Papers

We propose an alternative existence proof of global weak solutions to mean curvature flow and volume preserving mean curvature flow. We prove for the first time for a minimizing movements scheme the unconditional convergence towards a varifold solution, here a De Giorgi solution. The argument is purely variational and does not rely on comparison principles. The key novelty is an alternative proxy for the completely degenerate $L^2$ distance that is more robust than the one of Almgren-Taylor-Wang and Luckhaus-Sturzenhecker.

Positive operator-valued measures on a tree admit intrinsic local coordinates coming from the way each cylinder value splits into its children. We show that these local splittings, taken on the range spaces of the cylinder values, recover the measure and at the same time build an intrinsic direct limit dilation whose cylinder projections yield the minimal Naimark dilation. In these coordinates, the commutant of the dilation becomes a martingale calculus on the range spaces. This gives local descriptions of extremality and domination, and it also yields a bounded change-of-measure transform that updates the tree coordinates in a natural way. For self-adjoint range martingales we obtain a quadratic variation formula from the range space isometries, and the associated local variance terms detect the projection-valued case.

In this article, we show that for all finite irreducible Coxeter groups $G$ with a conjugation closed subset $C$, the conjugation quandle $\rm{Conj}(G,C)$ satisfies the Hayashi property.

Polar codes are proven to be capacity-achieving codes, being gradually practiced in wireless communications. However, their successive cancellation (SC) and successive cancellation list (SCL) decoding incur latency challenge especially for long codes. This paper proposes the generalized integrated interleaved (GII)-polar codes for block Rayleigh fading channels, yielding both reduced decoding latency and competent decoding performance. Under the GII paradigm, two consecutive polar codewords of length $N$ are virtually coupled through a nested codebook which has a lower dimension, capable of correcting more errors. The component polar codes are known as interleaves of a GII-polar code. If decoding of an interleave fails, it can be projected into the decoding of the nested one, enabling richer error patterns to be corrected by each interleave. Since decoding of the interleaves can be performed in parallel, GII-polar codes yield a reduced decoding latency over a single polar code of length $2N$. Our simulation results validate both the decoding latency and performance merits of the proposed coding scheme under block Rayleigh fading channel.

We study skew double-shifted plane partitions with three-element profiles using MacMahon's partition analysis. We present new generating function formulas for these partitions, incorporating an extra bound on the number of non-zero diagonal elements. These objects are closely related to the Göllnitz-Gordon and little Göllnitz identities. Moreover, we investigate some infinite hierarchies of $q$-series identities that stem from our formulas. We also observe certain palindromic properties within these generating functions. Finally, we examine these same objects from the perspective of linear partitions, which leads to interesting new $q$-series identities.

We develop a finite-field stratification for characteristic polynomials of deletion subarrangements of the full $m$-Catalan arrangement. It reduces the count to cyclic placements of rigid blocks and yields falling-factorial expansions for deletions indexed by graphs, digraphs, and gain-labeled digraphs. The coefficients are graphical Stirling numbers for zero-layer deletions, directed matching numbers when the deleted layer $\ell$ satisfies $1\le \ell\le\lfloor m/2\rfloor$, directed path-cover numbers when $\lfloor m/2\rfloor<\ell\le m$, and admissible gain-labeled arc sets for multilayer deletions. For $\ell=m$, a complementary path-cover expansion yields factorization consequences. The method also gives formulas for directed Ish-type arrangements in terms of path-cycle covers and outdegrees.

The efficient simulation of steady-state rarefied gas flows remains a significant computational challenge due to the high dimensionality of the collision integral and the severe numerical stiffness in the near-continuum regime. In this work, we propose a modified Newton method equipped with a macroscopic synthetic system (Newton-MS) for the steady-state Boltzmann equation with the quadratic collision operator. In Newton-MS, the modified Newton iteration is utilized as the outer nonlinear solver, while each Newton correction equation is solved by an inner source iteration, where the linearized collision operator is utilized to approximate the quadratic collision model, and it is reduced into a linear iteration. Moreover, a macroscopic synthetic system based on Chapman-Enskog closure is derived to accelerate the convergence of the linear inner iteration in the continuum limit. Besides, the fully discrete macroscopic synthetic system is deduced under the framework of the discontinuous Galerkin method to reduce computational cost compared to directly discretizing the continuous macroscopic synthetic system. Several numerical examples, including the 1D Fourier, Couette flow problem, and the 2D cavity flow and thermal-driven cavity flow, are studied to validate the high efficiency of Newton-MS.

In the first part of this series we have shown how the Schrodinger equation for a single particle and the corresponding non relativistic quantum observables can be obtained from a purely classical phonon model through the Newtonian equations of motion. In this work we focus instead on how the classical Hamiltonian formalism applied to the same phonon system allows to recover the feature of relativistic quantum mechanics for a single spinless particle. Using the classical nature of the phonon model, we naturally define continuous Poisson brackets between classical observables, which allows to recover the dynamics of such observables, i.e. the Ehrenfest relations associated to real-valued Klein-Gordon fields. The Poisson brackets also permits to obtain the generic form of constants of motions, thus generalizing the concept of inner products and momentum on Klein-Gordon fields. We then connect the formalism of real-valued classical functionals with that of hermitian operators and complex-valued wave functions. This is done through the introduction of a non-local complex-valued change of variables which allows to rewrite the real-valued Klein-Gordon equation in a form akin to the Schrodinger equation, and the classical observables as quantum expectation values. Then, we show how this change of variables allows to rewrite the classical Poisson brackets as commutators of hermitian operators. This points out the strict equivalence between the Heisenberg formalism and the formalism of classical Poisson bracket. Eventually, we illustrate how the Poisson brackets allows to recover the transformations of Poincare group in 1+1 dimension together with its algebra. The latter makes the link between the Lorentz invariant inner product of Mostafazadeh and the Casimir invariant associated to the mass of particle.

We investigate the computational complexity of estimating the operator norm distance ${\rm T}_{\infty}(\rho_0,\rho_1)$, defined via the operator norm $\|A\|_{\infty} = \sigma_{\max}(A)$, given ${\rm poly}(n)$-size state-preparation circuits of $n$-qubit quantum states $\rho_0$ and $\rho_1$. We provide efficient quantum estimators for the operator norm distance whose complexity is independent of the rank (and thus the dimension) of the states: 1. When one state is pure, we establish an optimal quantum estimator using $\Theta(1/\epsilon)$ queries to the state-preparation circuits. Consequently, for constant additive error, say $\epsilon=1/5$, our estimator runs in ${\rm poly}(n)$ time. Since the operator norm distance ${\rm T}_{\infty}(|\psi\rangle\!\langle\psi|,\rho)$ is exactly half of the trace distance ${\rm T}(|\psi\rangle\!\langle\psi|,\rho)$, our result also gives rank-independent query complexity for estimating both quantities, whereas the approaches due to van Apeldoorn, Cornelissen, Gilyén, and Nannicini (SODA 2023) and Wang and Zhang (TIT 2024) have query complexity scaling at least linearly with ${\rm rank}(\rho)$, which can be $\exp(n)$ in general. 2. For general quantum states, we also provide a quantum estimator using $\widetilde{O}(1/\epsilon^{3/2})$ queries to the state-preparation circuits, which shows that the corresponding promise problem is ${\sf BQP}$-complete and improves the ${\sf QMA}$ upper bound sketched by Liu and Wang (ESA 2025). Together with an $\Omega(1/\epsilon)$ quantum query complexity lower bound, this leaves only square-root room for improvement. The key intuition behind our estimators is that, when one state is pure, the pure state $|\psi\rangle$ has overlap at least $1/2$ with the top unit eigenvector of $|\psi\rangle\!\langle\psi|-\rho$, reflecting a structural feature specific to the operator norm distance.

In this work, we address a real-life course timetabling problem that incorporates several practical academic requirements, including the chronological precedence of theoretical sessions over practical ones, the enforcement of rest periods for lecturers, and the simultaneous assignment of lecturers and teaching assistants. The formulation involves a trade-off between minimizing course conflicts and maximizing schedule compactness by reducing the number of dispersed working days for lecturers under strict daily working hour constraints. To handle this trade-off, we develop a bi-objective mixed-integer linear programming (MILP) model solved using a weighted-sum objective approach. Computational experiments using a real-world dataset demonstrate that the proposed model reduces timetable conflicts by 13.33%, improves lecturer schedule compactness by 26%, and completely eliminates violations of theoretical-practical session precedence compared to the real-life timetable. Overall, the findings highlight the efficiency and robustness of the proposed MILP model for complex university timetabling problems.

The Schrodinger equation for a single spinless particle is formally obtained via a classical phonon model, namely the Frenkel-Kontorova model. Starting from a one-dimensional lattice of coupled harmonic oscillators, we show that the continuous limit of the corresponding Newtonian equation of motion yields the Klein-Gordon equation for a real-valued field. By introducing a complex-valued change of variables mixing the real-valued displacement and velocity fields, and by separating fast and slow time scales, the Klein-Gordon equation is written as the Schrodinger equation within the non-relativistic limit. This complex change of variable also allows to rewrite classical global observables of the phonon field, such as the total energy or momentum, as the corresponding quantum observables. Additionally, we show that when a friction force is incorporated into the classical model, the corresponding Klein-Gordon equation can be rewritten as a Schrodinger equation with a non-Hermitian Hamiltonian. While the global approach is limited here to the non-relativistic regime and does not address the measurement problem, quantization or relativistic effects, it nonetheless illustrates how quantum algebra and complex-valued wave functions can be exactly reproduced using classical dynamics. The relativistic regime for a spinless particle and the link between commutators and Poisson brackets is addressed in the second part of this series.

An edge of a graph dominates itself and each edge adjacent to it. A perfect edge dominating set is a subset of edges such that each edge outside the subset is dominated by exactly one edge of the subset. In this article, we characterize the extremal graphs on $n$ vertices in the classes of trees, forests, and chordal graphs with respect to the number of perfect edge dominating sets. Moreover, we derive linear-time algorithms for counting perfect edge dominating sets and for counting dominating induced matchings in generalized series-parallel graphs and chordal graphs.

In a paper from 2015, we determined all spherical affine homogeneous varieties for simple algebraic groups in arbitrary characteristic. The present paper extends this classification to semisimple groups. This generalizes work done independently by Brion and Mikityuk in characteristic zero. Our primary approach relies on a lifting lemma to characteristic zero, enabling us to directly apply Brion's and Mikityuk's results.

Function-Correcting Codes (FCCs) are a class of codes designed to protect the evaluation of a specific function of a message against channel errors at a higher level than the level of protection for the message, while requiring significantly less redundancy than conventional error-correcting codes. In this paper, we study function-correcting codes under the sum-rank metric, which is a natural generalization of both the Hamming metric and the rank-metric and also we derive general upper and lower bounds on the optimal redundancy of FCCs in the sum-rank metric. In particular, we establish a Plotkin-like bound for irregular-distance codes in sum-rank metric. Furthermore, we present explicit construction of function-correcting sum-rank metric codes (FCSRCs) for locally binary functions with optimal redundancy.

For graphs $F$ and $H$, let $\ex(n,F,H)$ denote the maximum number of copies of $F$ in an $n$-vertex $H$-free graph. Gerbner, Győri, Methuku and Vizer proved that $\ex(n,C_6,C_8)=\Theta(n^3)$ and predicted that the unrestricted problem should have the same first-order asymptotics as the bipartite one. We determine the exact value for all sufficiently large $n$, showing that \[ \ex(n,C_6,C_8)=6\binom{n-3}{3}+12(n-5). \] Moreover, the unique extremal graph is $K_3\vee (K_2\cup I_{n-5})$. The main new ingredient is a codegree decomposition for $C_8$-free graphs: a packing lemma for triangles in the linear-codegree graph recovers an almost spanning common neighborhood, and a defect-absorption argument upgrades this stability to the exact extremal graph.

For positive integers \(m,n\), the partial permutohedron $\mathcal{P}(m,n)$ is a lattice polytope constructed as the convex hull of vectors in $\{0, 1, \dots, n\}^m$ that have distinct non-zero entries. We prove that for $n \ge m-1$, the Ehrhart polynomial of $\mathcal{P}(m,n)$ is magic positive except for the single case \((m,n)=(2,1)\). In particular, the Ehrhart polynomial of the parking function polytope (integrally equivalent to $\mathcal{P}(m,m-1)$) is magic positive for $m \ge 3$. For $n<m-1$, we discuss the magic positivity of the Ehrhart polynomial of $\mathcal{P}(m,n)$ for $n=1,2,3$. There exist infinitely many counterexamples with $n<m-1$ showing that the Ehrhart polynomial of $\mathcal{P}(m,n)$ is not magic positive. This partially resolves an open problem proposed by Ferroni and Higashitani.

We study virtual element discretizations of a well-known variational formulation in $H^2$ of Hamilton-Jacobi-Bellman and Isaacs equations with Cordes coefficients. We show that the use of $H^2$ conforming virtual element spaces leads to a relatively simple analysis, bypassing the need for discrete Miranda-Talenti estimates that typically arises when using nonconforming schemes. We investigate how the polynomial degree of projection operators, especially for lower order terms, affects both the error analysis and robustness of the proposed schemes. We also show the possibility of imposing weakly the Dirichlet boundary condition, which simplifies the implementation of the method in some virtual element codes. Our results are complemented by numerical experiments in which we compare the convergence of different variants of the scheme for some test problems.

Stability is often assumed in learning and identification, yet it is rarely characterized directly from input–output data. We show that an input–output family admits a stable finite-dimensional state-linear realization iff it has finite Hankel-rank and its response decays uniformly with time; for state-linear realizable maps this decay is necessarily exponential. We extend these results to state-affine, LPV, and linear switched systems via suitable input-forgetting notions, and relate forgetting to decay of impulse responses (sub-Markov parameters). In all cases, the decay/forgetting rate determines the decay rate of every minimal realization.

This paper is devoted to the transverse stability problem for small-amplitude periodic traveling waves of the $b$-Novikov equation, which arises as a two-dimensional extension of the $b$-family hierarchy. The perturbations under consideration fall into two groups. One group matches the fundamental period of the underlying wave along the longitudinal direction, while the other consists of profiles that are bounded or localized in the transverse direction. We perform spectral analysis of the associated linearized operator and derive precise stability and instability thresholds. The resulting conditions exhibit intricate dependence on two key parameters including the cubic coupling strength $b$ and the wavenumber $k$.

We give an explicit non-archimedean SYZ construction for the Landau-Ginzburg mirror of $Gr(2,4)$. This work is complementary to the approach of Hong-Kim-Lau \cite{hong2023immersed} to SYZ mirror symmetry for Grassmannians, while we focus on a more concrete fibration-level realization of the SYZ picture. Starting from a Lagrangian fibration on the A-side, we explicitly construct a non-archimedean analytic mirror fibration inside the Berkovich analytification of the Langlands dual Grassmannian on the B-side. We show that the two fibrations have identical smooth and singular loci and induce the same integral affine structure on the smooth locus. Moreover, the natural disk-counting Landau-Ginzburg superpotential agrees with the Marsh-Rietsch superpotential. While the construction is guided by the family Floer viewpoint, the proof proceeds mainly through explicit geometric constructions and does not rely on Floer-theoretic arguments. Thus, the Langlands-dual mirror and its superpotential are realized explicitly within a single framework, providing concrete geometric evidence for the SYZ principle.

In this paper, we study regional input-to-state stability (regional ISS) for time-varying control systems evolving on connected and complete Riemannian manifolds with a prescribed open state domain. Using the Riemannian distance, we formulate the notion of a region of ISS and introduce an ISS-Lyapunov function. We prove that the existence of such a function implies regional ISS in an explicitly estimated geodesic ball and yields a corresponding admissible input bound. Under additional geometric and dynamical assumptions, this region can be enlarged to the largest geodesic ball centered at the equilibrium point and contained in the prescribed state domain. Examples under the Euclidean and hyperbolic metrics show that different Riemannian metrics may lead to different estimates for the same system.

In an article published a few years before the modularity of elliptic curves over $\Q$ was proved, Mazur \cite{maz} looked at modularity as a purely complex analytic phenomenon, defining a notion of an elliptic curve over $\Q$ having a hyperbolic uniformisation of arithmetic type. Such an elliptic curve (of conductor $N$, say) is necessarily geometrically modular, i.e. a quotient of the jacobian of the modular curve $X_0(N)$, by a morphism defined over $\Q$. We extend these ideas to elliptic curves over totally real fields of odd degree, using Shimura curves for quaternion algebras split at all finite places and one real place. In particular, we prove that the existence of a hyperbolic uniformisation of arithmetic type would imply geometric modularity.

In this study, we introduce a general energy-based modelling approach for viscous poroelastic materials that feature diffusive transport in both Lagrangian and Eulerian frames. Our research produces refined weak formulations by using the reference map concept within the Eulerian configuration. We propose and implement a novel structure-preserving discretisation strategy, utilising mixed finite element methods. This paper highlights the spatial and temporal numerical convergence of our methods through a comparative analysis of Lagrangian and Eulerian schemes, thereby proving the robustness and usability of our approach. Furthermore, in the context of Eulerian multiphase flow, specifically of the quasi-static Euler-Euler type, our study demonstrates the existence of solitary fluid waves within poroviscoelastic media. This energy-based approach forms a basis for a deeper understanding of thermodynamical modelling and corresponding discretisation schemes for coupled poroelasticity, flow, and diffusion.

The theory of affine cellular algebras $A$ is extended to incorporate their asymptotic algebras $\hat{A}$, clarifying unexpected differences between classical and affine situations and comparing with Lusztig's asymptotic Hecke algebras. The main new results are about a double centraliser property between $A$ and $\hat{A}$, about constructing $\hat{A}$ from cell modules of $A$, about existence of an embedding $A \rightarrow \hat{A}$ and about a faithful functor from torsionless $\hat{A}$-modules to $A$-modules as well as about the embedding being weakly spectrum preserving (in the sense of Baum and Nistor) and about non-zero endomorphisms of cell modules being injective, while there are no non-zero homomorphisms between non-isomorphic cell modules.

For compact convex sets $L,K \subset \mathbb{R}^n$, denote by $\lambda_K(L)$ the smallest size of a homothet of $K$ that contains $L$. We define a measure of symmetry based on the $n$-simplex $\Delta = \Delta^n \subset \mathbb{R}^n$ as the ratio \[ \rho_\Delta(L):=\frac{\lambda_{-\Delta}(L)}{\lambda_{\Delta}(L)}. \] We study this measure and deduce the following results: (1) The classical Minkowski measure of symmetry $m^*(L)$ can be defined as an affine-invariant version of $\rho_\Delta(L)$. (2) We improve the stability analysis for the Minkowski measure of symmetry; if $m^*(L)\ge n-\varepsilon$ then $L$ is $\tfrac{1}{1-\varepsilon}$-close to $\Delta$ in the Banach–Mazur distance. (3) We obtain a novel characterization of simplices as the only convex bodies $K$ for which the function $L \mapsto \lambda_K(L)$ is additive (a property we term “outer additivity”). (4) Motivated by the expressivity of ReLU neural networks, we study the depth complexity of polytopes in $\mathbb{R}^n$ under the two operations: Minkowski sum and convex hull of a union. We prove the sharp bound $\rho_\Delta(P) \leq 2^d -1$ for every polytope $P$ of depth complexity $d$. In other words, simplices cannot be approximated by low-depth polytopes.

We develop a reaction–diffusion–chemotaxis model for spatial tumour–immune–chemokine dynamics that couples logistic tumour growth, immune-mediated killing, chemokine-dependent immune recruitment, chemotactic migration, and signal production. For the nondimensional system, we establish local classical solvability, nonnegativity, a uniform tumour-density bound, and global mass estimates for the immune and chemokine components. The tumour-free equilibrium is stable precisely when the baseline immune-control index satisfies \(\sigma_0/\delta>1\), whereas positive homogeneous coexistence is characterized by a scalar nonlinear equation. Linearization in the Neumann Laplacian eigenbasis yields a mode-dependent cubic dispersion relation, showing that chemotaxis does not alter the tumour-invasion threshold but can destabilize homogeneous coexistence through a finite-wavelength oscillatory instability above a critical sensitivity \(\xi_c\). A conservative finite-volume discretization with upwind chemotactic fluxes and implicit backward differentiation formula time integration is used to test these predictions. Numerical experiments recover the analytical equilibria and growth rates, identify the dominant unstable mode, reproduce the transition to spatial heterogeneity, and quantify the effects of immune recruitment, decay, and diffusion on the stability boundary. Grid-refinement, mass-balance, residual, and nonnegativity diagnostics support the computational reliability of the results.

Time delays are inherent in many physical and engineered systems and can significantly affect their stability and performance. In this work, we investigate the dissipativity of a class of nonlinear time-delay systems with multiple discrete delays and derive sufficient conditions for both delay-dependent and delay-independent dissipativity using Bessel-Legendre inequalities. For linear systems, the resulting dissipativity conditions are expressed in terms of linear matrix inequalities (LMIs) which can be solved numerically to obtain Lyapunov-Krasovskii-type storage functions.

In this paper, we prove that Herz-type Sobolev spaces form a Banach algebra and establish a Rellich–Kondrachov compactness theorem for these spaces. These results extend the corresponding classical theory and further demonstrate that Herz-type Sobolev spaces provide a natural generalization of the classical Sobolev spaces.

Let \(\mathrm G\) be a complex simple algebraic group and let \(\mathrm G_0\subset \mathrm G\) be a closed connected subgroup containing a regular unipotent element of \(\mathrm G\), with semisimple rank at least \(2\). Using Dynkin's classification, we prove that the restriction of an irreducible finite-dimensional representation of \(\mathrm G\) to \(\mathrm G_0\) determines the representation up to an outer automorphism of \(\mathrm G\) preserving \(\mathrm G_0\). We extend this method to the diagonal embedding $\mathrm G_0\hookrightarrow \mathrm G\times \mathrm G$ for the specific pairs $(\mathrm{SO}_{2k}(\mathbb C) \times\mathrm{SO}_{2k}(\mathbb C),\,\mathrm{SO}_{2k-1}(\mathbb C))$, $(E_6\times E_6,\,F_4)$ and $(Spin_8(\mathbb C) \times Spin_8(\mathbb C), G_2)$ and show that uniqueness continues to hold. Finally, we give examples showing that, in the diagonal setting, restriction to the principal \(\mathrm{SL}_2(\mathbb C)\) alone is not sufficient to establish uniqueness.

Determining the hull of linear codes has long been an important topic in coding theory. Recently, non-generalized Reed-Solomon (in short, non-GRS) codes have attracted extensive research interest. The (L,P)-twisted generalized Reed-Solomon (in short, (L,P)-TGRS) code, which is an extension of the generalized Reed-Solomon (GRS) code, constitutes a well-studied calss of non-GRS codes.There are numerous works focusing on the Euclidean hull of (L,P)-TGRS codes, while only a few results on the Hermitian hull of (L,P)-TGRS codes. In this paper, we focus on a class of (L,P)-TGRS codes C_k(a). By taking a special class of the vector a with length i(q-1), and analyze the parity of i and the relation between i and q+1, we divide three cases to fully determine the Hermitian hull dimension of C_k(a). As an application, we construct two classes of entanglement-assisted quantum error-correcting codes.