Papers

Minimum-degree thresholds ask when excluding a fixed graph $H$ forces a dense graph to admit a simple global description. For each fixed chromatic number, the chromatic threshold has only three possible values. We show that this finite-spectrum phenomenon is special to chromatic threshold: already among $3$-chromatic graphs, both the homomorphism and VC-dimension thresholds have infinite spectra and are nonmonotone under taking induced subgraphs. For complete tripartite graphs with a singleton part, we prove $\delta_{\mathrm{hom}}(K_{1,s,t}) \ge \max\left\{\frac13,\frac{s}{1+s+t}\right\}$, with equality for an infinite range of $s,t$; in particular, $\delta_{\mathrm{hom}}(K_{1,s,s})=s/(2s+1)$ for every $s\ge2$. More generally, for every $r\ge3$, the value $(r-2)/(r-1)$ is an accumulation point of the homomorphism thresholds of $r$-chromatic graphs. For maximal $H$-free graphs, we determine the VC-dimension threshold of every complete tripartite graph and prove that it is positive for every nonbipartite $H$, yielding in particular the exact value for every odd cycle. We also classify the chromatic threshold under an a priori VC-dimension bound. Together with known blowup-threshold results, our theorems reveal that $\delta_\chi,\delta_{\mathrm{hom}},\delta_{\mathrm{VC}}$, and $\delta_{\mathrm B}$ are pairwise distinct: bounded colorability, homomorphic compressibility, neighborhood complexity, and exact blowup structure are genuinely different forms of global simplicity. The proofs develop random and grid-based obstructions to bounded homomorphic images, saturated gadgets that preserve high VC-dimension under maximal completion, and a core-orientation method for raising minimum degree while preserving $H$-freeness.

We develop a notion of standard bases for subgroups of the restricted direct product $G^{(\mathbb{N}^n)}$ that are stable under translation by $\mathbb{N}^n$, where $G$ is an arbitrary finite group. We construct an algorithm that computes standard bases for such subgroups and use them to solve several algorithmic problems, including membership, saturation, and variable elimination. Our approach is inspired by Buchberger's algorithm and the theory of Gröbner bases for ideals in polynomial rings. Building on the standard bases and our solutions to the algorithmic problems above, we prove that Subgroup Membership is decidable in wreath products $G \wr \mathbb{Z}^n$ for finite $G$ and $n \in \mathbb{N}$.

Suppose that $f : M \rightarrow B$ is a normal $(q-1)$-smoothing of a $2q$-manifold $M$ over some $(B,\xi)$, where $q$ is even and $B$ is simply-connected. A diffeomorphism of $M$ over $B$ induces an automorphism of the extended intersection form (or “Q-form") of $f$, which consists of the intersection form of $M$ and the induced homomorphism $H_q(f)$. Assuming that $H_q(B)$ is free, we determine precisely which automorphisms can be realised by such diffeomorphisms. In particular, if $H_{q-1}(B)$ is also free, then we show that every automorphism can be realised. These results are obtained by studying a “two-sided" version of the extended surgery obstruction, which was introduced in earlier work of the author. As an application, we show that for a complex $q$-dimensional complete intersection, with $q > 2$ even, every automorphism of the cohomology ring is realised by a diffeomorphism.

We prove Breiman's conjecture under the first-moment assumption. Let $Y_1,Y_2,\ldots$ be iid nonnegative random variables with $\mathbb P\{Y_1>0\}>0$, normalized by their sum. If the resulting randomly weighted sum converges to a nondegenerate law for one fixed integrable, nonconstant mark distribution, then the tail of $Y_1$ is regularly varying. More generally, any full-sequence limit for one such mark, including a constant limit, determines the asymptotic regime of the ranked weights: one big jump, a Poisson$\unicode{x2013}$Dirichlet partition, or dust. It consequently determines the limit for every integrable mark, with convergence in the $1$-Wasserstein metric, and the limits of independently marked empirical measures. The inverse step is based on a countable power-sum theorem: signed Fourier$\unicode{x2013}$Mellin identities extract a positive limiting expected power sum from one nondegenerate marked limit, without a moment of order greater than one. A ratio$\unicode{x2013}$Tauberian argument then recovers the tail index. The same method classifies ratios formed from the marked jumps of a nonzero, unkilled, driftless subordinator at zero and at infinity, assuming infinite activity at zero. A nondegenerate limit is equivalent to regular variation of the Lévy tail with index in $(-1,0]$. A constant limit is equivalent to disappearance of the largest normalized jump, or, analytically, to slow variation of the integrated Lévy tail. The latter condition need not imply regular variation of the Lévy tail with index $-1$. A Cauchy-mark example shows that the conclusion can fail without integrability of the mark.

Hybrid automatic repeat request with chase combining (HARQ-CC) improves the reliability of slow fluid antenna multiple access (sFAMA) through multi-round combining. However, existing analysis has not fully utilized the structure of densely spaced and highly correlated fluid antenna system (FAS) ports to derive tractable per-round characterizations, thereby maintaining a computationally intensive process. This paper re-investigates downlink HARQ-CC-aided sFAMA with densely-spaced and highly-correlated FAS configuration. Under a spatial block correlation model, we first formulate two validity-corrected high-correlation approximations for the per-round selected-port signal-to-interference ratio (SIR) distribution and its Laplace–Stieltjes transform (LST): a Marcum-Q-kernel route and a lower-complexity step-threshold route. Closed-form expressions are also obtained for block-representative antenna selection (BR-AS) and fixed-position antenna (FPA). Then, the per-round characteristics are used to evaluate the multi-round accumulated-SIR distribution through SIR-domain Stieltjes convolution and numerical LST inversion, yielding the outage probability, average number of transmissions, and payload throughput. Numerical results show close agreement between the two evaluation methods. The analytical FAS results are conservative relative to simulation but preserve the performance trends and receiver ordering. The FAS receiver consistently outperforms the benchmarks, while the payload-throughput gain from increasing the HARQ transmission limit becomes marginal under severe multiuser interference.

Let $k=\mathbb{F}_q$. A polynomial representative of a finite-set map is a tuple of polynomials inducing that map on the rational-point grid. We prove a sharp distinction between a finite-set map and the geometry of its representatives. If $n=1$ or $n=2$, every polynomial representative of a permutation of $k^n$ has algebraically independent coordinates. If $n\geq3$, every set map $k^n\to k^n$ has both an algebraically independent and an algebraically dependent representative; the latter may be chosen to satisfy \[ F_2^q-F_2=(F_1^q-F_1)F_3. \] More generally, every map $k^m\to k^n$ has an algebraically independent representative exactly when $n\leq m$, while every such map has a dependent representative when $n\geq3$. The dependent construction combines an Artin–Schreier interpolation theorem, producing prescribed values by polynomials $A,B$ with $A^q-A\mid B^q-B$, with a three-coordinate suspension. For the identity on $k^3$, the scheme-theoretic image may be chosen to be exactly \[ V^q-V=(U^q-U)W, \] a smooth geometrically integral rational surface. We also establish low-degree and extension-field criteria forcing algebraic independence. An exact exhaustive computation additionally proves that every $2$-reduced representative of a permutation of $\mathbb{F}_2^3$ has algebraically independent coordinates.

We study the connection between AAA (triple A) algorithm and multi-point Padé approximation when the approximated functions belong to either the Nevanlinna-Pick class or are Pólya frequency series.

In low-regularity spacetimes and Lorentzian length spaces, achronal causal curves play the role of null (pre-)geodesics. Because of the lack of a geodesic equation, they do not come with a canonical parametrization. In this context, we discuss a notion of affine parametrization via limits of affinely parametrized timelike geodesics. However, we point out a major drawback: an example where this approximation procedures gives a non-unique result, in a way that even completeness or incompleteness of the limit null geodesic is not well-defined. This example involves discontinuous gluing of two Lorentzian metrics across a null hypersurface to give a well-behaved Lorentzian length space and as such is, just like the parametrization problem itself, manifestly Lorentzian. In view of this, we explore possibilities for a Penrose-type singularity theorem using timelike geodesics.

We discuss estimates for the maximal operator associated with the non-linear Fourier transform of an $L^2$-function on the half-line. For potentials with bounded dyadic $L^1$ masses we prove weak-type maximal and maximal fluctuation estimates on the sets where the spectral densities are bounded away from zero and three classical maximal functions of the spectral data are uniformly small; such sets exhaust almost all of the real line as the parameters of their definition relax.

Given a first-order theory $T$ (in discrete or continuous logic) and a Borel-definable global Keisler measure $\mu$ in $T$, we show that the following conditions are equivalent: $(i)$ $\mu$ is a frequency interpretation measure; $(ii)$ $\mu$ is definable and its canonical "random extension" $r_\mu$ is generically stable in the randomization theory $T^R$; $(iii)$ $\mu$ is "self-averaging". This result establishes a robust notion of generic stability for Keisler measures, which resolves a long-term research objective from previous work. The implications $(i)\Rightarrow(ii)\Rightarrow (iii)$ were previously established by the authors (for $T$ discrete). The primary focus of this paper is the reverse implications $(iii)\Rightarrow (ii)\Rightarrow(i)$. We also prove that generically stable measures are closed under Morley products, answering another well-known question that was open even in the case of types. These results are obtained through the use of AI models.

We consider convex admissible viscosity solutions of $\sigma_2(D^2u)=f>0$ in an open subset of $\mathbb R^n$, where $n\ge2$, $0<\alpha<1$, and $f\in C_{\mathrm{loc}}^{0,\alpha}$. We prove that every such solution belongs to $C_{\mathrm{loc}}^{2,\alpha}$. For solutions in $B_2$, we also prove a uniform $C^{2,\alpha}(B_{1/4})$ estimate under an $L^\infty$ bound for $u$, a positive lower bound for $f$, and a $C^{0,\alpha}$ bound for $f$.

Probabilistic models of Directed Acyclic Graphs (DAGs) with latent variables impose equality constraints on the observed data distribution beyond ordinary conditional independencies. These so-called Verma constraints arise in nested Markov models associated with Acyclic Directed Mixed Graphs, the latent projection of latent-variable DAGs. While nested Markov models have been extensively studied from the perspectives of graphical representation and causal identification, their implications for semiparametric efficiency theory remain less understood. We develop results toward establishing a semiparametric framework for statistical models defined by Verma constraints. Our key observation is that nested Markov constraints admit weighted conditional-moment representations under post-fixing distributions induced by graphical fixing operations. We show that fixing induces weighted orthogonality relations in L2(P), thereby converting Verma constraints into explicit tangent-space restrictions. Building on this representation, we characterize the tangent-space orthocomplement for models defined by a single nested Markov constraint through residualized weighted moment functions. This geometric formulation yields Hilbert-space characterizations of semiparametric efficient influence functions and efficiency bounds via orthogonal projection and equivalent minimum-variance formulations. We further discuss extensions to models involving multiple nested Markov constraints, for which we characterize a subspace of the orthocomplement as sums of the corresponding weighted orthogonality relations, while leaving the complete tangent-space characterization open. More broadly, our results connect nested graphical structure with semiparametric Hilbert-space geometry and provide a foundation for a general efficiency theory for nested Markov models. We illustrate the framework through several latent-variable DAGs.

This paper concerns the shock formation problem for the 3D steady supersonic potential equation of polytropic gases. The potential equation is described by a second order quasilinear wave equation $\displaystyle\sum_{i=1}^{3}\big[(\partial_i\Phi)^2 - c^2(\rho)\big]\partial_i^2\Phi + 2\displaystyle\sum_{1\le i<j\le 3}\partial_i\Phi\partial_j\Phi\partial_{ij}^2\Phi = 0$, where $x = (x^1,x^2,x^3)$, $(\partial_1,\partial_2,\partial_3)=(\partial_{x^1},\partial_{x^2},\partial_{x^3})$, $c(\rho) = \sqrt{p'(\rho)}$ is the sonic speed with $p(\rho)=A\rho^{\gamma}$ ($\gamma>1$), and $\partial_3\Phi > c(\rho)$. For the short pulse boundary data $\Phi|_{x^3=0} = \delta^\nu\Phi_0\big(\frac{r-1}{\delta},\omega\big)$ and $\partial_3\Phi|_{x^3=0}=q_0+\delta^{\nu-1}\Phi_1\big(\frac{r-1}{\delta},\omega\big)$ with $r=\sqrt{(x^1)^2+(x^2)^2}$, $\omega=\big(\frac{x^1}{r},\frac{x^2}{r}\big)\in\mathbb{S}$, $1<\nu<2$ and small $\delta>0$, it is shown that a shock will be formed in a finite $x^3$-distance as long as the boundary data are supersonic and satisfy $(\Phi_0,\Phi_1)\not\equiv 0$. This coincides with physical phenomenon that strong compression of supersonic polytropic gases yields shocks. One of our main ingredients is to find a good unknown so that the previously imposed compatibility conditions on the short pulse initial data are removed as well as the required weighted energy estimates in the existing literatures are derived. It is expected that the method here will be applied to study the shock formation problem with general short pulse initial data for the 3D steady supersonic Euler equations of polytropic gases.

Collision counts in a dilute gas of hard particles reflect the dependence between successive encounters. We study whether these fluctuations can be represented by a positive linear Hawkes process, in which each event adds a nonnegative contribution to the rate of future events. We construct such a model for particle pairs, require invariance under particle relabeling, and derive its stationary mean and covariance. We prove that, for a stationary positive Hawkes process with finitely many event types and finite expected total family sizes, the long-time growth rate of the third centered moment of any count formed by selecting event types is at least that of its variance. More generally, the long-time growth rates are nondecreasing with cumulant order. These predicted growth rates therefore cannot be matched simultaneously to the ones found in literature in approximations that follow velocity and collision count for a particle in an equilibrium gas. Simulations support the covariance predictions and illustrate the dependence of the physical comparison on the observation window. This restriction motivates models that retain the state left by a collision or allow past events to reduce future rates.

The embedding dimension of categorical predictors is usually selected through heuristic tuning, although it directly affects model complexity, approximation quality, and finite-sample generalization. This paper formulates embedding dimension selection as a constrained allocation problem. The main contribution is to show that embedding capacity can be allocated across heterogeneous categorical predictors according to an explicit approximation-estimation tradeoff. We characterize approximation error through the singular-value tail of the latent category representation, while estimation error increases with total embedding complexity. Under a fixed global embedding budget and a tractable approximation model, this leads to a closed-form allocation rule in which the dimension assigned to each predictor is proportional to the square root of its approximation value relative to its parameter cost. Simulation experiments support the proposed approximation-estimation interpretation and show that the allocation rule improves budget efficiency relative to standard uniform and cardinality-based heuristics, particularly when the budget is binding and predictor heterogeneity is substantial. A real-data healthcare application further shows improvements in predictive accuracy and probabilistic calibration. Overall, the results establish embedding-dimension allocation as a principled finite-sample optimization problem rather than a purely heuristic modeling choice.

We study spectral and variational properties of the (possibly) fractional Dirichlet Laplacian $\left(-\Delta_{n+k}\right)^s\!$, $s>0$, on bounded subsets of $\mathbb R^n\times\mathbb R^k$ whose extent in one or more directions becomes much larger than in the remaining ones. We first investigate the asymptotic behaviour of the first eigenvalue on stretching (or thinning) tubes, covering in particular the case of integers $s\geq 2$, for which $\left(-\Delta_{n+k}\right)^s\!$ reduces to a polyharmonic operator. Then we compute the $G$-limit of the rescaled operators, and the $\Gamma$-limit of the corresponding rescaled quadratic forms.

We study the notion of exceptional clique, a subset of a ring such that the difference of any two distinct elements of the subset is invertible. Motivated by applications in cryptography, our main focus is to determine the largest size of an exceptional clique in the ring $Mat_{n\times n}(\mathbb{Z})$ of square $n\times n$ matrices over the integers, for every $n$. We obtain several results for the question above, both in the general case and the “commutative” case where we additionally require that the elements in the clique commute with each other. As highlights, we prove that, at least for some values of $n$, the largest exceptional cliques in $Mat_{n\times n}(\mathbb{Z})$ are necessarily non-commutative; we then show that for an infinite family of $n$, there are non-commutative exceptional cliques of size $n^2$, and that for every $n$ there are commutative exceptional cliques of size $\frac23 n+O(n^{\theta})$, for a constant $\theta>\frac{11}{20}$.

Let $\cH$ be a separable complex Hilbert space with $\dim\cH\ge3$. We characterize the bounded bijective complex-linear maps $T$ on $\BH$ for which there exists a bijection $\psi$ of the set of conjugations,preserving commutativity in both directions, such that $T$ maps the space of $C$-symmetric operators onto the space of $\psi(C)$-symmetric operators for every conjugation $C$. We show that this condition is equivalent to the existence of a bijection $\varphi$ of the set of all orthonormal bases such that $T$ maps the set of all $(e_n)$-diagonal operators onto the set of all $\varphi((e_n))$-diagonal operators. We also characterize the corresponding preservers in dimension two using Pauli coordinates, without assuming the commutativity condition on $\psi$.

For the Landau-Lifshitz equation, the discrete linear systems obtained by our semi-implicit method possess the following properties: they are large-sparse systems with non-Hermitian yet positive-definite coefficient matrices. To solve these systems efficiently, we apply the Hermitian/skew-Hermitian splitting (HSS) method and its inexact variant (IHSS). Numerical experiments in one and three dimensions show that the spectral radius of the HSS iteration remains below its theoretical upper bound and strictly below one for the tested grid resolutions and damping parameters. Moreover, the theoretical bound closely follows the actual spectral radius, providing an accurate estimate of the convergence behavior. The IHSS results demonstrate effective convergence for the tested cases and show that its efficiency is sensitive to the splitting parameter. Overall, the two semi-implicit schemes exhibit comparable convergence behavior.

This paper studies the longest increasing subsequence (LIS) problem for sequences generated by dyadic-type chaotic interval maps. Starting from a single point $x\in[0,1)$ chosen uniformly at random, we form the order pattern of the first $N$ points of its orbit, with the doubling map as the basic model. Let $\lambda_1^{(N)}$ be the LIS length, equivalently the length of the first row of the Young diagram obtained by Schensted's insertion. We show that $\mathbb E[\lambda_1^{(N)}]/\sqrt N\to 2$, matching the leading asymptotics in the classical Ulam–Hammersley problem for uniform random permutations.

Let $(\Omega,\mu)$ and $(\Lambda,\lambda)$ be complete atomless localizable semifinite measure spaces. Suppose that $E(\Omega,\mu)$ and $F(\Lambda,\lambda)$ are real rearrangement-invariant Banach function spaces with order-continuous norms, in the Banach-lattice sense, and that neither norm is proportional to the $L_2$-norm. Every surjective real-linear isometry $U:E(\Omega,\mu)\longrightarrow F(\Lambda,\lambda)$ has the form $Uf=w\Phi(f)$, where $w$ has full support and $\Phi$ is induced by a complete measure-class Boolean isomorphism. Both factors are uniquely determined by $U$. Let $(\mathcal{M},\tau)$ and $(\mathcal{N},\nu)$ be atomless semifinite von Neumann algebras, and let $E(\mathcal{M},\tau)$ and $F(\mathcal{N},\nu)$ be symmetric operator spaces satisfying the same assumptions on their norms. Every surjective real-linear isometry $V:E(\mathcal{M},\tau)_{\mathrm{sa}}\longrightarrow F(\mathcal{N},\nu)_{\mathrm{sa}}$ has the form $V(x)=hJ(x)$, where $J:\mathcal{M}\longrightarrow\mathcal{N}$ is a normal surjective Jordan $*$-isomorphism and $h\in LS(\mathcal{Z}(\mathcal{N}))_{\mathrm{sa}}$ is central with full support; again, the two factors are unique. We also identify the bounded skew-Hermitian operators on the real self-adjoint part and derive commutative and noncommutative isometric forms of Mityagin's question.

We study second-order asymptotics for optimal thresholds in secretary-type problems. Under a unimodality assumption and a local affine expansion of the discrete payoff increments, we establish a general principle showing that sufficiently accurate rational approximations to the leading threshold proportion yield exact optimal integer thresholds. Continued-fraction convergents provide an immediate application. We also derive an asymptotic rounding law and apply the results to several variants of the secretary problem with fixed and random horizons.

Persistent entropy is the Shannon entropy of a persistence-based probability measure defined on a persistence diagram. However, its cross-entropy version is not naturally defined because two persistence diagrams generally have different event spaces. To bridge these event spaces, we combine a similarity function with persistence weighting to define an induced probability. The induced probability reflects information from one diagram on the event space of the other diagram and assigns unexplained probability mass to the unexplained event. Using the induced probability, we extend cross entropy to persistence diagrams, called persistent cross entropy (PCE). We establish the main properties of both the induced probability and PCE and prove stability theorems for both. Through three numerical studies, we show that PCE distinguishes diagrams with the same persistent entropy, separates causal directions in dynamical systems without constructing a joint persistent diagram, and can be used as a directional topology loss for knowledge distillation.

Identities asserting that an antisymmetrizer admits a determinantal expression are central to several proofs of counting formulas for alternating sign matrices. Antisymmetrizers appear also frequently in symmetric function theory as, for instance, the Hall-Littlewood polynomials can be realized via antisymmetrizers. In this paper, we establish a new identity of this type, thereby proving a conjecture of Lukas Riegler and one of the authors. We also elaborate on a related antisymmetrizer that may be pivotal in resolving a beautiful conjecture of Colomo and Pronko, and we formulate a version of their conjecture tailored to the suggested approach.

Optimization problems with geometric constraints have a broad range of applications, including machine learning, finance, and control. A powerful algorithmic tool to resolve these geometric constraints are constraint dissolving methods. To this end, we propose a framework for constraint dissolving mappings for nonconvex geometric constraints. Leveraging these, we develop a constraint dissolving inexact penalty method to solve optimization problems with general set-membership constraints and possibly nonconvex geometric constraints. We establish the convergence of the proposed algorithm and prove that every feasible accumulation point is Mordukhovich stationary. Notably, we rely only on mild asymptotic Mordukhovich regularity, which is significantly weaker than the constraint qualifications adopted in the existing literature on constraint dissolving methods. Numerical experiments addressing classical equality-, complementarity-, sparsity-, and low-rank constrained optimization problems demonstrate that the proposed method is competitive with the safeguarded augmented Lagrangian method in terms of solution quality and significantly outperforms the penalty decomposition method.

Fu and Nijhoff's direct linearization scheme for the KP hierarchy and its reductions employs a system of evolution equations of an infinite matrix $U$ with quadratic nonlinearity. Part of the matrix elements of $U$ can be identified with affine coordinates $w_{ij}$ of the top cell of the Sato Grassmannian. The evolution equations of these matrix elements are identical to the evolution equations of $w_{ij}$ representing the KP hierarchy in geometric terms. This geometric interpretation can be extended to other matrix elements of $U$ by introducing negative flows to the evolution equations of $U$. The extended system turns out to be substantially equivalent to the two-component KP hierarchy. The Cauchy matrix approach to the KP hierarchy can be explained in this geometric perspective. Multi-component generalizations of Fu and Nijhoff's nonlinear system are related to the AKNS and ASDYM hierarchies. Multi-component Sato Grassmannians show up therein as the relevant geometric structure.

We first present an introduction to the differential geometry of noncommutative algebras that carry a representation of a triangular Hopf algebra. Their noncommutativity is encoded in the universal R-matrix. The differential geometry is canonically constructed from these data. We then develop a new approach to the Bianchi identities for curvature and torsion of arbitrary connections (not necessarily bimodule connections). Using the Cartan calculus for connections, we prove the equivalence of their global formulations in terms of exterior forms and tensor fields. In particular, we obtain the noncommutative analogues of the first and second Bianchi identities familiar from general relativity.

A formula for the $\alpha$-K-energy functional for the Kähler-Yang-Mills (KYM) equations is provided in this paper. Using this formula and Chen's $\epsilon$-geodesic equation on the space of Kähler potentials, we prove that if a solution exists to the KYM equations for a simple vector bundle on a Kähler manifold with discrete automorphism group, then it is unique and the $\alpha$-K-energy is bounded from below. Inspired by the study of constant scalar curvature Kähler metrics, we introduce a coupled J-equation to study the $\alpha$-K energy functional.

Let $n\ge2$ and let $u$ be a smooth 2-convex and semi-convex solution of \[ \frac{\sigma _2(D^2u)}{\sigma _1(D^2u)}=f(x). \] We prove an interior Hessian estimate depending on the Lipschitz norm of $f$. The proof combines the integral approach of Chen–Jian–Zhou with the algebraic reduction of the quotient equation to a $\sigma _2$ structure. The main new point is a shifted algebraic inequality that yields a shifted trace Jacobi inequality in divergence form for $\log(\Delta u+a)$. We work with the linearized operator $G=(\Delta u-f)I-D^2u$ of the equivalent equation $\sigma_2(D^2u)=f \Delta u$. The almost divergence-free identity \(\partial_iG_{ij}=-f_j\) enables us to control the \(\Delta f\) term by integration by parts solely in terms of the Lipschitz norm of \(f\). A Legendre–Lewy transformation converts the resulting degenerate divergence-form equation into a uniformly elliptic one. The estimate then follows from a mean-value inequality together with a weighted energy argument. As an application, in dimension two we obtain interior $C^2$ regularity for convex viscosity solutions with positive Lipschitz right-hand side. Moreover, our counterexamples show that the Lipschitz regularity required of the right-hand side is optimal.

Iterative diagonalization is the dominant cost of plane-wave density-functional theory (DFT), with search-space orthogonalization scaling particularly quickly with problem size and the number of target states. We present a randomized block Davidson-type eigensolver that replaces Euclidean orthogonalization with randomized Gram-Schmidt in a sketched inner product, requiring only a single pass over the basis while keeping its conditioning bounded independently of the input vectors. This modification changes only the Rayleigh-Ritz step, which becomes a definite generalized Hermitian eigenproblem. Ritz extraction remains exact, preserving true Ritz pairs and the interlacing property that makes each band energy an upper bound on the true one. The method is implemented in mixed precision for CPUs and GPUs from a single Julia code, interfaces matrix-free with DFTK, and is released in the open-source RandESC library. On sparse test problems with a fixed number of eigenpairs, the sketched solver overtakes its deterministic counterpart beyond matrix dimensions of about $2\times 10^4$ and is $25\%$ faster at $5\times 10^5$. In full self-consistent field DFT calculations, however, both Davidson variants outperform the locally optimal block preconditioned conjugate gradient (LOBPCG) reference only by $5$ to $11\%$ in total time, while the additional benefit of sketching is limited. As the number of requested states grows with system size, orthogonalization savings are offset by the generalized eigenproblem. Therefore, the regime in which sketching pays off is set by how the number of wanted states scales with the problem dimension, not by the eigensolver as such.