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We study projections in the bidual of a $$\rC^*$$-algebra $$B$$ that are null with respect to a subalgebra $$A$$, that is projections $$p\in B^{**}$$ satisfying $$|\phi|(p)=0$$ for every $$\phi\in B^*$$ annihilating $$A$$. In the separable case, $$A$$-null projections are precisely the peak projections in the bidual of $$A$$ at which the subalgebra $$A$$ interpolates the entire $$\rC^*$$-algebra $$B$$. These are analogues of null sets in classical function theory, on which several profound results rely. This motivates the development of a noncommutative variant, which we use to find appropriate `quantized' versions of some of these classical facts. Through a delicate generalization of a theorem of Varopoulos, we show that, roughly speaking, sufficiently regular interpolation projections are null precisely when their atomic parts are. As an application, we give alternative proofs and sharpenings of some recent peak interpolation results of Davidson and Hartz for algebras on Hilbert function spaces, also illuminating thereby how earlier noncommutative peak-interpolation theory may be applied. In another direction, given a convex subset of the state space of $$B$$, we characterize when the associated Riesz projection is null. This is then applied to various important topics in noncommutative function theory, such as the F.\& M. Riesz property, the existence of Lebesgue decompositions, the description of Henkin functionals, and Arveson's noncommutative Hardy spaces (maximal subdiagonal algebras). 

We initiate the theory of real noncommutative (nc) convex sets, the real case of the recent and profound complex theory developed by Davidson and Kennedy (2025). The present paper focuses on the real case of the topics from the first several sections of their memoir. Later results will be discussed in future papers. We develop here some of the infrastructure of real nc convexity, giving many foundational structural results for real operator systems and their associated nc convex sets, and elucidate how the complexification interacts with the basic convexity theory constructions. Several new features appear in the real case, including the novel notion of the complexification of a nc convex set. 

ABSTRACT We continue the development of real noncommutative (nc) convexity, building on the recent and profound complex theory of Davidson and Kennedy. This paper focuses on the theory of nc extreme points (and pure and maximal points) and the nc Choquet boundary in the real setting, as well as on the theory of real nc convex and semicontinuous functions and real nc convex envelopes. Our main emphasis is on how these notions interact with complexification. In particular, parts of the paper analyze in detail how various notions of “extreme” or “maximal” relate to our earlier concept of the complexification of a convex set. Several new features emerge in the real case, especially in the later sections, including the novel notions of the complexification of an nc convex function and of the complexification of the convex envelope of an nc function. With an Appendix by Russell. 

Loess-paleosol sequences in eolian deposits on the Snake River Plain, Idaho, western United States, preserve records of late Pleistocene to Holocene glacial-interglacial cycles. Here, we examine climate-driven changes in the timing and rate of loess accretion, soil formation, and pedogenic carbonate accumulation in the eastern and central Snake River Plain. High rates of loess deposition often correspond with dry, cold, and windy conditions and/or high sediment supply, while soil formation indicates landscape stability. Optically stimulated luminescence dating of four loess-soil sequences provides a record of the timing of loess deposition, and radiocarbon dating (14C) of soil inorganic carbon provides ages for soil formation and pedogenic carbonate precipitation. On the central Snake River Plain, marine isotope stage (MIS) 3 is marked by loess deposition between ca. 50 ka and 33 ka, and pedogenic carbonate accumulation between ca. 42 ka calibrated years before present (cal. B.P.) and 31 ka cal. B.P. During MIS 2, loess deposition between ca. 21 ka and 15 ka is contemporaneous with pedogenic carbonate precipitation. MIS 2 is characterized by diffuse pedogenic carbonate formation and rapid loess accumulation. Both the eastern and central Snake River Plain study sites experienced high rates of loess deposition during MIS 2; however, the eastern Snake River Plain had almost three times faster rates of loess accumulation (0.56−0.63 m/k.y. from ca. 25 ka to 17 ka) as compared to the central Snake River Plain (0.18−0.26 m/k.y. from ca. 21 ka to 15 ka). During MIS 1, loess deposition on the central Snake River Plain ca. 11 ka was coincident with pedogenic carbonate formation ca. 10 ka cal. B.P. MIS 3 and MIS 1 are characterized by A horizons and well-developed Bk horizons, indicating soil stability and lower rates of loess accumulation during these intervals. This study highlights the role of late Quaternary−scale climatic variability on rates and amounts of soil inorganic carbon storage. 

It has recently been realised that illumination by intensely powerful radiation is not the only path to a nonlinear optical response by a given material. As demonstrated for a layer of indium tin oxide (ITO), strong nonlinear effects can be observed in a material for illuminating fields of quite moderate strength in a neighbourhood of the wavelengths which render it an epsilon-near-zero (ENZ) material. Inspired by these observations we introduce, discuss and analyse a rather different formulation of the governing equations for the Capretti experiment with a view towards robust and highly accurate numerical simulation. By contrast to volumetric algorithms which are greatly disadvantaged for the piecewise homogeneous geometries we consider, surface methods provide optimal performance as they only consider interfacial unknowns. In this contribution, we study an interfacial approach which is based upon Dirichlet–Neumann operators (DNOs). We show that, for a layer of nonlinear Kerr medium, the DNO is not only well-defined, but also analytic with respect to all of its independent variables. Our method of proof is perturbative in nature and suggests several new avenues of investigation, including stable numerical simulation, and how one would include the effects of periodic deformations of the layer interfaces into both theory and numerical simulation of the resulting DNOs. This article is part of the theme issue ‘Analytically grounded full-wave methods for advances in computational electromagnetics’. 

As experiments advance to record from tens of thousands of neurons, statistical physics provides a framework for understanding how collective activity emerges from networks of fine-scale correlations. While modeling these populations is tractable in loop-free networks, neural circuitry inherently contains feedback loops of connectivity. Here, for a class of networks with loops, we present an exact solution to the maximum entropy problem that scales to very large systems. This solution provides direct access to information-theoretic measures like the entropy of the model and the information contained in correlations, which are usually inaccessible at large scales. In turn, this allows us to search for the optimal network of correlations that contains the maximum information about population activity. Applying these methods to 45 recordings of approximately 10,000 neurons in the mouse visual system, we demonstrate that our framework captures more information—providing a better description of the population—than existing methods without loops. For a given population, our models perform even better during visual stimulation than spontaneous activity; however, the inferred interactions overlap significantly, suggesting an underlying neural circuitry that remains consistent across stimuli. Generally, we construct an optimized framework for studying the statistical physics of large neural populations, with future applications extending to other biological networks.