Papers

Wang and Sitters' 11/6-approximation for graph balancing (each job allowed on at most two machines) is not one algorithm but a set of permitted executions. Step 1 may return any feasible solution of the fractional relaxation. Step 3 may match remaining jobs to the rounding's slots in many ways. We determine exactly what that latitude permits: ratios arbitrarily close to 11/6, none reaching it, so 11/6 is the least constant bounding every permitted run. We then determine the worst-case guarantee, against the optimum itself, as a function of the big-job threshold beta, the fractional share at or above which Step 2 assigns a job. On Wang and Sitters' own range 1/2 < beta < 1 the guarantee is exactly max{3/2 + beta/2, 5/2 - beta}. We extend the same eligibility rule to 0 < beta <= 1/2, outside the range they state. There a big job's two shares can both reach the threshold, so Step 2 acquires a third choice. We determine it too: exactly 3/2 + (1-beta)floor(1/beta), hence unbounded as beta falls. The worst-case ratio is therefore known at every threshold in (0,1), and attained at none. Consequently 2/3 is the unique optimal threshold, and the guarantee jumps at one half and again at every threshold 1/k below it, rather than degrading smoothly. A companion note examines changes that leave the constant unchanged.

Let a group $G$ act linearly on a finite-dimensional complex vector space $V$. The group-additive diameter of a subspace $U \leq V$ is the least number of translates of $U$ whose sum is all of $V$. Counting dimensions, it is at least $\dim V / \dim U$. We show that when $G$ is compact and $V$ is irreducible, the diameter of every nonzero subspace is at most $\lceil (\dim V / \dim U) \ln \dim V \rceil$, so the trivial bound is correct up to a logarithmic factor. We measure the discrepancy by the covering complexity $\mathsf{C}(V)$, the largest ratio between the diameter of any subspace and its trivial lower bound, so that $1 \leq \mathsf{C}(V) \leq 2 + \ln \dim V$, and we determine where in this range various representations lie. Every irreducible representation of $\mathrm{SL}_2(\mathbf{C})$ has $\mathsf{C}(V) = 1$. The logarithm can be genuinely present along families of symmetric and exterior powers of $\mathrm{SL}_n(\mathbf{C})$ with $n$ varying, and it is present for finite Heisenberg groups and $2$-transitive groups of small order such as $\mathrm{PSL}_2(\mathbf{F}_p)$. The complexity is bounded above by a constant on the conjugation representations of $\mathrm{SL}_n(\mathbf{C})$ and on the representations $\operatorname{Sym}^k \mathbf{C}^3$ of $\mathrm{SL}_3(\mathbf{C})$. On the other hand, every fixed connected reductive group has a family of irreducible representations whose complexities tend to at least the dimension of the flag variety. Finally, for the Lie algebra $\mathfrak{sl}_3(\mathbf{C})$ acting on $\operatorname{Sym}^k \mathbf{C}^3$, the monomial diameter with respect to $\operatorname{Sym}^k X$ for a plane $X \leq \mathbf{C}^3$ is optimal, while the corresponding $\mathrm{SL}_3(\mathbf{C})$ diameter is not.

We study the operators obtained by inserting copies of a new largest letter into multiset permutations. Let $G_r$ denote the operator which inserts $r$ copies of a new largest letter. After the change of variables $\delta=y-x$, $u=x/y$, and $E=u\partial_u$, we find that $$G_r=\frac{\delta^r}{r!}E(E+1)\cdots(E+r-1).$$ Its generating series acts by a rational substitution, which yields the composition law. Our main result gives a common symmetric-function explanation for the ordinary and major-index operators. For $N\geq 0$, define $\Phi_N(F_{N,S})=x^{|S|+1}y^{N-|S|}$. We prove that multiplication by the complete homogeneous symmetric function $h_r$ becomes the ordinary insertion operator: $\Phi_{N+r}(h_r f)=G_r\Phi_N(f)$, where $f\in\mathrm{QSym}_N$. There is a parallel specialization for the major index. A reverse finite principal specialization sends multiplication by $h_r$ to an operator $Q_r$, which is a polynomial in the $q$-shift $\Theta_qf(t)=f(qt)$. Thus the ordinary and major-index operators arise from the same multiplication operator $f\mapsto h_r f$. Since the functions $h_r$ freely generate the ring of symmetric functions, the assignment $h_r\mapsto G_r$ extends to an algebra homomorphism. We determine the kernel of this homomorphism and the image of every homogeneous component. The images of Schur functions satisfy the Littlewood–Richardson multiplication identities, and the one-row case gives an Eulerian form of the Pieri rule.

We prove a scaling-critical regularity criterion involving only one velocity component for finite-energy suitable weak solutions of the three-dimensional incompressible Navier–Stokes equations. Let $2<p<\infty$ and $m=3p/(p-2)$, so that $2/p+3/m=1$. We show that a singularity cannot occur provided \[ \sum_{j\in\mathbb Z} \|\dot\Delta_j u^3\|_{L^p(0,T;L^m(\mathbb R^3))}<\infty, \] that is, $u^3\in\widetilde L^p(0,T;\dot B^0_{m,1}(\mathbb R^3))$. The assumption is a spatial-frequency $\ell^1$ refinement of the still unresolved critical condition $u^3\in L^p_tL^m_x$ and is complementary to the Lorentz-in-time refinement $L^{p,1}_tL^m_x$ obtained by Wang, Wu, and Zhang. By Bernstein embedding, the result extends to $u^3\in\widetilde L^p_t\dot B^s_{q,1}$ on the nonnegative-smoothness critical line $s=-1+2/p+3/q\ge0$. The principal innovation is a frequency–scale matching scheme embedded in the local energy inequality. Each dyadic block of $u^3$ is retained until it is paired with the vertical scale selected by a one-dimensional backward heat kernel. Low frequencies gain from the slab thickness, high frequencies from transferring the projection to the localized flux and applying an inverse Bernstein estimate, and spatially separated pressure sources from harmonic decay. These mechanisms generate a two-sided $\ell^1$ kernel, converting spatial-frequency summability into summability of physical-scale energy increments. Consequently, we obtain a uniform Type-I local energy bound without a Lorentz refinement in time; compactness and one-component rigidity then exclude singular blow-up limits whose third velocity component vanishes.

The aim of this paper is to develop a framework for measure theory that avoids infinities and allows for the uniform treatment of positive and vector measures. Our approach is based on a modification of the notion of measure, which supplements the usual $\sigma$-additivity requirement with a suitable maximality condition. To each Hausdorff topological vector space $\mathfrak A$ and $\sigma$-ring $\mathcal{Q}$, we associate a vector space $\mathscr M(\mathcal{Q},\mathfrak A)$ of `infinite' $\mathfrak A$-valued measures corresponding to $\mathcal{Q}$. In particular, the positive elements of $\mathscr M(\mathcal{Q},\mathbb R)$ are naturally identified with the $\sigma$-finite positive measures defined on $\mathcal{Q}$, thus placing positive and signed measures within the same setting. Finally, extension results for group-valued contents due to Sion and Weber are reformulated and refined within this new framework.

Let $G$ be a definable group in an NIP theory. We prove that every minimal proximal definable $G$-flow is strongly proximal. Consequently, the universal minimal proximal definable $G$-flow $\Pi^{\mathrm{def}}(G)$ coincides with the minimal strongly proximal definable $G$-flow $\Pi^{\mathrm{def}}_{\mathrm{s}}(G)$. Furthermore, for a $p$-adic definable group $G$, we can compute $\Pi^{\mathrm{def}}(G)$ explicitly. We show that $\Pi^{\mathrm{def}}(G)$ is exactly $\Pi^{\mathrm{def}}(S)$ where $S$ is the semisimple part of the definably amenable-semisimple decomposition of $G$. In addition, $\Pi^{\mathrm{def}}(S)\cong S^*_\mathcal{F}(\mathbb{Q}_p)$, the space of types of full dimension on $\mathbf{F}$, where $\mathcal{F}=\mathbf{F}(\mathbb{Q}_p)$ for a flag variety $\mathbf{F}$ constructed from $S$.

We study a data-driven reflection control problem for a Brownian model with unknown drift and volatility. We first propose a learn-then-optimize (LTO) algorithm: it estimates the policy-relevant parameter during exploration, plugs the estimate into the optimality equation, and exploits the resulting policy—achieving an $O(\sqrt{T})$ finite-time expected regret bound. We further propose two algorithms, adaptive-updating (AU) and full-history adaptive-updating (AU-FH), which continuously update the estimator and reflecting level, attaining an improved $O(\log T)$ regret bound. Notably, AU-FH algorithms leverages all historical data, yielding better performance in numerical simulations. Our analysis decomposes regret into exploration, transient, and learning components. Transient regret from nonstationarity is bounded by the time-integrated deviation of the transition semigroup from stationarity evaluated on the holding cost, which can be further bounded via a Foster-Lyapunov inequality for exponential convergence of the controlled reflected Brownian motion (RBM). For learning regret, we establish local regularity properties together with consistency and mean-squared error bounds for the estimator, which control the stationary cost gap between the learned and optimal reflection policies. In addition, we leverage the monotonicity of the moving boundary Skorokhod map to derive moment bounds for the AU algorithms' switching states, via pathwise comparison with fixed-boundary RBMs.

This paper is concerned with the precise asymptotics, as time goes to infinity, of a transport problem in a half plane coupled with a nonlinear nonlocal boundary condition. This system arises from a class of models for the spatial spread of epdemics, its space independent version being the classical Kermack-McKendrick model. Using ideas pertaining to the study of nonlocal equations of the Fisher-KPP type, and exploiting the particular structure of the model, we prove that any initially localized solution will lag behind the minimal traveling wave, with a delay that grows logarithmically in time.

We classify a class of abstract Bose field models in quantum field theory up to unitary equivalence. The class includes abstract free Bose field models, abstract van Hove–Miyatake models, and infrared-renormalized van Hove–Miyatake models. Moreover, as an application of our classification, we classify quadratic interaction models. In particular, our results answer a question of A. Arai concerning the classification of infrared-renormalized van Hove–Miyatake models.

Retail supply chain operations rely on coupled decision modules that must adapt as requirements evolve. LLMs offer a natural-language interface for this task, but existing methods primarily focus on individual optimization models. Extending them to heterogeneous decision pipelines is challenging because a requirement may admit multiple intervention paths with different downstream effects. We formulate requirement-driven adaptation as the joint selection of an intervention route and an admissible module-level change, and propose a graph-constrained agentic framework in which domain agents expose admissible reformulation interfaces and a central processor searches over bounded intervention paths. Candidates are validated and compared using downstream KPIs. In collaboration with a large retail partner, we evaluate 100 warehouse requirements elicited from practitioner interviews, with GPT, Qwen, and DeepSeek as base LLMs. Relative to direct LLM reformulation, our framework improves correctness and end-to-end success across all three models, raising end-to-end success from 72–76% to 79–83%.

We study the asynchronous $\ell^p$ relaxation introduced by Amir, Nazarov, and Peres: at each step, a uniformly chosen vertex minimizes its incident $\ell^p$ energy. For the profile $f_t$ after $t$ updates, let $$ \mathsf T_p(G,1/2):= \sup_{\lVert f_0\rVert_\infty\le1} \mathbb{E}\bigl[\min\{t\ge0:\operatorname{osc}(f_t)\le1/2\}\bigr]. $$ For $1<p<\infty$, we obtain graph-dependent estimates for boxes, trees, and conductance expanders. On the nearest-neighbor box $[L]^d$, for $d,L\ge2$ and $n=L^d$, the answer is, up to logarithmic factors, $nd^{1/(p-1)}L^{p/(p-1)}$ for $1<p<2$ and $ndL^2$ for $p\ge2$. On bounded-degree trees, an explicit rerooting-invariant parameter $T_G$ determines the answer up to logarithmic factors; for arbitrary trees it gives upper and lower bounds that differ additionally by the maximum degree. If the volume conductance $h(G)\ge h_0>0$, then $\mathsf T_p(G,1/2)=\Theta_{p,h_0}(n\log n)$ without a degree assumption. At $p=\infty$, every connected graph satisfies $\mathsf T_\infty(G,1/2)\ge c nD^2/\Delta$, where $D$ and $\Delta$ are its diameter and maximum degree.

We prove the following inequality for Dirichlet polynomials: \[ \int_0^1 |f(1/2+\sigma)|\,d\sigma\lesssim \lim_{T\to\infty} \frac{1}{2T} \int_{-T}^T |f(it)| \, dt. \] In particular, for a Dirichlet series $f(s) = \sum_{n\geq 1} a_n n^{-s}$ belonging to the Hardy space $\mathscr{H}^1$ of Dirichlet series, \[ \left|a_1+\sum_{n=2}^\infty \frac{a_n}{\sqrt n\log n}\right| \lesssim \|f\|_{\mathscr{H}^1}. \] This answers a question raised previously in the literature and it proves that the multiplicative Hilbert matrix has a bounded symbol.

The Schrödinger-Lohe (SL) model is a coupled system of nonlinear Schrödinger equations describing the temporal-spatial evolution of the component wave functions, and it corresponds to the infinite-dimensional counterpart of the Lohe matrix model for quantum synchronization. In this paper, we study a rigorous mean-field limit of the SL model and provide a quantitative estimate on the fluctuations between one-marginal distribution and one particle distribution in 2-Wasserstein distance. For the derived kinetic SL equation, we present sufficient conditions leading to the complete and practical synchronizations.

We disprove McKean's conjecture, which asserts that the entropy dissipation is monotone nonincreasing along solutions, or equivalently that the entropy is convex in time, for the spatially homogeneous Landau-Coulomb equation. We provide an explicit counterexample which consists of a Maxwellian equilibrium under radially symmetric, smooth perturbation on an annulus of scale $R\gg 1$. Developing the derivative of the entropy dissipation in powers of $R$ reveals that the leading order terms can be positive for a suitably chosen perturbation. Remarkably, the counterexamples are close to equilibrium in relative entropy and standard Sobolev norms. Since the Landau equation arises from the Boltzmann equation in the grazing collisions limit, our counterexample also shows that McKean's conjecture is not true in general for the Boltzmann equation.

We study the Bergman embeddings of degenerating families of Kähler-Einstein manifolds of negative curvature. We show that when the central fiber has only simple normal crossing singularities, we can construct orthonormal bases so that the induced Bergman embeddings converge to the Bergman embedding of the limit space together with multi-bubbles.

We prove a Haglund–Haiman–Loehr type combinatorial formula for the torus fixed point classes $I_{\mu,w}$ in the equivariant $K$-theory $K_{\mathbb{C}^{*}\times\mathbb{C}^{*}}(\operatorname{PFH}_{n,n-k})$ of parabolic flag Hilbert schemes. Under the identification of Bechtloff Weising and Orr, our result gives an HHL type formula for modified partially symmetric Macdonald functions in terms of weighted partial Dyck paths. When $w=\emptyset$, it specializes to the classical HHL formula. The proof relies essentially on the Carlsson–Gorsky–Mellit action of the Carlsson–Mellit algebra $\mathbb{A}_{q,t}$.

For a fixed characteristic $p$, let $\delta(p)$ denote the smallest degree needed to guarantee the existence of an isogeny from any supersingular curve $E$ to its Frobenius conjugate $E^{(p)}$. We translate existence questions concerning $\delta(p)$ into questions about positive definite integral ternary quadratic forms and use Voronoi's theory of perfect forms in dimension three to characterize when $\delta(p)$ is nearly maximal. For sufficiently large primes, we show that $\delta(p)$ attains its upper bound precisely when $p$ is represented by one of nine explicit cubic polynomials, reducing the infinitude of such primes to whether one of these cubics takes prime values infinitely often. We also prove that, for almost all primes, $\delta(p)$ lies at least $p^{1/6-o(1)}$ below its upper bound.

The topological mating of two postcritically finite polynomials with dendritic Julia sets is encoded by a pair of circle laminations $\Lambda^{\pm}$, whose collapse produces a sphere-filling curve. When a leaf of $\Lambda^{+}$ and a leaf of $\Lambda^{-}$ share an endpoint they form a perfect fit. An unpublished proposition of Epstein, recorded by Petersen and Meyer, shows that for matings of honest degree-$d$ polynomials an infinite-diameter ray equivalence class, i.e. an infinite chain of perfect fits is impossible. We show that this finiteness is a genuinely holomorphic phenomenon. Allowing the dynamics to carry a periodic critical orbit, we construct combinatorially expanding Thurston maps-realized by no expanding rational map-that admit invariant sphere-filling curves yet whose laminations $\Lambda^{\pm}$ contain infinite chains of perfect fits, in fact infinitely many of them.

We study the boundedness and the compactness of Rhaly and generalized Hilbert operators when acting between distinct spaces of analytic functions in the unit disc of the complex plane.

The cohomology ring of the complement of a complex hyperplane arrangement is given by its Orlik–Solomon algebra. It is known that the defining ideal of the Orlik–Solomon algebra has a quadratic Gröbner basis in the standard presentation if and only if the intersection lattice is supersolvable; such algebras are automatically Koszul. In 1997, Shelton and Yuzvinsky posed the question as to whether all Koszul Orlik–Solomon algebras arise from supersolvable arrangements. We answer this question negatively using three related constructions that produce non-supersolvable arrangements whose Orlik–Solomon algebras are Koszul. Moreover, these arrangements may be chosen to be irreducible, realizable over $\mathbb{Q}$, and of any rank $\geq 3$. Our constructions rely on a result of Falk and Proudfoot which we strengthen and generalize. In two of the three cases, we show non-supersolvability using a corrected form of a result of Ziegler regarding supersolvability of parallel connections. We also construct Koszul Orlik–Terao algebras coming from non-supersolvable arrangements.

For a general class of two-dimensional non-interacting semi-metallic electron systems on a lattice, we derive an explicit formula for the whole conductivity matrix, by using Euclidean many-body formalism. The semi-metallic phase takes place whenever two Bloch bands touch at the Fermi energy with conical intersections and generally occurs at the transition between distinct integer quantum Hall phases. Unlike the longitudinal conductivity (which depends solely on the shape of the cones at the Fermi level), the transverse conductivity is remarkably determined both by the conical structures of the bands and by the nature of the two nearby topological phases. Generically, in the semi-metallic state, neither the longitudinal nor the transverse conductivities are quantized in integer multiples of a universal conductivity quantum; however, universality of the conductivity matrix is restored under the assumption of emergent rotational symmetry of the linearized Hamiltonian at the Fermi points. Using our formula, we compute the conductivity matrix for several physically relevant models of quantum Hall fluids at the topological phase transition, exhibiting cases where the transverse conductivity is quantized in half-integer multiples of $e^2/h$ and cases where it depends continuously on an external strain parameter, thus shedding light on its universality or non-universality features.

For square-integrable divergence-free vector field $\boldsymbol{v}$ on $\mathbb{R}^d$ we prove that the following properties are equivalent: 1) the operator $A_0 \rho = \boldsymbol{v} \cdot \nabla \rho$ (where $\rho \in C^\infty_c(\mathbb{R}^d)$) is essentially skew-adjoint on $L^2(\mathbb{R}^d)$; 2) square-integrable (with respect to spatial variables) generalized solutions of the continuity equation are renormalized; 3) generalized square-integrable (with respect to spatial variables) solutions of the Cauchy problem for the corresponding continuity equation are unique both forward an backward in time. We also construct a compactly supported bounded divergence-free vector field $\boldsymbol{v}\colon \mathbb{R}^3 \to \mathbb{R}^3$ for which square-integrable (with respect to spatial variables) solutions of the Cauchy problem for the corresponding continuity equation are unique forward, but not backward in time.

We study parabolic Poincaré inequalities for solutions to nonlinear systems of partial differential equations. Our main results show that these inequalities are self-improving. As applications, we establish reverse Hölder inequalities for the mean oscillation over parabolic cylinders and for the gradient. We also consider the corresponding Poincaré inequalities and self-improvement results on time intervals at fixed spatial points. These results are based on a pointwise maximal function estimate. As an application, we obtain regularity results in the time direction for solutions to nonlinear systems.

Given a polynomial dynamical system $\mathbf{x}'=\mathbf{f}(\mathbf{x},\mathbf{u})$ together with an observation function $y=g(\mathbf{x},\mathbf{u})$, where $\mathbf{x}=(x_1,\ldots,x_n)$, $\mathbf{u}=(u_1,\ldots,u_m)$ and $y$ are differential variables, and $\mathbf{f}=(f_1,\ldots,f_n)$, $g$ are polynomials with coefficients in a differential field, we study the problem of determining a minimal polynomial differential equation satisfied by the inputs $\mathbf{u}$ and the output $y$ which follows as a differential consequence of the system. We provide a characterization of a finite superset of the set of monomials appearing with non-zero coefficients in this input-output equation. Specifically, we establish an upper bound for the degree of the minimal polynomial and a family of inequalities that define a polytope containing its Newton polytope. These results extend recent work by Mukhina and Pogudin for systems with constant parameters, and enable the use of evaluation-interpolation techniques for the efficient computation of such eliminant polynomials.

Graded Möbius algebras were a key tool in the proof of the Dowling-Wilson Top Heavy Conjecture. They are commutative algebras whose Hilbert functions recover the Whitney numbers of the second kind, i.e. the number of flats of a given rank. The graded Betti numbers of the defining ideal of a graded Möbius algebra refine the Hilbert function and describe its minimal free resolution. In this paper we derive precise formulas for the graded Betti numbers of the defining ideals of graded Möbius algebra for any uniform matroid. We also study when the graded Möbius algebra of an arbitrary matroid is linearly presented.

An edge-colored graph is rainbow if all of its edges receive distinct colors. For a fixed graph $H$, an edge-colored graph $F$ is called weakly $H$-rainbow saturated if there exists an ordering $e_1,e_2,\ldots,e_{|E(\bar{F})|}$ of $E(\bar{F})$ such that, for any edge coloring $c$ of $E(\bar{F})$ with $c(e_i)\neq c(e_j)$, there is always a rainbow copy of $H$ that contains $e_i$ in $F+\{e_1,e_2,\ldots,e_i\}$. The weak rainbow saturation number $\operatorname{rwsat}(n,H)$ is the minimum number of edges in a weakly $H$-rainbow saturated graph on $n$ vertices. Li, Ma, and Xie [JGT, 2025] showed that $\lim_{n\to\infty} \frac{\operatorname{rwsat}(n,H)}{n}$ exists for every nonempty graph $H$. Paths and stars attain, respectively, the minimum and maximum ordinary weak saturation numbers among all trees of the same order. We determine their weak rainbow saturation numbers exactly. For all $\ell>30$, we show that $$ \ell+1=\s(n,P_\ell)< \s(n,S_\ell)=\binom{\ell}{2}-1$$ where $P_\ell$ and $S_\ell$ denote the path and star on $\ell$ vertices, respectively. Thus, their dependence on $\ell$ is linear for paths and quadratic for stars. We then focus on cycles. Li, Ma, and Xie asked whether $\operatorname{rwsat}(n,C_\ell)$ has leading term $\frac32n$ for every $\ell\ge4$. We answer this question negatively by giving an explicit construction showing that, for every $\ell\ge4$ and all sufficiently large $n$, $$\s(n,C_\ell)< \frac{\ell}{\ell-1}n+c_\ell,$$ where $c_\ell$ depends only on $\ell$. Since $\frac{\ell}{\ell-1}<\frac32$, this strictly improves the proposed leading coefficient for every cycle $C_\ell$ with $\ell\ge4$.

In this article, we investigate the two-species zero-range process, a multi-species generalization of the classical zero-range process. First, we analyze its condensation regime, which directly parallels its single-species counterpart. As our main result, we establish the dynamical metastable behavior of the location of the single condensate, showing that its motion is governed by a simple Markov chain on the accelerated time scale $N^{1+\alpha}$, where $N$ denotes the total number of particles in the system and the parameter $\alpha>1$ governs the attractivity of the system. Another novelty of this work lies in the proof technique, which integrates the recently developed resolvent approach with the $H^{1}$-approximation method to rigorously characterize metastability.

Let $U$ and $H$ be two separable Hilbert spaces and $T>0$. We consider a stochastic differential equation which evolves in the Hilbert space $H$ of the form \begin{align} \label{SDEa} dX(t)=AX(t)dt+B(t,X(t))dt+GdW(t), \quad t\in[0,T], \quad X(0)=x \in H, \end{align} where $A:D(A)\subseteq H\to H$ is the infinitesimal generator of a strongly continuous semigroup $(e^{tA})_{t\geq0}$, $W=(W(t))_{t\geq0}$ is a $U$-cylindrical Wiener process defined on a normal filtered probability space $(\Omega,\mathcal{F},\{\mathcal{F}_t\}_{t\in [0,T]},\mathbb{P})$, $G:U\to H$ is a linear bounded operator and $B:[0,T]\times H\to H_\beta$ is a locally $\theta$-Hölder continuous function with respect to the second variable, uniformly with respect to the first one, for some suitable $\theta\in(0,1)$. Here, $H_\beta$ is a Hilbert space which contains $H$ with continuous embedding, so that the drift term is singular: it is not well-defined from the ambient space $H$ into itself. The coefficient $\beta\geq0$ measures the order of such a singularity and the case $\beta=0$ corresponds to a drift term with values in $H$. We prove that, under suitable assumptions on the coefficients, weak and pathwise uniqueness hold true for equation \eqref{SDEa}. In particular, the conditions assumed on the coefficients cover the stochastic damped wave equation in dimension $1$ and the stochastic damped Euler–Bernoulli beam equation up to dimension $3$, even in the hyperbolic case.

We study a time fractional convection-diffusion-reaction equation in a bounded domain $\Omega\subset\mathbb{R}^2$. A stabilized numerical scheme satisfying a discrete maximum principle is constructed by combining the conforming linear finite element method with the algebraic flux correction method. The resulting semi-discrete scheme is nonlinear, and its well-posedness is established. Assuming nonsmooth initial data, we derive error estimates for the semi-discrete scheme using energy arguments. For the temporal discretization, we employ the L1 method, obtaining a fully discrete scheme for which we prove well-posedness and the discrete maximum principle. We also present numerical experiments that validate the order of convergence as well as we test our schemes to solutions that possess layers.

A labeling of a connected graph $G$ on $n$ vertices is a bijection $c:V(G)\to\{1,\dots,n\}$; writing $c(u,v)=|c(u)-c(v)|$, a vertex $u$ is list-distance consistent if $d(u,v)<d(u,w)$ implies $c(u,v)\le c(u,w)$ for all $v,w$. The maximum number of such vertices over all labelings is the list-distance consistency ldc$(G)$, introduced by Casselgren and Henricsson. We prove that in a tree, the consistent vertices of any labeling lie on a single path, along which the labels form a block of consecutive integers in increasing order (with respect to a suitable orientation of the path), no vertex off the path receiving a label from that block. We deduce that ldc equals $3$ for every complete $k$-ary tree except the binary tree of height two, and we determine ldc for all spiders.