Papers

In this paper, we reinterpret pairwise comparison theory through the lens of optimal transport. The main contribution is the connection established between pairwise comparison matrices, doubly stochastic scaling, and entropy-regularized transport problems. Using Sinkhorn scaling, we show how consistency, priority derivation, and inconsistency assessment can be studied within a common optimal-transport framework. As methodological consequences of this connection, we obtain a new characterization of consistency, a Sinkhorn-based prioritization method, and an entropy-based inconsistency index. We study these proposals both analytically and numerically and compare them with established methods from the literature

Local characteristic decomposition (LCD) is widely used in high-order numerical methods for hyperbolic systems of conservation laws to reduce spurious oscillations appearing in the computed solutions. The LCD implementation requires a representative average interface state, typically obtained using arithmetic or Roe-type averages of the nearly grid values. We propose an entropy-based LCD (ELCD), in which the states in the two cells adjacent to an interface are considered together with several nearby states in phase space. Among these candidates, we select the state that locally minimizes the entropy and use it as an average interface state for linearizing the flux Jacobian. We incorporate the ELCD into several second-order finite-volume and fifth-order finite-difference schemes. Numerical experiments for the two-dimensional Euler equations of gas dynamics show that the schemes, which utilize the ELCD procedure generally resolve complex wave structures more sharply than their counterparts, which use the arithmetic averages.

The fictitious domain method offers a powerful alternative to circumvent complex mesh generation process by embedding the complex physical domain into a simpler, non-matching background mesh. Among the various strategies to enforce the boundary condition on such a mesh, one possibility is to impose it weakly through a Lagrange multiplier. Existing approaches following this strategy typically require two independent uniform meshes for the domain and the boundary, whose mesh sizes are related by a prescribed ratio. In this work, we relax this requirement by constructing the boundary mesh directly from the trace of the background triangulation. To recover the uniform discrete inf-sup condition under this weaker mesh assumption, we enrich the discrete space of the solution with bubble functions. Our main result is the construction of a suitable operator that allows us to prove this uniform discrete inf-sup condition, thereby establishing the stability of the resulting scheme. As a consequence, we derive optimal a priori error estimates and provide a numerical experiment to validate the theoretical results.

In this paper, we present Lappan's five-point type criteria for $\varphi$-normality of sense-preserving harmonic mappings and families of harmonic mappings in the unit disc $\mathbb{D}$. We establish a normality criterion involving the higher-order spherical derivative of order $k$, and $k+2$ distinct test values under suitable conditions. We further reduce the number of test values to $\left[\frac{k}{2}\right]+2$ under an additional differential condition involving the analytic and co-analytic parts. We also prove a generalization in which the spherical derivative is replaced by a polynomial with non-negative real coefficients.

We propose GPU algorithms for high-fidelity simulation of incompressible turbulent flows. Discretization in space is performed with H(div)-conforming high-order Raviart-Thomas finite elements for the velocity and an $L^2$-conforming discontinuous Galerkin approximation for the pressure. In time, a consistent splitting scheme based on higher-order BDF time stepping is used, with convection treated explicitly. In this scheme, a pressure Poisson equation and a symmetric reaction-diffusion-type equation for the velocity need to be solved in each time step. We develop a solution framework with fast matrix-free operator evaluation for all ingredients, combined with multigrid solvers for the Poisson problem, and propose a robust mixed-precision algorithmic framework. A key to mixed-precision efficiency is a least-squares projection to generate accurate initial guesses for the iterative linear solvers, enabling us to work with relative residual tolerances of $10^{-3}$. In this regime, running the solvers entirely in single precision leads to almost no change in overall iteration counts and maintains the crucial turbulence statistics, while showing up to $1.7\times$ speedup over pure double-precision simulations.

In this paper, we prove the existence and far-field behavior of an axisymmetric solution $(\rho,\mathbf{u},p)$ to the steady Euler system in an infinitely long cylindrical nozzle with a perturbed wall $r=R(x_1)$, containing a contact discontinuity $r=g_D(x_1)$. The background flow carries a nonzero swirl component $\beta_0\mathbf{e}_\theta$, where $\beta_0$ is not necessarily small. Consequently, the background entropy, Bernoulli function, and angular momentum density depend on the radial variable, and their radial derivatives are not necessarily small.

Environment reconstruction for low-altitude integrated sensing and communications (ISAC) has largely focused on geometry-centric maps, overlooking material-dependent propagation effects. Material-labeled reconstruction is therefore a key step toward propagation-aware mapping, enabling more physically grounded channel prediction and uncrewed aerial vehicle (UAV) networking. However, constructing such maps from wireless multipath observations is challenging in outdoor multi-building scenarios because multipath components (MPCs) from different facades are mixed, path-to-facade attribution is uncertain, and UAV measurements arrive sequentially under time-varying observation geometries. To address these challenges, we propose a unified online probabilistic framework that represents each reflecting facade as a virtual anchor (VA) and couples Bayesian VA localization, multipath attribution, and material inference. The Bayesian front end estimates facade-level geometry and computes soft MPC-to-VA attribution probabilities using a speculardiffuse likelihood model, thereby accounting for both dominant specular paths and diffuse surface-interacted components. These attribution probabilities are used to construct attribution-aware MPC representations, which are aggregated in a VA-centric manner and mapped by a material inference network to facadelevel material evidence. The resulting evidence is recursively fused through an online Bayesian update to produce stable material posteriors and material-labeled environment maps. Ray-tracing simulations in a representative urban street scenario show that the proposed method substantially outperforms a no-attribution baseline, achieves 93.75% final facade-level material accuracy on a held-out UAV trajectory, and maintains accurate VA-based facade localization.

Orthogonal arrays (OAs) are combinatorial configurations with applications in experimental design, error-correcting codes, and quantum information. Local unitary (LU) equivalence provides a natural framework for classifying multipartite entangled states. Using the correspondence between OAs and quantum states, Goyeneche and Życzkowski [Phys. Rev. A, 2014, 90: 022316] posed the problem of determining when OAs are $LU$ equivalent. In this paper, we construct OAs from generator matrices and establish conditions for Fourier-based $LU$ equivalence of OAs and irredundant orthogonal arrays (IrOAs). For prime alphabets, we identify the Fourier partners of the linear OAs considered here with the arrays of their corresponding dual codes. This gives an explicit connection between $LU$ equivalence and coding theory. We also identify families of $LU$ equivalent OAs arising from specific classes of linear codes.

Let $H_{27}=\mathrm{UT}_3(\mathbb{F}_3)$ be the nonabelian group of order $27$ and exponent $3$. We prove that $\mathsf{d}(H_{27}\times C_3^r)=2r+6$ for every integer $r\geq0$. The proof combines an affine coefficient identity in the group algebra of an elementary abelian group with a decomposition of the nonorthogonality graph of $\mathbb{F}_3^2$ into eight edge-disjoint zero-sum triangles. It is uniform in $r$, does not use the value of the small Davenport constant for a smaller nonabelian group, and requires no computational enumeration.

We study associative algebras endowed with a trace map. We classify the two-dimensional trace-prime algebras over an arbitrary field. In addition, when the base field is finite, for each such algebra, we provide a finite basis for the T-ideal of trace polynomial identities.

We consider the dynamic scheduling problem in a multi-class M/G/N + M queue with convex delay costs and impatient customers that have exponential abandonment times. We apply the Whittle index approach to find a reasonable heuristic solution for this tricky problem. By assuming exponential abandonment times, we are able to make a very straightforward use of the results of the corresponding problem with patient customers presented in Queueing Systems, Vol. 110, Article no. 2, 2026. Our main theoretical achievements are proving that the closed version of the corresponding discrete-time problem is indexable and deriving an explicit expression for the Whittle index. The discrete-time results are utilized to develop the Whittle index policy for the original continuous-time scheduling problem.

In this paper, we consider the inequality $\Delta_pu+\Delta_qu+u^s|\nabla u|^t\leq 0$ on geodesically complete noncompact Riemannian manifolds. By a test function argument, we establish Liouville-type theorems under the upper bound of volume of geodesic ball. In the Euclidean space $\mathbb{R}^n$, we obtain new nonexistence results which extend the result of Bhakta-Biswas-Filippucci \cite{BBF}. In particular, we have addressed the influence of the higher order term for $s<0$. At last, we present some examples to illustrate sharpness in some cases.

Following a recent characterisation of hydrodynamic systems with second-order Hamiltonian structure in terms of alternating three-forms on the projective space, we present a complete classification of systems in 6 components. We use non-trivial group action techniques to obtain 5 canonical forms. A general result for every dimension is also obtained for one of these cases, whose geometric structure is strictly related to the symplectic canonical form.

The research studies the extended mean-field anticipated forward-backward stochastic delayed differential equations with common noise (extended MF-AFBSDDEswCN) by expanding the domination-monotonicity conditions. We generalize the conditions from a linear setting to a nonlinear one by including nonlinear adjoint functions which are key to guaranteeing the well-posedness of extended MF-AFBSDDEswCN. Utilizing this broader well-posedness framework for extended MF-AFBSDDEswCN, combined with other refined analytical tools, we examine two classes of stochastic optimal control problems. These include a linear-convex problem and a linear-quadratic problem with input constraints that can be both time-dependent and random. Moreover, the control elements regarding the initial values are path-dependent, which will cause some essential difficulties. For each problem, we establish the existence and uniqueness of optimal controls and provide their explicit, closed-form representations.

Alon and Mubayi asked whether, for every integer $k\ge1$ and every $\varepsilon>0$, there exists $r_0=r_0(k,\varepsilon)$ such that every $r$-regular graph on $n$ vertices with $r\ge r_0$ contains a $k$-regular subgraph covering at least $(1-\varepsilon)n$ vertices. Previously, the conjecture was known for $k\in\{1,2\}$ and for $k$ and $r$ both even. We answer this question affirmatively for general $k$ and $r$ with $\varepsilon=O_k(r^{-1/2})$. For $k=2$ and every odd $r\ge3$, we show that $\varepsilon=1/(r^2-3)$ suffices which is best possible.

We study metrics whose distances on distinct two-point subsets are algebraically independent over the rationals. We prove that every compatible metric on a strongly zero-dimensional metrizable space of cardinality at most continuum can be uniformly approximated by compatible metrics with this property. If the space is completely metrizable, the approximating metrics can also be chosen complete. For every $\sigma$-compact metrizable space, the metrics with algebraically independent distances form a $G_\delta$ set in the uniform topology. We also study rigid metrics, whose only bijective self-isometry is the identity. On every locally compact Polish space, rigid proper metrics form a $G_\delta$ set among proper compatible metrics. Using a theorem of Niemiec, we obtain uniform density of rigid metrics on compact metrizable spaces with at least three points. Passing to compact completions then yields rigid approximations of every totally bounded compatible metric on any space with at least three points.

Convergence of the transition density of the Euler-Maruyama scheme for a Brownian-driven McKean-Vlasov stochastic differential equation is established, the coefficients being uniformly elliptic, spatially smooth, and Lipschitz with respect to the law in the 2-Wasserstein distance. The density p_n of the n-step scheme is shown to converge to the density p of the McKean-Vlasov equation in a Gaussian-weighted supremum norm at rate O(n^{-1/2}), that is, of order one half in the step size, and hence also uniformly in space. The proof develops a discrete parametrix (Levi-Hadamard) expansion for the Euler chain and compares it with the continuous parametrix of the limiting equation along a hierarchy of measure-flow, time-discretisation and lattice corrections; the discrepancy between the discrete and the continuous measure flows is of order O(n^{-1/2}) in the 2-Wasserstein distance and does not degrade the final rate.

Coronoid systems, nanotubes and toroidal polyhexes (or fullerenes) can all be regarded as carbon networks composed of carbon atoms linked in hexagonal shapes. The resonance graphs of coronoid systems and nanotubes are not necessarily connected. For coronoid systems and elementary nanotubes, by using flow across cuts the present authors gave criteria for two perfect matchings lying in the same connected component of the resonance graph (Discrete Appl. Math. 395 (2026) 443-455). However, the sufficiency of such criterion does not hold for general nanotubes and toroidal polyhexes. In this paper we strengthen this requirement to obtain valid criteria for two perfect matchings of a nanotube (resp. toroidal polyhex) to lie in the same connected component of its resonance graph: they have the same flows across cuts along the $x$-axis (resp. longitude and latitude) and the same ladders. For toroidal polyhexes, our method uses homotopic classes of simple loops on the torus, and the above criterion can be simplified by using only simple flows, for the case in which two perfect matchings have alternating hexagons.

Let $f_m(x_0,\ldots,x_{2m-2}) =\det(x_{i+j})_{0\leq i,j\leq m-1}$ be the determinant of the generic symmetric Hankel matrix of size $m$. We determine its global Bernstein–Sato polynomial. The proof combines the nearby cycles of the corresponding secant hypersurface with an explicit Archimedean zeta integral over the cone of positive definite Hankel matrices. A Gauss–Radau parametrization reduces this integral to a Morris integral and hence to a quotient of Gamma functions. Its primitive poles are detected by test functions supported arbitrarily close to the vertex of the cone. Lichtin's theorem, applied to a secant-stratum log resolution, fixes the integer shift between these poles and the roots of the Bernstein–Sato polynomial. A relative $D[s]$-lattice realization of nearby cycles, together with the simplicity of the corresponding intersection complexes, gives uniqueness and multiplicity one in every monodromy class. The induction in the matrix size follows from the local product structure of secant varieties.

For a graph $G$ and a set $B\subseteq V(G)$, the spread $\mathrm{sp}(B)$ of $B$ is the difference between the largest and the smallest degree in $G$ of a vertex of $B$, and for an integer $k\geq0$ the parameter $\mathrm{sp}(G,k)$ is the largest cardinality of a set $B$ with $\mathrm{sp}(B)\leq k$. Caro, Lauri and Zarb derived a lower bound for $\mathrm{sp}(G,k)$ and, among several families of graphs, considered \[ \mathrm{MOP}(n,k)=\min \{\mathrm{sp}(G,k):G\text{ is a maximal outerplanar graph of order }n\} \] and determined $\mathrm{MOP}(n,k)$ up to an additive constant for every $k\not =2,$ leaving the case $k=2$ open, with the bounds $4n/9\leq \mathrm{MOP}(n,2)\leq (5n+19)/11$. We first prove a lower bound on $\mathrm{sp}(G,k)$ for an arbitrary graph $G$ in terms of its order $n$, its number of edges $m$ and its minimum degree $\delta $. This lower bound contains the bounds of Caro, Lauri and Zarb and, for $k=0$, the bound $\mathrm{rep}(G)\geq \left\lceil n/(2d-2\delta +1)\right\rceil $ of Caro and West, where $d=2m/n$. We determine when this lower bound is attained, exhibit explicit graphs attaining it, and show that it is exact for all graphs once $n\geq n_{0}(\delta ,k,d)$. We then apply the bound to maximal outerplanar graphs: adjusting the count to this class we prove \[ \mathrm{MOP}(n,2)\geq \left\lceil \frac{4n+10}{9}\right\rceil \qquad \text{for every }n\geq 14, \] with equality for $n\equiv 2\ (\mathrm{mod}\ 18)$, and $\mathrm{MOP}(n,2)=4n/9+O(1)$ for every $n$.

We prove a quenched functional central limit theorem for the random conductance model with ergodic translation-invariant, strictly positive nearest-neighbor conductances, assuming only finite first moments of the conductances and their inverses. This settles an open problem by reaching the critical first-moment threshold, which is sharp for a certain class of integrability assumptions. A key ingredient of the proof is a new Sobolev inequality that seems to be missing from the literature.

Particle-based physics simulations, including fluids, smoke, and granular media, are fundamental to visual realism in immersive VR and AR. With the growing adoption of social VR and digital twins, demand is increasing for shared experiences in which multiple users interact with the same simulation in real time. Realizing such experiences requires low-latency streaming of large-scale particle data from a server to each client, yet existing point cloud compression methods such as G-PCC (TMC13) and Draco assume static geometric structures; when applied to dynamic particle streaming, their encoding latency exceeds the frame period, failing to meet real-time delivery requirements. We propose DELUGE, a streaming compression architecture that exploits the temporal coherence and velocity predictability inherent in physics simulation particles, achieving sub-frame-latency encoding and decoding through three complementary techniques. Evaluation on dynamic point cloud datasets demonstrates that DELUGE achieves approximately $20\times$ faster decoding than G-PCC (TMC13) and $6\times$ faster encoding than Draco. We further build end-to-end client implementations for both web browsers and Apple Vision Pro, and confirm through a within-participants perceptual quality evaluation and a two-person collaborative task study on Vision Pro that the proposed method supports real-time collaborative experiences with hand-tracked fluid interaction.

We prove that the integral Chow groups $CH_i$ and mixed Hodge degree $H^{2i, (i, i)}$ cohomology groups of a really full rank (RFR) sink-recurrent cluster variety vanish for $i > 0$. In particular this applies to braid varieties and open projected Richardson varieties in any Lie type. Our main technical tool is the construction of a stratification of any RFR sink-recurrent cluster variety $\mathcal{A}(\Sigma)$ over any field $k$ into (affine space times) RFR sink-recurrent cluster varieties of seeds with fewer mutable vertices than $\Sigma$, with which we show that RFR sink-recurrent cluster varieties are linear in the sense of Totaro. We also show that cycle class maps are isomorphisms onto the lowest-weight part of rational Borel-Moore homology for any mixed Tate variety over a number field. Finally, we use our results to deduce vanishing theorems for the Khovanov-Rozansky homology groups of closures of positive braids and generation properties for the Chow groups of flag Richardson, projected Richardson, and brick varieties: namely, that these groups are generated integrally by the fundamental classes of their closed Richardson, positroid, and braid strata, respectively.

Locally commuting multiple radial Schramm-Loewner evolutions ($\mathrm{SLE}_{\kappa}$) are encoded by partition functions satisfying the radial Belavin-Polyakov-Zamolodchikov (BPZ) equations and a conformal Ward identity with spectral parameters $\lambda,\nu\in\mathbb{R}$. For $\kappa>0$ and $\lambda\in\mathbb{R}$, we show that the solution space of the radial BPZ system has dimension $2^n$, where $n$ is the number of variables. We then determine all admissible Ward parameters $\nu$ and the exact dimensions of the subspaces selected by the conformal Ward identity, covering both generic and degenerate cases. The classification reveals a parity difference: nonzero rotation-invariant solutions exist for every $\lambda$ when $n$ is even, but only at finitely many exceptional values when $n$ is odd. When $0<\kappa\leq4$, $\lambda>0$ for odd $n$ or $\lambda>-3/2$ for even $n$, we construct a basis of positive solutions using multiple SLE, providing global realizations of the locally commuting SLEs. At $\kappa=4$, we identify a family of explicit solutions as partition functions for level lines of a Gaussian free field with suitable boundary data and interior singularities.

Circulant digraphs form a prominent class of Cayley digraphs defined on finite cyclic groups. Building on the existing classification of $2$-arc-transitive circulant graphs, this paper presents a complete classification of $2$-distance-transitive circulant digraphs. Our main theorem establishes that every connected $2$-distance-transitive circulant digraph is isomorphic to one of the following: the undirected cycle $C_n$, the complete bipartite graph $\K_{\frac{n}{2},\frac{n}{2}}$, the complete multipartite graph $\K_{m[b]}$ with $m\geq 3,b\geq 2$, the graph $\K_{\frac{n}{2},\frac{n}{2}}-\frac{n}{2}\K_2$ for odd $\frac{n}{2}$, prime-order Paley graphs, the directed cycle $\overrightarrow{C}_n$, the oriented graph \(G(p^m,r)\) satisfying Condition~\ref{p-power-normal-2dt-cond}, the oriented graph \( C_r(b,1)\) with $r\geq 3,b\geq 2$ and $rb=n$, the lexicographic product oriented graph \( G(p^m,r)[\overline{\K}_d]\) where \(G(p^m,r)\) obeys Condition~\ref{p-power-normal-2dt-cond}.

We investigate the relationship between weak asymptotic symmetry ($\mathrm{WAS}$) and asymptotic symmetry ($\mathrm{AS}$) for quasisymmetric maps associated with planar quasilines. To this end, we introduce a formally weaker condition, called equidistant weak asymptotic symmetry ($\mathrm{EWAS}$), and prove that, for quasisymmetric embeddings of $\mathbb{R}$ into $\mathbb{C}$, the three conditions $\mathrm{AS}$, $\mathrm{WAS}$, and $\mathrm{EWAS}$ are equivalent. We then establish that $\mathrm{WAS}$ implies $\mathrm{AS}$ for quasisymmetric homeomorphisms from an arbitrary quasiline onto $\mathbb{R}$. More generally, if $h\colon\Gamma_1\to\Gamma_2$ is a quasisymmetric homeomorphism between quasilines and $\Gamma_1$ is asymptotically conformal, then $\mathrm{WAS}$ implies $\mathrm{AS}$. A formally dual statement holds when $\Gamma_2$ is asymptotically conformal, provided that $h$ is uniformly continuous. In the compact setting of bounded quasicircles, these results yield an answer to the $\mathrm{WAS}$-$\mathrm{AS}$ problem posed by Brania and Yang.

Talagrand asked whether the Rademacher cotype of an operator is universally controlled by the maximum of its Gaussian cotype and its $(q,1)$-summing norm. We give a negative answer already for $q=2$. The counterexample was discovered by ChatGPT (GPT-5.6).

We prove that a path maximizes the expected range of a uniformly chosen graph homomorphism into the integers, with one vertex pinned at zero, among all connected bipartite graphs of the same order. This establishes the expectation form of the Benjamini–Häggström–Mossel conjecture. The proof restricts and rescales a homomorphism on each bipartition class, then contracts the edges on which the resulting height function is constant. A quantitative estimate for the rank of these zero edges compensates for a parity term in the expected range of a simple random walk, allowing an induction on the number of vertices. We then prove that the BHM inequality implies the Loebl–Ne\v set\v ril–Reed inequality for uniformly chosen integer 1-Lipschitz functions on arbitrary connected graphs, and hence obtain the LNR conjecture as a corollary of BHM. The proof was obtained through interaction with OpenAI GPT-6 Astra and verified by the author. The main results have also been formalized and checked in Lean~4.

Let $\mu$ be a finite positive Borel measure on $[0,1)$ and let $\gamma>0$. We establish sharp mapping criteria for the generalized Cesàro operator \begin{equation*} \mathcal C_{\mu,\gamma}f(z) =\sum_{n=0}^\infty \mu_n \left(\sum_{k=0}^n \frac{\Gamma(n-k+\gamma)}{\Gamma(\gamma)(n-k)!}a_k\right)z^n,\qquad z\in \mathbb{D},\end{equation*} between Hardy spaces, including the source and target $H^\infty$ endpoints. For $0<p<q<\infty$, for $0<p=q<1$, and for $0<p\le1$ with $q=\infty$, the boundedness of $ \mathcal C_{\mu,\gamma}: H^p \to H^q$ is equivalent to $\mu$ being a $(\gamma+1/p-1/q)$-Carleson measure, without further restrictions on $\gamma$. For $1\le q<p\le\infty$, set $1/r=1/q-1/p$. In this range, boundedness and compactness are equivalent to \[ \int_0^1 \left(\frac{\mu([t,1))}{(1-t)^{\gamma-1/r}}\right)^r \frac{dt}{1-t}<\infty. \] This condition is also equivalent to $F_{\mu,\gamma}\in H^r$ and to $\sum_{n\ge0}(n+1)^{r\gamma-2}\mu_n^r<\infty$, where $F_{\mu,\alpha}=\mathcal C_{\mu,\alpha}(1)$ and $\mu_n=\int_{[0,1)}t^n\,d\mu(t)$. For $1<p<\infty$, boundedness and compactness from $H^p$ to $H^\infty$ are characterized by the shifted condition $F_{\mu,\gamma+1}\in H^{p'}$, where $p'=p/(p-1)$. We therefore obtain a complete boundedness classification of the generalized Cesàro operators $\mathcal C_{\mu,\gamma}$ between Hardy spaces $H^p$ and $H^q$ for the full range $0<p,q\le\infty$.

We study a natural way to deform area-preserving maps between compact Riemann surfaces. Specifically, we evolve the metrics on the two Riemann surfaces by the normalized Ricci flow and the graph of the area-preserving map by the mean curvature flow. We prove that the flow exists for all time, remains the graph of an area-preserving map, and converges smoothly and exponentially to a minimal Lagrangian graph with respect to the product of the limiting metrics. This generalizes earlier results of Wang and Smoczyk, in which the Riemann surfaces have constant scalar curvature.