We study the random variable that counts the number of specific arrangements of clustered consecutive numbers in permutations under the Mallows distribution. We provide an asymptotic expression for the expected value of this random variable. This result extends and tightens the previously known result by Pinsky (2022) concerning clustered consecutive numbers in Mallows permutations. Moreover, we identify a range of parameters for which the distribution of the number of arrangements of clustered consecutive numbers in Mallows permutations is close to a Poisson distribution.

We prove short-time well-posedness for the Muskat problem with surface tension in the full two-phase setting, allowing different viscosities, arbitrary density contrast, and rigid boundaries. In particular, no Rayleigh--Taylor sign condition on the density contrast is imposed. The interface is assumed to be a graph, uniformly separated from the fixed boundaries, and the initial data may be large in the log-critical class $\dot C^{1,\log^\varkappa}\cap H^1$, with $\varkappa>1$. Thus the result reaches the natural Lipschitz threshold up to a logarithmic correction. The main difficulty is that, in the presence of viscosity jump and boundaries, the interface equation is not given by a closed explicit contour dynamics law. Instead, the normal velocity is recovered through an elliptic transmission problem in moving domains, and the resulting evolution is a genuinely nonlocal quasilinear equation. We derive sharp Schauder-type estimates, adapted to the log-critical scale, for the transmission operators generated by the bulk Darcy flow. These estimates identify the third-order parabolic mechanism produced by surface tension and control the nonlinear coupling between the interface geometry and the elliptic transmission structure. The proof builds on the Schauder framework developed in Part~I of this series \cite{CHN1}, but requires a new analysis of the Muskat transmission problem in moving domains. Combining this elliptic theory with the contour formulation and time-weighted Hölder estimates, we obtain existence, uniqueness, smoothing, and stability for large interfaces in arbitrary dimension.

In this paper we prove the Markov property for the solution to stochastic partial differential equations driven by a cylindrical orthogonal martingale-valued measure. We assume our coefficients are time-dependent and satisfy some growth and Lipschitz conditions. We also prove that for time-independent coefficients and under mild assumptions on the cylindrical orthogonal martingale-valued measure, the solutions to our stochastic partial differential equations are Feller. Finally, in the case that the $C_{0}$-semigroup is quasi-contraction, we show that the solution to our stochastic partial differential equation possesses a càdlàg version.

A family of $m \times n$ matrices $\mathcal{F} \subseteq \mathbb{F}_q^{m \times n}$ is {$(t-1)$-intersection-free} if $\dim \ker(A-B) \neq t-1$ for all $A,B \in \mathcal{F}$. A \emph{forbidden $(t-1)$-intersection problem} for a collection of matrices asks for the size and structure of extremal $(t-1)$-intersection-free families within that collection. We solve this problem in $\mathrm{GL}(n,q)$ for all pairs $(n,t)$ such that $t<c\cdot n$ where $c$ is a universal constant. We show that the $t$-umvirates and their duals, are the only maximal $(t-1)$-intersection-free families $\mathcal{F} \subset \mathrm{GL}(n,q)$. Here, a $t$-umvirate is defined as the family of all matrices that agree on a fixed $t$-dimensional subspace, and its dual as those whose transposes agree on it. The best previously known result, due to Ellis, Kindler, and Lifshitz, established this bound under the assumption $n \geq e^{Ct\log t}$ for some constant $C>0$. We also give Frankl--Rödl-type constructions showing that this range of $t$ is almost the best possible: we show that for values of $t>n/2$ the extremal behavior changes and no clean analogue is expected. Our proof builds upon recent global hypercontractivity results for matrix spaces due to Evra, Kindler, and Lifshitz, and broadly applies to any sufficiently dense class of matrices.

We continue our local singularity analysis for Ricci flow initiated in arXiv:2006.16227. Building on that framework, we study Type I singular points in general Ricci flows, without assuming any global Type I curvature bound, and prove that the scalar curvature must blow up at a Type I rate at each such point in all dimensions. As a consequence, Ricci flows with bounded scalar curvature cannot develop Type I singular points. This extends earlier results of the first author with Enders and Topping and with Mantegazza that relied on a global Type I assumption. We then adapt the same local perspective to ancient Ricci flows and analyse the curvature behaviour as time goes to negative infinity, showing in particular that every ancient Type I point exhibits scalar curvature behaviour of ancient Type I order.

We show that, under a division condition, the tangential Cauchy--Riemann cohomology of a compact Lie group with a left-invariant CR structure can be computed on a suitable maximal torus. As a consequence, we conclude that the tangential Cauchy--Riemann cohomology is finite-dimensional. We also show that, for a class of CR structures, this division condition is necessary for the total cohomology to be finite-dimensional. The proof combines Fourier analysis on compact Lie groups, highest-weight representations and Lie algebra cohomology. This not only generalizes but provides completely new proofs for the analogous result due to Pittie and for its extensions to Levi-flat CR structures, obtained by Jacobowitz and Jahnke.

In this note we give a lower bound for the second moment of a ratio of zeta functions summed over the non-trivial zeros of the Riemann zeta function that is half the size of the conjectured value. Our result is conditional on the assumption of the Riemann Hypothesis and that all the non-trivial zeros of the zeta function are simple.

This work presents a finite element formulation for coupled chemo-mechano-thermodynamical large deformation contact. The formulation is based on the contact theory of Sauer et al. (2022) that contains six coupled (but separate) fields: the deformation and temperature of the two contacting bodies, as well as an interfacial bonding field and interfacial temperature. The latter is governed by the chemical and mechanical energy dissipation at the interface. Here the focus is placed on the evolution of bonding and debonding, and how it is coupled to the mechanical and thermal contact state. Several elementary models are proposed for this based on a quadratic contact potential. The resulting contact formulation becomes very general and versatile, which is illustrated by several challenging examples. They include pressure- and gap- depended bonding, exothermic bonding reactions, thermal hardening and thermal expansion, as well as simultaneous bonding and debonding. They are based on a monolithic finite element implementation using classical and isogeometric shape functions together with implicit time integration. Its full linearization, required for the Newton-Raphson solution method, is also provided. If bonding sites are material points, the bonding variable can be condensed-out locally.

In a recent paper, the authors introduced an infinite hierarchy of $q$-hypergeometric identities, of which the first three orders, $0$, $1$, and $2$, relate to the partition theorems of Euler, Lebesgue, and Capparelli, and stated a partition theorem at order 4 which lies beyond Capparelli's theorem. Here, we first state certain partition theorems that hold at all even orders beyond Capparelli and provide bijective proofs for these theorems. In doing so, we show that there is a fourfold infinite hierarchy of partition theorems that emanates from Capparelli's theorem, which is the base case. It is also shown that the equality of two of the four generating functions holds for all orders, odd and even. Lastly, a very general framework for the remaining two functions is constructed via the method of weighted words, encompassing all possible orders and yielding several infinite hierarchies with different dilations and translations.

We give a higher-order higher-dimensional Eckmann-Hilton argument that is entirely algebraic. First we give an explicit argument showing that if we have two monoidal structures on a category with suitable interchange, we can derive a braiding on either of the monoidal structures. Then we show that given third monoidal structure, with suitable pairwise interchange on any pair of monoidal structures, each canonical braiding is forced to be a symmetry. As a motivating example, we show that for $n \geq 3$ any $n$-degenerate semi-strict $(n + 1)$-category has three suitably coherent monoidal structures on its single hom-category, thus the hom-category has the structure of a symmetric monoidal category.

We establish structure theorems for polarised flag type quotients arising in the study of the moduli space of supersingular abelian varieties. In particular, we prove a refined normal form for the top level of these flags and derive an explicit, computable descent criterion for quasi-polarisations. These results provide a complete classification of the first and last step of these flags. As an application, we compute polarised flag type quotients in dimensions $g\leq 5$, describing the fibers of the truncation morphisms as classification objects of supersingular abelian varieties.

This paper presents an improved physics-informed neural network for simulating the spatio-temporal solution profile of the multidimensional coupled Burgers' equations with high Reynolds numbers. As time evolves, the sharp shock fronts emerge in the solution, creating significant computational challenges for the conventional mesh-based numerical methods. In particular, numerical methods such as finite differences and finite elements suffer from poor stability and strong mesh dependency when resolving the steep solution gradients. To address these challenges, the proposed framework employs a layerwise self-adaptive weighting strategy that dynamically adjusts the penalty weights for the physics residual, initial conditions, and boundary conditions throughout training. Moreover, the framework uses a dual-phase optimization strategy to achieve more stable convergence. To check the effectiveness and accuracy of the proposed framework, a set of numerical experiments is conducted to compare it with the standard Physics-Informed Neural Network (PINN) with and without Limited-memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) optimization. Numerical results exhibit that the proposed framework achieves higher accuracy in terms of relative $L_2-$ error norm than the standard PINN and is able to capture the development of sharp shock fronts as time evolves in the solution.

The problem of optimal sizing and power scheduling in microgrids subject to uncertainties is well known to the control community. Commonly, the optimal control problem is cast as a mixed-integer program to model the logical constraints arising in energy storage systems, and is then solved approximately using numerical methods such as the scenario approach. In this paper, we propose and compare two formulations of a robust microgrid sizing and power scheduling optimal control problem with logical constraints and uncertainties in the user's power demand, solar power generation, grid electricity prices and battery efficiencies. The first formulation uses binary variables and big-M constraints, leading to a mixed-integer linear program. The second formulation casts the problem as a continuous nonlinear program through an exact smooth reformulation of the logical constraints, consisting of additional modelling variables and non-convex constraints. We then propose a novel local reduction algorithm, extending an existing method, to solve both problems. The two formulations are compared by evaluating the solutions returned by local reduction using 100,000-sample Monte Carlo simulations and achieve promising results, with both averaging feasibility rates above 90%.

Inverse problems governed by partial differential equations (PDEs) are central to computational mechanics and are commonly solved by adjoint-based optimization, while physics-informed neural networks (PINNs) have emerged as a flexible alternative. Their relative performance remains difficult to assess because the two approaches are often compared under different formulations, parameterizations, optimizers, and regularization choices. We present a fair comparison of adjoint optimization and PINNs for PDE-constrained inverse problems. From a common abstract formulation, we instantiate both methods on identical domains, governing equations, observation models, and regularization terms, while matching the optimizer, unknown parameterization, and arithmetic precision wherever applicable. The benchmarks include unsteady Burgers, noisy Darcy permeability inversion, three-dimensional Allen--Cahn reaction identification, and unsteady Navier--Stokes viscosity identification. The results show that the representation of the unknown largely determines the preferred method: grid-based fields favor the discrete adjoint, whereas neural representations are native to PINNs and relevant for closure and constitutive modeling. For time-dependent problems, adjoint inversion can be dominated by trajectory storage and differentiation, while PINNs provide satisfactory reconstructions at lower cost. A PINN-warm-started adjoint strategy then recovers adjoint-level accuracy at substantially reduced cost.

We develop a large-signal stability analysis for a sampled-data, optimization-based secondary controller for inverter-interfaced distributed energy resources in virtual power plants.

In 1988, Las Vergnas conjectured that if $M$ is a binary matroid with bicycle dimension $d$, then for $0 \leq k \leq d$, the $k$th derivative of the diagonal Tutte polynomial $T(M;z,z)$ evaluated at $z=-1$ is an integer multiple of $2^{d-k}$. While this was rapidly disproved for binary matroids and for graphs in general, extensive computations strongly suggested that it might be true for planar graphs. In this paper we prove that this is indeed the case. To do this, we consider a stronger divisibility property that we call the LV property, and a larger class of graphs, namely the class of delta-wye-reducible graphs. By a detailed analysis of how a delta-wye exchange affects the coefficients of the diagonal Tutte polynomial, we show that delta-wye-reducible graphs have the LV property. That Las Vergnas' conjecture holds for planar graphs immediately follows because planar graphs are delta-wye reducible and the LV property is stronger than Las Vergnas' divisibility conditions.

Expectiles provide a class of asymmetric location functionals that incorporate the magnitude of deviations and admit a natural geometric interpretation. Building on their structural consistency with the convex stochastic order, this paper introduces a family of integrated expectile functionals for measuring risk, dispersion, and inequality. The proposed functionals admit analytical representations as integrals of expectiles across asymmetry levels. For a distinguished subclass of these constructions, a geometric representation is available: the resulting quantities can be expressed as weighted areas of star-shaped sets encoding the distributional asymmetry of a random variable. This approach yields a new class of expectile-based inequality indices, constituting a natural counterpart to classical Gini-type measures while preserving desirable monotonicity and consistency properties. Empirical counterparts are derived in closed form and admit explicit decompositions over finite samples. The framework extends naturally to multivariate settings through directional expectile constructions, leading to measures capable of capturing genuinely joint forms of multivariate dispersion and inequality.

A cyclic meander is an embedded oriented loop in the plane intersecting a fixed infinite line, or circle, transversely in a linearly ordered set of $2n$ points. By keeping track of the order in which the loop visits these points, the cyclic meander induces a cyclic permutation on these marked points. Correspondingly, given a permutation on $n$ letters, one can ask whether or not a cyclic meander induces the permutation in this manner, and if not, what is the most efficient way of doing so if we allow more points of intersection? This process gives a way of associating to a permutation on $n$ letters a measurement of complexity of the permutation in question. The principal result of this work shows that the maximum of this quantity, the \emph{meander number}, over all cyclic permutations on $n$ letters, is bounded above and below quadratically in $n$. This result resolves a conjecture of Schwartz~\cite{richtpss} in relation to his work on the topological salesman problem. We conclude this work by constructing families of cyclic permutations on $n$ letters whose meander numbers realize a continuum of growth rates between linear and quadratic.

The cooling number measures the speed at which a slow-moving influence or contagion spreads on a graph. In this paper, we investigate the cooling number of four classical graph products: the Cartesian product, the strong product, the lexicographic product, and the direct product. We also determine the cooling number of a disconnected graph in terms of the cooling numbers of its components. We conclude with open problems.

This note is a tutorial on the deterministic version of the Kalman filter (state estimator), which is formulated as finding the state trajectory consistent with the system's equations with the minimal amount of $L^2$ process and measurement uncertainty. As stated, this is an input signal design problem with linear dynamics and an objective that is affine-quadratic in the state and inputs. The first step is to convert this problem to one with a purely quadratic objective by embedding in a larger system using ``homogeneous coordinates''. This converts the problem to a purely quadratic (i.e. an LQR) problem, but with non-standard initial or final state constraints. This latter problem can then be solved using a version of the matrix Differential Riccati Equation (DRE) for the larger LQR problem. The second step is a partitioning of this larger problem, which then yields the optimal dynamic observer and the DRE of the traditional Kalman filter. For comparison, the solution of the traditional LQ-tracking (Servomechanism) problem is also treated using a similar construction.

In arXiv:2311.10069, Matolcsi et al. show that the fractional chromatic number of the plane is at least 4. Their proof uses a 27-vertex unit-distance graph in the Moser lattice, with geometric fractional chromatic number exactly 4. We show that this bound is tight for graphs in the Moser lattice by exhibiting geometric 4-colorings of the entire lattice. The same colorings also extend to the entire Moser ring.

This paper extends a previous work in which the rank-2 co-Higgs bundles on a Hopf surface are classified based on the data of the underlying vector bundle. The aim of the paper is to study the Poisson 3-folds that can be constructed from these co-Higgs bundles by describing their symplectic leaves.

We establish a canonical Bernstein--Zelevinsky filtration for Casselman--Wallach representations that is analogous to the $p$-adic case. In addition, we outline an approach to Casselman's comparison conjecture and prove it for general linear groups, as well as for quasi-split even orthogonal groups in some special cases. We also give some applications of the Bernstein--Zelevinsky filtration, such as to the study of highest derivatives and the indecomposability of mirabolic restrictions.

For any closed surface of genus at least $3$, equipped with any bounding spin structure, we show that the index difference, viewed as a map from the fundamental group of the space of Dirac-invertible Riemannian metrics to $\KO^{-4}(*)$, is non-trivial. For products of two such surfaces, equipped with any spin structure, we prove that the corresponding space of Dirac-invertible Riemannian metrics is not contractible. We discuss the relationship of this result to the existence of metrics with harmonic spinors in dimension $4$.

Let $G$ be a loopless multigraph with maximum degree $\Delta(G)$, average degree $\overline{d}(G)$, density $\Gamma(G)$, and chromatic index $\chi'(G)$. A multigraph $G$ is called edge-$\Delta$-critical if $\Delta(G)=\Delta$, $\chi'(G)=\Delta(G)+1$ and $\chi'(H) \le \Delta(G)$ for every proper subgraph $H\subset G$. Vizing conjectured that if $G$ is an edge-$\Delta$-critical simple graph on $n$ vertices, then $\overline{d}(G) \ge \Delta-1+\tfrac{3}{n}$. Motivated by this, we conjecture that every edge-$\Delta$-critical multigraph $G$ satisfies $\overline{d}(G) \ge \tfrac{2\Delta+2}{3}$, which is best possible. We first give a general lower bound in this direction. For any such graph $G$, \[ \overline{d}(G) \ge \begin{cases} \frac{\sqrt{17}-3}{2}(\Delta+1) & \text{if } \Delta \le 112;\\[4pt] \frac{\Delta+\sqrt{2\Delta-1}}{2} & \text{if } \Delta \ge 113. \end{cases} \] This bound can be further improved under an additional condition on the multiplicity $\mu$. In this case, \[ \overline{d}(G)\ge \min\left\{ \frac{2\mu\Delta+2\mu(2\mu-1)}{4\mu-1},\; \frac{\sqrt{17}-3}{2}(\Delta+1) \right\}. \] We also confirm the conjecture for $\Delta \in \{2,3,4,5,6,7,8\}$. As a consequence, Goldberg's conjecture~\cite{Goldberg1984} holds for $\Delta(G)\in\{2,3,4,5\}$, that is, every multigraph $G$ with $\chi'(G)\ge \Delta(G)+1$ satisfies $\Gamma(G)\ge \Delta(G)$.

In this short note, we improve the upper bound on the condition number of the univariate B-spline basis of order $k$ from $k 2^k$ to $O(\sqrt{k}\log k\,2^k)$ for all $k\ge 2$.

We study the finite-scale one-component degeneration problem for suitable weak solutions of the three-dimensional incompressible Navier--Stokes equations under a scale-invariant bound and smallness of the vertical component. Qualitative compactness gives convergence, in the harmonic-pressure quotient, toward the strict two-and-a-half-dimensional boundary, but it does not provide a quantitative rate. This paper proves, in an explicitly abstract trace-obstruction skeleton associated with the old observable closure, that the standard old observable package is insufficient to force a logarithmic or power selected-trace rate. The negative result is an envelope/skeleton theorem, not a Navier--Stokes counterexample. After excluding elementary high-frequency escape by parabolic trace drop and fixed-window analytic obstruction by finite-dimensional Lojasiewicz control, the remaining obstruction is an all-order finite-mode flat branch. We identify the Navier--Stokes-specific mechanism needed to control this branch: strict Schur trace-projectability may fail, but the resulting defect can be visible through the relaxed vertical-pressure channel. In active finite-window models, strict Schur phantoms are relaxed-visible. The final theorem is a conditional dichotomy: either relaxed anti-phantom closure holds and yields conditional logarithmic strict-shadow selection, or there exists an NS-realizable, cleaned, relaxed-invisible, unaligned left-singular cascade.

In this paper, we investigate a stochastic averaging principle for a large class of mechanical systems in the presence of random vibrations. We show that a known deterministic averaging principle for affine connection control systems with large-amplitude high-frequency inputs also holds for non-periodic stochastic inputs. The randomly vibrating mechanical system is described by a stochastic differential equation whose solutions do not depend on its interpretation in the sense of Ito, Stratonovich, or Sussmann. We also show that solutions of this stochastic differential equation can be directly computed from a single ordinary differential equation. We illustrate our theoretical results by the example of stochastic source seeking with a nonholonomic vehicle.

Given a finite collection of Lagrangian submanifolds $\mathscr L$ in a compact symplectic manifold $X$, we construct a cyclic, filtered, strictly unital curved $A_{\infty}$ category $\mathcal L$ and develop Floer theory of closed-open maps and open-closed maps. Using them, we prove that, whenever the map from the quantum cohomology of $X$ to the Hochschild cohomology of the Fukaya category $\mathcal L$ with objects $\mathscr L$ is injective, the following consequences follow: (1) any other Lagrangian submanifold equipped with a weak bounding cochain lies in the category split-generated by $\mathscr L$, and (2) the Hochschild homology and cohomology of the Fukaya category are isomorphic to quantum cohomology. In the exact case a similar result was obtained in [Ab]. We also provide some applications.

Diffuse-interface models are a widely used framework for interfacial dynamics in complex fluids, in which interfaces are represented through smooth transition layers and capillary effects are encoded by a free-energy functional. For incompressible mixtures with more than two phases, however, robust computation is substantially more difficult because the numerical method should preserve the balance structure of the continuum model, maintain the saturation constraint, dissipate energy, and treat all phases symmetrically even when density ratios are arbitrary. Existing structure-preserving methods are largely developed for binary flows or for formulations that distinguish a reference phase, so a genuinely symmetric N-phase discretization remains lacking. The practical problem is therefore to construct a fully-discrete method for N-phase incompressible Navier--Stokes--Cahn--Hilliard mixture models that retains the key thermodynamic and conservation properties of the continuum equations for arbitrary density ratios. Here we propose a symmetric fully-discrete method for the N-phase incompressible Navier--Stokes--Cahn--Hilliard mixture model with arbitrary density ratios. The method yields a fully-discrete problem in which every solution satisfies exact phase volume conservation, phase mass conservation, total volume conservation, total mass conservation, and a discrete energy-dissipation law. In addition, if the volume-saturation constraint holds for the initial data, then it is preserved at every time step. We numerically verify these structure-preserving properties and demonstrate the robustness of the method in representative multiphase flow problems. The resulting scheme provides a computational framework for incompressible N-phase mixture flows with complex interfacial dynamics and arbitrary density contrasts.