Prime period sequences can serve as the fundamental tool to construct arbitrary composite period sequences. This is a review study of prime period perfect Gaussian integer sequence (PGIS). When cyclic group {1,2,…,N−1} can be partitioned into k cosets, where N=kf+1 is an odd prime number, the construction of a degree-(k+1) PGIS can be derived from either matching the flat magnitude spectrum criterion or making the sequence with ideal periodic autocorrelation function (PACF). This is a systematic approach of prime period N=kf+1 PGIS construction, and is applied to construct PGISs with degrees 1, 2, 3 and 5. However, for degrees larger than 3, matching either the flat magnitude spectrum or achieving the ideal PACF encounters a great challenge of solving a system of nonlinear constraint equations. To deal with this problem, the correlation and convolution operations can be applied upon PGIS of lower degrees to generate new PGIS with degree-4 and other higher degrees, e.g., 6, 7, 10, 11, 12, 14, 20 and 21 in this paper. In this convolution based scheme, both degree and pattern of a PGIS vary and can be indeterminant, which is rather nonsystematic compared with the systematic approach. The combination of systematic and nonsystematic schemes contributes the great efficiency for constructing abundant PGISs with various degrees and patterns for the associated applications.

The realization of a perfect absorber A = 1 with transmittance and reflectance T=R=0 by a thin metasurface is one of the hot topics in recent nanophotonics prompted by energy harvesting and sensor applications (A + R + T =1 is the energy conservation). Here we tested optical properties of over 400 structures of metal-insulator-metal (MIM) metasurfaces for a range of variation in thickness of insulator, the diameter of a disc and intra-disc distance experimentally and numerically. Conditions of a near-perfect absorption A > 95% with simultaneously occurring anti-reflection property (R < 5%) were experimentally determined. Differences between the bulk vs. nano-thin film properties at mid-IR of the used materials can be of interest for plasmonic multi-metal alloys and high entropy metals.

By combining a lattice subgroup diagram of Z_m with a weighted poset metric, we introduce a new weighted coordinates poset metric for codes over Z_m , called LS-poset metric. When I is an ideal in a poset, the concept of I-perfect codes with LS-poset metric is investigated. We obtain a Singleton bound for codes with LS-poset metric and define MDS codes. When the poset in poset metric is a chain, we provide sufficient conditions for a code with LS-poset metric to be r-perfect for some r ∈ N .

All-dielectric metasurfaces offer a low-loss alternative to plasmonic metasurfaces. We proposed the configuration for high-reflectivity all-dielectric metasurfaces based on single-crystal diamond (SCD) resonators on fused silica substrate and conducted simulations to optimize and analyse such configuration via FDTD solver. We utilized GMR as design principles to select the configuration and the substrate material, and analyzed the scattering properties of a single SCD resonator by multipole decomposition. Then we have demonstrated that both the cylindrical resonators in square lattice and frustum-shaped resonators in hexagonal lattice can achieve near-unity reflectivity (> 99.99%) and ultra-low absorption (< 0.001%) at 795 nm, the typical alkali-metal laser wavelength. Additionally, we have demonstrated that such design is quite tolerant to fabrication errors and further supports its potential for realistic applications. To expand the functionality of such devices across multiple wavelengths, a dual-band high-reflectivity meta-surfaces at 744 nm and 828 nm has also been designed. Our work is quite useful for designing diamond-based highly reflective mirrors, paving the way for low-loss all-dielectric reflective metasurfaces in high-power laser applications.

As solar energy is a low-cost and clean energy source, there has been a great deal of interest in how to harvest it. To absorb solar energy efficiently, we have designed a broadband metamaterial absorber based on the principle of Fabry–Pérot (FP) cavity and surface plasmon resonances (SPRs). We propose a broadband perfect absorber consisting of a four-layer structure of silica-tungsten-silica-titanium (SiO2-W-SiO2-Ti) for the incident light wavelength range of 300–2500 nm. The structure achieves perfect absorption of incident light in the wavelength range of 351.8–2465.0 nm (absorption >90%), with an average absorption of 96.3%. The advantage of our proposed structure is that it combines the characteristics of both high and broadband absorption and has a high overall absorption efficiency for solar radiation. It is also independent of polarization and insensitive to incident angle. We investigated how absorption was affected by different structures, materials, geometrical parameters, and refractive indices for different dielectric materials, and we explored the reasons for high absorption. This structure is refractory and ultra-thin, and it offers a good trade-off between bandwidth and absorption. It therefore has premium application prospects and value.

This paper definitively settles the longstanding conjecture regarding odd perfect numbers. A perfect number is one whose sum of divisors equals itself doubled. While Euclid's method for constructing even perfect numbers is well-known, the existence of odd ones has remained elusive. By employing elementary techniques and analyzing the properties of the divisor sum function, we conclusively prove that no odd perfect numbers exist.

A (2,6)-fullerene F is a 2-connected cubic planar graph whose faces are only 2-length and 6-length. Furthermore, it consists of exactly three 2-length faces by Euler's formula. The (2,6)-fullerene comes from Do\v{s}li\'{c}'s (k,6)-fullerene, a 2-connected 3-regular plane graph with only k-length faces and hexagons. Do\v{s}li\'{c} showed that the (k,6)-fullerenes only exist for k = 2, 3, 4 or 5, and some of the structural properties of (k,6)-fullerene for k = 3, 4 or 5 are studied. In this paper, we study the properties of (2,6)-fullerene. We obtain that the edge-connectivity of (2,6)-fullerenes is 2. Every (2,6)-fullerene is 1-extendable, but not 2-extendable. F is said to be k- resonant (k>0) if the deleting of any i (0=< i<= k) disjoint even faces of F results in a graph with at least one perfect matching. We have that every (2,6)-fullerene is 1-resonant. An edge set S of F is called an anti-Kekule set if F-S is connected and has no perfect matchings, where F-S denotes the subgraph obtained by deleting all edges in S from F. The anti-Kekule number of F, denoted by ak(F), is the cardinality of a smallest anti-Kekule set of F. We have that every (2,6)-fullerene F with |V(F)|>6 has anti-Kekule number 4. Further we mainly prove that there exists a (2,6)-fullerene F having f_{F} hexagonal faces, where f_{F} is related to the two parameters n, m.

A simple design of an ultrathin six-band polarization-insensitive terahertz perfect metamaterial absorber (PMMA) composed of a metal cross-cave-patch resonator (CCPR) placed over a ground plane was proposed and investigated numerically. The numerical simulation results demonstrate that the average absorption peaks are up to 95% at six resonance frequencies with high quality-factors (>65). In addition, the absorption properties can be kept stability for both normal incident transverse magnetic (TM) and transverse electric (TE) waves. The physical mechanism behind the observed high level absorption is illustrated by the electric and power loss density distributions. The different absorption mainly originates from the higher order multipolar and multipolar plasmon resonance of the structure, which are sharp different to the most previous studies of the PMMAs. Furthermore, the resonance absorption of the PMMA can be tunable by varying the geometric parameters of the unit cell.

In this paper, we have numerically investigated a metasurface based perfect absorber design established on the impedance matching phenomena. The paper comprises of two parts. In the first part, the device performance of the perfect absorber is studied which is composed of a silicon Nano-cylindrical meta-atoms periodically arranged on the thin gold layer. The device design is unique which works for both x-oriented and y-oriented polarized light, besides independent of the angle of incidence. In the second part of the paper, CO2 gas sensing application is explored by depositing a thin layer of functional host material polyhexamethylene biguanide polymer on the metasurface. The refractive index of the host material decreases due to the absorption of the CO2 gas. As a result, the resonance wavelength of the perfect absorber performs a prominent blueshift. With the help of the proposed sensor design based on metasurface, the CO2 gas concentration range of 0-524 ppm is detected. The maximum sensitivity of ~17.3 pm/ppm is acquired for the gas concentration of 434 ppm. The study presented in this work explores the opportunity of utilizing metasurface perfect absorber for gas sensing applications by employing the functional host materials.

In this paper we discuss a problem of generalization of binomial distributed triangle, that is sequence A287326 in OEIS. The main property of A287326 that it returns a perfect cube n as sum of n-th row terms over k; 0 ≤ k ≤ n−1 or 1 ≤ k ≤ n, by means of its symmetry. In this paper we have derived a similar triangles in order to receive powers m = 5; 7 as row items sum and generalized obtained results in order to receive every odd-powered monomial n^2m+1; m ≥ 0 as sum of row terms of corresponding triangle.

We propose the Hamiltonian formalism of Bianchi type 1 cosmological model for perfect fluid. We have considered both the equation of state parameter ω and the cosmological constant Λ as the function of volume V(t) which is defined by the product of three scale factors of the Bianchi type 1 line element. We propose a Lagrangian for the anisotropic Bianchi type-1 model in view of a variable mass moving in a variable potential . We can decompose the anisotropic expansion in terms of expansion and shearing motion by Lagrangian mechanism. We have considered a canonical transformation from expanding scale factor to scalar field ø which helps us to give the proper classical definition of that scalar field in terms of scale factors of the mentioned model. This definition helps us to explain the cosmological inflation. We have used large anisotropy(as in the cases of Bianchi models) and proved that cosmic inflation is not possible in such large anisotropy. Therefore we can conclude that the extent of anisotropy is less in case of our universe. Otherwise the inflation theory which explained the limitations of Big Bang cannot be resolved.Part II is contained with some analysis of the lagrangian ; derived in Part I ; on the quintessence model.

There is a skin allergy epidemic in the western world, and the rate of deterioration has increased significantly in the past 5-10 years. It is probable that there are many environmental contributing factors, yet some studies have linked it primarily to the rise in the use of synthetic chemical ingredients in modern cosmetics. Our challenge, therefore, was to find a mechanism to determine the effect these substances have on skin health, and whether they really are a primary cause of long term damage to the skin. The first problem is the lack of any definitive way to measure skin health. Motivated by the overwhelming evidence for a link between deficient gut flora and ill health, we decided to look at whether our skin microbiota could similarly be used as an indicator of skin health. Our research illustrates how it is microbiota diversity alone that can predict whether skin is healthy or not, after we revealed a complete lack of conclusive findings linking the presence or abundance of particular species of microbe to skin problems. This phenomenon is replicated throughout nature, where high biodiversity always leads to healthy ecosystems. ‘Caveman’ skin, untouched by modern civilisation, was far different to ‘western’ skin and displayed unprecedented levels of bacterial diversity. The less exposed communities were to western practices, the higher the skin diversity, which is clear evidence of an environmental factor in the developed world damaging skin. For the first time we propose benchmark values of diversity against which we can measure skin to determine how healthy it is. This gives us the ability to be able to predict which people are more likely to be prone to skin ailments, and start to test whether cosmetic ingredients and products are a main cause of the skin allergy epidemic.

We delve into the axion-dilaton model within the supergravity framework, with a specific focus on the intricacies of domain wall construction and stability. Employing holographic vitrification, we unravel the dynamics of domain wall formation in gauge theories featuring periodic vacuum structures. Our model, incorporating a QCD-like axion term and a stabilizing dilaton, undergoes scrutiny for conductivity variations under weak disorder. The investigation reveals the model's resilience, manifesting near-perfect conductivity under mild disorder conditions. However, the rigorous mathematical motivation for the holographic setup demands further elucidation. The scattered nature of our results prompts the necessity for a more systematic interpretation of QCD phenomena and conductivity transitions. This study contributes to the mathematical understanding of the axion-dilaton model's behavior, highlighting the imperative for a refined holographic framework and a more coherent interpretation of observed phenomena.