading coefficient and nonzero discriminant, and let $F(x)=f(x)^m$ with $m\geq 2$. For $m\geq 3$ we prove that the number of invisible lattice points in $[1,N]^2$ has order $N\log N$, and when $m=2$ the number of invisible lattice points satisfies $N\log N \ll_F\#\mathrm{Invisible}_F(N)\ll_F N(\log N)^4$. These estimates refine a previous result of the authors.

We prove that exact quadratic energy cancellation alone does not force growth of the spatial analyticity radius. On $\mathbb{T}^3$, we construct an explicit symmetric, translation-invariant, first-order bilinear operator $Q$ that preserves the divergence-free class and, for every smooth real-valued divergence-free vector field $v$, satisfies $ \int_{\mathbb{T}^3} Q(v,v)\cdot v\,dx=0.$ For every prescribed sufficiently small time $T>0$, the equation $\partial_t u-\Delta u=Q(u,u)$ admits a global smooth, real-valued, mean-zero, divergence-free solution $u$ such that $\operatorname{rad}(u(0))=\operatorname{rad}(u(T))=1$. The construction reduces the dynamics on an invariant cyclic-shear class to viscous Burgers and tunes a Cole-Hopf heat profile so that its nearest complex zero returns to its initial distance from the real torus. For every $1<\alpha<2$ and every prescribed sufficiently small $T>0$, we also construct a symmetric sparse frequency set, its associated Fourier projection, and trigonometric-polynomial initial data for the projected dissipative surface quasi-geostrophic equation. The resulting unique global smooth solution $\theta$ has infinite analyticity radius initially but satisfies $0<\operatorname{rad}(\theta(T))\leq1$. An additively separated Fourier cascade yields coefficientwise exponential lower bounds, while uniform comparison estimates control the feedback interactions. Thus entire analyticity need not persist even from trigonometric-polynomial data. Together, the two constructions show that an exact energy identity alone does not determine the frequency geometry governing analyticity-radius growth.

We study linear codes for insertion and deletion (insdel) errors through the lens of evaluation codes. We develop a general framework for analyzing random puncturings of evaluation codes, where the edit distance is controlled by only the size of the evaluation domain and the maximum number of zeros of a nonzero function in the underlying function space. Our proof generalizes the results of Con, Guo, Li, and Zhang (ICALP 2025), and simultaneously simplifies their arguments by avoiding an in-depth analysis of longest common subsequences. We demonstrate the applicability of our core theorem by instantiating it with random puncturings of Reed–Muller codes. We then recover the result that random Reed–Solomon codes approach the half-Singleton bound over linear-sized fields while also improving the dependence on the additive gap $\varepsilon$ from $2^{O(1/\varepsilon^2)}$ to $2^{O(1/\varepsilon)}$. Finally, by applying the framework to algebraic geometry codes arising from asymptotically good towers of function fields, we show that there exist randomized families of structured linear codes over constant-sized fields that approach the half-Singleton bound.

Multiple-input multiple-output (MIMO) technology is fundamental to modern wireless communication. Physical layer security seeks to protect transmitted information by exploiting properties of the noisy communication channel. Dean and Goldsmith proposed a polynomial time reduction from lattice problems to MIMO decoding by adapting Regev's reduction for learning with errors (LWE). If valid, this reduction would give physical layer security a strong computational foundation based on the hardness of established lattice problems. Subsequent works presented attacks and counterexamples against the resulting construction, casting doubt on its security but leaving the precise validity and limitations of the underlying reduction incompletely understood. We provide a theoretical examination of the revised reduction and identify the structural features of the LWE reduction that fail to carry over to the non-modular MIMO setting, hence showing that its published proof does not establish the claimed hardness of MIMO decoding. Our results distinguish flaws in the hardness proof from direct attacks on particular parameter choices and clarify what would be required of any attempted repair. We do not rule out physical layer security for MIMO systems in general, but show that the claimed lattice hardness guarantee does not follow from the existing reduction.

In this paper we first introduce the \emph{infinite multi-row periodic pattern of shape $\alpha$} as a geometric realization of $\mathbf{C}_\alpha$. We then give an explicit formula for the coefficients $b^{\lambda}_{\alpha,\beta}$ appearing in the species decomposition \[ \mathbf{C}_\alpha \times \mathbf{C}_\beta = \sum_{\lambda \vdash n} b^\lambda_{\alpha,\beta}\,\mathbf{C}_\lambda. \]

Algebraic combinatorics often seeks bijections that explain identities between distributions object by object. Encoding combinatorial objects as words lets automata theory study such a bijection as a word-to-word computation and measure its memory, input access, and control of output order. This refines existence questions by asking which computational mechanisms a bijection requires. We develop this viewpoint for Dyck paths. Our motivating example is the $q,t$-Catalan polynomial. Let $D_n$ be the set of Dyck paths of semilength $n$, let $D=\bigcup_{n\ge 0}D_n$, and let $area, dinv, bounce \colon D\to\mathbb{N}$ be the standard statistics. Then, \[ C_n(q,t)=\sum_{P\in D_n}q^{area(P)}t^{bounce(P)} =\sum_{P\in D_n}q^{dinv(P)}t^{area(P)}. \] Haglund's zeta map $\zeta\colon D\to D$ gives a bijective proof: it preserves semilength and sends $(dinv,area)$ to $(area,bounce)$. By contrast, the full symmetry $C_n(q,t)=C_n(t,q)$ still lacks a direct explanation: no explicit, uniform, semilength-preserving bijection is known that swaps area and dinv on every Dyck path. Polyregular maps from automata theory provide a natural computational starting point, but we prove that neither $\zeta$ nor the classical height-sweep bijection witnessing Narayana symmetry is polyregular. The missing mechanism is global ordering by numerical levels whose range grows with the input. We call this a rank sort and introduce weighted-rank polyregular maps (WRP), extending polyregular maps by one such sort and containing both bijections. Nevertheless, WRP is a proper subclass of deterministic logspace. We prove that $\zeta^{-1}$ lies outside WRP and that no WRP map can realise a semilength-preserving area-dinv swap. Thus the rank-sorting strategy behind $\zeta$ cannot be extended within WRP to exchange the two statistics.

Let $\mathfrak{g}$ be a complex semisimple Lie algebra, and $\overline{Z}$ a nilpotent orbit closure in the projectivization $\mathbb{P}(\mathfrak{g})$. We investigate the space of tangent directions of lines on $\overline{Z}$ passing through a general point $z$, denoted by $F(\overline{Z},\,z)$. We first prove that every irreducible component of $F(\overline{Z},\,z)$ is an integral subvariety of the contact hyperplane in the projectivized tangent space. Next, we study Legendrian components of $F(\overline{Z},\,z)$ in two cases: the case where $\overline{Z}$ admits a Springer resolution; the case where $\overline{Z}$ is square-zero in a projectivized simple Lie algebra of classical type. As an application, two corollaries are presented: a characterization of Richardson orbits among square-zero orbits in terms of $F(\overline{Z},\,z)$; a description of $F(\overline{Z},\,z)$ for $\overline{Z}$ arising from stratified Mukai flops associated to irreducible Hermitian symmetric spaces.

Let $F_1=F_2=1$, $F_{n+2}=F_{n+1}+F_n$, and define the Fibonacci zeta function by $$ Z_F(s)=\sum_{n\ge1}F_n^{-s},\qquad \operatorname{Re}s>0. $$ We determine the exact right edge of the closure of the real parts of its zeros in the half-plane of absolute convergence. If $\sigma_F$ is the unique solution of $$ Z_F(\sigma_F)=4+2\,144^{-\sigma_F}, $$ then $$ \sigma_F=0.743163398726901648\ldots, $$ $Z_F(s)\neq0$ for $\operatorname{Re}s\ge\sigma_F$, while $$ \overline{\{\operatorname{Re}\rho:Z_F(\rho)=0,\ \operatorname{Re}\rho>0\}}=[0,\sigma_F]. $$ The edge is sharp in an almost-periodic sense: zeros occur with relatively dense ordinates near every admissible vertical line. We prove growing-dimensional phase locking near the edge and a Diophantine zero-free cusp, and describe the associated Jessen function and smooth mean vertical zero density. For every partial sum with $N\ge12$ we determine the corresponding exact closure edge $\sigma_N$, prove $\sigma_N\nearrow\sigma_F$, and obtain an exponential asymptotic for $\sigma_F-\sigma_N$. We also derive a finite-core theorem for positive integral Lucas zeta functions, with the Pell zeta function as an explicit example. Finally, using the known meromorphic continuation, we construct a natural $q$-Pochhammer completion that is entire of exact order $2$ and type $\log\varphi/4$.

A generalized polynomial is a function defined by an iteration of the operations addition, multiplication and the floor function. Equidistribution results of sequences given by generalized polynomials have been established by Håland and later by Bergelson and Leibman from an ergodic theoretic point of view. In the present paper we show an asymptotic distribution result which can be applied to certain generalized polynomials. In the second part we prove equidistribution results for generalized polynomials along prime numbers, including bounds for the discrepancy.

This article develops the notion of a computational category, motivated by the observation that many mathematical structures occurring in scientific computing already admit concrete algorithmic representations. Finite sets, finite-dimensional vector spaces, relations, and graphs may all be represented through elementary MATLAB and Octave data structures such as indexing vectors, matrices, logical arrays, and sparse matrices. The central idea is that a category may often be described through a category of canonical representatives indexed by the natural numbers. This viewpoint leads naturally to representative categories whose objects are dimensions and whose morphisms are finite computational data structures. The resulting framework unifies several examples, including matrix categories, index categories, categories of column spaces, row-based models of finite sets, and categories combining finite subsets with finite-dimensional vector spaces through mixed morphisms. The paper also introduces computational categories with distinguished non-finite representatives, extending the finite setting while retaining concrete coordinate descriptions. Throughout, categorical constructions are interpreted algorithmically and implemented using standard MATLAB and Octave operations. The work contributes to a computational perspective on category theory and, conversely, to a categorical understanding of matrix-oriented scientific computing.

For each fixed positive integer $h$, we study the divisibility relation $\sigma(n)\mid\sigma(n+h)$. We isolate an explicit regular family arising from integral quotients of shifted abundancy indices and show that the complementary set satisfies a subexponential saving; in particular, the number of solutions up to $x$ is $O_h(x/(\log x)^2)$. We also study the proportionality equation $\sigma(n+h)=\lambda\sigma(n)$. For every fixed nonzero integer $h$, uniformly for all real $\lambda>0$, the number of solutions up to $x$ is $O(x/\sqrt{\log\log\log x})$, with an absolute implied constant once $x$ exceeds an $h$-dependent threshold. Finally, we give an explicit family which, under Schinzel's Hypothesis $H$, produces infinitely many solutions of $\sigma(n+1)=2\sigma(n)$; the Bateman–Horn conjecture yields a precise asymptotic for the number of members of this family up to $x$. We conjecture that $\sigma(n+h)=k\sigma(n)$ has infinitely many positive integer solutions for every fixed $h,k\ge1$.

Tree-child networks are a useful class of binary phylogenetic networks. The Pons–Batle identity (Pons and Batle, Scientific Reports, 2021) states that the number $a_{n,k}$ of tree-child networks with $k$ reticulations on $n$ taxa satisfies \[ a_{n,k}=(n-k+1)a_{n,k-1} +\frac{n(2n+k-3)}{n-k}a_{n-1,k}. \] In this paper, we present a short combinatorial proof of this identity.

Let $1<p<\infty$, let $X$ be a UMD Banach space, let $1<p<\infty$, and let $L_d$ be the generator of the Ornstein–Uhlenbeck semigroup $(P_d(t))_{t\geq 0}$ on $L^p(\mathbb R^d,\gamma_d;X)$. We prove that the operators $-L_d$ are $R$-sectorial with a common angle strictly smaller than $\pi/2$ and with bounds independent of $d$. Combining this with the Hieber–Prüss transference theorem and the Kalton–Weis angle comparison, we deduce that the operators $-L_d$ admit bounded $H^\infty$-calculi of a common angle strictly smaller than $\pi/2$, again with dimension-free bounds. The corresponding Walsh $R$-analyticity estimates are proved first and transferred to the Ornstein–Uhlenbeck setting by a central limit argument.

This paper studies the relationship between the controllability of a quasi-linear parabolic equation and that of its linear counterpart. Under suitable assumptions, we prove that for sufficiently small initial data both systems are null controllable, and that the differences between their corresponding controls and states satisfy a higher-order error estimate. Unlike standard controllability problems, the controls constructed here not only steer the system to a prescribed target at a given terminal time, but are also designed to ensure that the deviation between the nonlinear and linear dynamics obeys the aforementioned estimate. This method is applicable to general nonlinear partial differential equations, and the approximation order for controllability is optimal.

A colorful graph relation enhances the minor relation by merging color sets along contractions and by allowing the removal of colors; it generalizes rooted minors and models problems on graphs with several, possibly overlapping, annotated vertex sets. A graph has the Erdős-Pósa property for minors if and only if it is planar, by a classical theorem of Robertson and Seymour. In this work we determine, for the colorful minor relation, exactly which colorful graphs have the Erdős-Pósa property. Our characterization takes three equivalent forms. The first is structural: the colorful graphs with the property are those that can be drawn with all their colored vertices on one face and whose colors are, in a precise sense, laid out along that face without interleaving. The second is given by an obstruction set: they are those excluding every member of an explicit infinite family $\mathcal{O},$ of which only $\mathbf{O}(|I|^{4})$ members have colors that are a subset of $I,$ for every finite set $I$ of colors. The third is grid-like: they are exactly the colorful minors of unions of particular families of segregated grids, the colorful analogues of the grids that drive the classical proof.