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In this paper, we show that C2a+1 × Pn and (C2n+1×Pn)+K¯p is harmonious for a ⩽ 1, and n is even.

Milnor computed the volumes of ideal hyperbolic prisms as part of an effort to construct 3-manifolds whose volumes are finite rational sums of the Lobachevsky function evaluated at rational multiples of pi. Motivated by these results and with an eye to related applications, we prove a volume formula for arbitrary ideal hyperbolic antiprisms, also called drums.

We introduce and study graded perfectoid rings as graded analogues of Scholze's (integral) perfectoid rings. We establish a categorical equivalence between graded perfectoid rings and graded perfect prisms, extending the Bhatt-Scholze's correspondence to the graded setting. We also construct the initial graded perfectoid cover of any graded semiperfectoid rings and prove a graded version of André's flatness lemma. These results lay the foundations for a graded theory of perfectoid rings.

We prove that for every \\(t\\in \\mathbb{N}\\), there exists \\(\\tau=\\tau(t)\\in \\mathbb{N}\\) such that every (theta, prism, \\(K_{1,t}\\))-free graph has tree independence number at most \\(\\tau\\) (where we allow \"prisms\" to have one path of length zero).

We introduce and study graded perfectoid rings as graded analogues of Scholze's (integral) perfectoid rings. We establish a categorical equivalence between graded perfectoid rings and graded perfect prisms, extending the Bhatt-Scholze's correspondence to the graded setting. We also construct the initial graded perfectoid cover of any graded semiperfectoid rings and prove a graded version of André's flatness lemma. These results lay the foundations for a graded theory of perfectoid rings.

Differential interference contrast (DIC) microscopy allows high-contrast, low-phototoxicity, and label-free imaging of transparent biological objects, and has been applied in the field of cellular morphology, cell segmentation, particle tracking, optical measurement and others. Commercial DIC microscopy based on Nomarski or Wollaston prism resorts to the interference of two polarized waves with a lateral differential offset (shear) and axial phase shift (bias). However, the shear generated by these prisms is limited to the rectilinear direction, unfortunately resulting in anisotropic contrast imaging. Here we propose an ultracompact metasurface-assisted isotropic DIC (i-DIC) microscopy based on a grand original pattern of radial shear interferometry, that converts the rectilinear shear into rotationally symmetric along radial direction, enabling single-shot isotropic imaging capabilities. The i-DIC presents a complementary fusion of typical meta-optics, traditional microscopes and integrated optical system, and showcases the promising and synergetic advancements in edge detection, particle motion tracking, and label-free cellular imaging. The authors present a metasurface-assisted isotropic DIC microscopy technique. It is based on an original pattern of radial shear interferometry, that converts rectilinear shear into rotationally symmetric radial shear, enabling single-shot isotropic imaging capabilities.

The spin-echo small-angle neutron scattering (SESANS) technique utilises a series of inclined magnetic fields before and after the sample to encode the scattering angle into the polarisation to obtain a much higher resolution than in conventional SANS. The analogous technique (spin echo modulated SANS (SEMSANS)) implements spin manipulations before the sample only to encode the scattering into an intensity modulation. The technique can be combined with SANS to expand the length scale range probed from 1 nm to microns. Using McStas we show that using a series of four magnetic Wollaston prisms in two orthogonal pairs with a 90° rotation can be utilised to create SEMSANS modulations in 2-D. These modulations can also be of different periods in each encoding direction. This method can be applied to anisotropic scattering samples. Also this allows for the simultaneous measurement at two orthogonal independent spin-echo lengths. This technique yields directly information about the structure of oriented structures.

This paper presents a new approach sensor, and its effectiveness as a chemical sensor using surface plasmon resonance (SPR) is evaluated. The suggested design consists of a ZnO-Silicon (Si) and silver (Ag) layer, above BaF 2 prism and a sensing layer that contains analytes on top. Si has a high refractive index (RI), a semiconducting nature, is compatible with microfabrication techniques and biocompatibility, and is suitable for a wide range of biological and chemical sensing applications. The angular interrogation method is used to analyze the Kretschmann configuration. Sellmeier equations are used to calculate reflectivity and other multilayer design parameters. In this sample, the sensitivity for the sensing medium 1.3264 (D 2 O), 1.35, and 1.36 (Acetone) is obtained to be 201.8°/RIU, 257.85°/RIU and 311°/RIU, respectively. The corresponding evaluated values for the figure of merit (FOM) are 60.54/RIU, 54.66/RIU, and 70.18/RIU, respectively. Moreover, the ZnO layer’s effect is utilized to compare the proposed sensor with other prisms like CsF, NFK51A, and BK7. In various applications, the proposed sensor performance is improved for RI of sensing medium detection between 1.3264 and 1.36.

Results and analysis of experimental data on testing of samples - prisms made of non-autoclaved cell concrete at non-multiply load are given. According to the test results, the equations for non-autoclave cell concretes, as well as the values of the coefficient of dynamic hardening and the coefficient of operating conditions mkr, taking into account the growth of durability of non-autoclave cell concrete, have been revealed.

Augmented reality (AR) display, which superimposes virtual images on ambient scene, can visually blend the physical world and the digital world and thus opens a new vista for human–machine interaction. AR display is considered as one of the next-generation display technologies and has been drawing huge attention from both academia and industry. Current AR display systems operate based on a combination of various refractive, reflective, and diffractive optical elements, such as lenses, prisms, mirrors, and gratings. Constrained by the underlying physical mechanisms, these conventional elements only provide limited light-field modulation capability and suffer from issues such as bulky volume and considerable dispersion, resulting in large size, severe chromatic aberration, and narrow field of view of the composed AR display system. Recent years have witnessed the emerging of a new type of optical elements—metasurfaces, which are planar arrays of subwavelength electromagnetic structures that feature an ultracompact footprint and flexible light-field modulation capability, and are widely believed to be an enabling tool for overcoming the limitations faced by current AR displays. Here, we aim to provide a comprehensive review on the recent development of metasurface-enabled AR display technology. We first familiarize readers with the fundamentals of AR display, covering its basic working principle, existing conventional-optics-based solutions, as well as the associated pros and cons. We then introduce the concept of optical metasurfaces, emphasizing typical operating mechanisms, and representative phase modulation methods. We elaborate on three kinds of metasurface devices, namely, metalenses, metacouplers, and metaholograms, which have empowered different forms of AR displays. Their physical principles, device designs, and the performance improvement of the associated AR displays are explained in details. In the end, we discuss the existing challenges of metasurface optics for AR display applications and provide our perspective on future research endeavors.