Papers

We construct smooth closed embedded minimal surfaces with everywhere negative Gaussian curvature in the unit round four-sphere. More precisely, for every sufficiently large integer $m$, our construction gives a minimal surface $\Sigma_m$ of genus $m^2+1$ such that $\operatorname{Area}(\Sigma_m)\to 4\pi^2$ as $m\to+\infty$. This result yields a complete solution to Problem 101 in Yau's 1982 list of open problems.

We establish a sufficient condition for infinite log-concavity and apply it to five families of combinatorial sequences. We give a new proof of infinite log-concavity for the Boros–Moll coefficient sequences $(d_k(n))_{k=0}^n$ for fixed $n$. For the transposed sequences $(d_\ell(\ell+k))_{k\ge0}$, we prove Zhao's conjecture on infinite log-concavity for every fixed integer $\ell\ge3$. For the normalized sequences in Euler's difference table, we confirm the infinite log-concavity conjecture of Chen, Gu, Ma and Wang under essential iteration, in which both endpoints are discarded after each step. We also confirm a conjecture of Medina, Moll and Rowland on the infinite log-concavity of the coefficient sequences of polynomials arising from iterated primitives of $\log(1+x)$. Finally, we answer a question of Brändén and Chasse by showing that, for positive integers $d$, the sequence $(k^d)_{k\ge0}$ is infinitely log-concave if and only if $d\ne2$. For every real $d\ge3$, all its iterates are positive at every index $k\ge1$. The proofs combine holomorphic estimates with finite computer-assisted verification.

Recently, a uniform $\frac{7}{8}$ zero-free half-plane for finite-order Hecke $L$-functions over $\mathbb Q(\sqrt{-3})$ and its transfer to Dirichlet $L$-functions was given by OpenAI in [Ope26a]. In its proof, there are several parameters $b=\frac{1}{8}, \ell=\frac{1}{6}$, and $\kappa$. In this paper, via varying $b$, $\ell$, and $\kappa$, following the same strategy, we slightly improve the bound $\frac{7}{8}=0.875$ to \[ B_{\mathrm{new}}=\frac{18\xi_{\mathrm{new}}+7}{18\xi_{\mathrm{new}}+9} =0.874957019420098946\ldots, \] where $\xi_{\mathrm{new}}$ is the unique root in $(0.38858,0.38859)$ of $1188\xi^3-2160\xi^2+171\xi+190=0$. Explicitly, with $\arccos\in[0,\pi]$, \[ \xi_{\mathrm{new}}=\frac{20}{33}+\frac{\sqrt{1391}}{33} \cos\!\left(\frac13\arccos\frac{28470}{1391^{3/2}} -\frac{2\pi}{3}\right). \] The limiting parameters are \[ \begin{gathered} b_{\mathrm{new}}=\frac{8(3\xi_{\mathrm{new}}-1)(12\xi_{\mathrm{new}}^2-40\xi_{\mathrm{new}}+15)} {9(2\xi_{\mathrm{new}}+1)(24\xi_{\mathrm{new}}^2-20\xi_{\mathrm{new}}+5)}, \ell_{\mathrm{new}}=\frac{5-6\xi_{\mathrm{new}}}{9+18\xi_{\mathrm{new}}},\qquad \kappa_{\mathrm{new}}=\frac{18\xi_{\mathrm{new}}+5}{18\xi_{\mathrm{new}}+9} =2B_{\mathrm{new}}-1. \end{gathered} \] We also prove its optimality within the explicitly specified exponent comparison system. This result has been formalized by Lean [Liu26].

Symmetric and unitary matrices are among the most important classes of structured matrices admitting efficient and stable eigenvalue algorithms. Notable examples include structure-preserving variants of the QR algorithm and their square-root-free counterparts. For unitary matrices, these algorithms were historically derived from recurrence relations for orthogonal polynomials on the unit circle. A matrix-based derivation using core chasing was later given in [3]; this family of algorithms is backward stable, but the underlying symmetries needed to cast out the square roots were not identified, and the question of structured backward stability was left open. In this paper we present a matrix-based derivation of the symmetric unitary core-chasing algorithm and use it to obtain a square-root-free variant. We give a matrix-based proof of backward stability and show that strict structured backward stability is impossible in general. Open-source Fortran code and numerical experiments demonstrate that the combination of exploiting the symmetry and removing the square roots reduces computation time without degrading accuracy.

The paper considers a system of unidirectional first-order hyperbolic partial differential equations (PDEs) coupled with ordinary differential equations (ODEs), modeling two-phase advective transport processes subject to both boundary and distributed input delays. To address the system defined over two spatial domains, we design a set of three backstepping transformations involving both Volterra- and Fredholm-type integral operators, which introduce eight kernel functions: five associated with the PDE states and three with the ODE states. Since the distributed delayed variable spans the two spatial domains, its transformation must integrate over all state variables, leading to three-dimensional kernel functions with singular initial conditions involving Dirac delta functions that serve as sampling operators to establish mappings between variables of different dimensions. By employing the method of characteristics and successive approximations, we prove the well-posedness of the kernel equations. On this basis, we prove the invertibility of the Fredholm transformation and establish the exponential stability of the closed-loop system in the $L^{\infty}$ norm under the delay-compensated controller. Simulation results for a linear example and a nonlinear screw extrusion model are presented to demonstrate the effectiveness of the proposed method.

We study normalized solutions to the 1-Laplace equation \[ -\mbox{div}\left(\frac{Du}{|Du|}\right)+\lambda \frac{u}{|u|}=f(u)+|u|^{1^*-2}u,\quad\mbox{in}~ \mathbb{R}^N \] subject to the constraint $\int_{\mathbb{R}^N}|u| \, \mbox{d}x=m$, where $N\geq2$, $1^*=\frac{N}{N-1}$ is the critical Sobolev exponent for the embedding from BV into $L^p$, $\lambda, m\in\mathbb{R}^+$ and $f:\mathbb{R}\to\mathbb{R}$ is a continuous nonlinear perturbation. We first establish a Pohozaev-type identity for the equation and use it to study the associated critical energy, and then prove the existence of normalized solutions via nonsmooth critical point theory. A key feature of our approach is the proof of the existence of a Palais-Smale sequence for a Lipschitz continuous functional under a Lipschitz continuous constraint. This is significantly different in methodology from previous problems with $C^2$ constraint.

In 1991, Denef introduced an explicit formula for Igusa's local zeta functions in terms of embedded resolutions of singularities over $p$-adic fields with good reduction. In this paper, by slightly generalizing the $p$-adic McKay correspondence given by the second author in 2017, we extend Denef's result to the setting of $\mathbb{Q}$-embedded resolutions of singularities. Consequently, the strong monodromy conjecture with respect to Igusa's local zeta functions holds for weighted homogeneous polynomials with isolated singularities. Moreover, we use our formula to prove Igusa's conjecture for exponential sums in the case of non-degenerate weighted homogeneous polynomials with isolated singularities.

Fourier coefficients of Eisenstein series induced from Speh representations are the kernels of the generalized doubling method of Cai, Friedberg, Ginzburg and Kaplan. Ginzburg and Soudry integrated these coefficients against cusp forms to obtain Eisenstein series induced from shorter Speh representations and arbitrary cuspidal data. We determine the corresponding local modules over a non-archimedean local field $F$ of characteristic zero. Let $q_F$ be the cardinality of its residue field. Let $J_s$ be the twisted Jacquet module of the representation induced from $\Delta(\tau,m+i)|\det|^s$. It carries an action of $G\times H$, where $G$ and $H$ are split symplectic or special orthogonal groups. Let $\sigma$ be an admissible representation of $G$ of finite length. We prove that $(J_s\otimes\sigma)_G$ is induced from $\Delta(\tau,i)|\det|^s$ and a fixed conjugate of $\sigma$, outside a finite set of values of $q_F^{-s}$ depending on $\tau$ and $\sigma$. No genericity assumption on $\sigma$ is needed. For irreducible $\sigma$, this determines all irreducible quotients of $J_s$ of the form $\sigma^\vee\boxtimes\pi$. The proof computes every orbit contribution, including those that vanish globally by cuspidality. It also gives an explicit finite set containing the exceptional parameters. This set is empty for irreducible supercuspidal $\sigma$ unless $G$ is the split group $\mathrm{SO}_2$. We prove the analogous result for the symplectic double cover. Over finite fields of large characteristic, we give the complete decomposition of the twisted Jacquet module when $\tau$ is cuspidal on $\mathrm{GL}_n$ and $n$ is larger than the rank of $G$.

Let $A$ be a unital, simple, separable, and pure $C^*$-algebra. We prove that, for every $a\in A_+$, the set of positive elements in $A$ having the same Cuntz class as $a$ is path-connected. This may be viewed as a generalization to positive elements of the classical homotopy theory of projection classes and extends a line of results on the topology of constant Cuntz classes that previously required stronger regularity hypotheses to the class of pure $C^*$-algebras, which notably contains all selfless algebras.

The second part of Hilbert's 16th problem is one of the most fundamental open problems in the qualitative theory of ordinary differential equations and dynamical systems. A central unresolved question is whether the Hilbert number $H(d)$ is finite for arbitrary polynomial degree $d$. (A very recent manuscript from OpenAI asserts the finiteness of $H(d)$; see the remark at the end of this paper.) In the absence of a general upper bound theory, most existing work constructs explicit perturbed polynomial systems to obtain improved lower bounds for $H(d)$, while rigorous upper bound estimates remain largely unavailable. This paper focuses on improving the quantitative lower bound for polynomial systems through a systematic Hierarchical Melnikov Realization framework. We construct perturbed dynamics based on an anisotropic Chebyshev Hamiltonian, where hierarchically selected perturbation coefficients generate simple zeros of the first-order Melnikov function simultaneously across multiple disjoint period annuli. Two distinct 3-adic filtrations supply the necessary logarithmic correction factors, and a Borel–Gauss analysis establishes the required local rank condition under sufficiently strong anisotropy. As the main result of this work, we prove that $H(d)=\Omega\bigl(d^2\ln^2 d\bigr)$, which improves the previously known lower bound by a factor of $\ln d$. The full construction of limit cycles and the resulting dynamical bounds are formally certified using Lean 4 formal verification.

Function-correcting codes are designed to protect the function values of a prescribed function against errors. Every error-correcting code that provides data protection inherently offers some degree of protection for functions defined on the data. In this work, we introduce a class of codes over $\mathbb{Z}_m$, termed function-correcting Lee metric codes with data protection (FCLMCs with data protection), which simultaneously provide error protection for both the data and the corresponding function values under the Lee metric. We consider codes that provide protection against a prescribed level of error for the function values that exceeds the level of protection guaranteed for the underlying data. We present a general construction of these codes and derive lower and upper bounds on the optimal redundancy, including a Plotkin-type lower bound. Furthermore, we derive explicit upper bounds on the redundancy of FCLMCs with data protection for several important classes of functions, including locally binary Lee functions, the Lee weight function, and the modular sum function. Finally, since the Lee metric coincides with the Hamming metric over $\mathbb{Z}_2$, all of our results remain valid over $\mathbb{Z}_2$ with respect to the Hamming metric.

A verifier that accepts or rejects whole answers reveals little: under a flat prior over $k$-bit answers, zero error needs $2^k-1$ verifications. The usual remedy is to solve related source tasks, either all first, as a curriculum does, or interleaved with verification. We price this remedy in information and in calls. With an exact verifier, the least causal information that any interleaving of source calls and $n$ verifications needs to succeed with probability $s$ is a list rate-distortion function, attained by one observation before any verification. It lower-bounds the expected number of binary source calls, which designed sources meet within $1+\log_25$ calls for unique answers and within a logarithmic term in general, where no additive constant suffices. With an exact verifier and fixed sources, moving every call before the first verification preserves all hard caps on calls, although interleaving can save unboundedly many expected calls; under a noisy verifier, source-first protocols can lose unbounded factors in information and in error. For linear banks over $\mathbb{F}_2$, optimal accuracy has a closed form, and after a polynomial-time reduction the budget profile is computable in time $2^{O(h^2)}\operatorname{poly}(J,k+h)$ for $J$ sources and nuisance dimension $h$. In these banks, for zero error under a hard cap, the calls beyond the rounded-up information price are exactly those spent on nuisance. Every numbered result apart from two clauses about the planner is machine-checked in Lean 4, assuming two published results. Used as a ruler, the frontier shows a small transformer using all delivered bits at latent dimension $5$ and none at $11$ within fixed training budgets; in a test with predictions recorded before training, low XOR degree of the target bits did not suffice for their use.

This note gives direct proofs that, under an effective isometric $T^2$ action, Einstein metrics on smooth homotopy four-spheres are round and complete Ricci-flat metrics on smooth manifolds homeomorphic to $\mathbb R^4$ are Euclidean or positive one-center Taub–NUT, up to isometry, scaling, and orientation reversal.

We prove a single measurement uniqueness result for interior sound-soft obstacles in the planar Schrödinger equation with an unknown surrounding potential. For any two configurations with different obstacles, we construct a boundary datum for which the corresponding Neumann measurements differ. The full Dirichlet-to-Neumann map uniquely determines both the obstacle and the potential. The obstacles may have several connected components, and no convexity or analyticity assumptions are imposed. Neither potential is required to be known near the measurement boundary.

For $s>0$, and $z\in\mathbb H=\{z=x+iy:y>0\}$, define $\theta(s;z)=\sum_{m,n\in\mathbb Z}e^{-\pi s\big(m^2y+\frac{(mx-n)^2}{y}\big)}$ and $ J(z;a,b)=\sum_{m,n\in\mathbb Z}e^{-\pi \big(m^2y+\frac{(mx-n)^2}{y}\big)} \cos\bigl(2\pi(ma+nb)\bigr)$ for the classical and shifted theta functions (Gaussian lattice sums), respectively. We prove that, up to the modular group, there exist three thresholds $0<\alpha_a<\alpha_b<\alpha_c<1$ such that \begin{equation}\aligned\nonumber \Minima_{z\in\mathbb H,\,(a,b)\in\mathbb R^2}\Big(\theta(1;z)+\alpha J(z;a,b)\Big)= \begin{cases} \;\big(e^{i{\pi}/{3}};1/3,1/3\big), &\hbox{if}\;\; \alpha\in[0,\alpha_a),\\ \;\big(e^{i{\pi}/{3}};1/3,1/3\big)\;\hbox{or}\;\big(e^{i{\varphi_{\alpha_a}}};1/2,1/2\big), &\hbox{if}\;\; \alpha=\alpha_a,\\ \;\big(e^{i{\varphi_\alpha}};1/2,1/2\big),\;\varphi_\alpha\in(\varphi_{\alpha_a},\frac{\pi}{2}),\;\; &\hbox{if}\;\; \alpha\in(\alpha_a,\alpha_b),\\ \;\big(i;1/2,1/2\big),\; &\hbox{if}\;\; \alpha\in[\alpha_b,\alpha_c],\\ \;\big(iy_\alpha;1/2,1/2\big),\; y_\alpha\in(1,\sqrt3],\;\;y_1=\sqrt3,\;\;&\hbox{if}\;\; \alpha\in(\alpha_c,1], \end{cases} \endaligned\end{equation} where the complex variable $z$ and vector $(a,b)$, represent the total vortex shapes and relative positions of the two-component Bose gas, respectively. Consequently, this completely proves the conjecture of Mueller and Ho~\cite{Mue2002} (2002) arising in Bose–Einstein condensates.

Multiparameter complex processes, based on the Costa-Farber model, provide a common framework for growing random simplicial complexes, including the Linial-Meshulam and random flag complex processes. We establish laws of large numbers for persistent Betti numbers of these processes and prove that, after a deterministic time change and normalization, their persistence diagrams converge weakly in probability to a deterministic finite measure. Its quadrant masses are determined by universal limits of normalized persistent Betti numbers, characterized by an explicit finite-dimensional optimization problem. We extend the diagram convergence to integrals of continuous functions with polynomial growth in the death coordinate and controlled singularities at the diagonal. These results yield convergence of additive and multiplicative lifetime measures and total persistence. We further deduce convergence in optimal transport distance. The proofs combine local weak convergence of simplicial pairs, spectral analysis of compressed Laplacians accounting for dependence through shared faces, and quantitative estimates for large death coordinates and short lifetimes.

We study arrangements of conics and lines in the complex projective plane from the viewpoint of 4-manifolds. We introduce combinatorial conic-line arrangements, the conic-line counterpart of rank-3 matroids, together with their topological and smooth realisations, and we ask which restrictions on a complex arrangement survive when its components are replaced by locally-flat spheres in the same homology classes with the same local singularity models. For an ADE combinatorial conic-line arrangement of even total degree admitting a topological realisation we prove a Hirzebruch-type inequality; the proof uses only branched covers and the local topology of rational double points. In the complex-algebraic category we show that equality holds exactly when the arrangement is a maximising curve, hence free with prescribed exponents. We then introduce divisible and odd combinatorial conic-line arrangements and obtain inequalities and congruences for the multiplicity sequence of their resolutions, via cyclic branched covers, the G-signature theorem, spin structures, and Furuta's 10/8-theorem.

To a polycube $\mathcal P$ in $\mathbb R^n$, we associate an ideal $I_{\mathcal P}$ generated by the join–meet relations of vertices that are corners of a cuboid of $\mathcal P$. For $n=2$ these are the polyomino ideals. In contrast with the planar case, we show that for $n\geq 3$ the ideal $I_{\mathcal P}$ is prime if and only if $\mathcal P$ is a cuboid. For convex polycubes in $\mathbb R^3$ of uniform thickness whose vertex set is a sublattice of $\mathbb N^3$, the defining relations form a quadratic Gröbner basis, and we characterize the Cohen–Macaulay ones in terms of a poset on the vertices of $\mathcal P$; in this case the order complex of the poset is vertex decomposable. Finally, as an application, we show that the polycube algebra of a thickened $L$-convex polyomino is Koszul, and we characterize when it is Cohen–Macaulay.

Let $\Rn$ be the variety of rings satisfying $x^n\eqid x$, where $n\geq2$, and let $\W$ be a variety of idempotent semirings with commutative addition. The equality $\Rn\circ\W=\Rn\vee\W$ holds, and this variety has subvariety lattice $L(\Rn)\times L(\W)$. Its subdirectly irreducible members that are not additively idempotent have exactly one nontrivial ring component, which is a finite field. Such a member is determined by this field and an idempotent semiring with a multiplicative identity; a separation condition on unary polynomial functions of the latter characterizes subdirect irreducibility. Every subvariety has a finite identity basis. A four-element semiring generates a variety with subdirectly irreducible members of unbounded cardinality. The results extend the Mal'cev product theorem of Wang and Shao from the absorption subvariety to arbitrary $\W\leq\Slp$.

We establish matching lower bounds for deterministic first-order optimization of globally Lipschitz geodesically convex functions on hyperbolic space. A global minimizer lies within distance $r$ of a known point, each query returns an exact function value and a selected Riemannian subgradient, and the target objective gap is $\varepsilon Lr$, where $L$ is the Lipschitz constant. For sectional curvature $-k^2<0$, let $\rho=kr$ and $\zeta=\rho/\tanh\rho$. The optimal worst-case query complexity $Q^\star$, taken uniformly over dimensions, satisfies $Q^\star(\rho,\varepsilon)=\Theta(\zeta\varepsilon^{-2})$ for every $\rho>0$ and $0<\varepsilon\le1/64$. The new lower bound covers arbitrary adaptive deterministic queries and arbitrary outputs, matching the projected-subgradient upper rate of Zhang and Sra (2016). It removes the geodesic-span and historical-halfspace restrictions of earlier matching lower bounds. The construction extends a radial source from a convex carrier with boundary. A small residual cut selects minimizer candidates while an exact hyperbolic persistence inequality keeps every historical contact valid under future carrier growth. One fixed globally convex objective and one fixed subgradient selection consequently realize all exact replies simultaneously. Each query consumes $O(\varepsilon^2)$ of a geometric potential whose available budget is $\Theta(\zeta)$, yielding the product lower bound in a dimension linear in the query horizon.

In 2008, Jacobi and Madden proved that $a^4+b^4+c^4+d^4=e^4$ has infinitely many primitive solutions with the linear relation $e=a+b+c+d$. We extend their result to relations with rational cubes $e=a+k^3(b+c+d)$ and prove infinitude for $k=3$ and $10$ other $k\ne 1$. We compile $30$ starting solutions. In the spirit of Jacobi and Madden's original paper, everything here is simple algebra except one theorem of Mazur.

Which pure states give uniform outcomes in several mutually unbiased measurements, and when can they form another measurement basis? We distinguish common unbiased states, their orthonormal completion, and further extension. For a displayed phase-deformed Fourier pair in dimension $2m$, a clock symmetry and skew antiunitary complete every prescribed common unbiased state to an explicit orthonormal basis. In dimension six, an amplitude-profile obstruction proves strong unextendibility of a given mutually unbiased triple when its original third-basis rays are invariant under the ternary clock. We give an exact nonproduct example and an application to a companion whose columns are localized at single Fourier frequencies. For every exact Hadamard matrix in a specified closed entrywise neighborhood of a fixed reference matrix, the coordinate and Hadamard bases have exactly 48 common unbiased rays but no third measurement basis, with a quantitative lower bound on the squared Gram defect of every six-column candidate. We also study auxiliary complex equations for common states, retaining algebraic multiplicities. For a flat unitary of order six with all $3\times3$ minors nonzero, the projective ray scheme is finite; each singular two-minor contributes two reduced isolated boundary points, and the affine coordinate algebra is finite-dimensional over $\mathbb{C}$. In every order, a critical-algebra identity preserves multiplicities; in dimension six it gives finiteness for a fixed phase fibre. Specialization over a discrete valuation ring controls complete reciprocal-orthogonal collections. A nonphysical Tao/Potts collection separates algebraic from physical solutions. These results do not determine the global maximum number of mutually unbiased bases or classify all triples.

In this work, we propose a unified localization framework, termed UNILocMP, that combines model-based geometry and channel charting (CC) for multi-panel massive multiple-input-multiple-output (MIMO) under clock asynchronism. Owing to the multi-panel architecture, users are classified into multi-line-of-sight (LoS) users, which maintain LoS links with at least two panels, and single/non-LoS users, which maintain a LoS link with only one panel or with none of the panels. For multi-LoS users, a joint position and clock bias estimation is developed; while for single/non-LoS users, an unsupervised CC model is trained with a two-stage data augmentation strategy, where a clock-bias aware dissimilarity metric is introduced. It is numerically validated that the proposed UNILocMP outperforms model-based and CC-based baselines and achieves acceptable performance compared with fully-supervised fingerprinting. Moreover, for a fixed total number of antennas, the multi-panel architecture significantly improves localization accuracy and robustness compared with a single-panel base station (BS) deployment, particularly in the presence of clock asynchronism.

We prove that every weakly pseudoconvex Oka manifold admitting a positive line bundle is elliptic in the sense of Gromov, thereby proving Gromov's ellipticity conjecture. The proof establishes a cohomological construction of local sprays. Given a bundle morphism $\varphi\colon E\to T_X$, we introduce quadratic $\varphi$-vector fields on $E$ whose flows yield local sprays with fibre derivative $\varphi$. Their existence is characterized by the vanishing of the symmetrized $\varphi$-Atiyah class of $E$. This gives a local dominating spray on every weakly pseudoconvex manifold admitting a positive line bundle, without any Oka assumption. The positivity hypothesis is essential even for local existence: blow-ups at a single point of complex tori of algebraic dimension zero, and Kummer surfaces of algebraic dimension zero, admit no local dominating spray. The torus examples give compact Kähler Oka manifolds that are not elliptic and show that ellipticity is not preserved under blowing up, and that it is neither open nor closed in holomorphic families of compact Oka manifolds. Combined with the work of Xie and Zhao, the Kummer examples yield Oka K3 surfaces that are not elliptic. Under the additional Oka assumption, we globalize the local dominating sprays constructed above. The proof combines Oka approximation with a gluing argument based on weighted $L^2$ estimates for the $\bar\partial$-equation.

This article establishes a sharp singular Moser-Trudinger type inequality in a new function space and explores a concentration-compactness principle in the Heisenberg group, inspired by P.\textcolor{blue}{-}L. Lions. We analyze the equivalence of sharp critical and subcritical singular Moser-Trudinger inequalities, focusing on their asymptotic behavior and connections between their suprema. Additionally, we apply the mountain pass theorem to demonstrate the existence of positive ground state solutions for zero mass $(p, Q)$-Laplace subelliptic equations with singular exponential nonlinearity for the following equation: $$ -\Delta_{H,p} u-\Delta_{H,Q} u=\frac{f(\xi,u)}{r(\xi)^\vartheta}\quad\text{in}\quad \mathbb{H}^n, $$ with $1<p<Q$, $\vartheta\in(0,Q)$, $Q=2n+2$, the function $r(\cdot)$ is called the Korányi norm in $\mathbb{H}^n$ and the nonlinearity $f:\mathbb{H}^n\times \mathbb{R}\to \mathbb{R}$ is a Carathéodory function, which behaves like $\exp{(\alpha|s|^{\frac{Q}{Q-1}})}$ as $|s|\to~+\infty$ for some $\alpha>0$.

This is the third paper in a series where we study arithmetic applications of the higher elliptic Gamma functions. In the first two articles in this series, we defined geometric families of these functions and proved that they satisfy coboundary and cocycle relations under the action of special linear groups and associated congruence subgroups. The main purpose of the present paper is to present a construction of conjectural higher elliptic units above number fields with exactly one complex place as special values of higher elliptic Gamma functions, upgrading the construction carried out by Bergeron, Charollois and García for complex cubic fields. These higher elliptic units are obtained by evaluating a multiplicative $(n-2)$-cocycle built from higher elliptic Gamma functions against a $(n-2)$-cycle associated to some group of totally positive units of a given number field of degree $n$ with exactly one complex place. We conjecture that these higher elliptic units are algebraic units which belong to prescribed abelian extensions of the base field where they are evaluated and that they satisfy a Kronecker limit formula which relates the logarithm of their modulus to values of derivatives of partial zeta functions at $s = 0$ in the base field. We showcase our conjecture on various examples for number fields of degree 3, 4 and 5.

On any bounded convex domain $\Omega\subset\mathbb{R}^n$ with a $d$-dimensional flat boundary piece $F\subset\partial\Omega$, we consider convex solutions with zero boundary values to the Monge-Ampère equations $$ \det\mathsf{D}^2 u= \mathrm{dist}(\,\cdot\,,\mathrm{Aff}(F))^\alpha,\quad\det\mathsf{D}^2 u=\mathrm{dist}(\,\cdot\,,\partial\Omega)^\alpha,\quad \det\mathsf{D}^2 u=(-u)^\alpha, $$ where $\mathrm{dist}$ stands for Euclidean distance and $\mathrm{Aff}(F)$ is the affine hull of $F$. We show that the threshold of the exponent $\alpha\in\mathbb{R}$ for $u$ to have infinite slope at $F$ is $\alpha\leq 2d-n$. For the first two equations, we also show that no solution exists when $\alpha\leq 2d-2n$. Moreover, when $\Omega$ satisfies the “exterior cone condition with a $d$-dimensional ridge” at $F$, both thresholds are sharp for the first equation, and the infinite-slope threshold is sharp for the last equation.

For a finite set $A\subseteq\mathbb Z^d$, we study the capacities of a sausage obtained by transporting $A$ along the Minkowski sum of simple or critical branching random walks. We first establish quantitative capacity estimates for the random walk sausage and the branching sausage. This in turn, gives us estimates for the probability that Minkowski sums of a mixture of walks hit a finite set in high dimensions. Our approach is based on an annular decomposition of a critical branching random walk into successive waves, which may have further applications.

Geodynamic simulations involve solving the Stokes equations, where boundary conditions play a crucial role in governing internal dynamics and surface topography evolution. A common approach is to model the interface in contact with air as a free surface. This interface is stress-free and deforms in response to internal force balances, which is essential for understanding processes ranging from rifted margins to subduction. However, a free surface can introduce instabilities and oscillations in the numerical solution, commonly known as the "drunken sailor" instability, forcing the use of small time steps. Geodynamic models therefore often employ stabilization terms based on the gravitational effect of free-surface motion. To date, these terms have been derived primarily on a heuristic basis. Here, we apply variational calculus to the energy-minimization formulation of incompressible Stokes flow. Using the Gateaux derivative, we derive the complete weak form for a deforming free-surface domain and show that the stabilization terms arise naturally from the free-surface assumption. We consider the linear-viscosity case and identify two boundary-integral contributions: one associated with internal viscous deformation and the other with gravitational body force. Two benchmark cases demonstrate the consistency of the derived formulation with previously proposed stabilization approaches. Our results provide a mathematical foundation for the commonly used gravity stabilization and identify an additional viscous boundary contribution.

We consider a once-excited random walk on $\mathbb{Z}^d$, $d \ge 2$, where the walk on its first visit has a bias of constant strength $\beta>0$ toward the origin and moves like a simple symmetric random walk on subsequent visits to that site. We show that the walk is recurrent and prove an almost sure spherical shape theorem: the trace after $n$ steps is asymptotically a Euclidean ball centered at the origin, with radius proportional to $n^{1/(d+1)}$, and the local times have an asymptotically conical profile. This result confirms a conjecture of Kozma (2007) on the shape of the trace for this model.