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Recent investigations suggest that the discrete linear unitary group $U\left(N\right)$can be represented by interlacing a finite sequence of diagonal phase operations with an intervening unitary operator. However, despite rigorous numerical justifications, no formal proof has been provided. Here, we show that elements of $U\left(N\right)$can be decomposed into a sequence of $N$-parameter phases alternating with one-parameter propagators of a lattice Hamiltonian. The proof is based on building a Lie group by alternating these two operators and showing its completeness to represent $U\left(N\right)$for a finite number of layers. This is numerically verified by using Haar random matrices as targets, showing a convergence for exactly $N$layers. As a specific application, we propose an integrated all-optical logic gate device that performs OR, NAND, XOR, and XAND tasks within a lossless and passive optical circuit design. 

ABSTRACT Intermittent streams are prevalent worldwide, yet the understanding of drivers of their changing flow patterns remains incomplete. We examined hydrological changes spanning four decades (1982–2020) in Kings Creek, an intermittent grassland stream within the Konza Prairie Biological Station in Kansas, USA. We analysed streamflow data from a US Geological Survey gauge on Kings Creek and three upstream Long Term Ecological Reasearch (LTER) sub‐watersheds with annual, biennial or quadrennial burn frequencies and linked trajectories of woody encroachment to increased evapotranspiration and changes in streamflow. Riparian woody cover doubled in the annually and biannually burned sub‐watersheds and sevenfold in the quadrennially burned watersheds. We observed significant decreases (84%) in daily discharge and number of annual flow days (55%) at the downstream USGS Kings Creek gauge, with similar changes in the LTER sub‐watersheds. The changing riparian cover, propelled by the regional expansion of woody plants, contributed to decreased streamflow by amplifying actual evapotranspiration (ET). Seasonal assessments underscored the critical influence of late summer conditions (July–September), under which increases in ET were linked to rising temperatures and increased evapotranspiration by riparian cover. Our results highlight the significant hydrological impacts of woody encroachment in grasslands and emphasize the importance of long‐term ecohydrological monitoring in unravelling the interplay between climate and vegetation as controls on the hyper‐variable flow patterns in this intermittent stream. Predicting and managing hydrological impacts on the flow of intermittent grassland rivers and streams worldwide requires accounting for the effects of accelerating woody encroachment. 

Abstract Detailed geochronology from two compositionally distinct generations of dikes and sills intruded into the Alta metamorphic aureole, north‐central Utah, complement previous geochronologic studies from the Alta stock, providing information on the timing of magmatism and the nature of emplacement. Uranium/thorium‐lead dates and chemistry were measured in zircon and monazite from these intrusions and associated reaction selvages in hornfels by split‐stream laser ablation techniques. Concordant zircon U‐Pb dates (n = 532) define a dispersed population of dates that range from ∼38 to 32 Ma. Monazite Th‐Pb dates (n = 888) from granodioritic compositions range from ∼40 to 32 Ma. Evaluation of²⁰⁸Pb/²³²Th and²⁰⁷Pb/²⁰⁶Pb‐corrected dates with respect to common Pb, U and Th/U values allows rigorous evaluation of the effects of excess²⁰⁶Pb in these young monazites, yielding concordant²⁰⁸Pb/²³²Th and²⁰⁷Pb/²⁰⁶Pb‐corrected dates in monazites from the granodiorite, consistent with zircon dates from the same thin sections. Leucogranite sills and dikes, which cross‐cut the older granodiorite, have younger monazite dates from ∼33 to 28 Ma. Elevated heavy rare earth element concentrations and trends of larger negative Eu anomalies in the youngest monazites suggest crystallization from an evolved melt. Integration of these new geochronology results and field relationships with prior results from the Alta stock indicate the granodiorite represents the oldest material emplaced in the Alta system. Leucogranite aplite/pegmatite dikes and sills in the inner Alta aureole were emplaced during the final stage of Alta stock construction by injection of evolved water‐rich magmas.