Papers

We develop a quaternionic approach to conformal superminimal surfaces in Euclidean four-space. The starting point is the classical Weierstrass representation: every conformal minimal immersion $X\colon M\to\mathbb{R}^4$ is recovered as $X = c + Re\int\Phi dz$, where $\Phi$ is a holomorphic null curve in $\mathbb{C}^4$, identified with the algebra of complex quaternions $\mathbb{H}\otimes\mathbb{C}$. The multiplicativity of the quaternionic symmetrized norm makes it natural to factor null curves as $\Phi=ALB$, where $A$ and $B$ are holomorphic maps of unit symplectic norm and $L$ is a holomorphic null element. We show that on simply connected domains the null factor can always be taken constant. The main result is an explicit quaternionic reformulation of the superminimality condition – the requirement that the curvature ellipse be a circle at every point. In the fixed-null gauge $L=1+\sqrt{-1} e_1$, superminimality is equivalent to the vanishing of a product of two holomorphic functions built from the left and right Maurer–Cartan forms of $A$ and $B$. On a connected domain, this forces one of the two components of the generalized Gauss map $[\Phi]\colon M\to Q^2\simeq\mathbb{CP}^1\times\mathbb{CP}^1$ to be constant, recovering in spinorial terms the classical ruling condition on the projective null quadric. We further provide a first-order ODE parametrization of the superminimal $ALB$-data, analyse the residual gauge freedom, prove a fixed-gauge rigidity statement for polynomial spinorial factors, and illustrate the theory with explicit examples.

We study the statistical and computational limits of learning bounded linear operators between Sobolev spaces from noisy input-output data. In wavelet coordinates, the problem is recast as an infinite-dimensional matrix regression problem with a heterogeneous two-sided multiscale structure. We establish minimax rates under Sobolev operator-norm loss and construct a finite-resolution blockwise least-squares estimator attaining these rates. The analysis reveals a nonuniform local estimation difficulty across scales, which can be exploited algorithmically: by assigning scale-adaptive sample sizes, the estimator achieves the optimal computational cost among dense least-squares implementations.

We study a backward inverse problem for a coupled Cahn-Hilliard-reaction-diffusion system. Our main result is a Carleman estimate for a fourth-order/second-order parabolic system with cross-diffusion terms, which allows us to derive conditional stability estimates for the reconstruction of past states from a single final-time observation: Hölder stability at positive times and logarithmic stability for the initial datum. We then apply the Carleman estimate to a phase-field tumour growth model coupling the tumour volume fraction with a nutrient concentration. In this setting, we obtain backward uniqueness and quantitative stability for the recovery of early tumour states, improving earlier results based on logarithmic convexity, which only yielded uniqueness under an additional smallness assumption on the chemotaxis coefficient. We also discuss how these results support Lipschitz stability on finite-dimensional admissible sets, which is relevant for ensuring convergence of iterative discretisation algorithms.

We study finite-order meromorphic functions representable as absolutely convergent sums of squares of Cauchy kernels and having only finitely many zeros. By earlier work of Baranov and the author, such functions admit a representation $f=P/g^2$, where $P$ is a polynomial and $g$ is entire, satisfying the differential equation $ Pg''-P'g'+Qg=0, $ where $Q$ is a polynomial. We show that the zeros of $g$ asymptotically accumulate along the Stokes rays. If $\mathrm{deg}\ Q>\mathrm{deg}\ P$, they approach these rays in the Euclidean metric, whereas in the borderline case $\mathrm{deg}\ Q=\mathrm{deg}\ P$ one obtains in general only localization in logarithmic neighborhoods of the Stokes rays, and this is sharp. We then characterize the existence of a decomposition $ P/g^2=\sum c_n (z-t_n)^{-2} $ in terms of the sectorial behavior of $g$ and, equivalently, in terms of the Laine condition for the corresponding Schwarzian equation. Finally, for fixed $P$ and fixed order, we identify the resulting families, modulo the natural equivalence relation, with finite-dimensional affine algebraic varieties.

Let $T$ be a power-bounded operator on a Banach space $X$. We treat the sequence of polynomials $(p_{n;a})_{n\ge 0}$ such that the entire group generated by the fractional power operator $-(I-T)^a$ is given by $$ e^{-t(I-T)^a}=e^{-t}\sum_{n=0}^\infty p_{n;a}(t)T^n, \qquad t\in \CC, \quad \Re a>0. $$ We provide a self-contained introduction to the polynomial family $(p_{n;a})_{n\ge 0}$, for $a\in \CC$, whose coefficients are determined by means of a suitable recurrence relation. The sequence $(n!p_{n;a})_{n\ge 0}$ forms a family of Sheffer polynomials. For $\Re a>0$ and $t\in \CC$, the sequence $(p_{n;a}(t))_{n\ge 0}$ belongs to the Lebesgue sequence space $\ell^1$ of absolutely summable sequences. Moreover, these polynomials are closely related to the Lévy density functions $(f_{t,\alpha})_{t>0}$ defined for $0<\alpha<1$. Finally, we discuss several particular cases corresponding to specific values of $a\in \CC$, as well as applications to fractional powers in the Banach algebra $\ell^1$, multiplication operators, and Cesàro means.

Dyer (1991) proved that, for each fixed length $k$, only finitely many isomorphism types of Bruhat intervals occur in finite Coxeter groups. In this short paper, we prove that this result holds also for affine Coxeter groups, and moreover that this characterizes the affine groups among all infinite Coxeter groups. We also show that the coefficients of $q$ in the Kazhdan-Lusztig polynomials of a Coxeter group are bounded if and only if the group is finite or affine.

Gaussian networks are degree-four symmetric interconnection networks defined over residue classes of Gaussian integers. Earlier work showed that when the generator $\alpha=a+bi$ satisfies $\gcd(a,b)=1$, the real and imaginary dimensions directly form two edge-disjoint Hamiltonian cycles. A later construction extended the result to the non-coprime case $\gcd(a,b)=d>1$, but its proof used long node-sequence tables and separate odd/even cases for $d$. This paper gives a unified closed-form construction that covers both $d=1$ and $d>1$, and also covers both odd and even $d$, without separate case tables. In the rectangular representation with $d$ rows and $r=(a^2+b^2)/d$ columns, the construction uses a constant-time local switch rule for each $q=1,2,\ldots,d-1$ at column $a_q=q-1$. Each switch removes two horizontal edges and inserts two vertical edges. The switched horizontal structure forms the first Hamiltonian cycle, while its edge-complement in the Gaussian network forms the second Hamiltonian cycle. Thus, the full edge set is partitioned into two edge-disjoint Hamiltonian cycles. The construction requires $O(d)$ switch-generation time and $O(N)$ time to list the two cycles, where $N=a^2+b^2$. Exhaustive validation for all $1\leq a\leq b\leq 100$, excluding only the degenerate $N=2$ network, and large-scale validation up to $N=3{,}250{,}000$ confirm the construction.

The winner-takes-all process takes place on an arbitrary graph. There is an agent on each vertex of the graph, and active agents at neighboring vertices play games. In each game, a randomly chosen agent wins, while the loser is eliminated from subsequent games. The games are played at random times, finishing instantaneously, and ceasing when each active agent has only losers among its neighbors. On the one-dimensional lattice, the fraction of winners in the final state is $e^{-1}$; we also determine the fractions $w_j$ of winners who won $j=0, 1, 2$ games. For finite segments, we determine statistics of the total number of winners (the average, the variance, and all higher cumulants), the probabilities of attaining the minimum or maximum possible number of winners, and establish the behavior near the boundaries. For infinite regular trees with vertices of degree $d$, i.e., Bethe lattices with coordination number $d$, we show that the fraction of winners is $(2/d)^{d/(d-2)}$.

In this paper, we give a complete classification of indecomposable Ekedahl–Oort strata of Shimura varieties associated to the unitary group $\mathsf{GU}(a, b)$ over an odd inert prime. We show that each indecomposable stratum is one of four types: unitary unicycle, unitary bicycle, Serre unicycle, or Serre bicycle; the latter two types are named for a tensor construction of abelian varieties developed by Serre. We provide an algorithm that translates the description of a stratum in terms of words in the alphabet $\{\texttt{f},\texttt{v}\}$ to the corresponding Weyl group coset representative. Finally, using a $p$-adic lift, we construct a `tautological' point in each Ekedahl–Oort stratum, and compute its Newton polygon. As an application, we show that the indecomposable Ekedahl–Oort strata corresponding to unitary unicycles and Serre unicycles always intersect the supersingular locus.

Let $G$ be a semisimple complex algebraic group. We study the $\mathbb C^\times$-action on the moduli space of strongly parabolic $G$-Higgs bundles over a punctured curve with weights in the interior of the standard Weyl alcove. We describe its fixed points and, for the components with generically regular Higgs field, we classify the fixed points that have closed Bialynicki-Birula upward flows, called very stable. The classification is stated in terms of a naturally associated divisor valued in the extended affine Weyl group of $G$ which encodes the information of the fixed point. Under this point of view, very stable fixed points naturally coincide with divisors where the coefficients are minimal under the Bruhat order. Furthermore, when the divisor is supported at the parabolic punctures, the corresponding very stable upward flow is a section of the Hitchin map. For these upward flows, we propose a conjectural mirror line bundle over the dual moduli space, and verify its validity for $G=\mathrm{PGL}_n(\mathbb{C})$.

We numerically study non-linear perturbations of the Einstein-de Sitter spacetime as a solution to the Gowdy-symmetric Einstein-Euler system for a polytropic equation of state. The results suggest that the Einstein-de Sitter spacetime is stable for sufficiently small but otherwise generic perturbations. This is in stark contrast to the well known instability of this spacetime when the matter model is dust. Moreover, this indicates a previously unknown stable regime of the Einstein-Euler equations with direct implications for cosmology.

We define study privileged local systems on a smooth quasi-projective complex variety $X$ with fixed quasi-unipotent monodromies around a boundary divisor. The notion is a generalization of Katz' physical rigidity on an open of $\mathbb P^1$. It is defined in any dimension and includes non-rigid local systems. In the latter case, Katz used the middle convolution procedure to classify all physically local systems and derived several consequences such as the $p$-curvature conjecture for local systems of this type. We present a new proof, which avoids the use of middle convolution. This yields examples in higher dimension of local systems which verify the $p$-curvature conjecture.

We develop a categorical framework for Bass-Serre theory, which recovers parts of classical and profinite Bass-Serre theories for tree quotients. More precisely, we show that in what we call a Bass-Serre category, a group object acting on a graph satisfying some assumptions can be recovered as the fundamental group of the associated graph of groups acting on the standard graph. Examples of Bass-Serre categories include the category of sets, the category of profinite spaces, and any Grothendieck topos.

Bu, Gu, and Jaffe introduced the quantum uniformity measures to analytically characterize the Clifford hierarchy. We prove that the quantum uniformity measures are the pullbacks of the matrix-valued uniformity functionals of Gowers and Hatami under the map $U\mapsto(a\mapsto W_aUW_a^*)$, which we call the Weyl-orbit embedding. We study the structure of the Weyl-orbit embedding, and among other results, establish a canonical decomposition of the ambient matrix-valued uniformity space into uniformity preserving twisted copies of the quantum uniformity space. The Weyl-orbit embedding identifies the Clifford hierarchy with Leibman polynomial maps. This identification yields a representation theoretic characterization of the Clifford hierarchy in terms of unitary representations of the universal polynomial groups introduced by the first author and Thom.

We provide a simplified proof of a recent theorem of Phagan (arXiv:2509.14083) relating the number of orbits of a given length of an action $G \curvearrowright S$ of a cyclic group with the number of orbits of the induced action $H \curvearrowright S$ of a subgroup $H \subseteq G$. This combinatorial fact has applications to notions of arithmetic similarity of number fields, as investigated by Phagan (op. cit.).

We produce the dimensional reduction to 10D IIA supergravity (SuGra), via cyclification, of the remarkable result that full 11D SuGra is put on shell just by imposing the duality-symmetric Bianchi identities on C-field super-flux densities over supertorsion-free superspace. Generally, we highlight that when duality-symmetric superspace Bianchi identities are characterized by Whitehead bracket $L_\infty$-algebras $\mathfrak{l}\mathcal{A}$ of a classifying space $\mathcal{A}$, their dimensional reduction is characterized by the cyclic loop space $\mathrm{Cyc}(\mathcal{A})$. We promote this to a general mechanism of dimensional reduction on super-spacetime, compatible with the global (infrared) completion of supergravity theories by flux quantization in non-abelian cohomology with coefficients in $\mathcal{A}$ and $\mathrm{Cyc}(\mathcal{A})$, respectively. In the case of 11D SuGra, the characteristic $L_\infty$-algebra is $\mathfrak{l}S^4$ and hence we obtain that full on-shell 10D IIA SuGra is equivalent to $\mathfrak{l}\mathrm{Cyc}(S^4)$-Bianchi imposed identities on NS/RR super-flux densities over supertorsion-free 10D super-spacetime. This implies that any space which is $\mathbb{R}$-rationally equivalent to $\mathrm{Cyc}(S^4)$ classifies an admissible flux quantization law, which provides a global completion of 10D IIA SuGra that admits oxidation to 11D.

We present an experimental study of data interleaving for terrestrial free-space optical communication over a 4.6~km urban testbed. Results demonstrate a two-order-of-magnitude reduction in outage probability. A dependency between measured turbulence strength, interleaver length, and achievable data rate is revealed, enabling robust system design.

This work presents a novel digital image correlation (DIC) framework for full-field measurements of displacement, strain, and damage, based on a phase field (PF) approach. The idea is to take advantage of the ability of the PF method to track complex crack morphologies and to provide a natural way in DIC to perform damage and crack measurements from experimental speckle images, in addition to displacement and strain fields. Moreover, incorporating the damage variable into DIC can improve the displacement accuracy near the crack tip, and can avoid the need of user-defined masks when dealing with cracked samples, which is advantageous when cracks become complex and the manual application of masks becomes challenging. The theoretical formulation of the proposed framework, namely PF-DIC, was presented in detail in the paper, along with a finite element implementation. Numerical examples have demonstrated the capability of the proposed PF-DIC in terms of capturing different types of cracks while providing similar measurement accuracy to that of masked DIC. Additionally, it is shown that the PF-DIC can be easily adapted to selectively identify critical cracks under specific loading conditions or mechanisms for damage assessment and diagnostic purposes. The proposed DIC framework can be used to characterize material defects, support structural health monitoring, and enable a potential unification of PF simulations and experimental fracture measurements

The exact time-space correlation function of the $1D$ Glauber-Ising model, quenched to temperature $T=0$ and on a semi-open lattice of finite size $N$, is obtained. This also allows to deduce the exact empty-interval probability of the dual $1D$ coagulation-diffusion process on a periodic finite ring and to reproduce the long-time decay of the particle concentration. These results are consistent with the generic expectations of dynamical finite-size scaling theory.

We revisit the classical proof by Kuranishi on the existence of a locally complete family of complex deformations on a compact complex manifold using Hamilton's Nash-Moser implicit function theorem. This is relevant to existing and future extensions of this theorem to generalized complex structures on Courant algebroids.

We introduce selection structures, a topological framework that extends the role played by color classes and matroids in discrete geometry and KKM theorems. Selection structures allow us to extend classic results to genuinely non-matroidal examples, including chessboard complexes and matching complexes. We show that several matroidal versions of classic results can be generalized to selection structures. These include McGinnis' version of Komiya's KKMS theorem, Holmsen's version of Carathéodory's theorem, Kalai and Meshulam's version of Helly's theorem, and Sadovek's version of the Goodman–Pollack transversal theorem.

Block Markov superposition transmission (BMST) codes provide a flexible framework for constructing codes with near-capacity performance and low-complexity sliding-window decoding. However, existing BMST variants show contrasting performance limitations: recursive BMST (rBMST) codes suffer from error propagation but avoid high error floors, whereas non-recursive BMST codes exhibit the opposite behavior. Motivated by these complementary characteristics, we combine recursive and non-recursive components through parallel and serial concatenation, yielding mixed BMST (mBMST) codes. The proposed framework subsumes existing BMST variants and enables new BMST structures. Simulations show that these structures improve FER and BER performance with lower memory requirements than rBMST.

Inspired by work of Altmann and Christophersen, we study the graded pieces of the cotangent cohomology $T^i_{S_{\mathcal{K}}}$, $i\geq 3$ of the Stanley-Reisner ring $S_{\mathcal{K}}$ associated to a simplicial complex $\mathcal{K}$. We prove a localization formula allowing one to reduce to the case of negative weights. Our results give a complete description of $T^3$ and $T^4$ in terms of the topology of $\mathcal{K}$ whenever $\mathcal{K}$ is a flag complex. As an application, we give a sufficient criterion for the vanishing of $T^3$ for simplicial spheres, classify two-spheres that have vanishing $T^3$, and show that the boundary complex of the dual associahedron has vanishing $T^3$. Our results make use of the arborescent resolutions considered by Hancharuk, Laurent-Gengoux, and Strobl. We give an alternative and self-contained treatment of these resolutions that may be of independent interest.

We analyze Column-Row Impartial Merge (CRIM), an impartial combinatorial game played on integer partitions. A move in CRIM consists of removing an arbitrary row or column from the corresponding Young diagram, with the remaining parts reattaching to form a single partition. We define rectairs – a common generalization of rectangles and staircases – and characterize their $\mathcal{P}/\mathcal{N}$-status. We define the meld operation on partitions and show that the meld of losing rectairs is losing. We introduce Odds-Are-Even (OAE) and Evens-Are-Odd (EAO) partitions, proving that all OAE partitions are $\mathcal{P}$-positions and characterizing the losing positions within EAO partitions. We determine the $\mathcal{P}/\mathcal{N}$-status for staircases and for $2$- and $3$-part partitions. We evaluate CRIM and its restrictions to certain partition families within the Conway-Gurvich-Ho classification scheme, establishing that CRIM is neither returnable nor domestic. We conjecture that every losing partition has even rank.

This paper investigates mathematical programs with second-order cone complementarity constraints (SOCMPCCs), which extend classical mathematical programs with complementarity constraints (MPCCs) by incorporating second-order cone structures. SOCMPCCs present significant theoretical and computational challenges, primarily due to the failure of standard constraint qualifications (such as Robinson's constraint qualification) at all feasible points. This difficulty hinders the direct application of classical nonlinear programming theories and algorithms. Motivated by the success of the augmented Lagrangian method (ALM) in solving MPCCs, we explore its extension to SOCMPCCs. The ALM, known for its matrix-free implementation and strong local convergence properties, is well suited for handling the intricate interplay between complementarity and second-order cone constraints. In this paper, we propose a tailored ALM algorithm framework for SOCMPCCs and establish its feasibility and convergence properties. We show that, under bounded ALM penalty parameters or bounded augmented Lagrangian functions, the generated sequence converges to feasible points of the SOCMPCC. Moreover, under the feasibility and an additional SOCMPCC-weak-Robinson's constraint qualification, we prove convergence to K-stationary points, which constitute a fundamental optimality condition for SOCMPCCs. Numerical experiments, including both illustrative examples and high-dimensional problems, are conducted to demonstrate the effectiveness and practical applicability of the proposed algorithm in addressing the challenges inherent in SOCMPCCs.

Let $\K$ be an algebraically closed field of characteristic zero. We study isotropy groups of simple derivations on Danielewski-type algebras \[ A_{c,q}=\K[x,y,z]/(c(x)z-q(x,y)). \] More precisely, motivated by the recent result of Mendes and Pan for simple derivations of $\K[x,y]$ (\cite[Theorem 1]{MP}), we ask whether every simple derivation $D$ of $A_{c,q}$ has trivial isotropy group. We prove that this property holds generically: for each fixed pair of degrees $\deg(c)\geq2$ and $\deg_y(q)\geq2$, there exists a nonempty Zariski open subset of the parameter space of reduced pairs $(c,q)$ such that every simple derivation of $A_{c,q}$ has trivial isotropy group. On the other hand, we construct a special Danielewski-type algebra admitting a simple derivation with nontrivial isotropy. Thus, the Mendes-Pan phenomenon holds generically for Danielewski-type algebras, but fails in full generality. Over $\K=\mathbb C$, we also discuss the associated foliations and formulate questions about their polynomial and birational symmetries.

In this paper we observe when saturation of ideals in a local ring R commutes with extension along ring maps and initial ideals. We give a characterization in terms of Cohen Macaulayness for when this happens along the map from R to R modulo its nilradical. We give several examples and non examples where this happens, and we demonstrate an application of this condition to epsilon multiplicity. Additionally, we show that saturation commutes with extension along a flat injection of local rings if and only if the closed fiber of the injection is Artinian.

In 2026, Li and Zhan characterized the nonhamiltonian graphs of order $n$ with the maximum number of paths of length $k$, where $n$ and $k$ are integers satisfying $1\leq k\leq n-1$. This work solves and generalizes a problem proposed by Erdős in 1980. In this paper, we further determine the nonhamiltonian graphs of order $n$ attaining the maximum number of cycles of length $k$ for given integers $n$ and $k$ with $3\leq k\leq n-1$. As a corollary, we determine the generalized Turán number $\mathrm{ex}(n,C_k,C_n)$ for every $3\leq k\leq n-1$.

In [Problem 3.2.23, CBMS Regional Conference Series in Mathematics 83, 1994], Kenig asked whether the vanishing Carleson condition is the necessary and sufficient for the logarithm of the Poisson kernel of a perturbation of the Laplacian on the unit ball in $\mathbb{R}^n$ belonging to the space VMO. The sufficiency was proved by Escauriaza [Israel J. Math. 1996] and extended by Milakis, Pipher, and Toro [Contemp. Math. 2014] to more general setting. In this article, using the technique of bi-Lipschitz mappings, we construct a counterexample to show that the vanishing Carleson condition is not necessary and hence give a negative answer to the aforementioned problem.

This paper establishes time decay estimates for the following two-dimensional beam (plate) equation with a decaying real-valued potential $V$: \begin{equation*} \partial_t^2 u + (\Delta^2 + V) u = 0, \qquad u(0,x)=f(x),\quad \partial_t u(0,x)=g(x). \end{equation*} When zero is a regular point or a first-kind resonance of $H=\Delta^2+V$, we first prove sharp $L^1\to L^\infty$ estimates for the solution operators: \begin{align*} \left\|\cos(t\sqrt{H})P_{\mathrm{ac}}(H)\right\|_{L^1\to L^\infty} + \left\|\frac{\sin(t\sqrt{H})}{t\sqrt{H}}P_{\mathrm{ac}}(H)\right\|_{L^1\to L^\infty} \lesssim \frac{1}{|t|}, \end{align*} and obtain an enhanced decay $(|t|\log|t|)^{-1}$ in logarithmically weighted spaces $L^1_\omega\to L^\infty_{-\omega}$ with $\omega(x)=\log(2+|x|)$. For second-kind resonances of $H$ (the bi-Laplacian $\Delta^2$ belongs to this class), a non-zero trace moment $\langle |x|^2V,\phi\rangle\neq0$ for some second-kind resonance function $\phi$ induces severe threshold singularities, worsening the $L^1\to L^\infty$ estimate to $|t|^{-1}(\log|t|)^2$. Finally, for third-kind resonances or a zero eigenvalue, we prove that the presence of $d$-wave resonance leads to the worst $L^1\to L^\infty$ decay rate $\sim(\log|t|)^{-1}$. Several improved estimates are also obtained without a $d$-wave resonance. In particular, in the pure eigenvalue case (i.e., neither $d$-wave nor $p$-wave resonance), both propagators recover the optimal unweighted $L^1\to L^\infty$ estimate $|t|^{-1}.$