Let $M$ be a relatively compact open subset of a complex manifold $M'$ with smooth boundary $X$ and let $L$ be a holomorphic line bundle over $M'$. Assuming that condition $Z(q)$ holds, we establish the semi-classical asymptotic behavior of $e^{-\frac{t}{k}\Box^{q}_k}$ near the boundary $X$ as $k\to\infty$, where $\Box^{q}_k$ is the $\bar{\partial}$-Neumann Laplacian acting on $(0,q)$-forms on $M$ with values in $L^k$. Our results extend the seminal work of Bismut to complex manifolds with boundary. As applications of our results, we provide a heat kernel-based proof of the holomorphic Morse inequalities for complex manifolds with boundary and derive a semi-classical Weyl law for the $\bar{\partial}$-Neumann Laplacian.

This paper studies the numerical approximation of a singularly perturbed convection-diffusion problem over a bounded polygonal domain in $\mathbb{R}^d$ ($d=2,3$), where the velocity field is modeled by a log-uniform random field, a setting typical in uncertainty quantification. We introduce a novel numerical framework for computing the expected value of the linear functionals of the solution. The approach combines a finite element discretization of the problem, a truncated Karhunen--Loève expansion to represent the stochastic velocity field, and a lattice-based quasi-Monte Carlo (QMC) method to estimate expectations over the parameter space. We provide a rigorous error analysis of the proposed scheme, establishing bounds on the mean squared error and demonstrating that the QMC method achieves a nearly linear optimal convergence rate, with a constant independent of the integration dimension. Furthermore, the convergence rate is shown to be independent of the singular perturbation parameter.

Let $\Gamma$ be a geometrically finite thin subgroup of an arithmetic lattice $\Gamma_0 < G:= \operatorname{SO}(n, 1)$ and consider the congruence covers of $\Gamma \backslash G$. In the breakthrough work of Bourgain-Gamburd-Sarnak, the expansion machinery was used to establish a uniform spectral gap in the setting $(G, \Gamma_0) = (\operatorname{SL}_2(\mathbb{R}), \operatorname{SL}_2(\mathbb{Z}))$ when the critical exponent satisfies $\delta_\Gamma > \frac{1}{2}$. The main applications are affine sieve for $\Gamma$-orbits and uniform resonance-free half-planes for the resolvent of the Laplacian. These results were generalized in subsequent works by Mohammadi-Oh, Oh-Winter, the author, and Edwards-Oh. Yet, the region $\delta_\Gamma \in \bigl(\frac{1}{2}, n - 2\bigr]$ for $n \geq 3$ remains to be treated when there are cusps. The purpose of this paper is to fill in this gap in the literature. The difficulty lies in working with a countably infinite coding due to the presence of cusps. In particular, we incorporate new tools to prove the Zariski density and full trace field properties of the return trajectory subgroups.

In this paper, we study $(-1)$-homogeneous steady solutions to the Euler equations on $\mathbb{R}^n \setminus \{0\}$. In low dimensions $n=2,3$, such flows are known to be essentially trivial. In contrast, we show that in higher dimensions $n \ge 4$, every $(-1)$-homogeneous Euler flow is a geodesible vector field with constant Bernoulli function. Moreover, any $(-1)$-homogeneous geodesible field is induced by a geodesible field on the sphere $\mathbb{S}^{n-1}$. In particular, in the case $n=4$, every $(-1)$-homogeneous Euler flow is obtained as an extension of a Beltrami field on $\mathbb{S}^{3}$.

We study infinite exchangeable sequences with Gaussian one-dimensional marginals. We formulate the conjecture that joint Gaussianity of a single pair of coordinates forces the entire sequence to be a Gaussian process. Although this conjecture remains open, we prove that joint Gaussianity of the first four coordinates is sufficient. We also establish the corresponding two-point criterion under the additional assumption that the directing measure is almost surely infinitely divisible.

Motivated by the Kodaira-Spencer local stability theorem for Kahler structures and by C. Li's study of Kahler structures on holomorphic submersions between compact complex manifolds, we establish an equivalent characterization of the Kahlerness of the total space near a Kahler fiber, in an optimal manner. The proof combines a (1,1)-type lifting argument for flat sections of a local system, observations on torsion-freeness and a Kahler neighborhood criterion for compact Kahler submanifolds.

This work proposes a Dual Time Phase Field Inversion (DTPFI) method for recovering unknown potential functions in phase field models. The reconstruction is formulated as an optimization problem that minimizes the mismatch between model predictions and observed fields at the final measurement time. We prove the differentiability of the measured field with respect to the unknown potential and establish the local convexity of the regularized objective function, thereby ensuring the existence of a local optimal solution. For numerical implementation, automatic differentiation is employed to compute gradients, avoiding the expensive evaluation of analytic gradients. Extensive numerical experiments demonstrate that DTPFI accurately reconstructs both polynomial and logarithmic potentials and remains robust under measurement noise. The framework is further extended to inverse problems involving field dependent mobility and joint parameter identification in coupled Cahn-Hilliard-Allen-Cahn systems.

We investigate the global well-posedness of a class of generalized Hillen--Painter systems -- specifically, supercritical volume-filling chemotaxis models -- in $\mathbb{R}^2$ under the influence of Couette flow. It is well established that, in the absence of fluid flow, solutions to this system may develop finite-time singularities (blow-up) for arbitrary initial cell mass. It is proved that the introduction of a Couette flow with sufficiently large amplitude guarantees the global existence of solutions for all initial masses. By employing a novel frequency decomposition technique, we successfully remove the mass threshold limitation presented in previous studies on the domain $\mathbb{T}\times\mathbb{R}$ (Wang et al., Commun. Contemp. Math.), thereby establishing global regularity in the whole space without any smallness assumptions.

This is the second of a series of three papers, in which we prove a formula expressing the modular height of a unitary Shimura variety over a CM number field in terms of the logarithmic derivative of the Hecke L-function associated with the CM extension. The main idea of our proof is to compare the holomorphic projection of the derivative of a certain mixed Eisenstein-theta series and the arithmetic degree of a generating series of divisors on unitary Shimura varieties. In this paper, we define the arithmetic generating series of divisors on unitary Shimura varieties, compute the corresponding arithmetic intersection numbers, and derive the modular height formula for unitary Shimura curves as well as the height formula for a CM point on them.

We present a new classification of the class of all the hypergeometric orthogonal polynomial sequences. The classification uses properties of the coefficients $\alpha_n$ and $\beta_n$ in the three-term recurrence relation satisfied by the orthogonal polynomial sequences. Such coefficients are rational functions of $n$ and are determined by a set of six parameters. The partial fraction decomposition of $\alpha_n$ is a linear combination of five linearly independent functions of $n$ with coefficients $d_j$, for $0 \le j \le 4$, that are polynomials in our six parameters. For each value of a parameter $r$, associated with the eigenvalues, we classify the $\alpha_n$ according to the set of coefficients $d_j$ that are nonzero. There are certain particular values of $r$ that must be considered separately. Our classification yields a collection of 53 disjoint classes and it is different from the Askey scheme in several aspects. It is similar to the classifications proposed recently by Koornwinder.

We give a complete recipe for constructing extremal van der Geer--van der Vlugt curves over finite fields of odd characteristic. The input data consist of a nonzero element of the base field together with a linear subspace satisfying a certain trace condition. This construction may be viewed as an odd-characteristic analogue of the one previously obtained by the authors in characteristic two.

We introduce a closure operator for valuated matroids and prove that it yields a cryptomorphic definition of valuated matroids. As an application, we introduce a class of modules over the tropical semifield (geometric modules) and prove that there is one-to-one correspondence between projective equivalence classes of simple valuated matroids and isomorphism classes of finitely generated geometric modules. We further illustrate how this correspondence can be lifted to simple infinite valuated matroids and geometric modules.

The scalar auxiliary variable (SAV) method constructs linear, unconditionally energy-stable time discretizations of gradient flows. In a first-order SAV step, eliminating the auxiliary variable shows that the state equation is a semi-implicit update augmented by a rank-one positive semidefinite correction from the previous nonlinear force. The multiple-SAV (MSAV) method produces this correction componentwise, yielding a correction of rank up to the number of energy components. This separates two mechanisms usually coupled in MSAV: the number of scalar variables tracking the nonlinear energy and the rank of the correction applied to the state equation. We introduce a pullback-corrected SAV (PB-SAV) family that keeps a single scalar auxiliary variable but replaces the rank-one SAV correction by the pullback correction induced by an admissible component decomposition. The correction remains positive semidefinite, has rank at most the number of components, and may change from step to step without changing the scalar energy tracker. We prove modified-energy dissipation laws for fixed and step-dependent decompositions, derive a refinement identity whose gain is an explicit weighted variance, and give a Sherman-Morrison-Woodbury implementation of the low-rank perturbation of the standard semi-implicit solve. We also show, in finite dimensions, that the pullback correction is the Gauss-Newton matrix of a least-squares representation of the nonlinear energy. Numerical experiments on finite-dimensional gradient flows, Allen-Cahn dynamics, and nonlocal Cahn-Hilliard models illustrate regimes in which PB-SAV mainly changes the first-order error constant and regimes in which it substantially improves trajectory accuracy.

Streetlights shining across rippled water often produce tall, narrow reflections with strikingly parallel sides. These light pillars appear almost architectural, yet the water surface is neither vertical nor smooth. We develop a geometric optics model that explains the phenomenon using the specular reflection rule, projection geometry, and the physics and statistics of surface slopes. A rippled water surface can be viewed as an ensemble of small facets (tangent planes on waves) acting as tiny mirrors. As one looks farther across the water (coordinate x), the physical width of the region whose facets reflect the light into the eye (or camera/pinhole) increases, but the pinhole projection onto the image plane compresses this widening by a factor proportional to 1/x. The two effects cancel, producing a reflection whose image width remains constant. This paper appears to be the first to formally document this. (Deviations from the strict cancellation at both ends of the image are also explained.) The analysis clarifies how earlier qualitative treatments failed through a lack of identifying the role of projection geometry. The model also qualitatively explains the brightness variation along the pillar, and why extended objects such as buildings do not produce elongated reflections. The treatment is intended for scholars of introductory optics and for those interested in the physics and mathematics of everyday visual phenomena. We conclude by outlining how the same framework extends naturally to more complex viewing situations, such as the setting sun viewed from a cliff top over the sea.

Different kinds of definable patterns in the models of a first-order theory, such as the order property, the tree property, or the ($n$-)strict order property, allow us to distinguish theories according to their logical complexity. The complexity distinctions given by these definable patterns play a central role in model theory. However, a rigorous definition of the notion of a model-theoretic patterning property has yet to be established. We start by discussing different proposals from the literature for making the notion of a model-theoretic patterning property rigorous. Some examples will include the straight definability of Shelah, which will describe properties definable by a pattern of consistency and inconsistency in a formula and its negation, and the poset definability of Garcia and Mennuni, covering properties definable by interpreting a partial order embedding a given poset. We will also introduce a higher-arity version of straight definability. In our first main result, we will answer open questions of Bailetti and Garcia-Mennuni, showing that the $n$-strict order property $\mathrm{SOP}_{n}$ is straightly definable and poset definable even for integers $n \geq 4$. This will complete the categorization of all of the classical classification-theoretic properties as straightly definable. Our other main result will concern properties that are straightly definable without negation: the positively straightly definable properties defined by Bailetti. We will show using Saracino's theorem and results of Bodirsky, Bodor and Marimon that, in any countably categorical theory, implications between positively straightly definable properties must be exhibited at the level of $\exists\forall$-formulas. This will have special consequences under the assumption that $\mathrm{SOP}_{2}$ is equal to $\mathrm{SOP}_{3}$.

We prove that there is no local quasiconformal map between the limit set of a convex-cocompact Kleinian group and the Julia set of a postcritically finite rational map, provided that both are Sierpiński carpets. This contrasts with the recent results by Y. Luo, D. Ntalampekos and Y. Luo, M. Mj, S. Mukherjee for gasket and tree-like spaces, respectively.

We define a homological Conley index for a class of hybrid dynamical systems. This is achieved by factoring through the hybrid suspension semiflow, which views a class of hybrid dynamics as a continuous semiflow. We establish that this invariant is well-defined, invariant under continuation, and satisfies the Ważewski property and other reconstruction results. We compute the index for several examples, showcasing how classical reconstruction theorems apply in this setting to characterize hybrid dynamics.

In this paper, we show that there are $57$ nonnegative integer solutions $(a,b,x)$ to the inequality $1\le \lvert x^{2} - 2^{a}\cdot 3^{b}\rvert < 3\max\{a, b\}$ and we list them explicitly. The inequality is converted into a statement about how closely $x/q$ approximates irrational number $\sqrt{d}$ for $d\in\{2,3,6\}$, where $q$ is an integer which is $3$-smooth, after which Worley's theorem on rational approximations via continued fractions is applied to parametrise the solutions and a $p$-adic lower bound for a linear form in logarithms due to Bugeaud and Laurent is applied to find a rather large bound on $\max\{a,b\}$. We finish with an application of the LLL algorithm to reduce this bound.

We show that finite morphisms of smooth algebraic varieties naturally give rise to Cartan--Coxeter type structures. Starting from the self-fiber product $X\times_YX$ of a finite morphism $Q:X\to Y$, we construct local symmetry operators and intrinsic Cartan-type invariants from the geometry of its non-diagonal irreducible components. This provides a mechanism for reconstructing root-theoretic structures directly from algebraic correspondences rather than from reflection groups. A central part of the theory is a rank-two geometry associated with pairs of non-diagonal components. We establish a rank-two reduction theorem, derive explicit trace and determinant formulas for the corresponding operators, and obtain a classification into elliptic, parabolic, and hyperbolic transport types. These results yield intrinsic analogues of Cartan matrices, Coxeter transformations, exponents, and Dynkin diagrams associated with finite morphisms. We further prove rigidity theorems showing that the structures arising from a single finite morphism are highly constrained. To obtain richer geometries, we introduce transport atlases of compatible local finite covers equipped with connection data, leading to nonlinear Cartan fields with variable local geometry. This places classical Weyl and complex reflection geometries within a broader correspondence-based root theory extending beyond finite reflection groups.

We prove a fixed finite-window structural theorem for sharp localized Navier-Stokes packages, formulated as both a local-to-clean detection theorem and an anti-phantom principle. The result addresses whether a defect visible in the baseline quotient geometry can disappear after pressure-tail enrichment, residual transfer, quotienting, and clean-to-local detector comparison. Under synchronized representatives, baseline-to-tail visibility, component comparison, residual-ledger closure, detector comparison, chart visibility, and a clean quotient gap, any baseline-visible defect is either detected by the localized detector or charged to an explicit quotient-residual ledger. Quantitatively, M_Lambda^loc(D-zeta_*) >= c_{Lambda,0} Dist_{loc,intg,0}(D,Gamma^{intg}*{Lambda,adm}) - E^{quot}*{Lambda,0}(D). The proof assembles three modules: pressure-tail visibility, componentwise residual-ledger closure, and detector comparison. The anti-phantom interpretation is that a baseline-visible defect cannot be simultaneously detector-silent and residual-cheap. We also record provenance for the imported quotient interface, finite-dimensional pressure-tail models, explicit matrix realizations of the structural inputs, NS-generated coordinate realizability, compactness criteria for clean pressure images, and reduced pressure/tax kernel-free criteria.

In bilevel optimization, optimistic and pessimistic follower behaviors are the most commonly used forms to define how the follower ties-breaks among multiple optimal solutions. In this work, we go beyond these extreme tie-breaking behaviors and investigate the intermediate bilevel optimization program (I-BO), where the follower's selected optimal response is a decision-dependent random event, with a probability measure influenced by the leader's decision. We formally introduce a class of such endogenous measures, including the special case of strong-weak decision-dependent I-BO. We reformulate the I-BO as a Transformed I-BO (T-I-BO) with exogenous uncertainty by defining inverse and Markov-chain transformations, which represent the follower's response as a function of the leader's decision and exogenous randomness. We handle the T-I-BO's uncertainty via sample-average approximation (SAA), and we propose tailored approaches for its SAA program according to the chosen transformation. Computationally, our methods solve reasonable-sized instances efficiently and outperform the deterministic equivalent when available. Furthermore, experiments stress the critical need to accurately model follower tie-breaking behavior, particularly depending on its alignment with the leader's objective, as misspecification leads to suboptimal leader decisions.

In 1921, Kuratowski gave the now-standard definition of ordered pair in the context of set theory. This paper studies the problem of defining ordered tuples as sets in the hierarchy of simple types, of which Kuratowski's construction is a special case.

Dirac asked in 1970 whether for every k >= 4 there is a k-vertex-critical graph without critical edges; Jensen settled all k >= 5, and only k=4 remains open. Following Skottova and Steiner, call a graph G a (4,1)-graph if chi(G)=4, chi(G-v)=3 for every vertex v, and chi(G-e)=4 for every edge e; they proved delta(G) >= 6 and lambda(G) >= 6 for every (4,1)-graph and asked whether a 6-regular (4,1)-graph exists. We prove three results about this 6-regular case. Theorem A (computational): there is no 6-regular 4-vertex-critical graph on n <= 15 vertices, except for a unique graph (up to isomorphism) on n=13, whose 13 critical edges form a Hamilton cycle; hence any 6-regular (4,1)-graph has at least 16 vertices. Theorem B: in a 6-regular (4,1)-graph every 6-edge-cut is either the edge star of a vertex or has both shores of size at least 15; consequently every 6-regular (4,1)-graph on at most 29 vertices is super-6-edge-connected. Theorem C (all sizes): no shore of a nontrivial 6-edge-cut in a 6-regular (4,1)-graph induces a bipartite graph; more generally, a shore whose deficiency is concentrated on two vertices forces them to receive equal colours in every proper 3-colouring. The proof of Theorem B rests on an exact classification of the 3x3 cut matrices of 6-edge-cuts in (4,1)-graphs (exactly 21 matrices, five types up to row/column permutations) together with a boundary-shortfall lemma; the unique near-miss is K_{3,3,3} minus a rainbow 3-matching. Several supporting lemmas are machine-checked in Lean 4/Mathlib.

As a promising solution for extremely large-scale arrays, reconfigurable holographic surfaces (RHS) can be integrated with integrated sensing and communication (ISAC) to form the holographic ISAC paradigm, where significant beamforming gains provided by RHS can improve both communication and sensing performance. However, most existing works on holographic ISAC assume that RHS elements can control the amplitudes of radiated ISAC signals in a continuous manner, which is hard to implement in practice. In this paper, we investigate how the limited radiation amplitudes of the RHS influence ISAC system performance. Specifically, we first derive closed-form lower bounds for both communication rate and sensing SINR to establish a tight upper bound on the minimum radiation amplitude quantization bits. Based on this, we further explore how the communication-sensing performance tradeoff impacts the quantization bits. Numerical results validate our theoretical analysis.

We develop a spectral based coefficient of determination to measure how well the spectral density of a stationary process is represented by the class of MA($q$) models. Using periodogram-based estimators, we establish asymptotic normality, derive tests for the MA($q$) hypothesis, and construct procedures for determining the smallest order $q$ achieving a prescribed approximation quality.

We study the geometry of polynomial neural networks with monomial activation functions and no bias. We introduce a general framework of algebraic networks, together with their realization maps and associated affine neurovarieties. In this setting we define morphisms, subnetworks, symmetry groups and quotient parameter spaces and we discuss geometric notions of identifiability and reducibility. Our main goal is to relate the singularities of neurovarieties to degenerations of the underlying architecture. For fully connected networks, we define the architectural degeneracy locus as the locus of functions admitting a representation by parameters with a rank-deficient layer or an inactive hidden neuron. We prove that, for fully connected networks with non-increasing widths and scalar output, full parameters give smooth points of the corresponding neurovariety under explicit layerwise regularity assumptions. In particular, for these architectures, the singular locus is contained in the architectural degeneracy locus.

In this paper, we use the recently introduced notation and terminology for magnetic systems, which provide a natural framework for extending techniques and results from the theory of geodesic flows to magnetic flows. In this context, we prove a magnetic analogue of the classical Freire--Mañé theorem and obtain some applications. As a consequence, we show that for an energy level of a magnetic system without focal points, either the metric entropy is positive or the magnetic curvature is identically zero.

This paper investigates a class of reflected McKean-Vlasov Stochastic Differential Equations driven by both Brownian motion and a compensated Poisson random measure. We establish the existence and uniqueness of solutions and provide moments estimates for the state processes.

We establish an unconditional analogue of Sylow's first theorem for finite skew braces, and deduce an analogue of Cauchy's theorem. We also prove an analogue of the existence part of Hall's theorem for finite skew braces with soluble additive and multiplicative groups.

Cholesky-QR is among the fastest algorithms for computing the thin QR factorization of tall-and-skinny matrices on GPUs, relying entirely on BLAS-3 operations. However, it is numerically unstable: forming the Gram matrix squares the condition number, causing breakdown when $\kappa_2(\boldsymbol{A}) \gtrsim 10^8$. We present MRCQR (Mixed-Precision Randomized Cholesky-QR), a stable GPU algorithm that addresses this limitation. MRCQR uses a subsampled randomized trigonometric transform to construct a preconditioner $\boldsymbol{R}_s$ that reduces $\kappa_2(\boldsymbol{A}\boldsymbol{R}_s^{-1})$ to near unity with high probability, then applies Cholesky-QR in double precision to the preconditioned matrix. The key insight -- supported by perturbation analysis -- is that the preconditioner requires far less accuracy than the final result: single (FP32) precision suffices when $\kappa_2(\boldsymbol{A}) \lesssim 10^8$, and half (FP16) when $\kappa_2(\boldsymbol{A}) \lesssim 10^4$. MRCQR produces an explicit orthogonal factor $\widehat{\boldsymbol{Q}}$ satisfying $\|\boldsymbol{I} - \widehat{\boldsymbol{Q}}^\top\widehat{\boldsymbol{Q}}\|_2 = \cal O(\mathbf{u})$ ($\mathbf{u} \approx 10^{-16}$, double-precision unit roundoff) for condition numbers up to $10^{16}$, far beyond the $10^8$ limit of CholQR2. Experiments on an NVIDIA H100 GPU show that MRCQR (FP16) outperforms rand-cholQR by $1.4$--$1.8\times$ across all tested column counts and is $1.8$--$13.5\times$ faster than cuSOLVER geqrf, while the FP16 sketch (used when $\kappa_2(\boldsymbol{A}) \lesssim 10^4$) is $2\times$ cheaper than FP64 at no accuracy cost.