Rayleigh quotient minimization deals with optimizing a quadratic homogeneous function over a sphere. Its critical points correspond to the normalized eigenvectors of the symmetric matrix associated with the quadratic form. In this paper, we consider a homogeneous polynomial objective function f over a sphere, a projective algebraic variety X, and we study the X-eigenpoints of f, which are classes of critical points of f constrained to the sphere and the affine cone over X. The number of X-eigenpoints of a generic polynomial f is the Rayleigh-Ritz degree of X. This invariant is a version of the Euclidean distance degree of a Veronese embedding of X. We provide concrete formulas in various scenarios, including those involving varieties of rank-one tensors.
Since its original formulation by Isaac Newton in 1685, the problem of determining bodies of minimal resistance moving through a fluid has been one of the classical problems in the calculus of variations. Initially posed for cylindrically symmetric bodies, the problem was later extended to general convex shapes, as explored in \cite{BK93}, \cite{BFK95}. Since then, this broader formulation has inspired a number of articles dedicated to the study of the geometric and analytical properties of optimal shapes, with particular attention to their structure, regularity, and behavior under various constraints. In this article, we provide a comprehensive overview of the principal results that have been established, highlighting the main theoretical advancements. Furthermore, we introduce some new directions of research, some of which were described in \cite{P12}, that offer promising perspectives for future investigation.
In this paper we study the class of split Nakamura manifolds, which are a type of solvmanifolds generalizing Nakamura's threefold, defined as quotients of the semidirect product & Copf;n & rtimes;rho & Copf; by a lattice. We discuss their de Rham and Dolbeault cohomology, with emphasis on the degeneration of the Fr & ouml;licher spectral sequence and the 88 & strns;-Lemma, and their deformations. Finally, we describe their automorphism group in detail. (c) 2026 The Author(s). Published by Elsevier Masson SAS. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
Decentralised Finance (DeFi) protocols are usually built by composing a variety of smart contracts. Although composability is a key driver of DeFi innovation, it also introduces security risks: adversaries may exploit interactions between newly deployed contracts and the existing blockchain state to cause economic losses. We introduce a formal security notion for DeFi composability, called MEV non-interference, which requires that interactions with the existing blockchain state do not increase the Maximal Extractable Value (MEV) obtainable from a set of newly deployed contracts. To formalise this notion, we define local MEV, a new measure of economic attacks that quantifies the losses incurred by a given set of victim contracts. We study MEV non-interference under two adversarial models, corresponding to bounded and unbounded adversarial wealth, and establish sufficient conditions and locality principles that enable modular reasoning about secure composability. We apply our framework to representative DeFi compositions including exchanges, automated market makers, options, lending pools, routers, and arbitrage contracts, and show that it distinguishes secure compositions from vulnerable ones. Our results provide a formal foundation for analysing the economic security of DeFi systems built through smart contract composition.
In this work, we establish a sharp form of a nonlocal quantitative isoperimetric inequality involving the barycentric asymmetry for convex sets. This result can be seen as the nonlocal analogue of the one obtained by Fuglede in 1993.
