Real Matrices Research Articles

Periodic two-dimensional descriptor systems

In this note, we analyze the compatibility conditions of 2D descriptor systems with periodic coefficients and we derive a special coordinate system in which these conditions reduce to simple matrix commutativity conditions. We also show that the compatibility of the different trajectories in such a periodic 2D descriptor system can elegantly be formulated in terms of so-called matrix relations of regular pencils, which were introduced in [Benner and Byers. An arithmetic for matrix pencils: Theory and new algorithms. Numer. Math., 103(4):539-573, 2006]. We then show that these ideas can be extended to multidimensional periodic descriptor systems and briefly discuss the difference between the case of complex and real coefficient matrices.

Wafer-Scale Graphene Field-Effect Transistor Biosensor Arrays with Monolithic CMOS Readout.

The reliability of analysis is becoming increasingly important as point-of-care diagnostics are transitioning from single-analyte detection toward multiplexed multianalyte detection. Multianalyte detection benefits greatly from complementary metal-oxide semiconductor (CMOS) integrated sensing solutions, offering miniaturized multiplexed sensing arrays with integrated readout electronics and extremely large sensor counts. The development of CMOS back end of line integration compatible graphene field-effect transistor (GFET)-based biosensing has been rapid during the past few years, in terms of both the fabrication scale-up and functionalization toward biorecognition from real sample matrices. The next steps in industrialization relate to improving reliability and require increased statistics. Regarding functionalization toward truly quantitative sensors, on-chip bioassays with improved statistics require sensor arrays with reduced variability in functionalization. Such multiplexed bioassays, whether based on graphene or on other sensitive nanomaterials, are among the most promising technologies for label-free electrical biosensing. As an important step toward that, we report wafer-scale fabrication of CMOS-integrated GFET arrays with high yield and uniformity, designed especially for biosensing applications. We demonstrate the operation of the sensing platform array with 512 GFETs in simultaneous detection for the sodium chloride concentration series. This platform offers a truly statistical approach on GFET-based biosensing and further to quantitative and multianalyte sensing. The reported techniques can also be applied to other fields relying on functionalized GFETs, such as gas or chemical sensing or infrared imaging.

Hybrid multi-core shell magnetic nanoparticles for wet peroxide oxidation of paracetamol: Application in synthetic and real matrices

Clean water availability is becoming a matter of global concern in the last decades. The responsible entities for wastewater treatment do not have the proper facilities to deal with a wide range of pollutants. Special attention should be given to emerging contaminants, whose presence in water bodies may cause adverse effects on the aquatic ecosystem and human health. Most studies in the literature do not consider the development of their solution in real matrices, which can hinder the applicability of the explored alternative in the real scenario. Therefore, in this work, we demonstrate the applicability of hybrid magnetic nanoparticles for removing paracetamol (PCM) from simulated and real matrices by catalytic wet peroxide oxidation (CWPO). To achieve carbon coating, the nanoparticles were prepared via the traditional route (resorcinol/formaldehyde, CoFe@CRF). A new methodology was also considered for synthesizing thin-layered carbon-coated magnetic nanoparticles (phloroglucinol/glyoxalic acid, CoFe@CPG). TEM images revealed a multi-core shell structure formation, with an average carbon layer size of 7.8 ± 0.5 and 3.2 ± 0.3 nm for resorcinol/formaldehyde and phloroglucinol/glyoxalic acid methodology, respectively. Screening the materials’ activity for PCM oxidation by CWPO revealed that the nanoparticle prepared by phloroglucinol/glyoxalic acid methodology has higher performance for the degradation of PCM, achieving 63.5% mineralization after 24 h of reaction, with similar results for more complex matrices. Iron leaching measured at the end of all reactions has proven that the carbon layer protects the core against leaching.

Spectral Applications of Vertex-Clique Incidence Matrices Associated with a Graph

Using the notions of clique partitions and edge clique covers of graphs, we consider the corresponding incidence structures. This connection furnishes lower bounds on the negative eigenvalues and their multiplicities associated with the adjacency matrix, bounds on the incidence energy, and on the signless Laplacian energy for graphs. For the more general and well-studied set S(G) of all real symmetric matrices associated with a graph G, we apply an extended version of an incidence matrix tied to an edge clique cover to establish several classes of graphs that allow two distinct eigenvalues.

A Glass-Fiber-Optic Turbidity Sensor for Real-Time In Situ Water Quality Monitoring.

Turbidity is an important water quality parameter, especially for drinking water. The ability to actively monitor the turbidity level of drinking water distribution systems is of critical importance to the safety and wellbeing of the public. Traditional turbidity monitoring methods involve the manual collection of water samples at set locations and times followed by laboratory analysis, which are labor intensive and time consuming. Fiber-optic measurement permits real-time, in situ turbidity monitoring. But the current technology is based on plastic fibers, which suffer from high optical attenuation and hence are unsuitable for large-scale remote monitoring. In this paper, we report the demonstration of a fiber-optic turbidity sensor based on multi-mode glass fibers. The system uses a single fiber to both deliver laser light into the water sample and collect the back-scattered light for detection. A balanced detection scheme is utilized to remove the common-mode noise to enhance the turbidity sensitivity. Highly linear turbidity responses are obtained and a turbidity resolution as low as 0.1 NTU is achieved. The test unit is also shown to have excellent reproducibility against repeated measurements and good stability against temperature changes. Turbidity measurement in real environmental matrices such as tap water and pond water is also reported with an assessment of the impact of flow rate. This work demonstrates the feasibility of future large-scale distributed fiber-optic turbidity monitoring networks.

Heterogeneous Activation of Persulfate by Nickel Oxide/Strontium Carbonate Composite for Sulfamethoxazole Degradation in Water

The development of efficient heterogeneous persulfate activators is one of the main research topics in the wastewater treatment area. The present work deals with the heterogeneous activation of sodium persulfate (SPS) using nickel oxide/strontium carbonate (NiO/SrCO3) for the degradation of sulfamethoxazole (SMX), a representative compound from the group of antibiotics. Results showed that NiO/SrCO3 exhibited high performance towards the activation of SPS, leading to SMX elimination in brief time spans. The impact of SPS (25–100 mg/L), NiO/SrCO3 (50–250 mg/L), and SMX (0.25–3.00 mg/L) concentration, and initial pH on the decomposition of SMX was further examined. Experiments were also conducted in real matrices such as secondary effluent and bottled water, revealing the existence of retarding phenomena compared to ultrapure water. This behavior was further investigated with the addition of bicarbonates, chlorides, or humic acid in ultrapure water. It was found that organic matter significantly hampered SMX removal. The role of the main radicals (hydroxyl and sulfate radicals) was determined using appropriate radical traps (methanol and tert-butanol). These quenching experiments combined with the conducted electrochemical measurements revealed that both a radical and a non-radical mechanism contribute to the decomposition of SMX.

Two-parameters formulas for general solution to planar weakly delayed linear discrete systems with multiple delays, equivalent non-delayed systems, and conditional stability

Weakly delayed planar linear discrete systems with multiple delaysx(k+1)=Dx(k)+∑l=1nHlx(k−ml),k=0,1,… are considered where m1,m2,…,mn are fixed integers, D, H1, …, Hn are nonzero 2×2 real constant matrices and x:{−mn,−mn+1,…}→R2. Formulas for general solutions are found and simplified, equivalent non-delayed planar linear discrete systems are constructed and conditional stability is analyzed. Results are illustrated by examples.

Dimensions of projected sets and measures on typical self-affine sets

Let T1,…,Tm be a family of d×d invertible real matrices with ‖Ti‖<1/2 for 1≤i≤m. For a=(a1,…,am)∈Rmd, let πa:Σ={1,…,m}N→Rd denote the coding map associated with the affine IFS {Tix+ai}i=1m. We show that for every Borel probability measure μ on Σ, each of the following dimensions (lower and upper Hausdorff dimensions, lower and upper packing dimensions) of π⁎aμ is constant for Lmd-a.e. a∈Rmd, where π⁎aμ stands for the push-forward of μ by πa. In particular, we give a necessary and sufficient condition on μ so that π⁎aμ is exact dimensional for Lmd-a.e. a∈Rmd. Moreover, for every analytic set E⊂Σ, each of the Hausdorff, packing, lower and upper box-counting dimensions of πa(E) is constant for Lmd-a.e. a∈Rmd. Formal dimension formulas of these projected measures and sets are given. The Hausdorff dimensions of exceptional sets are estimated.

On Novel Results about the Algebraic Properties of Symbolic 3-Plithogenic and 4-Plithogenic Real Square Matrices

Symbolic n-plithogenic sets are considered to be modern concepts that carry within their framework both an algebraic and logical structure. The concept of symbolic n-plithogenic algebraic rings is considered to be a novel generalization of classical algebraic rings with many symmetric properties. These structures can be written as linear combinations of many symmetric elements taken from other classical algebraic structures, where the square symbolic k-plithogenic real matrices are square matrices with real symbolic k-plithogenic entries. In this research, we will find easy-to-use algorithms for calculating the determinant of a symbolic 3-plithogenic/4-plithogenic matrix, and for finding its inverse based on its classical components, and even for diagonalizing matrices of these types. On the other hand, we will present a new algorithm for calculating the eigenvalues and eigenvectors associated with matrices of these types. Also, the exponent of symbolic 3-plithogenic and 4-plithogenic real matrices will be presented, with many examples to clarify the novelty of this work.

Phase-mediated mercury removal from water using multilayered molybdenum disulfide: The key role of redox mechanism

It is crucial to distinguish the performance of different phases of molybdenum disulfide (MoS2) for Hg(II) removal owing to its distinct activity and atomic arrangement. Herein, we directionally synthesized three nanocrystalline MoS2 with metallic 1T, semiconducting 2H, and 1T/2H mixed M phase, and elucidated Hg(II) adsorption behaviors. Kinetics experiments suggested that the 1T-MoS2 exhibited the most promising performance toward Hg(II) removal over other two counterparts (2H- and M-MoS2), including highest mass transfer coefficient of 1.85 × 10−6 m s−1, an extraordinary selectivity of 3.89 × 108 mL g−1 and deep removal capability with residual concentration lower than 0.35 μg L−1. Combining various characterization tools, the phase-mediated removal mechanism was unveiled for Hg(II) onto 1T-MoS2. Specifically, Hg(II) species could be selectively captured by 1T-MoS2 via Hg-S interactions, enabling a redox reaction involving electron transfer along conductive lamellas, where Hg(II) was reduced to Hg2Cl2 crystals. Moreover, the 1T-MoS2 also demonstrated a high adsorption capacity (1622 mg g−1), a broad pH working range (1–12) and superb selectivity for Hg(II) removal. Highly practical applicability of 1T-MoS2 was confirmed for deep removal performance in various real water matrices. Great economic efficiency of the methodology was synergistically demonstrated by high adsorbent reusability (removal efficiency >99.9% over 3 cycles) and resources recovery feasibility (e.g., recyclable Hg2Cl2 products). Overall, this work has demonstrated high applicability of 1T-MoS2 for mercury removal and provides guidance for designing advanced nanoadsorbents from the perspective of crystal phase regulation.

Peroxymonosulfate activation by Ni-Fe (hydr)oxides through radical and nonradical pathways for efficient trichloroethylene degradation

Peroxymonosulfate (PMS)-based advanced oxidation processes have become promising water treatment technologies in recent years. The redox cycle between transitional metals in heterogeneous catalysis is essential for continuous activation of PMS and efficient removal of pollutants. In this work, we prepared Fe3O4 using precipitation method and subsequently added Ni aqueous solution to synthesize Ni-Fe (hydr)oxides. Ni-Fe (hydr)oxides show efficient PMS activation ability, causing the degradation of 96.8 % trichloroethylene (TCE) after 40 min with the dosage of Fe, Ni and PMS at 0.2, 0.1 and 0.2 mM, respectively. The catalyst concentration normalized TCE degradation rate constant of Ni-Fe (hydr)oxides/PMS system reaches 1.48 L g−1 min−1. Nonradical pathway is dominant at the initial stage of the reaction which generates high-valent metals, SO5•− and indirectly produces 1O2. 1O2 and dissolved O2 together participate in the O2 cycling, •OH generation, and electron transfer involving the reduction of trivalent metals. Divalent metals react with PMS to generate strong oxidative •OH and SO4•− through radical pathway in later stage. TCE is degraded and dechlorinated by multiple reactive species in Ni-Fe (hydr)oxides/PMS system. The optimum Ni/Fe ratio of catalysts under the studied conditions was 0.50. Ni-Fe (hydr)oxides show high reactivity and stability in real water matrices, demonstrating Ni-Fe (hydr)oxides are promising catalysts for PMS activation to degrade pollutants in contaminated water.

Facile synthesis of 2D/0D Bi2O3/MnO2 Z-scheme heterojunction for enhanced visible light-assisted photocatalytic degradation of acetaminophen

The construction of heterojunctions is a prominent strategy for improving the photocatalytic activity of semiconductor nanoparticles. In this study, a novel visible-light-driven 2D/0D Bi2O3/MnO2 (BMO) Z-scheme heterojunction was fabricated through a facile room-temperature solution-phase synthesis scheme to achieve enhanced photocatalytic degradation of acetaminophen (ACT). The BMO nanocomposite achieved 94.3% degradation efficiency at 0.0202 min−1 in 120 min, approximately 3.5 and 3.8 times higher than the degradation rate of the individual MnO2 and Bi2O3 photocatalysts, respectively. Based on electron paramagnetic resonance spectra and Mott-Schottky measurements, a Z-scheme heterojunction appeared between the Bi2O3 nanoflakes and MnO2 nanoparticles. The effective charge separation and electron transfer efficiency due to Z-scheme heterojunction and a narrow band gap of 2.10 eV were responsible for the enhanced photocatalytic performance. Further, the intermediates and end products of ACT degradation were identified, and plausible degradation pathways were established. Additionally, the performance of BMO in various real water matrices was evaluated. The degradation efficiency was highest in deionized water, followed by tap water, municipal, hospital, and pharmaceutical industry wastewater. Eventually, a mathematical model based on an artificial neural network has been developed to predict the photocatalytic process. Besides, the excellent photocatalytic performance of the BMO nanocomposite remained intact even after five consecutive cycles. The results of this study elicit crucial insights into synthesizing highly efficient and stable Z-scheme heterojunction photocatalysts to degrade emerging contaminants.

Influence of the aqueous matrix on the degradation of cyanotoxins by CWPO: A study on the Iberian Peninsula freshwaters

The increasing occurrence of toxic cyanobacterial blooms worldwide represents a critical health and environmental risk. Catalytic wet peroxide oxidation (CWPO) has emerged as an efficient and environmentally friendly technology for the removal of cyanotoxins in water. Nevertheless, its effectiveness has just been demonstrated in deionized water or simple synthetic aqueous matrices. In this work, the effect of the different components of the aqueous matrix on the CWPO of cyanotoxins was deeply evaluated considering the widespread properties of the Iberian Peninsula freshwaters. The presence of Cl-, HCO3- and SO42- ions reduced the oxidation rate of cylindrospermopsin (CYN) up to 70–80% at the highest concentrations tested (2000, 250 and 500 mg L−1 for Cl-, HCO3- and SO42-, respectively) due to their hydroxyl radical scavenging capacity. The presence of natural organic matter (NOM) resulted in a similar outcome (oxidation rate reduction up to 90% at the highest concentration tested, 20 mg L−1), but in this case due to the consumption of hydroxyl radicals in competition with CYN oxidation. The presence of NO3- and H2PO4- did not show any significant effect on CYN oxidation. Similarly, the presence of cyanobacteria (Chrysosporum ovalisporum, 50 µg chlorophyll-a L−1) did not appreciably affect the CYN oxidation rate. These results were consistent with those obtained by evaluating the impact of real aqueous matrices from drinking water treatment plants (DWTPs) located in Castilla y León and Extremadura regions (Spain) on CWPO performance. This knowledge is key to the implementation of the technology for the treatment of surface waters affected by toxic cyanobacterial blooms.

Removal of benzotriazole derivatives by biochar: Potential environmental applications

Benzotriazole and its derivatives (BTAs) are commonly present in wastewater due to their extensive use in industrial processes, yet their removal is still unexplored. Here, we test the removal of these pollutants using two functionalised biochars, synthesised from wild plum (WpOH) and apricot (AsPhA) kernels. The aim of this work was to optimise the adsorption process against various BTAs (i.e., benzotriazole (BTZ), 4-hydroxy-1H-benzotriazole (OHBZ), 4-methyl-1H-benzotriazole (4 MBZ), 5-methyl-1H-benzotriazole (5 MBZ), 5-chloro-1H-benzotriazole (ClBZ), 5,6-dimethyl-1H-benzotriazole (DMBZ)), and determine the adsorption mechanisms at play, using real wastewater matrices. Batch studies showed that the optimal adsorption pH ranged between 4 and 6 for WpOH and AsPhA, respectively, and equilibrium was reached after 240 min. The kinetic models that best described the adsorption process were in the following order: Elovich model > pseudo-second order model > pseudo-first order model. The equilibrium data showed the highest correlation with the Freundlich isotherm, indicating multilayer adsorption. The maximum adsorption capacity obtained in mixtures was 379 mg/g on WpOH and 526 mg/g on AsPhA. The mechanistic work revealed that the BTAs became bound to the biochar primarily through H-bonding, n-π and π-π EDA interactions. In wastewater, obtained before and after conventional treatment, the concentration of OHBZ and BTZ was reduced by >40%, while the concentration of the other compounds studied fell below the detection limit (∼2.0–90 ng/L). Finally, using a Vibrio fischeri assay, we showed that adsorption onto AsPhA significantly reduced the relative toxicity of both raw and treated wastewater.

Role of polyamide microplastics as vector of parabens in the environment: An adsorption study

The prevalence of microplastics in the aquatic environment has become a global problem. Their capacity to adsorb pollutants may influence in their environmental fate, bioavailability, and toxicity to biota. This work provides a systematic study to characterize the role of polyamide (PA) as vector of parabens (PBs) in the environment. The effect of PA and influence of environmental factors in the adsorption process were investigated. The amount of PBs adsorbed onto 50μm PA at 30 mg/L of PBs follows the order: butylparaben (1.440 mg/g) > propylparaben (1.321 mg/g) > ethylparaben (0.995 mg/g) > methylparaben (0.543 mg/g), which is positively correlated with their log Kow and length of the ester alkyl chain. Physical adsorption forces such as hydrophobic interaction, pore filling and hydrogen bond dominated the adsorption mechanism The size of the PA particles has been resulted in a significant factor; a higher adsorption capacity was remarkable when decreasing PA particle size (from 10.3% for 3 mm to 79.5% for 50 μm (0.5 mg/L PBs)). The adsorption percentage increases with pH until the PBs pKa value and decrease significantly at pH>pKa values due to repulsion forces. Salinity increases adsorption capacity until at 2% NaCl content while the dissolved organic matter negatively affects and leads to low adsorption. Our results reveal that PBs were potentially adsorbed onto PA in real water matrices (average of 82% in tap water, 72% in surface water, 76% in effluent wastewater, and, 53% in influent wastewater (0.5 mg/L PBs)), which may have important environmental implications.

A Cubic Class of Iterative Procedures for Finding the Generalized Inverses

This article considers the iterative approach for finding the Moore–Penrose inverse of a matrix. A convergence analysis is presented under certain conditions, demonstrating that the scheme attains third-order convergence. Moreover, theoretical discussions suggest that selecting a particular parameter could further improve the convergence order. The proposed scheme defines the special cases of third-order methods for β=0,1/2, and 1/4. Various large sparse, ill-conditioned, and rectangular matrices obtained from real-life problems were included from the Matrix-Market Library to test the presented scheme. The scheme’s performance was measured on randomly generated complex and real matrices, to verify the theoretical results and demonstrate its superiority over the existing methods. Furthermore, a large number of distinct approaches derived using the proposed family were tested numerically, to determine the optimal parametric value, leading to a successful conclusion.

Using the discriminant in a numerically stable symmetric 3 × 3 direct eigenvalue solver

AbstractAnalytical methods for solving eigenvalue problems involving real symmetric matrices are computationally efficient compared to iterative approaches, but not numerically robust when two of the eigenvalues coalesce. Analysis of the associated characteristic polynomial reveals an alternative form for the definition of the discriminant in terms of a sum of squares which is not susceptible to the numerical difficulties of conventional methods. Expressions for the angle used in the definition of the eigenvalues are derived in terms of simple combinations of the matrix invariants and the discriminant. The proposed method improves over previously proposed robust analytical schemes and is comparable in speed with the standard analytical method, making it a viable alternative.

New finite-type multi-indexed orthogonal polynomials obtained from state-adding Darboux transformations

Abstract The Hamiltonians of finite-type discrete quantum mechanics with real shifts are real symmetric matrices of order N + 1. We discuss the Darboux transformations with higher-degree (&amp;gt;N) polynomial solutions as seed solutions. They are state-adding and the resulting Hamiltonians after M steps are of order N + M + 1. Based on 12 orthogonal polynomials ((q-)Racah, (dual, q-)Hahn, Krawtchouk, and five types of q-Krawtchouk), new finite-type multi-indexed orthogonal polynomials are obtained, which satisfy second-order difference equations, and all the eigenvectors of the deformed Hamiltonian are described by them. We also present explicit forms of the Krein–Adler-type multi-indexed orthogonal polynomials and their difference equations, which are obtained from the state-deleting Darboux transformations with lower-degree (≤N) polynomial solutions as seed solutions.

Lower bounds for the smallest singular values of generalized asymptotic diagonal dominant matrices

This article presents three classes of real square matrices. They are models of coefficient matrices of linearized Galerkin’s equations. These Galerkin’s equations are derived from first order nonlinear delay differential equation with smooth nonlinearity. This paper shows results of computer experiments stating that the minimum singular values of these matrices are unchanged even if orders of matrices are increased. A time variant Hutchinson equation is also considered. A computer experimental result is shown about this equation. This result shows that “the minimum singular values invariant-ness” holds also for the coefficient matrices of linearized Galerkin’s equations of this equation. A remark is presented that this property can be seen experimentally for a wide class of nonlinear delay differential equation with smooth nonlinearity. A theorem is presented based on the Schur complement. Through it, tight lower bounds are derived for the minimum singular values of such three matrices. It is proved that these lower bounds are unchanged even if orders of matrices are increased.

Banach Fixed Point Theorems in Generalized Metric Space Endowed with the Hadamard Product

In this paper, we prove some Banach fixed point theorems in generalized metric space where the contractive conditions are endowed with the Hadamard product of real symmetric positive definite matrices. Since the condition that a matrix A converges to zero is not needed, this produces stronger results than those of Perov. As an application of our results, we study the existence and uniqueness of the solution for a system of matrix equations.