Papers

The present work proposes a numerical approach for solving coupled flow and mechanics problems in fractured porous media, represented as mixed-dimensional domains. In this formulation, the elements of the 3D mesh are allowed to arbitrarily intersect the fractures. Displacements are discontinuous across fractures through the use of the eXtended Finite Element Method (XFEM) on the 3D mesh. The mechanical problem is formulated as a saddle-point problem, in which Lagrange multipliers are used to enforce displacement continuity across the fractures. The resulting Lagrange multipliers represent the stress field acting on the fracture surfaces. Likewise, the pressure field is allowed to be discontinuous across fractures through the XFEM formulation on the non-conforming mesh and is computed using an optimization-based domain decomposition strategy specifically designed for mixed-dimensional problems. The fixed-stress splitting scheme is employed to decouple the flow and mechanics subproblems, while the mixed-dimensional pressure problem is solved at each fixed-stress iteration using the Conjugate Gradient (CG) method. The combination of the fixed-stress scheme and the CG solver proves to be highly effective for this class of problems.

In this paper, we study critical points of second Neumann eigenfunctions on convex domains in two-dimensional space forms from three complementary perspectives: spectral and geometric criteria for the absence of interior critical points, quantitative localization of possible critical points, and explicit bounds for the hot spots constant. Precisely, we first establish monotonicity-radius localization principles in $\mathbb S^2$ and $\mathbb H^2$. As a consequence, we obtain a unified diameter criterion for convex domains in these two space forms: if $\mu_2(\Omega)D^2\le j_{1,1}^2,$ then every second Neumann eigenfunction on $\Omega$ has no interior critical points. Moreover, when interior critical points may exist, we derive explicit quantitative location restrictions in terms of the domain diameters in $\mathbb{S}^{2}$ and $\mathbb{H}^{2}$. Finally, we develop an analytic approach to study the hot spots constant $\mathfrak{C}(\Omega)$ on convex domains. For planar convex domains, we improve the Euclidean upper bound to $\mathfrak{C}(\Omega)<1.48$. We further obtain the corresponding hot spots constants for convex domains in non-Euclidean space forms, that is, $\mathfrak{C}(\Omega) < 4$ for $\Omega\subset\mathbb{S}^{2}$ contained in a hemisphere; $\mathfrak{C}(\Omega) < 11.2$ for $\Omega\subset\mathbb{H}^{2}$. Our proofs combine the properties of Bessel and Legendre functions, eigenvalue estimates, and Green's identity. Our results quantitatively measure “how wrong” the hot spots conjecture can be.

We prove the valuative criteria of $\Theta$-reductivity and $S$-completeness for the moduli problem of $t$-K-semistable adjoint Fano foliated structures. We develop a mixed Ding theory for arbitrary linearly bounded multiplicative filtrations, prove inversion of adjunction with arbitrary ideals for adjoint foliated structures, and establish a relative extraction and finite generation theorem. Together, these results yield the required relative extension theorems for families. As applications, we prove uniqueness of $t$-K-polystable degenerations, reductivity of the automorphism group of $t$-K-polystable adjoint Fano foliated structures, and finiteness of the automorphism group in the $t$-K-stable case.

Future wireless networks are expected to support networked artificial intelligence (AI) services, where multiple devices transmit learned features to an edge server for distributed inference. This setting calls for task-oriented physical-layer optimization, where wireless transmission should preserve useful information for inference rather than only maximize the rate or reconstruct the transmitted signals. A key physical-layer control variable is the multiple-input multiple-output precoder, which determines how device features are shaped and combined over wireless channels. Existing task-oriented precoding methods typically adapt the precoder to instantaneous channel state information at the transmitter (CSIT). However, in multi-device MIMO systems, acquiring the aggregate channel, feeding back CSI or optimized precoders, and reoptimizing across coherence blocks introduce substantial overhead. This paper develops a random-matrix-theoretic framework based on statistical CSIT that designs a slow-timescale precoder from channel covariance statistics and training-set feature statistics, without requiring instantaneous CSIT. We adopt maximal coding rate reduction (MCR${^2}$) to measure the class separability of the received features, yielding a task-aware utility for MIMO precoder design. Since this utility still depends on random small-scale fading, we derive a deterministic approximation that converts it into a fixed-point objective depending only on long-term statistics and large-system dimension ratios via random matrix theory. A projected block-coordinate ascent and successive convex approximation algorithm is developed to optimize this deterministic objective under per-device power constraints. Experiments on ModelNet10 verify the approximation and show that the proposed statistical precoder improves task-aware mode allocation and inference performance over competitive benchmarks.

We show that the Belopol'skaya-Daletskii formulation of stochastic differential equations on a Riemann manifold offers an elementary way to construct equivariant representations of finite-dimensional approximations to the path measure of a diffusion. The key ingredient is the use of the exponential map to describe increments of the diffusion.

Known integrable systems with noncommutative dependent variables are typically formulated over free associative algebras, quantum algebras, or Grassmann algebras. For differential-difference integrable equations, we identify a new class of noncommutative algebras that is compatible with the dynamics and can be positioned between quantum and free algebras. In this brief communication, we consider reductions of the nonabelian Volterra hierarchy to new algebras. This approach extends to a broad class of integrable systems, including the Toda lattice, the Ablowitz-Ladik system, and many others.

Many prediction and decision-making problems in operations research involve path-valued covariates – data that evolve over time – for which path signatures have become a canonical feature representation. Their use is justified by a universal approximation theorem, but this is an existence result: it guarantees that a finite-level signature can approximate any continuous path functional, without quantifying how fast the approximation error decreases as the truncation level grows. This paper develops approximation and statistical theory for signature-based path regression. We establish an \(L^2\) approximation rate for smooth functionals of Itô diffusions and show that it is minimax optimal. We then propagate the truncation error through three statistical learning procedures – Signature-OLS, Signature-LASSO, and Signature-Logistic – and establish their consistency. Three real-data applications show that signatures provide informative finite-dimensional representations of path-valued covariates and can improve prediction relative to handcrafted features, in the context of finance – foreign exchange realized volatility forecasting from intraday price paths; energy – battery end-of-life prediction from early diagnostic current-voltage pulse paths; and medicine – epileptic seizure detection from short electroencephalogram windows.

Global linear stability analysis of bluff body wake flows is performed using the matrix-forming method based on finite-difference discretization. Particular emphasis is placed on the influence of outflow boundary conditions, with the aim of minimizing the required computational domain size without degrading accuracy or inducing spurious oscillations near the outlet. This study focuses on incompressible wakes behind bluff bodies such as cylinders and airfoils at high angle of attack, especially in regimes where global modes exhibit downstream spatial amplification. It is shown that below the critical Reynolds number – where the global mode remains linearly stable – significant spatial growth can persist far downstream, even when the wake is nearly absent. This behavior underscores the importance of imposing a physical boundary condition at the outlet. Several commonly used outflow boundary conditions are evaluated, including Dirichlet, Neumann, extrapolation, stress-free, sponge layer, and the Robin condition that incorporates predictions from local linear stability analysis at the outlet. The results demonstrate that, for different $Re$ cases, the Robin condition enables robust convergence of global modes within substantially truncated domains, thereby improving the efficiency of global stability analysis. These findings highlight the broader applicability of the matrix-forming approach for complex stability analyses, including Floquet analysis of time-periodic flows and extensions to compressible configurations.

In this paper, we consider the almost sure nonlinear smoothing, the almost sure spatial decay and the almost sure uniform convergence of the stochastic mKdV equation and the stochastic cubic KdV-Benjamin-Ono equation. Firstly, for initial data $g\in H^{s}(\mathbb{R})(s\geq\frac{1}{4})$ and $\Phi_{2}\in L_{2}^{0,s}$, we prove the local well-posedness for the stochastic cubic KdV-Benjamin-Ono equation. Secondly, we establish the almost sure nonlinear smoothing of the stochastic mKdV equation and the stochastic cubic KdV-Benjamin-Ono equation. Finally, by using the almost sure nonlinear smoothing, we obtain the almost sure spatial decay and the almost sure uniform convergence of the integral term in the pathwise solutions to the stochastic mKdV equation and the stochastic cubic KdV-Benjamin-Ono equation. More precisely, we have the following results: for the stochastic mKdV equation, let $s>\frac{1}{3}$, $f\in H^{s}(\mathbb{R})$ and $\Phi_{1}\in L_{2}^{0,s}$. Then, the local pathwise solution $u$ satisfies \begin{eqnarray*} &&\mathbb{P}\Big(\Big\{\omega: \lim_{t\rightarrow0}\Big\|u-U(t)f-\int_{0}^{t}U(t-s)\Phi_{1}dW(s)\Big\|_{L_{x}^{\infty}}=0\Big\}\Big)=1,\\ &&\mathbb{P}\Big(\Big\{\omega: \forall t\in[0,T_{\omega}], \lim_{|x|\rightarrow\infty}\Big(u-U(t)f-\int_{0}^{t}U(t-s)\Phi_{1}dW(s)\Big)=0\Big\}\Big)=1. \end{eqnarray*} For the stochastic cubic KdV-Benjamin-Ono equation, let $s>\frac{1}{3}$, $g\in H^{s}(\mathbb{R})$ and $\Phi_{2}\in L_{2}^{0,s}$. Then, the local pathwise solution $v$ satisfies \begin{eqnarray*} &&\mathbb{P}\Big(\Big\{\omega: \lim_{t\rightarrow0}\Big\|v-V(t)g-\int_{0}^{t}V(t-s)\Phi_{2}dW(s)\Big\|_{L_{x}^{\infty}}=0\Big\}\Big)=1,\\ &&\mathbb{P}\Big(\Big\{\omega: \forall t\in[0,T_{\omega}],\lim_{|x|\rightarrow\infty}\Big(v-V(t)g-\int_{0}^{t}V(t-s)\Phi_{2}dW(s)\Big)=0\Big\}\Big)=1. \end{eqnarray*}

An affinization of the notion of a Poisson algebra is presented. This is termed a Poisson affgebra and consists of an affine space together with an associative bi-affine multiplication and a bi-affine Lie bracket that acts as an affine derivation for the associative product. The constructive relation between Poisson affgebras and Poisson algebras is described and several low-dimensional examples are studied in detail.

This paper investigates the Dirichlet problem for the anisotropic minimal surface equation in a bounded domain. Under the natural assumption of non-negative boundary anisotropic mean curvature, we establish the unique solvability of the Dirichlet problem for continuous boundary data. To achieve this, an essential a priori gradient estimate is established, which also allows us to prove a weak version of Bernstein's theorem for entire solutions under a sharp, one-sided linear growth assumption. Moreover, using the direct method in the calculus of variations, we prove the existence and local Lipschitz regularity of generalized minimizers in $BV(\Omega)$ with $L^1(\partial\Omega)$ boundary data. We also find that this variational formulation naturally yields a Neumann-type boundary condition, geometrically explaining why the Neumann problem requires no boundary curvature constraints.

We present a numerical construction of cubature rules for area integrals of analytic functions over planar domains with rectifiable Jordan boundary. The starting point is the Cauchy–Green identity. Given a weight $w$, we choose a $\bar\partial$-antiderivative $W$ and reduce the area integral to a contour integral involving the boundary values of $W$. These values are then approximated by a rational function with free poles, computed by the AAA algorithm. The poles inside the domain become cubature nodes, the corresponding residues become weights, and the boundary residual controls the error through an a posteriori estimate, rigorous once the continuous boundary residual is bounded. The same rule admits a dual reading, as the exact integral of a rational interpolant to the integrand, the area analogue of the one-dimensional interpolatory viewpoint. The numerical examples recover the disk mean-value rule and the focal-segment rule of the ellipse to machine precision, reproduce the exact finite quadrature identities of quadrature domains with both separated and confluent nodes, and evaluate logarithmic and Cauchy volume potentials from boundary data alone. The interior poles trace analytic skeletons that we identify tentatively with the mother bodies of potential theory, along with image points that appear without being imposed; for the square the observed convergence is root-exponential.

We study the Köthe dual $\mathcal{B}(D,\ell_2)^K$ of the matricial Bloch space. A 2015 conjecture [Publ. Math. Debrecen 87 (2015), 351–370] proposed that this space coincides with the dyadic mixed-norm space determined by the operator norms of the diagonals. We disprove the conjecture by revealing a structural obstruction: membership in $\mathcal{B}(D,\ell_2)^K$ is sensitive to the placement of the entries within the diagonals and cannot be detected from diagonal data alone; in particular the trace-norm variant fails as well. However, testing against Toeplitz matrices exactly recovers the trace-norm variant, a matricial analogue of the Anderson–Shields theorem, proved via analytic majorants. Finally, we establish two-sided estimates: row-wise and column-wise $\ell(2,1)$ conditions are sufficient, while the trace-norm dyadic condition is necessary; the latter inclusion is strict.

In this article we completely characterize the Bergman number of general domains. Namely, we prove that the Bergman number of a hyperbolic unbounded domain can be calculated in terms of the asymptotic behavior of its hyperbolic metric near infinity. To obtain the proof of this result we first work in the setting of growth spaces, defining the growth number of a domain. Later, we prove an equality relating the Bergman number and the growth number of domains. We also provide examples of domains with prescribed Bergman number which have zero Hardy number, solving a question posed previously by Betsakos and Cruz-Zamorano. At the end, a similar idea is treated for the case of Bloch-type spaces.

We consider time-independent solutions with constant vorticity of the free boundary Euler equations for a 3D liquid drop with capillarity. A rigidity result for the solutions of this problem has been recently proved with variational methods: if a certain quantity involving the vorticity parameter, the capillarity coefficient and the area of the equatorial section of the drop is below a certain value, then the solution has necessarily cylindrical symmetry, the shape of the drop is an oblate spheroid, flattened at the poles and bulged at the equator, and each fluid particle moves along a horizontal, circular trajectory with constant angular velocity. In this paper we develop a perturbation analysis of the problem for fluid domains whose equatorial section is close in C2 norm to a disc, and we show that a rigidity result holds also above the threshold obtained with variational methods.

Let $R(T_n,C_m)$ denote the Ramsey number of a tree $T_n$ on $n$ vertices versus a cycle $C_m$ of length $m$. Burr, Erdős, Faudree, Rousseau, and Schelp (1982) asked for the least function $f(m)$ such that $R(T_n,C_m)=2n-1$ for every odd $m\ge 3$ whenever $n\ge f(m)$. They proved that $f(m)\le 756m^{10}$. This bound was later improved to $25m$ by Brennan (2016) and to $4m-8$ by Fan and Lin (2025). In this note, we show that $f(m)\le 2m-4$ by using a different method and conjecture that $f(m)=\lceil (2m-1)/3\rceil$.

For a topological space $X$, the space of finite subsets $\mathrm{Sub}_n X$ consists of non-empty subsets of $X$ of cardinality at most $n$. We compute the Euler characteristic of $\mathrm{Sub}_n X$ for any space $X$ having the homotopy type of a finite CW complex. We obtain the explicit formula $\chi(\mathrm{Sub}_n X) = \sum_{i=1}^n \binom{\chi(X)}{i}$.

This thesis investigates two central directions in algebraic graph theory, with an emphasis on spectral methods: spectral determination of graphs and transitivity properties of generalized-Hamming graphs and their complements. The first part focuses on graphs that are determined by the spectra of associated matrices. We study spectral determination with respect to the adjacency, Laplacian, signless Laplacian, and normalized Laplacian matrices, with particular emphasis on the adjacency spectrum. We survey existing results on graphs determined by their spectrum and develop new proof techniques for establishing spectral uniqueness. In particular, we present new proofs for the spectral characterization of complete bipartite graphs and Turán graphs, as well as some new results related to the spectral characterization of the important family of strongly regular graphs. In addition, we introduce a new family of graphs, called the graphs of pyramids, and prove that they are determined by their adjacency spectrum using tools from matrix analysis, such as Cauchy's interlacing theorem and Schur complements. The second part of the thesis studies generalized-Hamming graphs, a family of Cayley graphs that generalize the sub-family of Hamming graphs, and their complements. We classify the parameters for which these graphs are edge-transitive or even distance-transitive. Our analysis combines spectral methods, group-theoretic arguments, and techniques from the theory of association schemes. As an application, we derive closed-form expressions for the Lovász $\vartheta$-function of generalized-Hamming graphs and their complements whenever either the graph or its complement is edge-transitive. Overall, the results demonstrate how spectral methods provide powerful tools for understanding the structure and symmetry of graphs, and they suggest several directions for further research.

We prove that the reduced Hao-Ng isomorphism problem has an affirmative answer for every generalised gauge action of a locally compact Hausdorff group on a product system over a unital subsemigroup of a discrete group, without any assumptions on the product system or the subsemigroup. We show that the Fock C*-algebra of the induced product system is canonically isomorphic to the reduced crossed product of the Fock C*-algebra, by identifying the Fock space of the induced product system with the reduced crossed product of the Fock space. This also yields the corresponding completely isometric isomorphism for the tensor algebras. Using the representations of the fixed point algebra that detect Sehnem's strong covariance relations, we show that the canonical representation of the induced product system in the reduced crossed product of the reduced strong covariant C*-algebra is strong covariant.

We revisit established Krylov subspace methods for linear combinations of matrix $\varphi$-function actions from the viewpoint of the block triangular formulation of Al-Mohy and Liu [SIAM J. Sci. Comput., 48 (2026), pp. A726–A747]. In algorithms such as KIOPS [J. Comput. Phys., 372 (2018), pp. 236–255], one uses an augmentation approach based on evaluating the exponential of a slightly larger matrix that contains the operant vectors in its off-diagonal block, and its field of values may therefore grow substantially with these vectors. Typical convergence estimates for Krylov subspace methods result from bounding the error of polynomial approximations for the exponential on the field of values, so that only very pessimistic convergence estimates are available for these methods, in spite of their good practical performance. In contrast, the larger block formulation established by Al-Mohy and Liu involves an operator whose field of values is independent of the operant vectors, leading to more favorable convergence bounds. We work out the details of how these two approaches are connected to each other, which allows us to transfer the convergence bounds from the latter to the former, thus better explaining the observed performance.

We show a Liouville theorem for domains in graphs with Dirichlet boundary conditions. More specifically, we characterize the non-existence of non-zero bounded harmonic functions. Since Dirichlet boundary conditions give rise to Laplacians with a positive killing term, we can first characterize the validity of the Liouville theorem by the fact that the Green operator applied to the killing terms is equal to 1, or in other words, that the constant function $1$ is a potential. Secondly, we derive a characterization in terms of stochastic completeness at infinity and and total loss of heat. Thirdly, we give a characterization in terms of a Green formula for superharmonic potentials. Finally, we investigate the Liouville property in terms of recurrence and transience of the graph without killing term. As an application, we consider subsets of the Euclidean space such as cones and percolation clusters, and weakly spherically symmetric graphs.

The celebrated Bollobás-Eldridge-Catlin packing conjecture states that every $n$-vertex graph $G$ with minimum degree at least $\big(1-\frac{1}{\Delta+1}\big) n$ contains every $n$-vertex graph $H$ of maximum degree at most $\Delta$. Despite considerable attention, the conjecture remains widely open. We show that for bipartite $H$ this threshold can be greatly improved: there is an absolute constant $c>0$ such that every $n$-vertex graph $G$ with minimum degree at least $ \big(1-c\frac{\log\Delta}{\Delta}\big)n $ contains every $n$-vertex bipartite graph $H$ of maximum degree at most $\Delta$, provided $\Delta$ is not too large compared to $n$. Moreover, we prove that this logarithmic improvement is best possible up to the value of the constant.

An eigenvalue of the signless Laplacian $Q(G)$ is $Q$-main if its eigenspace is not orthogonal to the all-ones vector. We characterize graphs with exactly $\ell\ge3$ $Q$-main eigenvalues, one of which is zero. The case $\ell=3$ reduces to non-semiregular bipartite graphs satisfying a vertexwise signed degree-sum identity. For each integer $k\ge0$, we construct infinitely many pairwise nonisomorphic graphs of cyclomatic number $k$ and unbounded diameter, all with exactly three $Q$-main eigenvalues including zero. These families provide counterexamples to the stated classifications of trees, unicyclic graphs, and bicyclic graphs of Javarsineh and Fath-Tabar.

We first establish estimates for the projective and injective dimensions for quotients of perfectoid algebras by radical ideals. As applications, we obtain homological characterizations of modules of finite injective dimension, (big) Cohen–Macaulay modules, Gorenstein local rings, and regular local rings in mixed characteristic in terms of perfectoid algebras. These are mixed characteristic analogues of the corresponding characterizations via Frobenius morphisms in positive characteristic.

We investigate the relation between minimizers and weak solutions for a class of singular functionals arising from Born–Infeld type theories $\mathcal{L}(s)$. In the setting of an electrostatic field $s=\frac{1}{2}|\nabla\phi|^2$, $\mathcal{L}(s)$ satisfies $\lim_{s\to(1/2)^-}\mathcal{L}(s)=+\infty$, which naturally enforces the finite gradient bound $|\nabla\phi|\le 1$, also called the truncation threshold. For a prescribed extended charge density $\rho$, we consider the relation between the weak solution of the system \begin{equation} \begin{cases} -div\left(b\left(\frac12|\nabla\phi|^2\right)\nabla\phi\right)=\rho,& \text{in }\mathbb{R}^N,\\ b(s)=\mathcal{L}'(s),\quad\lim_{s\to\frac12^-} b(s)=+\infty,\\ \lim_{|x|\to\infty}\phi(x)=0 \end{cases} \end{equation} and the minimizer $\phi_0$ of the singular functional. We propose a monotonic approximation method to handle the intrinsic singularities of $\mathcal{L}(s)$. We prove that the gradient of the minimizer never touches the singular boundary $|\nabla\phi|^2=1$; this structural result yields a key integrability property, the existence and uniqueness of the minimizer, and the corresponding variational inequality. Under the additional assumption that $\rho$ is radially distributed, we show that the minimizer is the unique weak solution. Furthermore, we establish the $C^1$ and $C^2$ regularity of the minimizer under suitable integrability conditions on $\rho$, and provide a uniform estimate for the strict spacelikeness condition $|\nabla\phi_0|\le 1-\epsilon$, where the parameter $\epsilon>0$ is explicitly characterized in terms of the spatial dimension, the spatial region, and $\rho$. Our results extend the classical Born–Infeld theory to a general class of singular Born–Infeld type theories, thereby providing a unified framework for the variational analysis and regularity of such singular functionals and systems.

Shi-Catalan and affine Weyl arrangements are obtained in a similar way: they consist of a Weyl arrangement embedded into a higher dimensional space together with shifted copies. We consider the more general case when the Weyl arrangement is replaced by an arbitrary finite arrangement of hyperplanes. Such an arrangement is called an arrangement with special vertex. As our main result we prove that the number of parallel classes of lines in certain simplicial arrangements with special vertex is at most 12.

Nonlinear state estimation under structural model uncertainty remains a fundamental challenge in autonomous Guidance, Navigation, and Control (GNC) systems. Conventional Bayesian filtering separates state propagation from measurement correction, allowing model mismatch to accumulate during propagation, resulting in proposal–likelihood inconsistency, particle degeneracy, and degraded estimation accuracy. Existing approaches primarily improve proposal distributions or weighting strategies without explicitly incorporating measurement geometry into state propagation. This paper introduces a geometry-consistent Bayesian filtering framework that incorporates measurement geometry directly into the propagation process. The nominal drift is projected onto the measurement-consistent subspace, yielding a geometry-consistent proposal while preserving the Bayesian posterior through a rigorous change-of-measure formulation. A Geometric Projection Particle Filter (GPF) is developed together with a geometric co-state that quantifies instantaneous dynamics–measurement inconsistency. Theoretical analysis establishes existence and uniqueness of the projected dynamics, posterior preservation, standard Monte Carlo convergence of the particle approximation, and robustness under structural model uncertainty. The framework is validated using lunar descent navigation under partial observability and persistent model uncertainty. Compared with the bootstrap particle filter and conventional Gaussian filtering methods, GPF consistently achieves higher effective sample size and lower estimation error. These results demonstrate that geometry-consistent propagation provides a principled, computationally efficient, and theoretically grounded framework for robust nonlinear Bayesian filtering.

We use partially ordered sets (posets) to provide a canonical parameterization for the Zariski closure of the image of a semialgebraic set under a rational map whose coordinate functions are polynomials with nonnegative integral coefficients. The resulting poset parametrization of such a unirational variety allows us to translate several well-studied problems into combinatorics; e.g. reducing the problems to describing the poset associated to the variety. These problems include, the implicitization problem from algebraic geometry, the toric reparameterization problem, the computation of the linear span of the variety, and the problem of distinguishing two semialgebraic subsets of the same ambient space. The technique applies to instances of these problems in several fields, including algebraic geometry, algebraic combinatorics, statistics and applied algebra. We demonstrate the technique on examples from each field, including degenerate subvarieties of secant varieties, matroid flat varieties – which generalize toric varieties of edge polytopes, as well as varieties arising in multivariate data analysis and evolutionary biology.

Value-function-type reformulations have generated a broad class of methods for bilevel optimization. However, the corresponding value-function-type constraints are inherently degenerate and generally fail to satisfy standard constraint qualifications, so the associated multiplier sequences may be unbounded and bounded-multiplier convergence analyses become inapplicable. We study this issue for the regularized gap-function reformulation of bilevel problems with constrained convex lower-level programs. We prove that accumulation points of approximate stationary sequences are C-stationary for the corresponding Karush-Kuhn-Tucker-based mathematical program with complementarity constraints (MPCC), even when the multiplier sequence associated with the regularized gap-function constraint is unbounded. The result holds under Mangasarian-Fromovitz constraint qualification (MFCQ) for the upper- and lower-level constraint systems and MPCC-MFCQ at the limiting MPCC point, without any constraint qualification on the regularized gap-function constraint itself. We further provide an example showing that approximate stationary points of the standard regularized gap-function reformulation may converge to a point that is C-stationary but not M-stationary. To guarantee M-stationarity, we introduce a slack-based two-parameter penalty formulation preserving exact multiplier-slack complementarity and establish M-stationarity under a domination condition on the penalty parameters. We develop an inexact slack-penalty method with adaptive penalty updates and feasibility correction, whose accumulation points are M-stationary under the stated assumptions.

We study the threshold for the existence of exactly one unbounded cluster for Poisson-Voronoi percolation on the $d$-dimensional hyperbolic space $\mathbb{H}^d$ for $d\geq 3$. By recent results of Grebík and Recke and d'Achille et al., this "uniqueness threshold" $p_u(\lambda)$ tends to zero as the intensity $\lambda$ of the underlying Poisson point process tends to zero, for Poisson-Voronoi percolation defined on an ambient space from a family of geometric spaces that includes Cartesian products $\mathbb{H}^{d_1}\times\dots\times\mathbb{H}^{d_k}$ with $k,d_1,\dots,d_k\geq 2$. In contrast, for Poisson-Voronoi percolation on the hyperbolic plane $\mathbb{H}^2$, Benjamini and Schramm have shown that $p_u(\lambda)$ tends to one as $\lambda$ tends to zero, and $p_u(\lambda)>1/2$ for all $\lambda>0$. An unpublished argument of D'Achille and Curien shows that for Poisson-Voronoi percolation on $\mathbb{H}^d$ with $d\geq 3$, the uniqueness threshold satisfies $p_u(\lambda)\leq 1/2$ for all $\lambda>0$. Here we will show that $\inf_{\lambda>0}p_u(\lambda)>0$ for Poisson-Voronoi percolation on $\mathbb{H}^d$ with $d\geq 3$. This answers a question of Grebík and Recke.