Papers

This paper explores variations of vertex-coloring problems defined on graph visibility properties. It introduces and studies the dual, outer, and total mutual-visibility chromatic numbers, which partition the vertex set of a graph into color classes that preserve specific mutual-visibility conditions called dual, outer or total. The work provides structural conditions under which these chromatic parameters are finite or infinite, and establishes that deciding whether a graph admits a dual, outer, or total mutual-visibility coloring using a given number of colors is NP-complete, even when restricted to two colors. Exact formulas and tight bounds for these chromatic parameters are established across several fundamental graph classes. For block graphs, complete characterizations are provided for the outer and dual mutual-visibility chromatic numbers based on structural invariants such as cut vertices and specific forbidden subgraph structures. On Hamming graphs, the dual and total mutual-visibility chromatic numbers are shown to equal the smaller dimension of the factors, while the outer mutual-visibility chromatic number is proven to equal the star arboricity of a corresponding complete bipartite graph. Finally, the paper examines strong grid graphs, determining exact values for their total, outer, and dual mutual-visibility chromatic numbers. These results demonstrate how the parameter behaviors range from finite constants to infinity depending on the grid dimensions.

Let $p>3$ be a prime, and let $b,c$ be integers with $p\nmid b+2c$. We obtain an explicit congruence modulo $p^2$ for the sum of $\binom{2k}{k}T_{2k}(b,c^2)/(4^k(k+1)(b+2c)^{2k})$ from $k=0$ to $(p-1)/2$, where $T_n(b,c)$ denotes the generalized central trinomial coefficient. This gives a parametric formula for the sum considered in a question of Wang and Cui. The proof uses a constant term representation and a logarithmic generating function to reduce the sum to two coefficients of an algebraic generating function. A differential identity and a coefficientwise polynomial congruence then yield the desired evaluation. The formula also applies when $p$ divides $b$ or $b-2c$.

A smooth complete intersection $X\subset\mathbb{P}^{n+2}_{\mathbb{C}}$ of two quadrics admits an interpretation as a moduli space of bundles. For $n=2$, it parametrizes stable rank $2$ parabolic bundles on $\mathbb{P}^1$ with five marked points, while for $n=3$ it parametrizes stable rank $2$ bundles of odd degree with fixed determinant on a genus $2$ curve. For general $n$, it can be interpreted as a moduli space of semistable twisted $\mathrm{Spin}$-bundles. We study $X\setminus W$, where $W$ is the wobbly locus, consisting of bundles that admit a nonzero nilpotent Higgs field. We prove that $W$ is an irreducible divisor for $n\geq 3$ and compute $H_1(X\setminus W,\mathbb{Z})$. Using a root-stack description, we identify $\pi_1(X\setminus W)$ with the kernel of a monodromy homomorphism from an orbifold mixed braid group. The Reidemeister-Schreier method then yields an explicit presentation in every dimension. For $n=2$, where $X$ is a degree $4$ del Pezzo surface and $W$ is the union of its sixteen $(-1)$-curves, we obtain a presentation with $10$ generators and $25$ relators and prove that both numbers are minimal among all finite presentations. For $n=3$, we show that $\pi_1(X\setminus W)$ admits a presentation with $4$ generators and $78$ relators.

Manufacturing firms face uncertainty in both supply and demand. They implement process flexibility, enabling plants to produce multiple products, to hedge against supply–demand mismatches. Assigning more products to a plant results in efficiency losses at the plant, whereas limited flexibility may improve efficiency because of the specialization effect. Furthermore, the flexibility design determines the sourcing locations of products, thereby affecting demand due to the country-of-origin effect and exposure to tariffs. We incorporate these mechanisms into the supply network flexibility design problem by modeling supply and demand uncertainties as endogenous, that is, as functions of the first-stage design decisions. We propose a two-stage stochastic program and solve it via a decomposition scheme that relies on distribution-specific optimality cuts. To avoid exhaustive enumeration of all distributions, we derive distribution-free formulations that are valid for all distributions and for subsets of supply and demand distributions. Through extensive computational experiments, we demonstrate the superior performance of our solution approach and distribution-free formulations relative to exhaustive distribution enumeration and state-of-the-art inequalities, respectively. Moreover, we explain how the problem structure affects the effectiveness of the proposed formulations. From a managerial perspective, we show the trade-offs between endogenous effects and how they affect the optimal flexibility design.

We analyze a general class of difference operators $H_\varepsilon = T_\varepsilon + V_\varepsilon$ on $\ell^2(\varepsilon\mathbb{Z}^d)$, where $V_\varepsilon$ is a multi-well potential and $\varepsilon$ is a small parameter. We construct approximate eigenfunctions in neighbourhoods of the different wells and give weighted $\ell^2$-estimates for the difference of these and the exact eigenfunctions of the associated Dirichlet-operators.

For any finite-dimensional algebra $A$ over an algebraically closed field, we prove that if $A$ is $E$-infinite, then the Second brick-Brauer-Thrall Conjecture holds for $A$. In fact, we show that if there is a rational ray outside the $\tau$-tilting fan of $A$, then $A$ admits an infinite family of $\theta$-stable bricks of dimension $d$, for some positive integer $d$ and weight $\theta$. To show our main result, from any $\tau$-regular component of $A$ with no dense orbit, we obtain a brick component that also has no dense orbit and whose points in general position are $\theta$-stable modules for a given weight $\theta$. As an important consequence, we conclude that $A$ is stably-discrete (also called multiplicity-free) if and only if $A$ is $E$-finite. Moreover, we prove that Demonet's (lattice point) Conjecture implies the stable Second brick-Brauer-Thrall Conjecture. For $E$-infinite algebras, our results settle several open conjectures. Together with our earlier work and recent developments, this leads to important reductions in the study of some challenging open problems.

Let $X=\mathbb{P}^n$, where $p=n+1$ is prime. Consider the orbit of the standard numerical exceptional basis in $K_0(X)$ under mutations, sign changes, and isometries of the Euler form. We construct an invariant of this orbit: for every basis from the orbit, the square of rank of all basic vectors art congruent to $1$ modulo $p$. For $\mathbb{P}^4$ and $\mathbb{P}^6$, we present explicit numerical exceptional bases violating this congruence; hence the action is not transitive. We also establish a stronger obstruction modulo $p^2$ and use Polishchuk's theorem to show that no vector of the constructed bases is the class of an exceptional object. Finally, we prove that for every prime $p$, the category $D^b(\operatorname{Coh}\mathbb{P}^{p-1}_{\mathbb C})$ does not contain a pair of mutually orthogonal exceptional objects.

This proof-of-concept paper demonstrates how the Kalman filter, as a brain imaging method, can be enhanced by incorporating raw diffusion tensor imaging (DTI) data without tractography, in addition to EEG recordings. An efficient algorithm is designed to streamline the otherwise tedious process of inferring connectivity between brain regions from DTI data. The DTI-based evolution model is applied to the Kalman filter and the Standardized Kalman filter, and these are compared with their conventional counterparts using the random walk evolution model. The numerical experiments use synthetic somatosensory and auditory evoked potentials. The results show that the DTI-modelled Kalman filter can obtain deep activity in the brainstem and thalamus. Moreover, the connection model limits the estimated spreads to the correct or nearest anatomical brain region.

The number of unlabeled regular graphs can be expressed as an average of fixed-point counts over vertex permutations, but evaluating each fixed-point count still requires the degree constraints to be enforced. We give an exact recurrence that processes one complete permutation cycle at a time and records the remaining cycles only by their lengths and residual degrees. The recurrence combines internal edge orbits with orbits joining distinct cycles, while binomial and multinomial coefficients retain the multiplicities of choices that lead to the same remaining state. We prove that this state description is sufficient under complete-cycle elimination and derive bounds on the number of states and transitions. For every fixed degree, the resulting algorithm has an $\exp(O(\sqrt n))$ upper bound in the number of vertices, including integer-arithmetic costs. The quartic case requires only four positive residual-degree classes for each cycle length. Small-instance comparisons with a separately implemented, vertex-indexed edge-orbit calculation verify both regular and nonuniform residual-degree inputs. The quartic calculation gives unrestricted and connected counts through order 50, including 22 orders beyond the corresponding reference tables through order 28. Connected counts are recovered by the inverse Euler transform, and all 22 identity-permutation contributions for orders 29–50 agree with the published labeled counts. The nonidentity fixed-point terms at these orders have not been independently recomputed.

This Letter transitions neutrino mixing from an empirical fit to a geometric law by analytically deriving the PMNS mixing angles and mass spectrum from an $A_5$ topological manifold. Solving the Quantum Yang-Baxter Equation forces a geometric fracture at $M_0 \approx 3.88 \times 10^{16}$ GeV. Projecting this to the Electroweak scale yields parameter-free targets: maximal CP violation $\delta_{CP} = \pm \pi/2$, a mass sum $\sum m_i \approx 72.48$ meV, with $m_1 < m_3$, and effective masses $m_\beta \approx 12.47$ meV and $m_{\beta\beta} \approx 8.20$ meV, setting definitive experimental thresholds.

This paper investigates the periodic homogenization of a two-phase elasto-plastic material in the framework of $\Gamma$-convergence. It establishes the link between the homogenized functional and that which was previously obtained in [12] in the context of two-scale convergence. Although there is no gap between the two formulations and that involving two-scale has the hallmark of a (two-scale) elasto-plastic problem, we are unable to positively ascertain the elasto-plastic character of the homogenized energy except in a one-dimensional setting.

In this paper, we apply Nash-Moser iteration and Saloff-Coste's Sobolev inequalities to derive gradient estimates for positive solutions to a class of nonlinear parabolic equations of the form \begin{equation*} u_{t}=\Delta u^{m}+a(x, t)u\log u+b(x, t)u, \quad 1-\frac{2}{n}<m<1 \quad \text{or}\quad 1<m<1+\frac{1}{1+\sqrt{2n}}, \end{equation*} on a complete Riemannian manifold $(M,g)$ of dimension $n$, where $a(x, t)$, $b(x, t)$ are $C^{1}$ functions. For $a=b=0$, this equation becomes the porous medium equation (PME) or the fast diffusion equation (FDE) when $m>1$ or $m<1$, respectively. As consequences, we obtain Liouville type theorems for the corresponding constant coefficient equations when \begin{equation*} 1-\frac{1}{\sqrt{n-1}}<m<1\quad \text{and}\quad 1<m<1+\frac{1}{\sqrt{n-1}}. \end{equation*}

We show that for every $n$ unit vectors $v_1, \dots, v_n$ in the $d$-dimensional Euclidean space, there exist signs $\varepsilon_1, \dots, \varepsilon_n \in \{\pm 1\}$ such that $\lVert \varepsilon_1 v_1 + \dots + \varepsilon_n v_n \rVert \ge \sqrt{2n - d}$, and we characterize the equality cases.

A group is called an $n$-$\bfc$-group if each of its conjugacy classes is finite and contains at most $n$ elements. By a theorem of B.~H.~Neumann the derived subgroup of such a group is finite of $n$-bounded order, whereas the order of the group itself need not be bounded, as extraspecial $p$-groups show. We prove that an $n$-$\bfc$-group with trivial centre is finite of order at most $n^{64(\log n)^{5}}$. Passing to the quotient by the hypercentre, we deduce that no assumption on the structure of the group is needed at all: for an arbitrary finite $n$-$\bfc$-group $G$ one has $|G:Z_\infty(G)|<n^{64(\log n)^{5}}$. The finiteness of the group is required here only for the correct definition of the hypercentre.

We formulate and prove a simple criterion for the sphericity of a CR manifold (that is, for its equivalence to its model surface). We discuss the differences between the general case, in which the model surface is weighted homogeneous of arbitrary degree, and the case of Levi-nondegenerate manifolds, whose model surface is quadratic.

Optimization of mathematical models involving integer decisions and neural networks with ReLU activation (ReLU ANNs) is a challenging task. Nevertheless, such models are an enabling technology in many application domains. A prominent example is superstructure optimization in chemical engineering, where ReLU ANNs are frequently employed as surrogate models for complex nonlinear processes. We survey recent developments in this area. We argue that in addition to network size and training options of the ANNs, the ReLU activation geometry and the number of linear regions on the domain of interest have a strong impact on computational optimization performance. While standard model compression approaches such as structured pruning reduce network size, they do not explicitly address geometric considerations. Therefore, we propose a novel region-reduced model compression approach that combines the stabilization of unstable neurons and the merging of redundant neurons to reduce the number of linear regions while maintaining predictive accuracy through error compensation. We evaluate our method against standard compression approaches on multiple optimization use cases. First, the two-dimensional peaks function for which we can visualize the activation geometry. Second, on optimization over individual surrogate ReLU ANNs for three chemical processes, and third, on a hybrid superstructure optimization problem that involves the three ReLU ANNs, additional process submodels, and binary variables. The results for the superstructure problem demonstrate the large potential of region reduction with a decrease of 40% to 50% in computational time, yielding solutions closer than 1% to the reference at negligible effort of obtaining the compressed model.

We consider the problem of controlling a three-dimensional falling liquid film using same-fluid blowing and suction through the substrate. Building upon recent results for the corresponding two-dimensional problem, we employ a reduced-dimensional weighted-residual model for the film height and downstream flux as the basis for controller design. Linear stability analysis is used to estimate the number of actuators required to stabilise the film, while a linear-quadratic regulator (LQR) framework is used to construct feedback controls for the linearised system. The resulting gain matrix is then combined with observations from direct numerical simulation (DNS) of the three-dimensional Navier-Stokes equations, allowing controls derived from the reduced-dimensional model to be applied directly to the full flow. Through systematic numerical tests, we demonstrate that these controls successfully stabilise the flat-film solution across a physically relevant range of Reynolds numbers and domain configurations. We further show that both the number and placement of actuators play a crucial role in control performance, with spanwise instabilities requiring additional actuator coverage in the cross-stream direction. Although the extension from two to three dimensions substantially increases the cost of controller construction, this cost is incurred entirely offline during the computation of the gain matrix, making the overall framework computationally tractable. Our results demonstrate that LQR feedback controls derived from a three-dimensional weighted-residual model retain predictive value when applied to fully resolved DNS, extending previous two-dimensional control methodologies and helping bridge the gap between theoretical control design and experimentally relevant falling-film flows.

We establish local Hamilton-type gradient estimates for positive $C^{2,1}$ solutions of $u_t=\Delta u^m$ on complete Riemannian manifolds whose Ricci curvature is bounded from below. For every fixed $m>1$ and every fixed $0<m<1$, there exist $\beta=\beta(m,n)>0$ and $C=C(m,n)>0$ such that a solution $0<u\leqslant A$ in $B_{2R}(x_0)\times(t_0-T,t_0]$ satisfies the following local gradient estimate \begin{equation*} \sup_{B_R(x_0)\times(t_0-T/2,t_0]}|\nabla u^\beta| \leqslant C A^\beta \left(\frac1R+\sqrt{k}+\frac{A^{(1-m)/2}}{\sqrt T}\right), \end{equation*} where $\mathrm{Ric}_M\geqslant-k$. The proof uses an intrinsic quantitative alternative to locate, around each prescribed positive point, a cylinder on which the solution has a controlled upper-to-lower ratio. A local gradient estimate on this cylinder is combined with a stopping argument. As a consequence, every uniformly bounded positive ancient solution on a connected complete manifold with nonnegative Ricci curvature is constant.

We study a fully discrete approximation theory for the two-dimensional incompressible Euler equations on $\mathbb{T}^{2}$, in vorticity form, driven by Kraichnan-type transport noise and with $\mathbb{H}^{-1}(\mathbb{T}^2)$-valued initial vorticity. The main new ingredient is a finite-dimensional coercivity estimate for the torus Kraichnan model, which captures at the discrete level the regularizing mechanism induced by the rough transport noise. Together with a new compactness strategy based on continuous equation-based interpolants and localized discrete estimates, this coercivity estimate gives tightness of the approximations and shows that subsequential limit is a weak martingale solution of the Euler equations.

In this work, we define $(3+1)$-dimensional topological quantum field theories (TQFTs) and their lattice realizations with input data given by a pivotal fusion 2-category; more precisely, our framework applies in wider generality to presemisimple locally fusion pivotal tensor $2$-categories, which are equipped with a so-called shadow trace. We introduce a decoration of triangulated manifolds by the input data, and develop a $20j$-symbol formalism for computing 4-simplex scattering amplitudes. We verify the topological invariance of 20$j$-symbol summations under the 4-dimensional Pachner moves. Furthermore, we also construct explicitly a 3-dimensional Levin-Wen-type lattice Hamiltonian model realizing the state-sum invariant. Our construction can be understood as a higher-dimensional analogue of the spin-networks construction for the Turaev-Viro invariant. Examples arising from $2$-groups are discussed from the perspective of higher gauge theory.

Distributionally robust optimization (DRO) provides a principled framework for learning under distribution shift, but its practical use is hindered by the difficulty of evaluating worst-case risks for nonconvex loss functions. We study a penalized DRO formulation in which the adversary may choose any distribution but incurs a Wasserstein penalty for deviating from the empirical distribution. We show that the adversary's problem can be reformulated as an optimization problem over transport maps that push empirical samples to adversarial ones, and we prove that optimal maps are cyclically monotone. We also show that standard adversarial training—based on per-sample local optimization—violates cyclical monotonicity and wastes transport costs unless the adversary is severely restricted. We propose two remedies. First, we introduce multi-start particle ascent, which alternates parallel gradient ascent with reassignment to enforce cyclical monotonicity across samples. Second, we parameterize adversarial maps as gradients of input-convex neural networks, which guarantees cyclical monotonicity by construction. Experiments on robust regression, image classification, and robust control show that our methods consistently outperform standard adversarial training and state-of-the-art baselines, achieving improved robustness and better generalization under distribution shift.

Motivated by Brouwer's conjecture, Ashraf, Omidi and Tayfeh-Rezaie proposed the following Brouwer-type conjecture that for every graph $G$ on $n$ vertices with $m$ edges, the sum $S_k^+(G)$ of its $k$ largest signless Laplacian eigenvalues satisfies $S_k^+(G)\le m+\binom{k+1}{2}$ for $k=1, \ldots, n$. In this paper, we prove that the above conjecture holds. Moreover, the equality holds if and only if $k=1$ and $G$ is either star $K_{1,a}$ or triangle $K_3$ with adding some isolated vertices. For split graphs, properties of block signless Laplacian matrices based on clique and independent set are adapted. While for non-split graphs, some spectral graph substructure are used to control the sum of signless Laplacian eigenvalues.

We study the Wasserstein lift of nonlinear Poisson systems with inhomogeneous Neumann boundary conditions. We establish the existence of minimizers using new estimates involving the $p$-moments of measure-valued traces. We also derive the optimality conditions, which turn out to be matrix-valued generalizations of pressureless Euler equations. While deterministic minimizers exist under suitable convexity assumptions and for one-dimensional source domains, we show that genuinely measure-valued minimizers arise in higher dimensions. In particular, we construct examples in which the lifted and classical variational problems have different minimum values, thereby precluding the solutions to the lifted problem from being deterministic, despite the absence of any a priori mechanism enforcing measure-valued behavior. To the best of our knowledge, this phenomenon is new in the nonlinear elliptic theory.

The goal of this paper is to refine methods in computable analysis and to employ them in the study of solutions of an optimization problem posed by Bellman. This problem asks how to find optimal (shortest) paths which do not fit into given plane figures. We show that each instance of Bellman's problem has an arbitrarily small uniformly computable perturbation for which the minimal length of path is computable, and thus the class of optimal paths is a $\Pi_1^0$ class, and there are uniformly computable paths which escape with arbitrarily small length greater than the minimum possible.

Gradient-based optimization (GBO) of problems constrained by partial differential equations is central to parameter identification and structural design in solid mechanics. Lattice Boltzmann method (LBM) schemes for linear elastic solids have only recently been derived. Combined with automatically generated discrete adjoint collision kernels, they make GBO possible with a purely LBM-based primal and adjoint solver. We propose, to our knowledge, the first optimization benchmarks for solid LBM: three inverse problems, each with a manufactured solution of the forward problem and a known reference control. These are the identification of two scalar amplitudes of a body force field on a periodic domain, the distributed control of the same field with up to $12800$ degrees of freedom, and the identification of Young's modulus on an elliptic plate with Dirichlet and Neumann boundaries. The primal scheme converges with second order on the periodic domain and with first order on the elliptic plate. The recovered controls follow these rates, with experimental orders of convergence of $1.92$ and $1.06$, respectively. Gradients obtained via forward-mode automatic differentiation (f-AD) and the discrete adjoint LBM (ALBM) require the same number of optimization steps and yield control errors agreeing to at least eleven significant digits. The ALBM gradient is further validated node-wise against a finite difference quotient, with deviations decreasing under mesh refinement. The benchmarks are implemented in the open-source LBM framework OpenLB and provide a verification basis for optimization in LBM-based solid mechanics and its extension to fluid-structure interaction.

The diffusion synthetic accelerated (DSA) source iteration is a standard solver for the radiative transfer equation. Using Fourier analysis, a convergence rate \(\rho_\infty(c)\le0.2247\,c\) with \(c\) the maximum ratio of scattering to total cross section has been established for an infinite homogeneous medium. For slab geometry with inflow boundary conditions and arbitrary bounded cross sections we prove that the spectral radius of the DSA iteration is at most \(\rho_\infty(c)\). We show that this convergence rate carries over to a variational discretization of the DSA iteration on every conforming tensor-product Galerkin space whose angular factor contains the constants. The analysis rests on an exact min–max characterization of the spectral radius. From it we derive a checkable sufficient condition for a given rate. We verify this condition using a suitable energy-stable projection, a weighted angular average whose weight is chosen so that the condition holds with the rate \(\rho_\infty(c)\). Since the projection maps discrete spaces into discrete spaces, the argument applies verbatim to the discrete iteration. As a consequence, the condition number of the preconditioned system is uniformly bounded by \(1+0.29\,c\), which implies rapid convergence of the preconditioned conjugate gradients method.

This article is concerned with the asymptotic shape of quantile surfaces, defined as the set of quantiles at a given level $\alpha$ generated by a controlled one-dimensional distribution. Specifically, when the distribution arises as a linear combination of log-normal random variables and the control is a vector of positive coefficients, we prove that quantile surfaces are globally concave in the left tail ($\alpha\to0$) and globally convex in the right tail ($\alpha\to1$). Moreover, these surfaces exhibit asymptotic separation of scale and shape.

We prove that at any nondegenerate neck pinch singularity of Lagrangian mean curvature flow of surfaces the mean curvature becomes unbounded, with control on the blow-up rate.

Place a queen in each successive column of an infinite $\mathbb{N}\times\mathbb{N}$ chessboard, always choosing the lowest row such that no two queens may attack one another. We prove that the row $q_n$ occupied by the queen in the $n$th column satisfies $q_n=n\phi+O(1)$ or $q_n=n/\phi+O(1)$, where $\phi=(1+\sqrt5)/2$ is the golden ratio.

We describe, with proof, an infinite class of examples of finite sets of integers $A$ for which $\|\widehat{1_A}\|_1\geq (c_\mathcal{R}-o_{|A| \rightarrow \infty}(1))\sqrt{|A|}$ where $c_\mathcal{R}=0.953\dots$.