Papers

We prove that every bounded-to-one action of a finitely generated commutative monoid has finite asymptotic dimension, without a freeness assumption. If the group completion has torsion-free rank $r$, we give an explicit upper bound $(3^{r+1}-3)/2$. This answers a question of Shinko, Weilacher and Yu, and extends a theorem of theirs.

Our main result is a construction of a class H of pathwise connected, locally connected, compact Hausdorff spaces such that (i) if X,Y are distinct spaces in H then a continuous, injective mapping from X to Y does not exist; (ii) H contains 2^k spaces of weight k and size k for every cardinal number k not smaller than the cardinality of the continuum; (iii) if k,l are infinite cardinals and l is not greater than k then H contains 2^k spaces of weight k and size k^l; (iv) the metrizable continua in H are subspaces of the plane.

Drifting methods have recently been introduced as a promising new paradigm for training generative models, and have attracted a lot of attention so far. In this work, we focus on how the optimization procedure shapes their convergence properties, shedding light on the underlying machinery that may explain their empirical success. More specifically, when the drift field is a Wasserstein gradient, optimizing the drifting loss via plain Euclidean gradient descent seems to forfeit any hope of exploiting convexity and therefore of obtaining any convergence guarantees. Changing either the drift field or the optimization procedure, however, may lead to different dynamics: we show that keeping the Wasserstein gradient as drift field but replacing gradient descent with a Gauss-Newton scheme amounts to performing a natural gradient descent in parameter space. This connection allows us to provide global convergence guarantees for the optimization procedure under suitable convexity assumptions.

Spectral Turán-type problems ask how the absence of prescribed subgraphs constrains the spectral radius of a matrix associated with a graph. Given a family of graphs $\mathcal{F}$, a graph is called $\mathcal{F}$-free if it contains no member of $\mathcal{F}$ as a subgraph. The theta graph $\theta(l_1,\ldots,l_k)$ consists of $k$ internally disjoint paths of lengths $l_1,\ldots,l_k$ with two common end vertices. In this paper, we study two spectral Turán-type extremal problems for the signless Laplacian spectral radius. First, among all $\{C_3,C_4\}$-free graphs of fixed order with no pendant vertices, we determine the maximum signless Laplacian spectral radius and uniquely characterize the extremal graph attaining it. The extremal structure exhibits a parity phenomenon: odd and even orders give rise to two distinct graph families. These results, in particular, sharpen a recent general upper bound for this class given by Liu and Wang (2026). Next, we obtain the corresponding extremal results for all $\{\theta(1,2,2),\theta(1,2,3)\}$-free graphs of fixed size with no pendant vertices when the size is congruent to $1$ modulo $3$ and $2$ modulo $3$, again obtaining unique but structurally different maximizing graphs in the two cases. Together with the previously known result for sizes congruent to $0$ modulo $3$ by Liu and Wang (2026), this completes the fixed-size problem across all three congruence classes modulo $3$.

We introduce d-minimal and b-minimal extensions, two weakenings of minimality based respectively on eventual domination and infinitely often domination. We prove that the generic extensions obtained by the full Mathias forcing and by Mathias forcing relative to Ramsey ultrafilters are d-minimal, whereas Hechler extensions are b-minimal. We also construct a d-minimal extension that is weakly $\omega^\omega$-bounding but neither minimal nor $\omega^\omega$-bounding, and realize every constellation of these properties compatible with their natural implications.

In this paper, we study local and global properties of positive solutions to the equation $-\Delta u=u^p|\nabla u|^q$ in a domain $\Omega$ of $\mathbb R^N$, where $p$ and $q$ are parameters. We introduce a linear operator to construct the differential inequality to obtain gradient estimates, and further establish Liouville-type theorems. As an application, we derive universal estimates for local solutions. Some of our results are new, as we extend the condition $p+q<(N+3)/(N-1)$ considered by He, Hu and Wang [Math. Z. 313 (2026), No. 6] to a wider range of parameters.

We construct relative Du Bois and relative intersection Du Bois complexes for smooth-factorizable morphisms to a smooth complex base using relative de Rham complexes of mixed Hodge modules and Grothendieck duality. For a smooth morphism, these complexes recover the relative Kähler differentials. Comparisons with the relative Du Bois complexes of Kovács and Ning remain open. We construct finite filtrations on the absolute Du Bois and intersection Du Bois complexes whose graded pieces are the corresponding relative complexes tensored with differential forms from the base. We also study restriction to fibers and its relation to symbolic $m$-Du Bois and weak $m$-rational singularities.

We study an optimal intervention problem for linear threshold models. This is a popular class of dynamical network systems whereby a number of agents, identified with the nodes of a graph, strategically change their binary action (0 or 1) according to a threshold rule. Specifically, an agent adopts action 1 if and only if the fraction of its neighbors in the interaction graph that do so is greater than or equal to a prescribed threshold. Assuming that a planner can modify the agents' thresholds at a cost equal to the aggregate threshold increase, we study the minimum intervention cost needed to ensure global convergence to the all-1 configuration. Our main contribution is the introduction of a new graph-theoretic quantity, called oriented path number, that is the minimum number of disjoint paths needed to cover the graph that can be oriented to form a directed acyclic graph. When thresholds are all equal to 1/2, the optimal cost is shown to coincide with the oriented path number, whereas, in the general case, it turns out to be the main ingredient of a bound on the optimal intervention cost.

Let $\mathcal{S}^ r$ denote the Schatten–von Neumann class and let $S_{\Psi_{f,\lambda}}$ be the Schur–Hadamard multiplier whose symbol is the divided-difference matrix of $f$ along $\lambda$. Let $0<\alpha,r<\infty$, let $f\in C^1([-1,1])$ satisfy $f(0)=0$, $|f'(t)|\lesssim |t|^\alpha$, and let $\lambda\in\ell^r$ be real with $\|\lambda\|_{\ell^\infty}\leq1$. We determine the pairs $0<p,q\leq\infty$ for which $S_{\Psi_{f,\lambda}}:\mathcal{S}^ q\to\mathcal{S}^ p$ is bounded for every such $f$ and $\lambda$. Our principal new positive estimates treat $q=\infty,1$ and $0<p<1$. Precisely, we show that the boundedness holds exactly when \[ \frac1p\leq\frac{\alpha}{r} +\min\!\left\{1,\frac1q\right\}. \] In particular, this resolves the untreated cases in [Arazy, PAMS, 1982] and [Potapov, Sukochev, Tomskova, Adv. Math., 2015]. Our method also delivers a new proof of the main results in just cited papers.

We develop a theory of Orlicz and Lorentz–Zygmund sequence spaces. We study in detail elementary properties of both structures. In particular, we characterize the values of governing parameters for which the corresponding spaces satisfy the axioms of rearrangement-invariant Banach sequence spaces, we evaluate the asymptotic behavior of their fundamental functions, and we characterize their associate spaces. In our main result, we characterize all cases in which a Lorentz–Zygmund sequence space coincides with some Orlicz sequence space.

We establish a second main theorem with truncated counting functions for algebraically nondegenerate meromorphic mappings into a projective variety and a family of hypersurfaces in subgeneral position. The above bound of the total defect obtained from our result is better than that of the previous results. Moreover, in our result, the truncation level of the counting functions is estimated explicitly and independently of the number of hypersurfaces. Especially, we do not need the assumption that the family of hypersurfaces must satisfy the Bezout properties as imposed in some earlier studies, but only a weak Bezout property.

Chen and Ngo showed that the Hitchin morphism for Higgs bundles on a projective manifold $X$ factors through a closed subscheme of the Hitchin base called the spectral base, and the induced map is called the spectral morphism. In this paper, we investigate local Cohen-Macaulay obstructions to the surjectivity of spectral morphisms via a refined spectral correspondence. We show that every affine normal variety with a local ring that admits no rank one maximal Cohen-Macaulay modules can be geometrically realized as a local obstruction to the surjectivity of spectral morphisms, and that these local obstructions can be compactified via the Kawamata covering trick. As an application of these constructions, we provide counterexamples to the Chen-Ngo conjecture on the surjectivity of spectral morphisms for $GL_r$-Higgs bundles in dimensions $n\geq 9$. Moreover, we construct counterexamples to the Chen-Ngo conjecture for $SL_2$-Higgs bundles in every dimension $n\geq 2$ and for semistable $GL_3$-Higgs bundles on a smooth projective surface.

We reconstruct a spatially varying material coefficient in a scalar elliptic equation from finitely many boundary excitations, each producing a full Dirichlet trace. To mitigate the ill-posedness and the cost of repeated PDE solves, we restrict the coefficient to a low-dimensional family, specified analytically or learned from samples, and solve the inverse problem in its latent coordinates. We consider a \(C^1\) parametrization with \(m\) latent coordinates and full-rank derivative at a reference point. If the continuous linearized Neumann-to-Dirichlet map is injective on the corresponding tangent space, at most \(m\) excitations suffice for local injectivity and Lipschitz stability. Convergence of the coefficient sensitivities then transfers this stability to conforming finite element discretizations. For sufficiently fine meshes, the stability constant and neighborhood can be chosen independently of the mesh size. Under a local residual-comparison condition, uniform accuracy of the surrogate forward map yields coefficient-error bounds separating representation error, data noise, finite element error, and surrogate error. Derivative accuracy additionally preserves the surrogate's own local stability. All stability statements are local to a reference coefficient. Two-dimensional numerical experiments combine analytic and learned representations of inclusions and crack-like coefficients with neural forward surrogates. They illustrate latent-space reconstruction, reduced online cost, and further improvement from optional FEM-based refinement.

In 1989, Dimitrić showed that any pseudo-umbilical biharmonic submanifold in a Euclidean space is minimal when the dimension of the submanifold is different from four. However, the four-dimensional case has remained open for nearly four decades. This difficulty comes from the fact that the tangential part of the biharmonic equation vanishes identically in dimension four. Therefore, we consider biminimal submanifolds, whose defining equation generalizes the normal part of the biharmonic equation. In this paper, we show that any connected pseudo-umbilical 0-biminimal submanifold of codimension two and dimension at least three in a Riemannian manifold of constant sectional curvature $c$ has constant mean curvature. Furthermore, when $c$ is non-positive, we prove that it is minimal. As a corollary, we show that any four-dimensional pseudo-umbilical biharmonic submanifold in a six-dimensional Riemannian manifold of non-positive constant sectional curvature is minimal. These results also provide partial affirmative answers to Chen's conjecture, generalized Chen's conjecture for non-positive constant sectional curvature, and the Balmuş-Montaldo-Oniciuc conjecture.

This paper systematically investigates the structural, analytical, and operator-theoretic properties of the Nevanlinna class $\mathcal{N}_u$ of Dirichlet series, introduced by Brevig and Perfekt [Adv.\ Math., 2021] and further developed by Guo et al.\ [Ann.\ Inst.\ Fourier (Grenoble), 2025]. First, we examine the topological structure of $\mathcal{N}_u$. We then prove a Littlewood–Paley type identity for functions in $\mathcal{N}_u$, establish an equivalent characterization via vertical limit functions, and demonstrate that the interchange of limits in this identity is permissible. In addition, we provide an alternative proof of this identity using potential-theoretic approach. Applying these analytical tools, we study composition operators $C_\Phi$ acting on $\mathcal{N}_u$ and characterize those symbols $\Phi$ that induce bounded composition operators $C_\Phi$. Utilizing Carleson measure techniques on half-planes and infinite-dimensional tori, we characterize the symbols $\Phi$ that generate bounded and compact composition operators. In particular, we prove the complete equivalence between the (vanishing) Carleson embedding condition and the geometric (vanishing) Carleson condition on Carleson squares. We conclude with two function-theoretic applications to functions in $\mathcal{N}_u$.

We prove that groups acting freely and cocompactly on (possibly exotic) affine buildings of type $\tilde{A}_2$ and $\tilde C_2$ are mixed-identity free and have selfless reduced $C^*$-algebras. These results follow from a strong form of ping-pong dynamics that we call 'transversal contractivity'. Our main geometric result regards domesticity properties of elements in the associated polygons at infinity: we prove that, in our context, the opposite geometry of a hyperbolic element is topologically large.

For a nontrivial finite abelian group $G$, let $\nu(G)$ be the smallest {nonnegative} integer $\ell$ such that every zero-sum free sequence $T$ over $G$ of length at least $\ell$ has the following property: all nonzero elements of $G$ that do not occur as a subsequence sum of $T$ lie in a proper coset of some subgroup of $G$. It is easy to check that $\mathsf d (G)-1 \le \nu (G) \le \mathsf d (G)$, where $\mathsf d (G)$ is the small Davenport constant of $G$. A conjecture by Gao from the year 2000 stated that equality should always hold at the lower bound. This conjecture has since been confirmed for many families of groups (including all p-groups and groups of rank at most two). In this paper, we disprove the conjecture.

This paper investigates nonlinear eigenvalue problems governed by a family of operators that combine two distinct diffusion mechanisms: a classical local diffusion, which accounts for short-range interactions, and a fractional nonlocal diffusion, which captures long-range effects. The analysis is carried out in the Heisenberg group, a non-Euclidean geometric setting in which motion is constrained to a distinguished set of horizontal directions. As an application of the obtained spectral theory, we establish multiplicity results for perturbed singular problems in both the purely nonlocal and mixed local–nonlocal settings. For the mixed case, a gradient convergence theorem plays a crucial role in deriving these multiplicity results. A key feature of our analysis is the treatment of variable singular exponents, allowing the singularity to vary throughout the domain. To the best of our knowledge, this is the first systematic study of nonlinear eigenvalue problems and singular problems with variable singularity exponents for mixed local-nonlocal operators in the Heisenberg group. The results therefore open a new direction in the spectral theory of subelliptic operators and provide a rigorous mathematical foundation for applications ranging from anomalous diffusion and nonlocal phase transitions to image analysis and control theory on nonholonomic systems.

We design and analyse an arbitrary-order numerical scheme for the Reissner–Mindlin plate problem on general polygonal meshes. The scheme is derived from the Hodge–Laplacian associated with a discrete Bernstein–Gelfand–Gelfand (BGG) twisted complex and exploits a discrete $H_2$-based construction for the transverse displacement. We establish a discrete Korn inequality for the underlying Discrete de Rham method (DDR) spaces and prove the convergence of the method. At the lowest order, the analysis yields an error estimate that is uniform with respect to the plate thickness, showing that the scheme is locking-free. Numerical experiments on several families of polygonal meshes support the theoretical results and illustrate the benefits of the enhanced continuity of the transverse displacement discretisation.

Partial sums of the Taylor expansions of functions from the Neil algebra are uniformly bounded by their uniform norms on the ball of radius $r$ centered at the origin, where $r$ denotes the unique real root of \[ 4x^3 + x - 2 = 0. \] Moreover, this radius $r$ is optimal.

The forced colouring function of a graph gives the probability that a random assignment of colours to a random subset of vertices can be extended, by a simple local process called forcing, to give a proper colouring of the whole graph using the same set of available colours. This is a polynomial for each fixed number of colours, and was introduced as a subject for research on the general theory of graph polynomials. In this paper we establish its fundamental properties and give combinatorial interpretations of its derivatives at two particular points. We also prove that the problem of computing its value at any specific point in a certain interval is #P-hard.

We construct a smooth projective family of rational surfaces over $\mathbb{G}_{m,\mathbb{Z}}$. The Mori dream property of a fiber is determined by the torsion of the normal bundle of an anticanonical cycle. Over $\mathbb{C}$, the locus of Mori dream fibers is Zariski dense. For every prime $p$, every geometric fiber over a closed point of the reduction modulo $p$ is a Mori dream surface, whereas the geometric generic fiber is not a Mori dream space. In either setting, no restriction to a nonempty open subset is a Mori dream morphism. We also prove that, over any algebraically closed field, a projective fibration becomes a Mori dream morphism after shrinking the base whenever the set of points with Mori dream fibers is not contained in a countable union of proper closed subsets.

The purpose of this paper is the identification of high-fidelity digital twin data models from numerical code outputs by non-intrusive techniques (i.e., not requiring Galerkin projection of the governing equations onto the reduced modes basis). In this paper the author defines the concept of the digital twin data model (DTM) as a model of reduced complexity that has the main feature of mirroring the original process behavior. The significant advantage of a DTM is to reproduce the dynamics with high accuracy and reduced costs in CPU time and hardware for settings difficult to explore because of the complexity of the dynamics over time. This paper introduces a new framework for creating efficient digital twin data models by combining two state-of-the-art tools: randomized dynamic mode decomposition and deep learning artificial intelligence. It is shown that the outputs are consistent with the original source data with the advantage of reduced complexity. The DTMs are investigated in the numerical simulation of three shock wave phenomena with increasing complexity. The author performs a thorough assessment of the performance of the new digital twin data models in terms of numerical accuracy and computational efficiency.

This paper develops an embedded-hybridized discontinuous Galerkin (E-HDG) method combined with a first-order implicit-explicit scalar auxiliary variable (IMEX-SAV) time discretization for variational initial-state data assimilation governed by the unsteady incompressible Navier-Stokes equations. We adopt an optimize-then-discretize strategy. The spatial discretization uses discontinuous piecewise polynomials of degrees $k$ and $k-1$ for the element velocity and pressure, respectively, a continuous degree-$k$ velocity trace, and a discontinuous degree-$k$ pressure trace. The resulting state and adjoint velocities, as well as the reconstructed initial velocity, are globally divergence-free. The forward IMEX-SAV state scheme is unconditionally energy stable. Under suitable regularity and local-trajectory assumptions whose bounds introduce no explicit negative powers of the viscosity, the mesh-dependent condition $\Delta t\lesssim h^2$, and a contractivity condition on the Tikhonov parameter, we establish local existence and uniqueness of the fully discrete OTD optimality system and Reynolds-semi-robust $L^2$ error estimates of order $O(h^k+\Delta t)$ for the state, adjoint and reconstructed initial velocities; the constants contain no explicit negative powers of the viscosity. Numerical experiments confirm convergence on smooth tests and demonstrate machine-precision discrete incompressibility, effective nonlinear-solver behavior, and stable performance in small-viscosity regimes.

We prove the generalized Cartan-Hadamard conjecture, also known as the Aubin conjecture, in the small volume regime under a lower Ricci curvature bound, for any dimension $n\geq 2$. More precisely, we show that if $(M^n,g)$ is a Cartan-Hadamard manifold satisfying $\mathrm{Sec}_g\leq\bar k\leq 0$ and $\mathrm{Ric}_g\geq (n-1)\underline k$, then its isoperimetric profile is at least that of the model space of curvature $\bar k$ for small enough volumes. We also reduce the proof of the generalized Cartan-Hadamard conjecture to establishing rigidity in the equality case.

The choice of Riemannian metric can strongly influence the convergence of gradient-based optimization over covariance matrices. Euclidean, Bures-Wasserstein and affine-invariant metrics are common choices, but their relative effectiveness depends on the objective. We introduce a two-parameter family defined by $X^{p}LX^{q}+X^{q}LX^{p}=U$, solved for $L$ at each tangent vector $U$, that contains all three as exact members, at $(0,0)$, $(1,0)$ and $(1,1)$, and extends past them. We treat the choice of member as a particular way of preconditioning for a given problem. To this end, we analyze the conditioning of the Riemannian Hessian at the solution. We show that it obeys a lower bound that depends on $(p,q)$ only through the exponent $r=p+q$. When the Euclidean Hessian is a pure power that mixes no eigendirections, the member $p=q=r/2$ attains that bound, and a closed-form criterion identifies the other members that do. We discuss ways to tune $r$ for a given problem. Experiments on real covariance data confirm the predicted conditioning and the benefit of tuning $r$. A task covariance example shows a further gain from tuning the shape.

If ZFC is consistent, then so is ZF+DC+BPI+there is no Vitali set.

We continue to develop a theory of Besov and Triebel-Lizorkin spaces associated with weighted subcoercive operators on a real connected Lie group, specializing to the case of groups with polynomial volume growth. We consider the full scale of spaces and consider equivalent definitions, inclusions, discretization, and duality.

In the present paper, we study the maximization problem of the $k$-Torsional rigidity under quermassintegral constraint, with particular emphasis on the role of the ball. Our main result shows a phenomenon which, as far as we know, has not previously been observed: the ball, rather than being an extremal shape, is a saddle point. More precisely, for suitable quermassintegral constraints, we show that the $k$-torsional rigidity admits both increasing and decreasing perturbations around the ball. The proof is based on a second-order analysis of the corresponding variational problem. To the best of our knowledge, this is the first use of a second-order shape derivative approach in the study of variational problems for $k$-Hessian operators. We also establish an analogous saddle-point behavior for a related curvature functional under a volume constraint, highlighting a structural distinction between the intermediate Hessian regime and the classical Laplace and Monge-Ampère cases.

The purpose of this paper is to extend an article by Hilhorst, Mimura and Schätzle [18] about the limit as the latent heat coefficient tends to zero of a two-phase Stefan problem arising in biology. We introduce a rather general additive noise white in time and colored in space, and search for the limit of the solution of the corresponding stochastic Stefan problem as the latent heat coefficient vanishes. We first prove the existence and uniqueness of the weak solution of this problem, and then study the limit of the solution as the latent heat coefficient tends to zero. Unlike in [18], our method of proof is based upon an error estimate between the solution of the Stefan problem with positive latent heat and that of the Stefan problem with zero latent heat, which seems to be novel even in the deterministic case when no noise is added.