We investigate the regularity of the viscosity solutions to a class of degenerate/singular fully nonlinear elliptic equations with Hamiltonian terms. To overcome the difficulty caused by the simultaneous presence of the general degenerate/singular gradient terms and Hamiltonian terms, we analyze the coupled interplay between the degeneracy/singularity law and the growth of Hamiltonian terms and establish lower regularity results. Finally, we obtain sharp interior C^1,α regularity estimates via a geometric tangential method.
In this study, we explore a robust testing procedure for the high-dimensional location parameters testing problem. Initially, we introduce a spatial-sign based max-type test statistic, which exhibits excellent performance for sparse alternatives. Subsequently, we demonstrate the asymptotic independence between this max-type test statistic and the spatial-sign based sum-type test statistic (Feng and Sun, 2016). Building on this, we propose a spatial-sign based max-sum type testing procedure, which shows remarkable performance under varying signal sparsity. Our simulation studies underscore the superior performance of the procedures we propose.
Talagrand's correlation inequality [25] provides quantitative lower bounds on the covariance of two increasing Boolean functions in terms of their coordinate influences, but, in general, a logarithmic loss is necessary. Motivated by a question of Kalai, Keller and Mossel [14, Problem 6.1], we identify a natural log-free regime. We prove that if two increasing Boolean functions on {0, 1}nare either both submodular or both supermodular, then E[fg]-E[f]E[g] >= 1/4 center dot Sigma(n)(i=1) Infi[f]Inf(i)[g], where the constant 1/4 is optimal. We also prove a real-valued extension: for two functions with the same second-difference sign, the covariance is bounded below by the sum of products of their Level-1 Fourier coefficients. As a consequence, we verify the Friedgut-Kahn-Kalai-Keller spectral conjecture [11, Conjecture 5.8] in this structured setting. The proofs combine a heat-semigroup representation based on second-order discrete derivatives with an independent induction argument for the Boolean case.
In this paper, by using a dilation technique on Schauder frames, we extend Godefroy and Kalton's approximation theorem (1997), and obtain that a separable Banach space has the lambda-unconditional bounded approximation property (lambda-UBAP) if and only if, for all epsilon > 0, X isometrically embeds as a 1-complemented subspace of a Banach space Y epsilon with a (lambda+epsilon)-unconditional finite-dimensional decomposition ((lambda + epsilon)-UFDD). As applications on ball-covering property (BCP) (Cheng, 2006) of operator spaces, also based on the relationship between the a-UBAP and block unconditional Schauder frames, we prove that if X-& lowast;, Y are separable and (1) X or Y has the a-reverse metric approximation property (lambda-RMAP) for some lambda > 1; or (2) X or Y has an approximating sequence {Sn}(infinity)(n=1) such that lim(n) ||id-2S(n) ||<3/2, then the space of bounded linear operators B(X,Y) has the uniform ball-covering property (UBCP). Actually, we give a uniformly quantitative estimation for the renormed spaces. We show that if X-& lowast;, Y are separable and X or Y has the (2-epsilon)-UBAP for any epsilon> 0, then for all 1-epsilon/2 < alpha <= 1, the renormed space Z(alpha) = (B(X, Y ), ||& centerdot;|| (alpha)) has the (2 alpha, 2 alpha + epsilon - 2-sigma)-UBCP for all 0 < sigma < 2 alpha + epsilon - 2. Furthermore, we point out the connections between the UBCP, u-ideals and the ball intersection property (BIP). (c) 2026 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
The Helmholtzian matrix of a graph G=(V(G),E(G)) is a graph-theoretic analogue of the vector Laplacian (or Helmholtz operator) introduced in S. Li, L. Lu and J. F. Wang, A graph discretization of vector Laplacian, Discrete Appl. Math. 379 (2026), 446–460. Motivated by applications of the graph Helmholtzian to simplicial networks, we investigate its basic spectral properties. The matrix is naturally indexed by the edges of G and is positive semidefinite for every choice of orientations. We characterize when an edge orientation makes it entrywise nonnegative and relate its irreducibility to an associated signed graph with loops. We prove that the Helmholtzian spectrum is independent of the edge and triangle orientations and establish eigenvalue interlacing inequalities for induced subgraphs. We also show that the nonzero Helmholtzian spectrum, with multiplicities, is the multiset union of the nonzero Laplacian spectrum of G and the nonzero spectrum of a shifted adjacency matrix of the signed graph on the triangles of G. These results connect the Helmholtzian spectrum with ordinary Laplacian spectra, signed graphs, and combinatorial Hodge theory.
