This paper is concerned with representing sums of finite products of Chebyshev polynomials of the second kind and of Fibonacci polynomials in terms of several classical orthogonal polynomials. Indeed, by explicit computations each of them is expressed as linear combinations of Hermite, generalized Laguerre, Legendre, Gegenbauer and Jacobi polynomials which involve the hypergeometric functions ${}_1 F_1$ and ${}_2 F_1$.

We use a recently proposed formulation of quantum mechanics based, not on potential functions but rather, on orthogonal energy polynomials. In this context, the most important building block of a quantum mechanical system, which is the wavefunction at a given energy, is expressed as pointwise convergent series of square integrable functions in configuration space. The expansion coefficients of the series are orthogonal polynomials in the energy; they contain all physical information about the system. No reference is made at all to the usual potential function. We consider, in this new formulation, few representative problems at the level of undergraduate students who took at least two courses in quantum mechanics and are familiar with the basics of orthogonal polynomials. The objective is to demonstrate the viability of this formulation of quantum mechanics and its power in generating rich energy spectra illustrating the physical significance of these energy polynomials in the description of a quantum system. To assist students, partial solutions are given in an appendix as tables and figures.

Using an algebraic method for solving the wave equation in quantum mechanics, we encountered a new class of orthogonal polynomials on the real line. One of these is a four-parameter polynomial with a discrete spectrum. Another that appeared while solving a Heun-type equation has a mix of continuous and discrete spectra. Based on these results and on our recent study of the solution space of an ordinary differential equation of the second kind with four singular points, we introduce a modification of the hypergeometric polynomials in the Askey scheme. Up to now, all of these polynomials are defined only by their three-term recursion relations and initial values. However, their other properties like the weight function, generating function, orthogonality, Rodrigues-type formula, etc. are yet to be derived analytically. This is an open problem in orthogonal polynomials.

In this paper, we proposed the numerical method called the variational iteration orthogonal collocation method (VIOCM), for the approximate solution of the deadly Corona virus model using Mamadu-Njoseh polynomials as basis functions. The proposed method is an elegant mixture of the variational iteration method (VIM) and the orthogonal collocation method (OCM). It was observed that the proposed method converges rapidly to the exact solution even as N increases. This suggests that the use of orthogonal polynomials as trial functions for the SEIR model is indeed an effective approximant as it produces the analytic solution at just few iterations. Resulting numerical evidences were compared with the standard variational iteration method as available in literature. All computational frameworks were executed with MAPLE software.

The orthogonal matrix with cube be symmetric is a common class of matrices with important properties. We pay attention to this kind of matrices. By using the close relationship between the eigenvalues of a matrix and the trace of its power, we obtain the algorithm for its all possible different eigenvalues and multiplicities. The calculation formula is expressed only by the trace of a matrix and its power, avoiding solving the characteristic polynomial. The method is simple and practical. Furthermore, a new essential characterization for the sum of orthogonal matrix pairs being orthogonal is given as well.

Kronecker products of matrices have some striking operation properties, one of which is called the mixed-product property $(A \!\otimes\! B)(C \!\otimes\! D) = AC \!\otimes\! BD$. In view of this property, the two-term Kronecker product $A_1 \otimes A_2$ can be rewritten as $A_1 \otimes A_2 = (A_1 \otimes I_{m_2})(I_{m_1} \otimes A_2)$ of dilation forms of $A_1$ and $A_2$, and three-term Kronecker product $A_1 \otimes A_2 \otimes A_3$ can be rewritten as the products $A_1 \otimes A_2 \otimes A_3 = (A_1\otimes I_{m_2} \otimes I_{m_3})(I_{m_1} \otimes A_2 \otimes I_{m_3})(I_{m_1} \otimes I_{m_2} \otimes A_3)$ of the dilation forms of $A_1$, $A_2$, and $A_3$, respectively, where the matrices on the right-hand sides of the two factorizations are commutative. In this note, we approach the commutative Kronecker products on the right-hand sides of the two factorization equalities, and present a variety of new and useful analytical formulas for calculating the ranks, dimensions, orthogonal projectors, and ranges of matrices composed of these Kronecker products.

Frames are more stable as compared to bases under the action of a bounded linear operator. Sums of different frames under the action of a bounded linear operator are studied with the help of analysis, synthesis and frame operators. A simple construction of frames from the existing ones under the action of such an operator is presented here. It is shown that a frame can be added to its alternate dual frames, yielding a new frame. It is also shown that the sum of a pair of orthogonal frames is a frame. This provides an easy construction of a frame where the frame bounds can be computed easily. Moreover, for a pair of orthogonal frames, the necessary and sufficient condition is presented for their alternate dual frames to be orthogonal. This allows for an easy construction of a large number of new frames.

In view of increasing concerns over non-renewable resource depletion and waste management, this study aimed to apply the Taguchi optimization technique to determine the process conditions for producing bricks by incorporating thin film transition liquid crystal displays (TFT-LCD) waste glass powder with reservoir sediments. An orthogonal array L₁₆(4⁵) was adopted, which consisted of five controllable four-level factors (i.e., cullet content, drying method, preheat time, sintering temperature, and error). Moreover, the analysis of variance method was used to explore the effects of the experimental factors on the density, water absorption, shrinkage ratio, loss of ignition, porosity, and compressive strength of the fired bricks. The microstructures of the fired specimens were investigated by scanning electron microscopy. Then, the large-scale production techniques for fired bricks containing recycled TFT-LCD glass cullet and reservoir sediments was developed in a commercially available tunnel kiln. The test results showed that the structure of fired specimen was loose at a sintering temperature ranged from 900–950 °C. However, the fired specimen showed a significant densification at the sintering temperature of 1050 °C. In addition, Taguchi method is a feasible approach for optimizing process condition of brick using recycled TFT-LCD glass cullet and reservoir sediments and it significantly reduces the number of tests. On the other hand, the characteristics of fired bricks developed in the tunnel kiln were in compliance with Chinese National Standards class Ι building bricks criteria.

Let G be a finite group. The prime graph Γ(G) of G is defined as follows: The set of vertices of Γ(G) is the set of prime divisors of |G| and two distinct vertices p and p' are connected in Γ(G), whenever G has an element of order pp'. A non-abelian simple group P is called recognizable by prime graph if for any finite group G with Γ(G)=Γ(P), G has a composition factor isomorphic to P. In [4] proved finite simple groups ²D_n(q), where n ≠ 4k are quasirecognizable by prime graph. Now in this paper we discuss the quasirecognizability by prime graph of the simple groups ²D_2k(q), where k ≥ 9 and q is a prime power less than 10⁵.

Limonene is a valuable monoterpenoid that with diverse applications. Microbial biosynthesis offers a sustainable alternative to plant-based extraction, which is prone to hinderance by climate change and acreage. In the present study, the effects on limonene production of carbon source, inoculum, working volume, temperature, pH, dodecane coverage, and media were investigated with engineered strains of Rhodosporidium toruloides. Through single-factor and orthogonal experiments, the limonene titer was improved from 52.2 to 358.1 mg/L on 250-mL flask, a 586% increase. The optimized medium was MM, with a working volume of 50 mL, inoculum OD600=0.6, 22 °C, pH 6, and dodecane coverage (v/v) of 20%. The results here demonstrated that the R. toruloides could be explored as an efficient limonene producer and facilitate other monoterpenoids biosynthesis with R. toruloides.

We propose two methods for obtaining estimations on the minimum distance and covering radius of orthogonal arrays. Both methods are based on knowledge about the (feasible) sets of distance distributions of orthogonal arrays with given length, cardinality, factors and strength. New bounds are presented either in analytic form and as products of an ongoing project for computation and investigation of the possible distance distributions of orthogonal arrays with parameters in doable ranges.

Smart materials represent an elegant class of (macro)-molecules endowed with the ability to react to chemical/physical changes in the environment. Herein, we prepared new photo responsive azobenzenes possessing halogen bond donor groups. The X-ray structures of two molecules highlight supramolecular organizations governed by unusual noncovalent bonds. In azo dye I-azo-NO2, the nitro group is engaged in orthogonal H···O···I halogen and hydrogen bonding, linking the units in parallel undulating chains. As far as compound I-azo-NH-MMA is concerned, a non-centrosymmetric pattern is formed due to a very rare I···π interaction involving the alkene group supplemented by hydrogen bonds. The Cambridge Structural Database contains only four structures showing the same I···CH₂=C contact. For all compounds, an ¹⁹F NMR spectroscopic analysis confirms the formation of halogen bonds in solution through a recognition process with chloride anion, and the reversible photo-responsiveness is demonstrated upon exposing a solution to UV light irradiation. Finally, the intermediate I-azo-NH2 also shows a pronounced color change due to pH variation. These azobenzenes are thereby attractive building blocks to design future multi-stimuli responsive materials for highly functional devices.

The security of several full homomorphic encryption (FHE) schemes depends on the hardness of the approximate common divisor (ACD) problem. The analysis of attack and defense against the system is one of the frontiers of cryptography research. In this paper, the performance of existing algorithms, including orthogonal lattice, simultaneous diophantine approximation, multivariate polynomial and sample pre-processing are reviewed and analyzed for solving the ACD problem. Orthogonal lattice (OL) algorithms are divided into two categories (OL-$\land$ and OL-$\vee$) for the first time. And an improved algorithm of OL-$\vee$ is presented to solve the GACD problem. This new algorithm works well in polynomial time if the parameter satisfies certain conditions. Compared with Ding and Tao's OL algorithm, the lattice reduction algorithm is used only once, and when the error vector $\mathbf{r}$ is recovered in Ding et al.'s OL algorithm, the possible difference between the restored and the true value of $p$ is given. It is helpful to expand the scope of OL attacks.

Non-orthogonal multiple access (NOMA) has been investigated to support massive connectivity for Internet-of-things (IoT) networks. However, since most IoT devices suffer from limited power and decoding capabilities, it is not desirable to pair a large number of devices simultaneously, which encourages two-user NOMA grouping. Additionally, most existing techniques have not considered the diversity in the target QoS of IoT devices, which may lead to spectrum inefficiency. Few investigations have partially considered that issue by using an order-based power allocation (OPA) approach, where the power is allocated according to the order to the user's target throughput within a priority-based NOMA (PNOMA) group. However, this does not fully capture the effects of diversity in the values of the users' target throughputs. In this work, we handle both problems by considering a throughput-based power allocation (TPA) approach, that captures the QoS diversity, within a three-users PNOMA group as a compromise between spectral efficiency and complexity. Specifically, we investigate the performance of a time-division PNOMA (TD-PNOMA) scheme, where the transmission time is divided into two-time slots with two-users per PNOMA group. The performance of such TD-PNOMA is compared with a fully PNOMA (F-PNOMA) scheme, where the three users share the whole transmission time, in terms of the ergodic capacity under imperfect successive interference cancellation (SIC). The results reveal the superiority of TPA compared with OPA approach in both schemes, besides that the throughput of both schemes can outperform each other under imperfect SIC based on the transmit signal-to-noise ratio and the deployment scenarios.

This note shows that the well-known reverse order law $(AB)^{\dag} = B^{\dag}A^{\dag}$ for the Moore--Penrose inverse of matrix product is equivalent to many other equalities for that are composed of multiple products $(AB)^{\dag}$ and $B^{\dag}A^{\dag}$ by means of the definition of the Moore--Penrose inverse and orthogonal projector theory.

Large scale coherent structures in atmospheric boundary layer (ABL) are known to contribute to the power generation in wind farms. In the current paper, we perform a detailed analysis of the large scale structures in a finite sized wind turbine canopy using modal analysis from three dimensional proper orthogonal decomposition (POD). While POD analysis sheds light on the large scale coherent modes and scaling laws of the eigenspectra, we also observe a slow convergence of the spectral trends with the available number of snapshots. Since the finite sized array is periodic in the spanwise direction, we propose to adapt a novel approach of performing POD analysis of the spanwise/lateral Fourier transformed velocity snapshots instead of the snapshots themselves. This methodology not only helps in decoupling the length scales in the spanwise and the streamwise direction when studying the energetic coherent modes, but also provides a detailed guidance towards understanding the convergence of the eigenspectra. In particular, the Fourier-POD eigenspectra helps us illustrate if the dominant scaling laws observed in 3D POD are actually contributed by the laterally wider or thinner structures and provide more detailed insight on the structures themselves. We use the database from our previous large eddy simulation (LES) studies on finite-sized wind farms which uses wall-modeled LES for modeling the Atmospheric boundary layer laws, and actuator lines for the turbine blades. Understanding the behaviour of such structures would not only help better assess reduced order models (ROM) for forecasting the flow and power generation but would also play a vital role in improving the decision making abilities in wind farm optimization algorithms in future. Additionally, this study also provides guidance for better understanding the POD analysis in the turbulence and wind farm community.

We consider the second order differential equation y'' + ω^(2)ρ(x)y = 0 where ω is a positive parameter. The principal concern here is to find conditions on the function ρ^(−1/2)(x), which ensure that the consecutive differences of sequences constructed from the zeros of a nontrivial solution of the equation are regular in sign for ω sufficiently large. In particular, if c_(νk)(α) denotes the k-th positive zero of the general Bessel (cylinder) function C_(ν)(x; α) = J_(ν)(x) cos α−Y_(ν)(x) sin α of order ν, and if |ν| < 1/2, we prove that (−1)^(m)∆^(m+2)c_(νk)(α) > 0 (m = 0, 1, 2, ...; k = 1, 2, ...), where ∆a_(k) = a_(k+1) − a_(k). This type of inequalities was conjectured by Lorch and Szego in 1963. We also show that the differences of the zeros of various orthogonal polynomials with higher degrees possess the sign-regularity.

Deep neural networks (DNNs), founded on the brain's neuronal organization, can extract higher-level features from raw input. However, complex intellect via autonomous decision-making is way beyond current AI design. Here we propose an autonomous AI inspired by the thermodynamic cycle of sensory perception, operating between two information density reservoirs. Stimulus unbalances the high entropy resting state and triggers a thermodynamic cycle. By recovering the initial conditions, self-regulation generates a response while accumulating an orthogonal, holographic potential. The resulting high-density manifold is a stable memory and experience field, which increases future freedom of action via intelligent decision-making.

Based on the satellite attitude control method, this paper proposes an attitude control method based on neural network disturbance compensation. The paper firstly analyzes the neural network algorithm and proposes an orthogonal least squares algorithm to implement network learning. In this paper, a set of high-precision directional neural network compensation controllers is designed for the attitude control of acupuncture small satellites. The feasibility of the improved orthogonal least-squared algorithm combined with the neural network supplementary control method in satellite attitude control is verified by experiments.

Given a quasi-definite linear functional u in the linear space of polynomials with complex coefficients let us consider the corresponding sequence of monic orthogonal polynomials (SMOP in short) (Pn)n≥0. For the Christoffel transformation u˜=(x−c)u with SMOP (P˜n)n≥0, we are interested to study the relation between u˜ and u(1)˜, where u(1) is the linear functional for the associated orthogonal polynomials of the first kind (Pn(1))n≥0 and u(1)˜=(x−c)u(1) is its Christoffel transformation. This problem is also studied for the Geronimus transformations.

Reverse order laws for generalized inverses of matrix products is a classic object of study in the theory of generalized inverses. One of the well-known reverse order laws for a matrix product $AB$ is $(AB)^{(i,\ldots,j)} = B^{(s_2,\ldots,t_2)}A^{(s_1,\ldots,t_1)}$, where $(\cdot)^{(i,\ldots,j)}$ denotes an $\{i,\ldots, j\}$-generalized inverse of matrix. Because $\{i,\ldots, j\}$-generalized inverse of a singular matrix is unique, the relationships between both sides of the reverse order law can be divided into four situations for consideration. This paper provides a thorough coverage of the reverse order laws for $\{i,\ldots, j\}$-generalized inverses of $AB$, from the development of background and preliminary tools to the collection of miscellaneous formulas and facts on the reverse order laws in one place with cogent introduction and references for further study. We begin with the introduction of a linear mixed model $y = AB\beta + A\gamma + \epsilon$ and the presentation of two least-squares methodologies to estimate the fixed parameter vector $\beta$ in the model, and the description of connections between the two types of least-squares estimators and the reverse order laws for generalized inverses of $AB$. We then prepare some valued matrix analysis tools, including a general theory on linear or nonlinear matrix identities, a group of expansion formulas for calculating ranks of block matrices, two groups of explicit formulas for calculating the maximum and minimum ranks of $B^{(s_2,\ldots,t_2)}A^{(s_1,\ldots,t_1)}$, as well as necessary and sufficient conditions for $B^{(s_2,\ldots,t_2)}A^{(s_1,\ldots,t_1)}$ to be invariant with respect to the choice of $B^{(s_2,\ldots,t_2)}A^{(s_1,\ldots,t_1)}$. We then present a unified approach to the 512 matrix set inclusion problems associated with the above reverse order laws for the eight commonly-used types of generalized inverses of $A$, $B$, and $AB$ through use of the definitions of generalized inverses, the block matrix method (BMM), the matrix rank method (MRM), the matrix equation method (MEM), and various algebraic calculations of matrices.

Floods and droughts are driven, in part, by spatial patterns of extreme rainfall. Heat waves are driven by spatial patterns of extreme temperature. It is therefore of interest to design statistical methodologies that allow the identification of likely patterns of extreme rain or temperature from observed historical data. The standard work-horse for identifying patterns of climate variability in historical data is Principal Component Analysis (PCA) and its variants. But PCA optimizes for variance not spatial extremes, and so there is no particular reason why the first PCA spatial pattern should identify, or even approximate, the types of patterns that may drive these phenomena, even if the linear assumptions underlying PCA are correct. We present an alternative pattern identification algorithm that makes the same linear assumptions as PCA, but which can be used to explicitly optimize for spatial extremes. We call the method Directional Component Analysis (DCA), since it involves introducing a preferred direction, or metric, such as `sum of all points in the spatial field'. We compare the first PCA and DCA spatial patterns for US rainfall anomalies on a 6 month timescale, using the sum metric for the definition of DCA in order to focus on total rainfall anomaly over the domain, and find that they are somewhat different. The definitions of PCA and DCA result in the first PCA spatial pattern having the larger explained variance of the two patterns, while the first DCA spatial pattern, when scaled appropriately, has a higher likelihood and greater total rainfall anomaly, and indeed is the pattern with the highest total rainfall anomaly for any given likelihood. In combination these two patterns yield more insight into rainfall variability and extremes than either pattern on its own.

Millimeter wave wide-band imaging is widely studied for a variety of applications. However real-time millimeter wave wide-band imaging at frequencies above 30GHz for moving targets in a large field of view has not been realized commercially. A 2D sparse array with transmitter multiplexing is a promising solution to this problem. In this article, a method combining compressed sensing and orthogonal coded multiplexing was proposed, and the imaging performance was analyzed for different reconstruction algorithms and observation matrices by imaging simulation for a continuous object. Also the influence on the dynamic range of the original signal introduced by orthogonal coded multiplexing was studied. This work demonstrated that the proposed method was effective in reconstructing the image with a real-time capability. It is shown that different algorithms and matrices resulted in distinct performances, while the evaluation parameter selection also played a role. This work provided useful instructions for both the hardware and software design of a real-time 3D millimeter wave imaging system in the future.

In order to solve the interference of narrowband communication system to ultra-wideband (UWB) system and meet the design requirements of high isolation of multiple-input multiple-output (MIMO) antenna, two MIMO antennas with double notch structures are designed. Firstly, a two-element antenna was placed symmetrically in parallel, and two rectangular branches were loaded to improve the isolation. U-shaped slots and inverted U-shaped slots are etched on the radiating patch and feeder to achieve notch characteristics in WIMAX and ITU band, respectively. Then, in order to meet more application scenarios, a four-element MIMO antenna is designed, where each element is placed orthogonally to each other, and the isolation is improved by loading a cross-shaped branches in the middle of elements. Both antenna samples are tested to verify the design. Measured results show that the working bandwidth of the two-element UWB MIMO antenna is 2.45-14.88 GHz, the isolation is greater than 17 dB, the peak gain is 5.7 dB, the maximum radiation efficiency can reach 96%, the envelope correlation coefficient (ECC)is less than 0.02. Meanwhile, the working bandwidth of the four-element antenna is 2.14-14.95 GHz, the isolation is greater than 20 dB, the peak gain is 5.9 dB, the maximum radiation efficiency can reach 95%, the ECC is lower than 0.05. They both are with very good omnidirectional characteristic, and meet the requirements of the UWB application.

Gauss quadrature integral approximation is extended to include integrals with a measure consisting of a continuous as well as a discrete component. That is, we give an approximation for the integral of a function plus its sum over a discrete weighted set.

We compare several different measures of non-Gaussianity for two families of four-photon superpositions of the Fock states: even vacuum squeezed states and orthogonal-even coherent states.

In this paper, we introduce the evolve-then-filter (EF) regularization method for reduced order modeling of convection-dominated stochastic systems. The standard Galerkin projection reduced order model (G-ROM) yield numerical oscillations in a convection-dominated regime. The evolve-then-filter reduced order model (EF-ROM) aims at the numerical stabilization of the standard G-ROM, which uses explicit ROM spatial filter to regularize various terms in the reduced order model (ROM). Our numerical results based on a stochastic Burgers equation with linear multiplicative noise. It shows that the EF-ROM is significantly better results than G-ROM.

In the current era of ‘big data’, scientists are able to quickly amass enormous amount of data in a limited number of experiments. The investigators then try to hypothesize about the root cause based on the observed trends for the predictors and the response variable. This involves identifying the discriminatory predictors that are most responsible for explaining variation in the response variable. In the current work, we investigated three related multivariate techniques: Principal Component Regression (PCR), Partial Least Squares or Projections to Latent Structures (PLS), and Orthogonal Partial Least Squares (OPLS). To perform a comparative analysis, we used a publicly available dataset for Parkinson’ disease patien ts. We first performed the analysis using a cross-validated number of principal components for the aforementioned techniques. Our results demonstrated that PLS and OPLS were better suited than PCR for identifying the discriminatory predictors. Since the X data did not exhibit a strong correlation, we also performed Multiple Linear Regression (MLR) on the dataset. A comparison of the top five discriminatory predictors identified by the four techniques showed a substantial overlap between the results obtained by PLS, OPLS, and MLR, and the three techniques exhibited a significant divergence from the variables identified by PCR. A further investigation of the data revealed that PCR could be used to identify the discriminatory variables successfully if the number of principal components in the regression model were increased. In summary, we recommend using PLS or OPLS for hypothesis generation and systemizing the selection process for principal components when using PCR.rewordexplain later why MLR can be used on a dataset with no correlation

In this article, we prove an orthogonal decomposition theorem for real inner product gyrogroups, which unify some well-known gyrogroups in the literature: Einstein, M\"{o}bius, Proper Velocity, and Chen's gyrogroups. This leads to the study of left (right) coset partition of a real inner product gyrogroup induced from a subgyrogroup that is a finite-dimensional subspace. As a result, we obtain gyroprojectors onto the subgyrogroup and its orthogonal complement. We construct also quotient spaces and prove an associated isomorphism theorem. The left (right) cosets are characterized using gyrolines (cogyrolines) together with automorphisms of the subgyrogroup. With the algebraic structure of the decompositions, we study fiber bundles and sections inherited by the gyroprojectors. Finally, the general theory is exemplified for the aforementioned gyrogroups.

With the proliferating of video surveillance system deployment and related applications, real-time video analysis is very critical to achieve intelligent monitoring, autonomous driving, etc. It is non-trivial to achieve high accuracy and low latency video stream analysis through the traditional cloud computing. In this paper, we propose a non-orthogonal multiple access (NOMA) based edge real-time video analysis framework with one edge server (ES) and multiple user equipments (UEs). A cost minimization problem composed of delay, energy and accuracy is formulated to improve the QoE of UEs. In order to efficiently solve this problem, we propose the joint video frame resolution scaling, task offloading, and resource allocation algorithm based on the Deep Q-Learning Network (JVFRS-TO-RA-DQN), which effectively overcomes the sparsity of the single-layer reward function and accelerates the training convergence speed. JVFRS-TO-RA-DQN consists of two DQN networks to reduce the curse of dimension, which respectively select the offloading and resource allocation action, the resolution scaling action. Experimental results show that JVFRS-TO-RA-DQN can effectively reduce the cost of transmission and computation, and have better performance in convergence compared to other baseline.

This paper focuses on parametric model order reduction (PMOR) of guided ultrasonic wave propagation and its interaction with damage in a fiber metal laminate (FML). Structural health monitoring in FML seeks to detect, localize and characterize the damage with high accuracy and minimal use of sensors. This can be achieved by the inverse problem analysis approach which employs the signal measurement data recorded by the embedded sensors in the structure. The inverse analysis requires to solve the forward simulation of the underlying system several thousand times. These simulations are often exorbitantly expensive and triggered the need for improving their computational efficiency. A PMOR approach hinged on the proper orthogonal decomposition method is presented in this paper. An adaptive parameter sampling technique is established with the aid of a surrogate model to efficiently update the reduced-order basis in a greedy fashion. A numerical experiment is conducted to illustrate the parametric training of the reduced-order model. The results show that the reduced-order solution based on the PMOR approach is accurately complying with that of the high fidelity solution.

In this article, the usual factorials and binomial coefficients have been generalized and extended to the negative integers. Basing on this generalization and extension, a new kind of polynomials has been proposed, which led directly to the non-classical hypergeometric orthogonal polynomials and the non-classical second-order hypergeometric linear DEs. The resulting polynomials can be used in non-relativistic and relativistic QM, particularly, in the case of the Schrödinger equation, and Dirac equations for an electron in a Coulomb potential field.

Data integration has been proven to provide valuable information. The information extracted using data integration in the form of multiblock analysis can pinpoint both common and unique trends in the different blocks. When working with small multiblock datasets the number of possible integration methods is drastically reduced. To investigate the application of multiblock analysis in cases where one has few number of samples, we studied a small metabolomic multiblock dataset containing six blocks (i.e. tissue types), only including common metabolites. We used a single model multiblock analysis method called Joint and unique multiblock analysis (JUMBA) and compare it to a commonly used method, concatenated PCA. These methods were used to detect trends in the dataset and identify underlying factors responsible for metabolic variations. Using JUMBA, we were able to interpret the extracted components and link them to relevant biological properties. JUMBA shows how the observations are related to one another, the stability of these relationships and to what extent each of the blocks contribute to the components. These results indicate that multiblock methods can be useful even with a small number of samples.

Adaptive Sliceable-Bandwidth Variable Transceivers (S-BVTs) are key enabler for future optical networks. In particular, those based on Discrete MultiTone (DMT) modulation and Direct Detection (DD) can be considered a flexible solution suitable to address the cost efficiency requirement of optical metro networks. In this paper, we propose to use this cost-effective S-BVT option/implementation in optical metro networks adopting switching nodes based on Semiconductor Optical Amplifier (SOA) technology. Bit loading (BL) and power loading (PL) algorithms are applied to the Digital Signal Processing (DSP) modules, to maximize the performance and/or the capacity as well as enhance the flexibility and adaptability of the system. Our analysis considers switching nodes based on SOAs with and without filtering elements and fiber spans of 25 km. We present the results up to 100 km, with and without SOA-based nodes. Firstly, we analyze the adaptive BVT transmission using the Margin Adaptive (MA) BL/PL algorithm at a fixed bit rate of 28 Gb/s. The possibility of controlling the SOAs current is a key factor to face the transmission impairments due to the fiber and the filtering elements. We also analyze the system considering Rate Adaptive (RA) transmission at a fixed target BER of 3.8·10⁻³, showing that a maximum capacity above 34 Gb/s can be achieved for a single span of 25 km. Although the cascading of filtering elements still constitutes a limiting factor, we show that an improvement of the net bit rate performance can be obtained thanks to the combined use of S-BVT and SOA technology at the switching nodes, resulting in a promising approach for designing future optical metro networks.

The extensive use of engineered nanoparticles (ENPs) such as zinc oxide nanoparticles (ZnO NPs) in various commercial fields has spurred significant concern about its possible toxicological effects of ZnO NPs on human health and eco-system. Therefore, it is substantial to understand the aggregation phenomena which could attenuate this effect. This data article provides information on aggregation of ZnO NPs under the influence of various parameters of water chemistry. Moreover, Taguchi orthogonal array L27(313) design matrix was used to evaluate the effect of multiple parameters in aqueous solution. The data were obtained from single beam UV-Vis spectrophotometer (Optizen 2120 UV, Mecasys, Korea) using Optizenview 4.2.5 PC interference software, Fourier transform infrared spectrometer (FT/IR-4700, spectroscopy, JASCO Analytical Instruments, Easton, USA) and particle electrophoresis (NanoZS, Zetasizer, Malvern Instruments Ltd, Worcestershire, UK). The dataset draws attention on dominant parameters influencing aggregation of ZnO NPs in water. The analysis of variance (ANOVA) data revealed that electrolyte concentration, type and natural organic matter (NOM) concentration were the most significant parameters. Furthermore, FT-IR data presents a possible mechanism of ZnO NPs stabilization in the presence of different NOM. This data will be helpful for the development of environmental risk assessment strategies and prediction of fate and mobility of other ENPs in the aquatic environment.

Cooperative non-orthogonal multi access communication is a promising paradigm for the future wireless networks because of its advantages in terms of energy efficiency, wider coverage and interference mitigating. In this paper, we study the secrecy performance of a downlink cooperative non-orthogonal multi access (NOMA) communication system under the presence of an eavesdropper node. Smart node selection based on feed forward neural networks (FFNN) is proposed in order to improve the physical layer security (PLS) of a cooperative NOMA network. The selected cooperative relay node is employed to enhance the channel capacity of the legal users, where the selected cooperative jammer is employed to degrade the capacity of the wiretapped channel. Simulations of the secrecy performance metric namely the secrecy capacity ($C_S$) are presented and compared with the conventional technique based on fuzzy logic node selection technique. Based on our simulations and discussions the proposed technique outperforms the existing technique in term the of secrecy performance.

For the in-house validation of a droplet digital PCR method, a factorial experimental design was implemented. This design serves different purposes. On the one hand, it is an efficient design in relation to the workload involved in achieving a desirable level of reliability of variance estimates. On the other hand, it allows a partitioning of total variance into different components, thus providing information regarding the dominant sources of random variation. The statistical modelling reflects the actual measurement mechanism, establishing relationships between nominal target DNA copies per well, the range of variation of copy numbers per droplet, probability of detection values, and estimated numbers of copies.

In this paper, the author utilizes Saddle Theorem and variational methods to deduce existence of at least six stationary solutions for reaction-diffusion Gilpin-Ayala competition model (RDGACM). To obtain the global stabilization of the positive stationary solution of the RDGACM, the author designs a suitable impulsive event triggered mechanism (IETM) to derive the global exponential stability of the the positive stationary solution. It is worth mentioning that the new mechanism can exclude Zeno behavior and effectively reduce the cost of impulse control through event triggering mechanism. Besides, compared with existing literature, the restrictions on the parameters of the RDGACM are relaxed so that the methods used in existing literature can not be applied to the relaxed case of this paper, and so the author makes comprehensive use of Saddle Theorem, orthogonal decomposition of Sobolev space $H_0^1(\Omega)$ and variational methods to overcome the mathematical difficulty. Numerical examples show the effectiveness of the methods proposed in this paper.

Compressive sensing (CS) is a technique that allows a signal vector to be reconstructed with a number of linear measurements far fewer than its size. In this study, optical hybrid cryptography with simultaneous compressive sensing based on orthogonal matching pursuit (OMP) is proposed as an alternative solution to the two important problems in communications, namely, effective / efficient / signal processing and secure encryption/decrypton. The whole system is able to both compress and encrypt data with attack immunity.