Papers

Let $\mathbb F$ be an algebraically closed field of characteristic $0$. Let $k\geq 2$ be an integer, and let $n\in \mathbb F[x]\setminus\{0\}$. We study generalized Diophantine tuples $A\subset \mathbb F[x]$ with property $D_k(n)$, meaning that $ab+n$ is a $k$-th power in $\mathbb F[x]$ for all distinct elements $a,b\in A$. For $k\ge18$, we prove that every such tuple satisfies $|A|\le6$, except for the necessary exceptional family in which $n=s^2$ is a $k$-th power and $A\subset s\mathbb{F}$. This bound is absolute: it is independent of both $n$ and $\operatorname{deg} n$. Our proof develops a new method for studying polynomial Diophantine tuples, combining a determinant criterion, generalizations of the Mason–Stothers theorem, and the Combinatorial Nullstellensatz. We also record a conditional analogue for generalized Diophantine tuples over the integers.

In a series of papers in the 1980's Alexander Holevo proved a classification theorem for continuous quantum measurement processes, or, as they would today be called, stationary quantum trajectories in continuous time. His main tools were functional analytic in character: starting from a Bochner-type inequality he employed dilation techniques for positive definite kernels. Here we give an alternative, more probabilistic proof: we use weak convergence of measures and employ Levy's Continuity Theorem. We clarify the boundedness conditions in Holevo's theorem, and supply a simple example from quantum optics.

In this paper, we study blowup solutions of an important class of Chern-Simons systems. We first show that when blowup of mean-field type occurs, the corresponding blowup solution is unique under natural geometric assumptions. We also establish the non-degeneracy of the linearized system around these blowup solutions. To prove these main results, we carry out a precise blowup analysis, so that the asymptotic description of the solutions reveals the curvature information needed for the uniqueness and non-degeneracy results. Compared with related work on similar problems, our estimates are more delicate and technically involved.

Very recently, using Meshulam's lemma, Blagojević proved a constrained version of the colorful Carathéodory theorem for joins of bipartite spanning trees and wedge of spheres. Our main contribution extends his result from joins of bipartite spanning trees with wedges of spheres to joins of spanning \(k\)-trees with wedges of spheres. Our proof is elementary and avoids the topological machinery. We also discuss a homological variation of spanning \(k\)-trees and some Carathéodory-type results for them.

Loewner chains are ubiquitous in the theory of slit mappings, and hence in the study of bounded conformal maps. They have attracted new interest in the past decades through their applications to statistical physics and fractal geometry, particularly in contexts involving randomness. In this article, we delve into topological features of the growing hulls obtained from Loewner chains with a general local growth property, inspired by the classical works of Loewner and Pommerenke. We first revisit Loewner's theorem, associating to each locally growing collection of hulls a real-valued driving function W, possibly discontinuous. We then investigate the points chronologically added to the growing hulls, which may be part of a simply connected swallowed “bubble”, or a compact connected boundary set. For continuous driving functions, the Loewner chain can often be associated with a continuous curve (dubbed “generating curve”). Motivated by this, we introduce a more general notion of a “generating function” for the Loewner chain, and characterize when there exists such a function {\eta} (which can be continuous, càdlàg, càglàd, or neither). We then investigate the necessity of left and right limits for {\eta} from the point of view of the topology of the growing hulls. We find in particular that left-continuity implies path-connectedness and local connectedness of the hulls, as well as the existence of right limits, whereas failure of left-continuity leads to pathological boundary behavior.

In this paper, we investigate the combinatorial structure arising from the $(p, q)$-deformed generalized Weyl algebra generated by variables $X, Y$, and $Z_p$, satisfying the $(p, q)$-commutation relations $XY-qYX=h Y^sZ_{p}, XZ_p=pZ_pX$, and $Z_pY=pYZ_p$, where $s\in \mathbb{N}_0$. Our primary objective is to use the normal ordering process defined by these relations to develop a novel model of $(p, q)$-deformed rook theory. Specifically, we introduce a new framework of $(p, q)$-deformed $s$-rook numbers derived from this normal ordering process. Utilizing these combinatorial models, we provide explicit combinatorial interpretations for the associated $(p, q)$-generalized Stirling numbers via rook placements on staircase boards. Our results extend several classical and recent formulations in the literature to the general $p\neq 1$ setting.

We prove the Ashbaugh–Benguria reciprocal-gap conjecture for the Dirichlet Laplacian in every dimension $N\ge2$. Specifically, if $\Omega\subset\mathbb R^N$ is a bounded domain and $$ 0<\lambda_1(\Omega)<\lambda_2(\Omega)\le\lambda_3(\Omega)\le\cdots $$ are its Dirichlet eigenvalues, then $$ \sum_{i=1}^{N} \frac{\lambda_1(\Omega)} {\lambda_{i+1}(\Omega)-\lambda_1(\Omega)} \ge \frac{N}{j_{N/2,1}^2/j_{N/2-1,1}^2-1}, $$ where $j_{\mu,1}$ denotes the first positive zero of the Bessel function $J_\mu$ of the first kind of order $\mu$. We also characterize the equality case: equality holds precisely when $\Omega$ agrees with a Euclidean ball up to a set of Sobolev $H^1$-capacity zero. In particular, among bounded Lipschitz domains, equality holds if and only if $\Omega$ is a Euclidean ball.

The generalized honeymoon Oberwolfach problem (HOP) asks whether it is possible to seat $2n$ participants consisting of $n$ newlywed couples at a conference with $s$ tables of size $2$ and $t$ "round” tables of sizes $2m_1, 2m_2, \ldots, 2m_t$, where $n = s + \sum_{i=1}^{t} m_i $ with all $m_i \geq 2$, over several nights so that each participant sits next to their spouse every time and next to each other participant exactly once. We denote this problem by $HOP(2^{\langle s \rangle}, 2m_1, \ldots, 2m_t)$. In this paper, we provide a complete solution to the generalized HOP with one round table, showing that the obvious necessary conditions for $HOP(2^{\langle s \rangle}, 2m)$ to have a solution are also sufficient.

Operator learning for partial differential equations (PDEs) on arbitrary geometries builds fast neural surrogates for large-scale simulation. While recent geometry-adaptive neural operators have made substantial progress, they are mainly designed for forward problems in which inputs and outputs share the same spatial domain. This limits their applicability for boundary value problems (BVPs) and inverse problems, where inputs and outputs may live on different domains. We introduce the Geometry-Adaptive Integral Autoencoder (GAIA), an operator learning model that encodes the domain boundary and the interior field distribution into geometry tokens, and conditions integral transform layers on these tokens via cross-attention, allowing the kernel to adapt locally to geometric features. This yields an architecture capable of learning solution operators for forward and inverse problems on arbitrary domains, solving each in a single pass at inference without per-domain optimization, iterative refinement, or graph construction. We evaluate GAIA on seven 2D and 3D benchmarks, four of which are new or substantially extended benchmarks for inverse problems and BVP: electrical impedance tomography, optical tomography, 3D Darcy flow on varying geometries, and a modified setting of the Poisson BVP on mechanical components benchmark (MCB). GAIA sets new state-of-the-art results on every inverse and BVP task, reducing median relative $L^2$ error by 64% on airfoil flow reconstruction and 22% on EIT relative to the next best amortized method, and outperforming all baselines on every shape category of MCB. On other forward problems, GAIA is competitive with specialized solvers while maintaining stable accuracy across point resolutions on which transformer-based baselines degrade.

Using modifications to work of Klagsbrun, Mazur, and Rubin, we study (assuming the Extended Riemann Hypothesis) the distribution of Selmer ranks of twist families of some given even-dimensional Galois modules satisfying some mild technical conditions. As a corollary, we study the probability with which a fixed elliptic curve gains (or does not gain) rank in $p$-cyclic extensions, obtaining bounds for this distribution. Likewise, for some superelliptic curves $C$, we bound the average size of $C(L)$ as $L$ ranges over $p$-cyclic extensions over a number field $K$ containing primitive $p$-th roots of unity. Lastly, we study the probability with which a fixed hyperelliptic curve gains (or does not gain) rank in quadratic extensions, also obtaining bounds for this distribution. In all three cases, the extensions under consideration are ordered by the product of ramified primes.

Zeroth-order (ZO) optimization enables fine-tuning large language models when backpropagation is unavailable or memory-prohibitive, but existing methods often perturb full model weights or randomly constructed low-dimensional subspaces, yielding high-variance estimates and limited performance. We propose ZO-Act, an activation-informed ZO fine-tuning method that restricts perturbations to a fixed low-rank subspace derived from input activations. For each linear layer, ZO-Act computes a small activation basis once at initialization and optimizes only lightweight coefficient matrices using forward-only loss evaluations. This reduces the effective perturbation dimension, exposes explicit trainable variables compatible with momentum-based optimizers such as Adam, and naturally supports quantized LLM fine-tuning by keeping low-bit weights frozen. We analyze ZO-Act as zeroth-order optimization over a restricted coefficient space and show that perturbing the low-dimensional coefficients reduces both the variance-dependent convergence term and the finite-difference error of the ZO estimator, at the cost of a controlled subspace approximation bias that is mitigated by the low-rank structure of LLM activations and gradients. Experiments on Llama-3-8B, OPT-13B, and INT4 Llama-3-8B show consistent gains over strong ZO fine-tuning baselines across language understanding, question answering, and commonsense reasoning.

The theory of boundary quadruples and boundary triples is well-studied for symmetric and skew-symmetric operators and in general for dual-pairs. This paper adapts a suitable version for abstract Friedrichs operators and addresses the following questions: which parameters yield bijective realisations, and which parameters yield $m-$accretive realisations. We study a boundary-quadruple framework in which closed realisations are parametrised by closed relations in a boundary space. This yields the intrinsic criterion \[T_\Theta\ \rm{is} \ \rm{bijective}\iff \lK=\Theta\dotplus \Gamma(\ker T_1) \;.\] For bounded operator parameters $\phi:\lK_1\to \lK_0$ in the boundary space, we introduce the reference operator \[Q_0=\Gamma_1(\Gamma_0|_{\ker T_1})^{-1}\,,\] prove that $\|Q_0\|< 1$, and obtain the exact criterion \[T_\phi \ \rm{is} \ \rm{bijective} \iff \I_{\lK_0}-\phi Q_0\ \rm{is} \ \rm{bijective}\;.\] Consequently, every non-expansive parameter gives a bijective realisation with signed boundary map, which is also $m-$accretive. An existence criterion for boundary quadruples and boundary triples is established in terms of (V)-boundary conditions. The multiplicity of $M-$operators associated with a fixed (V)-boundary condition is addressed in an explicit way and a parametrisation of such operators is given. The theory is illustrated by a first-order ordinary differential operator and by the stationary diffusion equation, where $Q_0$ is identified as a Cayley transform of the Dirichlet-to-Neumann operator.

We consider the Hasegawa-Mima equation in its “Euler-like” velocity form: \[\partial_t(u-\Delta^{-1}u)+(u\cdot\nabla)u-u^\perp\log n_0=0,\] $n_0$ being the time-independent function appearing in the particle count $n=n_0e^{\frac{e\varphi}{T}}$, and $u$ being the drift velocity $\nabla^\perp\varphi=-\nabla\varphi\times\hat z$. Adapting the notion from Lions' book on the Euler equations, we prove the existence of dissipative solutions for this equation for any $L^2$ divergence free initial condition $w\in L^2(D)$, for $D=\mathbb T^2$ and $D\subset\mathbb R^2$ a bounded $\mathcal{C}^1$ domain.

We present a symmetric Lanczos method for computing charge-changing QRPA strength functions in atomic nuclei. Starting from the finite-amplitude-method formulation of the QRPA linear-response problem, we derive equivalent spectral representations and, in the real case, a reduced eigenvalue problem involving the matrix products $MK$ and $KM$, where $M\equiv A+B$ and $K\equiv A-B$ are formed from the usual QRPA matrices $A$ and $B$. The resulting formulation enables a matrix-free Lanczos approximation of the Lorentzian-smeared strength function over a broad energy interval from a single Krylov run, in contrast to conventional frequency-by-frequency response calculations. Numerical tests for $^{112}$Sn and $^{150}$Nd first show that GMRES reproduces the converged iterative FAM strength profiles while requiring fewer iterations. Using GMRES as the frequency-by-frequency reference, we then show that the Lanczos approximation reproduces the same strength profiles with reduced overall cost. These results indicate that symmetric Lanczos projection provides an efficient and accurate approach for QRPA strength-function calculations when spectral information is required over an extended frequency range.

We introduce the intersection theory of the moduli space of curves and its tautological ring. We survey open questions about the tautological ring and sketch techniques for proving the Chow ring is or is not generated by tautological classes.

We study the expressive power of the simplest modal-temporal language, obtained by adding Prior's temporal operators \(G\) and \(H\) to the basic modal language with \(\Box\). This language is the standard bimodal combination of modal and tense logic; under its functional interpretation it is denoted \(L_{T\times W}\). To analyse its definability across five order types, we consider two semantic readings of the temporal operators: the standard reading (\(G,H\)), which includes the current instant, and the strict reading (\(G^{\ast},H^{\ast}\)), which always excludes it. We examine nine functional properties – totality, non-totality, injectivity, surjectivity, monotonicity, strict monotonicity, antitonicity, strict antitonicity, and constancy – over preorders, strict preorders, partial orders, linear orders, and strict linear orders. Our analysis reveals two different levels of expressive power. In the original multiflow setting, the language is quite weak and the two readings coincide. When we restrict the semantics to minimal functional frames (the \(O^{2}\) family), many properties become definable, and the choice of reading becomes crucial: the strict reading can define properties such as injectivity even in reflexive orders. The same definability patterns appear with indexed languages and with the Uniform Domain condition on the semantics of \(L_{T\times W}\). That three such different ways of controlling functional multiplicity lead to identical definability patterns indicates that the expressive limitations of the original framework come from the uncontrolled multiplicity of functions, not from any weakness of the operators. Even after controlling functional multiplicity, a set of properties remains undefinable in all non-linear orders, showing that the lack of connectivity is a fundamental obstacle.

Vizing's conjecture, dating back to 1963, asserts that \[ \gamma(G\mathbin{\square}H) \geq \gamma(G)\gamma(H) \] for all finite graphs $G$ and $H$, where $\gamma$ denotes the domination number and $\square$ denotes the Cartesian product. In 2000, Clark and Suen proved the universal bound \[ \gamma(G\mathbin{\square}H) \geq \frac{1}{2}\gamma(G)\gamma(H). \] Recently, Steiner obtained the first constant-factor improvement of the Clark–Suen bound, proving that \[ \gamma(G\mathbin{\square}H) \geq \frac{5+\sqrt{73}}{24}\gamma(G)\gamma(H) \approx 0.5643\,\gamma(G)\gamma(H). \] In this paper, we further improve the universal constant by proving that \[ \gamma(G\mathbin{\square}H) \geq \frac{2}{3}\gamma(G)\gamma(H) \] for all finite graphs $G$ and $H$. Thus, we raise the best known universal constant in the approximate form of Vizing's conjecture from $(5+\sqrt{73})/24$ to $2/3$.

We construct new multiparticle models of $\mathcal{N}=4$ supersymmetric mechanics with spin degrees of freedom by employing nonlinear indecomposable supermultiplets ${\bf (1,4,3){\supset\hspace{-1.1em}+}(4,4,0)}$. These systems are proper deformations of those associated with the standard irreducible $d=1$, $\mathcal{N}=4$ multiplets. In this way we find a deformation of $\mathcal{N}=4$ supersymmetric U$(2)$-spin rational Calogero system invariant under $d=1$ superconformal group OSp$(4|2)$. One more deformed model reproduces $\mathcal{N}=4$ supersymmetric U$(2)$-spin hyperbolic Calogero system, up to a shift of the Hamiltonian by some U$(1)$ generators.

We prove that if an $n\times n\ (n > 3)$ companion matrix $A$ with the spectrum $\sigma(A) = \{ a \}$ has a circular numerical range, then $A$ is the Jordan block. This problem can be described by examining zeros of the Laurent polynomial arising from geometric properties of the numerical range. The difficulty is that the relevant Laurent coefficients involve both the repeated eigenvalue $a$ and the radius parameter $\lambda$, so direct coefficient comparison does not isolate $a$. We address this by decomposing the relevant matrix into a tridiagonal Toeplitz part plus a rank-two update and using Chebyshev polynomials of the second kind. This reduction yields an explicit Laurent-coefficient formula whose vanishing under the circularity condition gives $a=0$. Furthermore, we extend this result when the spectrum is $\sigma(A) = \{0, a\}$ with algebraic multiplicities $n-m$ and $m$, respectively.

We use the Laplacian in n-dimensional spherical coordinates (n>1) to write the divergence of a vector field defined on radially symmetric domains in the context of vector calculus. We apply straightforward equations of vector calculus with the nabla operator and the transformation matrices from Cartesian to spherical polar coordinates. One needs the divergence of a vector field e.g. to prove that vector fields are eigenfunctions of the Stokes operator on n-dimensional annuli and balls. Our divergence formula in partial derivatives in n-dimensional spherical polar coordinates is an important step in a future verification of further Stokes eigenfunctions on those domains.

We investigate Gaussian process (GP) bandit optimization with quantum kernels, assuming the mean reward function lies in the reproducing kernel Hilbert space (RKHS) induced by the quantum kernel. This setting is motivated by NISQ-era tasks such as quantum control, state preparation and variational quantum algorithms. While quantum kernels can offer a `quantum advantage' via domain-specific inductive biases, naïvely using full, high-dimensional kernels increases model complexity and information gain, leading to higher cumulative regret and poor learnability. To address this, we propose projected quantum kernels and classical kernel approximation techniques that reduce feature dimensionality while preserving key quantum properties. Using these approximate kernels, we develop misspecified GP bandit algorithms and derive regret bounds that characterize the trade-off between approximation error and information gain. The regret bounds provide principled guidance for selecting the optimal model complexity. Empirically, our methods outperform full quantum kernels in sample efficiency, while substantially reducing computational overhead, enabling scalable GP optimization for quantum-native applications.

Let $G$ be a quaternionic form of a $p$-adic classical group ($p$ odd). We construct a Bushnell-Kutzko-Stevens type for every Bernstein block of the category of smooth complex representations of $G$. Further we construct a system of compatible $\beta$-extensions, i.e. a family of $\beta$-extensions parametrised by the points of a chamber of the Bruhat-Tits building of the centralizer $G_\beta$ which are related via transfer.

We study the supersaturation problem in its edge-spectral form. Let $\lambda(G)$ be the adjacency spectral radius of $G$. Nikiforov proved that every $K_{r+1}$-free graph $G$ with $m$ edges satisfies $\lambda (G)\le \sqrt{(1\!-\!1/r )2m}$. Recently, Li, Liu and Zhang proved the same bound for every $F$-free graph $G$, where $F$ is any color-critical graph with $\chi(F)=r+1\ge4$, with equality only for regular complete $r$-partite graphs. It is then natural to ask how many copies of $F$ are forced once $\lambda (G)$ exceeds this threshold. Fang, Lin and Zhai answered this at the threshold itself, and conjectured that for any fixed $C>0$, the condition $\lambda (G)\ge \sqrt{(1\!-\!{1}/{r})2m} +C$ forces $\Omega\!\left(m^{(f-1)/2}\right)$ copies. In this paper, we answer this question with the best possible constant. Building on the proof framework of Fang, Lin and Zhai, we prove that for every color-critical graph $F$ with $\chi(F)=r+1\ge4$, there exists $\delta_F>0$ such that if $m$ is sufficiently large, $0<q\le\delta_F\sqrt m$, and $G$ is an $m$-edge graph with $\lambda^2(G)\ge 2\left(1-\tfrac1r\right)m+q$, then \[ N_F(G)\ge\bigl(B_F-o(1)\bigr)\,q\, m^{{(f-2)}/{2}}, \quad \text{where}~~ B_F:=\tfrac{\alpha_F}{4} (\tfrac{2r}{r-1} )^{{f}/{2}}, \] and the constant $B_F$ is best possible. Our result can be viewed as an edge-spectral counterpart of Mubayi's theorem, since it converts the spectral surplus $q$ into a linear number of copies of $F$, and it solves the conjecture of Fang, Lin and Zhai in a stronger form.

Wave scattering by localized constraints in microstructured solids is strongly influenced by the interplay of material length scales, dispersion and geometry. This work investigates plane-strain scattering of time-harmonic P and SV waves by clusters of rigid point constraints embedded in an infinite strain gradient elastic medium. A closed-form dynamic Green's tensor is derived for the plane-strain problem. Unlike the classical elastodynamic Green's tensor, the strain gradient Green's tensor remains bounded at the source, enabling point constraints to be introduced directly through superposition of fundamental solutions. The multiple-scattering problem is reduced to a finite-dimensional algebraic system for the pin reaction amplitudes. A frequency-domain procedure is developed to identify resonance-like amplification and trapping. Candidate resonant frequencies are associated with local minima of the Green matrix determinant, while higher-order curvature criteria distinguish trapping-dominated resonances from non-localized scattering responses. The results show that the response is governed primarily by the ratio of the microinertial and energetic strain gradient lengths. In the anomalous dispersion regime, sharp resonances produce strong displacement localization, including perimeter-localized trapping modes in dense circular arrays. In the normal dispersion regime, these resonances are strongly attenuated and the pins behave as weak scatterers, producing a cloaking-type response in which the incident field is only weakly perturbed. The influence of Poisson's ratio, incidence angle and compound pin configurations is also examined, demonstrating how intrinsic material lengths and geometric arrangement can be used to tune scattering, trapping and wave-screening mechanisms in microstructured elastic media.

We prove $L^{p(\cdot)}$-to-$L^{q(\cdot)}$ bounds for variable versions of the fractional maximal $M^{\alpha(\cdot)}$ and Riesz potential $I^{\alpha(\cdot)}$ operators. The changing fractionality in these operators is given by averaging the function $\alpha(\cdot)$ over balls. The bounds for $M^{\alpha(\cdot)}$ are in terms of a three-exponent Muckenhoupt condition relating $p(\cdot),q(\cdot),$ and $\alpha(\cdot)$, while the bounds for $I^{\alpha(\cdot)}$ are in terms of the boundedness of $M^{\alpha(\cdot)}$ and a packing condition on $\alpha(\cdot).$ These bounds hold under Hardy–Littlewood maximal function boundedness and Muckenhoupt conditions on the individual exponents $p(\cdot),q(\cdot),\alpha(\cdot).$ The proofs are based on an adaptation of sparse domination to variable fractionality and an embedding into variable sequential spaces.

A topological space \(X\) is called a \(Q\)-space if every subset of \(X\) is a \(G_\delta\)-set, and \(X\) is a \(\Delta\)-space if for any decreasing sequence \(\{D_n : n \in\omega\}\) of subsets of \(X\) with empty intersection there is a decreasing sequence \(\{U_n : n \in \omega\}\) of open sets with empty intersection such that \(D_n \subseteq U_n\) for all \(n \in\omega\). Our main result shows that the following statements are equiconsistent: (1) There exists a measurable cardinal; (2) There exists a crowded Baire \(T_1\) \(\Delta\)-space; (3) There exists a crowded Baire \(T_4\) \(Q\)-space; (4) There exists a \(T_1\) \(\Delta\)-space admitting a strictly positive probability measure vanishing on points; (5) There exists a \(T_3\) \(Q\)-space admitting a strictly positive probability measure vanishing on points. This provides complete answers to some problems and partial answers to other problems that have recently appeared in the literature. We also prove a new result concerning Lindelöf \(Q\)-spaces: if \(X\) is a \(T_3\) Lindelöf \(Q\)-space with \(w(X)\leq \mathfrak c\), then \(|X|<\operatorname{cf}(\mathfrak c)\). This yields a number of nonexistence results for large Lindelöf, locally compact, compact, and countably compact \(Q\)-spaces.

We introduce double transposed Poisson algebras, a noncommutative analogue of the transposed Poisson algebras of Bai, Bai, Guo and Wu that is compatible with the Kontsevich–Rosenberg principle. We first consider a simplified version which we call id-adapted double transposed Poisson algebras and then explore the general definition. We prove that every such structure on a unital associative algebra $\mathbb{A}$ is governed by a single derivation $\mathbb{A}\to\mathbb{A}\otimes\operatorname{S}(\mathbb{A}/[\mathbb{A},\mathbb{A}])$. Furthermore, this induces a $\operatorname{GL}_N$-equivariant transposed Poisson structure on each representation algebra $\mathbb{A}_N=\Bbbk[\operatorname{Rep}_N(\mathbb{A})]$. We also define $H_0$-transposed Poisson structures, the transposed counterpart of Crawley-Boevey's $H_0$-Poisson structures, and use the trace map to obtain a transposed Poisson structure on the ring of $\operatorname{GL}_N$-invariants $\mathbb{A}_N^{\operatorname{GL}_N}$.

In the fair allocation problem under conflict constraints, the goal is to partition the vertices of a graph among agents in a fair manner, such that no two adjacent vertices are assigned to the same agent. We study this problem for agents with common preferences through the lens of three fairness criteria: stochastic-dominance envy-freeness up to one item for preference orders (SD-EF1), envy-freeness up to one item for monotone additive valuations (EF1), and envy-freeness up to one item from each side for general additive valuations (EF[1,1]). To do so, we introduce a hierarchy of variants of the strong chromatic number, a graph quantity introduced independently by Alon and Fellows in the early nineties. Our results reveal a close connection between fair allocation under conflict constraints and the first two levels of this hierarchy, providing a unified route to both existential and algorithmic results. For SD-EF1, we fully characterize the number of agents needed to guarantee a fair allocation of a given graph for every common preference order. For EF1 and EF[1,1], we provide analogous sufficient conditions, extending a result on path graphs due to Equbal, Gurjar, Igarashi, Kumar, Manurangsi, Nath, Saxena, Vaish, and Yoneda. We also show that, unlike in the SD-EF1 setting, the sufficient conditions for EF1 and EF[1,1] are not necessary in general. Our framework yields existential and algorithmic consequences in terms of the maximum degree. We obtain that every graph with maximum degree $\Delta$ admits SD-EF1, EF1, and EF[1,1] allocations for common preferences whenever the number of agents is at least $3\Delta-1$. We further provide, for any $\varepsilon>0$, deterministic polynomial-time algorithms that find such allocations whenever the number of agents is at least $(3+\varepsilon)\Delta$. These guarantees strengthen earlier work by Barman and Viswanathan on equitable colorings.

Silting mutation in triangulated categories, both at the level of objects and of subcategories, was introduced in arXiv:1009.3370, and later generalized to extriangulated categories in arXiv:2303.08125. It simultaneously encompasses the mutation theories of cluster-tilting objects in cluster theory and of compact 2-term silting complexes and support $\tau$-tilting modules in $\tau$-tilting theory. In this article, we develop an infinite-dimensional analog of silting mutation in extriangulated categories with set-indexed (co)products, which we then apply to obtain a theory of mutation for $n$-cosilting complexes over an arbitrary ring, as well as for infinite-dimensional $n$-(co)tilting modules over a ring of finite global dimension. The former theory is also shown to reinterpret the cosilting mutation introduced in arXiv:2201.02147.

We study a broad class of graphical models whose independencies correspond to vertex separation in mixed graphs with directed, undirected, and bidirected edges, that are capable of encoding independence structures arising from feedback, latent and selection mechanisms. In particular, we introduce separable graphs, in which each missing edge implies the existence of a separating set for its endpoints, and essentially separable graphs, those graphs separation equivalent to a separable graph. We show that these models include many existing graph families used to define graphical models an provide several characterizations of separable graphs and essentially separable graphs. We also provide multiple characterizations of separation equivalence for separable graphs. One is a graphical characterization in terms of ordinary graph properties, extending earlier results for specific subfamilies Another is a separational characterization depending only on graph separation properties. Finally, we provide a canonical representation for the equivalence classes of essentially separable graphs and develop an algorithm that, under suitable assumptions, identifies the equivalence class of any essentially separable graph.