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The Balancing number n and the balancer r are solution of the Diophantine equation 1 + 2 + ⋯ + (n - 1) = (n + 1) + (n + 2) + ⋯ + (n + r) . It is well known that if n is balancing number, then 8 n 2 + 1 is a perfect square and its positive square root is called a Lucas-Balancing number. Let k ≥ 2 . A generalization of the well-known Fibonacci sequence is the k-Fibonacci sequences. For these sequence the first k terms are 0 , ... , 0 , 1 and each term afterwards is the sum of the preceding k terms. In this manuscript, our main objective is to find all k-Fibonacci numbers which are the product of two Balancing or Lucas-Balancing numbers. [ABSTRACT FROM AUTHOR] more...

Let F n n ≥ 0 and M n n ≥ 0 be the Fibonacci and Mulatu sequences. Let b be a positive integer such that b ≥ 2. In this paper, we prove that the following Diophantine equation F n + ϵ M m = ρ (b ± 1) b ℓ ± 1 , where (ϵ , ρ) ∈ { (1 , 1) , (- 1 , 1) , (- 1 , - 1) } have only finitely many solutions in non-negative integers n, m, b and positive integer ℓ. Additionally, we present a method to determine all solutions within the range 2 ≤ b ≤ 12. All this is done using linear forms in logarithms of algebraic numbers and Dujella–Pethő’s reduction method. [ABSTRACT FROM AUTHOR] more...

We are concerned with the following nonlinear Hartree system − Δ u + P 1 (| x |) u = α 1 | x | − 1 ∗ u 2 u + β | x | − 1 ∗ v 2 u in R 3 , − Δ v + P 2 (| x |) v = α 2 | x | − 1 ∗ v 2 v + β | x | − 1 ∗ u 2 v in R 3 , where P1(r) and P2(r) are positive radial potentials, α1 > 0, α2 > 0 and β ∈ R is a coupling constant. We first study nondegeneracy of ground states for the limit system of the above problem. As applications, we show that the nonlinear Hartree system has infinitely many non-radial positive synchronized solutions, whose energy can be arbitrarily large. [ABSTRACT FROM AUTHOR] more...

For any positive integer k > 2, let (P (k) n )n>2-k be the k-generalized Pell sequence which starts with 0, ... 0, 1 (k terms) with the linear recurrence ...Let (Nncript>)n>0 be Narayana's sequence given by N0 = N1 = N2 = 1 and Nn+3 = Nn+2 + Nn The purpose of this paper is to determine all k-Pell numbers which are sums of two Narayana's numbers. More precisely, we study the Diophantine equation ... in nonnegative integers k, p, n and m [ABSTRACT FROM AUTHOR] more...

As is well known, one of the main challenges in dealing with the time‐fractional diffusion‐wave equation is that the solutions have weak regularity at the initial values. This necessitates the use of nonuniform grids for error analysis. However, the theoretical analysis of nonuniform grids is relatively complex and lacks efficient and stable numerical methods. The main research problem addressed in this article is the time‐fractional diffusion‐wave equation on nonuniform grids. We use the order reduction method and discrete complementary convolution kernel to discretize the Caputo derivative of the equation and obtain the L1‐type finite element method on nonuniform grids. The properties related to discrete complementary convolutional kernels are often used for algorithm convergence analysis, which simplifies the process of finite element theory analysis and is efficient and innovative. This article demonstrates the solvability, stability, and convergence of the L1‐type finite element schemes on nonuniform grids. Finally, the consistency between the proposed finite element scheme and the convergence order results obtained from theoretical analysis was verified through experiments. [ABSTRACT FROM AUTHOR] more...

• The stronger electronegativity of Ag in Ag/Co 3 O 4 led to richer surface Co3+. • Ag/Co 3 O 4 gained by Ag-reduction method showed higher surface Co3+/Co2+ ratio. • The reducibility of Co 3 O 4 was greatly improved by the loading of Ag via reduction method. • Ag/Co 3 O 4 prepared by Ag-reduction method showed higher catalytic activity. Silver (9 wt.%) was loaded on Co 3 O 4 -nanofiber using reduction and impregnation methods, respectively. Due to the stronger electronegativity of silver, the ratios of surface Co3+/Co2+ on Ag/Co 3 O 4 were higher than on Co 3 O 4 , which further led to more adsorbed oxygen species as a result of the charge compensation. Moreover, the introducing of silver also obviously improved the reducibility of Co 3 O 4. Hence the Ag/Co 3 O 4 showed better catalytic performance than Co 3 O 4 in benzene oxidation. Compared with the Ag/Co 3 O 4 synthesized via impregnation method, the one prepared using reduction method (named as AgCo-R) exhibited higher contents of surface Co3+ and adsorbed oxygen species, stronger reducibility, as well as more active surface lattice oxygen species. Consequently, AgCo-R showed lowest T 90 value of 183°C, admirable catalytic stability, largest normalized reaction rate of 1.36 × 10−4 mol/(h·m2) (150°C), and lowest apparent activation energy (E a) of 63.2 kJ/mol. The analyzing of in-situ DRIFTS indicated benzene molecules were successively oxidized to phenol, o-benzoquinone, small molecular intermediates, and finally to CO 2 and water on the surface of AgCo-R. At last, potential reaction pathways including five detailed steps were proposed. [ABSTRACT FROM AUTHOR] more...

We consider linear bounded operators acting in Banach spaces with a basis, such operators can be represented by an infinite matrix. We prove that for an invertible operator there exists a sequence of invertible finite-dimensional operators so that the family of norms of their inverses is uniformly bounded. It leads to the fact that solutions of finite-dimensional equations converge to the solution of initial operator equation with infinite-dimensional matrix. [ABSTRACT FROM AUTHOR] more...

An enormous quantity of titanium slag has caused not merely serious environment pollution, but also a huge waste of iron and sulfur resources. Hence, recycling iron and sulfur resources from titanium slag has recently been an urgent problem. Herein, hematite nanoparticles were fabricated through a pyrite reduction approach using as-received titanium slag as the iron source and pyrite as the reducing agent in an nitrogen atmosphere. The physicochemical properties of the hematite nanoparticles were analyzed using multiple techniques such as X-ray diffraction pattern, ultraviolet–visible spectrophotometry, and scanning electron microscopy. The best synthesis conditions for hematite nanoparticles were found at 550 °C for 30 min with the mass ratio of 14:1 for titanium slag and pyrite. The results demonstrated that hematite nanoparticles with an average particle diameter of 45 nm were nearly spherical in shape. The specific surface area, pore volume, and pore size estimated according to the BET method were 19.6 m2/g, 0.117 cm3/g, and 0.89 nm, respectively. Meanwhile, the fabricated hematite nanoparticles possessed weak ferromagnetic behavior and good absorbance in the wavelength range of 200 nm-600 nm, applied as a visible light responsive catalyst. Consequently, these results show that hematite nanoparticles formed by the pyrite reduction technique have a promising application prospect for magnetic material and photocatalysis. [ABSTRACT FROM AUTHOR] more...

Reducing environmental pollution, including air pollution, contributes to improving people's health and quality of life, which is one of the goals of sustainable development. One of the important air pollutants is ammonia, which is mainly emitted from the agriculture sector. This sector is responsible for over 81% of global ammonia emissions. The aim of this research was a preliminary assessment of the implementation status of methods for reducing ammonia emissions on farms and to learn the views and awareness of agricultural producers on reducing emissions of pollutants into the air. The research was conducted using a survey questionnaire that was made available to farmers in various ways. Based on the results, it can be concluded that farmers have knowledge of environmental protection and agree that people have an impact on the environment. Low-emission practices to reduce ammonia emissions from agricultural sources are not widely used. The best situation is considering reduction practices in the storage of natural fertilizers and the use of low-emission fertilizer application techniques. The results of this type of monitoring research may be useful in determining the level of ammonia emission reduction. In the future, the data may be used during air pollution inventories conducted by state institutions. [ABSTRACT FROM AUTHOR] more...

Let Pn be an n-th Padovan number, En an n-th Perrin number, and Nn an n-th Narayana number. In this paper, we solve the Diophantine equations Pn = (2a - 1)(2b -1), En = (2a - 1)(2b - 1), and Nn = (2a ± 1)(2b ± 1), in positive integers n, a and b. Therefore, we determine the Padovan or Perrin numbers that are products of two Mersenne numbers and the Narayana numbers that are products of two Mersenne or Fermat numbers. [ABSTRACT FROM AUTHOR] more...

This paper presents an analytic study of determining all the possible solutions of the Diophantine equations such that qk = JmJn and Jk = qmqn. These give intersections of the Modified Pell and Jacobsthal numbers too for the case where m = 1 or n = 1. [ABSTRACT FROM AUTHOR] more...

Disasters are unforeseen occurrences requiring extensive transport deployment to support and relieve victims. Sometimes, this transportation is not feasible directly from some supply points to some destination points. Due to this tragedy, it is unclear precisely what is available at supply points, what is needed at destinations, how much transportation capacity there is, and what the routes are like. In this study, we investigate a two-stage multi-item fixed charge four-dimensional transportation problem using the concept of big data theory under the two-fold uncertainties. Here, the model’s parameters such as unit transportation costs, availabilities of items at the suppliers, fixed charges, capacities of conveyances, and demands of the items at the retailers are considered type-2 zigzag uncertain variables. Using big data theory and based on uncertain programming theory, two novel uncertain models are developed such as chance-constrained programming and expected value programming model. These two uncertain models transformed into the deterministic form via uncertainty inverse distribution theory. A critical value based reduction method with three categories (i.e., expected value, pessimistic value, and optimistic value) is applied to reduce the type-2 zigzag uncertain variable to the type-1 zigzag uncertain variable. The genetic algorithm and particle swarm optimization techniques have been proposed to find the optimal solution for the two deterministic models. The efficiency of our proposed approach is demonstrated with a real-life numerical example. more...

This study presents a methodology aimed at enhancing the performance of coherent systems through the application of survival signature analysis, focusing on the calculation of reliability equivalence factors (REFs). In the context of system improvement, the selection of reliability improvement strategies, such as reduction and duplication, depends on various factors like space limitations, costs, and other constraints. The importance of REF lies in their ability to quantify the extent of reliability improvement, providing a clear metric for decision-makers to assess the cost-effectiveness of various enhancement strategies. The analysis focuses on two distinct types of REFs, namely, mean reliability equivalence factors (MREFs) and survival reliability equivalence factors (SREFs), targeted at reliability enhancement via strategies including component failure rate reduction and the implementation of warm standby duplication. Both perfect and imperfect switching scenarios in warm duplication are examined, with survival signature analysis applied to determine the system's survival function and mean time to failure (MTTF). The methodology's effectiveness is illustrated through a case study of a six-unit bridge system, where the components are modeled using exponential and Weibull distributions. REFs are evaluated for sequential upgrades in either individual components or entire component types. The study also conducts a comparative analysis between the reliability and MTTF of the original and improved systems across different improvement techniques. more...

Let (P n) n ≥ 0 and (Q n) n ≥ 0 be the Pell and Pell–Lucas sequences respectively. Let b be a positive integer such that b ≥ 2 . In this paper, we investigate the solutions of the following two Diophantine equations: P n = b d Q m + Q k and Q n = b d P m + P k , where n, m, k and d are non-negative integers with a condition on d. More precisely, we prove that the above equations have only finitely many solutions in integers (m, n, k, b, d). We explicitly determine these solutions for the cases 2 ≤ b ≤ 10 as an application. [ABSTRACT FROM AUTHOR] more...

In this paper, we designed and investigated a reduction-based method to synthesize controllably monodisperse superparamagnetic nano Fe3O4 colloidal clusters for magnetically responsive photonic crystals. It was shown that the addition of ascorbic acid (VC) to the system could synthesize monodisperse superparamagnetic nano Fe3O4 and avoided the generation of γ-Fe2O3 impurities, while the particle size and saturation magnetization intensity of nano Fe3O4 gradually decreased with the increase of VC dosage. Nano Fe3O4 could be rapidly assembled into photonic crystal dot matrix structures under a magnetic field, demonstrating tunability to various diffraction wavelengths. The nano Fe3O4 modified by polyvinylpyrrolidone (PVP) and silicon coated could be stably dispersed in a variety of organic solvents and thus diffracted different wavelengths under a magnetic field. This is expected to be applied in various scenarios in the field of optical color development. [ABSTRACT FROM AUTHOR] more...

In this paper, we consider the following nonlinear Schrödinger equation: 0.1−Δu+V(|x|)u=Q(|x|)up,u>0inℝN,u∈H1ℝN,$$ -\Delta u+V\left(|x|\right)u=Q\left(|x|\right){u}^p,\kern0.30em u>0\kern0.20em \mathrm{in}\kern0.20em {\mathrm{\mathbb{R}}}^N,u\in {H}^1\left({\mathrm{\mathbb{R}}}^N\right), $$where N≥3,p∈1,N+2N−2$$ N\ge 3,p\in \left(1,\frac{N+2}{N-2}\right) $$, and V,Q$$ V,Q $$ have the algebraical decay that V(|x|)=V0+a|x|m+O1|x|m+κ,Q(|x|)=Q0+b|x|n+O1|x|n+θ,$$ V\left(|x|\right)={V}_0+\frac{a}{{\left|x\right|}^m}+O\left(\frac{1}{{\left|x\right|}^{m+\kappa }}\right),\kern0.30em Q\left(|x|\right)={Q}_0+\frac{b}{{\left|x\right|}^n}+O\left(\frac{1}{{\left|x\right|}^{n+\theta }}\right), $$as |x|→∞$$ \mid x\mid \to \infty $$, where V0,Q0,κ,θ,a>0,b∈ℝ$$ {V}_0,{Q}_0,\kappa, \theta, a>0,b\in \mathrm{\mathbb{R}} $$, and m,n>1$$ m,n>1 $$. By introducing the Miranda theorem, via the Lyapunov–Schmidt finite‐dimensional reduction method, we construct infinitely many multi‐bumps solutions of (0.1), whose maximum points of bumps lie on the top and bottom circles of a cylinder provided m0$$ m0 $$, or b≤0$$ b\le 0 $$. This result complements and extends the ones in [Duan and Musso, JDE, 2022] for a slow decaying rate of the potential function at infinity from m>max2,4p−1$$ m>\max \left\{2,\frac{4}{p-1}\right\} $$ to m>1$$ m>1 $$. [ABSTRACT FROM AUTHOR] more...

This paper proposes a new highly effective method for determining the optimal control of the stress-strain state of spatially multi-connected composite bodies using a stationary temperature field. The proposed method is considered based on the example of a stationary axisymmetric thermoelastic problem for a space with a spherical inclusion and cavity. The proposed method is based on the generalized Fourier method and reduces the original problem to an equivalent problem of optimal control, in which the state of the object is determined by an infinite system of linear algebraic equations, the right-hand side of which parametrically depends on the control. At the same time, the functional of the cost of the initial problem is transformed into a quadratic functional, which depends on the state of the equivalent system and parametrically on the control. The limitation on the temperature distribution is replaced by the value of the control norm in the space of square summable sequenc es. In fact, this paper considers for the first time the problem of optimal control of an infinite system of linear algebraic equations and develops a method for its solution. The proposed method is based on presenting the solutions of infinite systems in a parametric form, which makes it possible to reduce equivalent problem to the problem of conditional extremum of a quadratic functional, which explicitly depends on the control. A further solution to this problem A further solution to this problem is found by the Lagrange method using the spectral decomposition of the quadratic functional matrix. found by the Lagrange method using the spectral decomposition of the quadratic functional matrix. The method developed in this paper is strictly justified. For all infinite systems, the Fredholm property of their operators is proved. As an important result necessary for substantiation, for the first time, an estimate from below of the module of the multi-parameter determinant of the resolving system of the boundary value problem of conjugation – space with a spherical inclusion – was obtained when solving it using the Fourier method. The theorem that establishes the conditions for the existence and uniqueness of the solution of equivalent problem or optimal control problem without restrictions in the space of square summable sequences is proved. The numerical algorithm is based on a reduction method for infinite systems of linear algebraic equations. Estimates of the practical accuracy of the numerical algorithm demo nstrated the stability of the method and sufficiently high accuracy even with close location of the boundary surfaces. Graphs showing the optimal temperature distribution for various geometric parameters of the problem and their analysis are provided. The proposed method extends to boundary value problems with different geometries. [ABSTRACT FROM AUTHOR] more...

In this article, we consider the following elliptic system of Hamiltonian-type on a bounded domain:−Δu=K1(∣y∣)∣v∣p−1v,inB1(0),−Δv=K2(∣y∣)∣u∣q−1u,inB1(0),u=v=0on∂B1(0),\left\{\begin{array}{ll}-\Delta u={K}_{1}\left(| y| ){| v| }^{p-1}v,\hspace{1.0em}& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{B}_{1}\left(0),\\ -\Delta v={K}_{2}\left(| y| ){| u| }^{q-1}u,\hspace{1.0em}& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{B}_{1}\left(0),\\ u=v=0\hspace{1.0em}& \hspace{0.1em}\text{on}\hspace{0.1em}\hspace{0.33em}\partial {B}_{1}\left(0),\end{array}\right. where K1(r){K}_{1}\left(r) and K2(r){K}_{2}\left(r) are positive bounded functions defined in [0,1]\left[0,1], B1(0){B}_{1}\left(0) is the unit ball in RN{{\mathbb{R}}}^{N}, and (p,q)\left(p,q) is a pair of positive numbers lying on the critical hyperbola 1p+1+1q+1=N−2N.\frac{1}{p+1}+\frac{1}{q+1}=\frac{N-2}{N}. Under some suitable further assumptions on the functions K1(r){K}_{1}\left(r) and K2(r){K}_{2}\left(r), we prove the existence of infinitely many nonradial positive solutions whose energy can be made arbitrarily large. Our proof is based on the reduction method. The most ingredients of the article are using the Green representation and estimating the Green function and its regular part very carefully. For this purpose, some more extra ideas and techniques are needed. We believe that our method and techniques can be applied to other related problems. more...

In this article, we study the following nonlinear Schrödinger system −Δu1+V1(x)u1=αu1u2+μu1,x∈R4,−Δu2+V2(x)u2=α2u12+βu22+μu2,x∈R4,\left\{\begin{array}{ll}-\Delta {u}_{1}+{V}_{1}\left(x){u}_{1}=\alpha {u}_{1}{u}_{2}+\mu {u}_{1},& x\in {{\mathbb{R}}}^{4},\\ -\Delta {u}_{2}+{V}_{2}\left(x){u}_{2}=\frac{\alpha }{2}{u}_{1}^{2}+\beta {u}_{2}^{2}+\mu {u}_{2},& x\in {{\mathbb{R}}}^{4},\end{array}\right. with the constraint ∫R4(u12+u22)dx=1{\int }_{{{\mathbb{R}}}^{4}}\left({u}_{1}^{2}+{u}_{2}^{2}){\rm{d}}x=1, where α>0\alpha \gt 0 and α>β\alpha \gt \beta , μ∈R\mu \in {\mathbb{R}}, V1(x){V}_{1}\left(x), and V2(x){V}_{2}\left(x) are bounded functions. Under some mild assumptions on V1(x){V}_{1}\left(x) and V2(x){V}_{2}\left(x), we prove the existence of normalized peak solutions by using the finite dimensional reduction method, combined with the local Pohozaev identities. Because of the interspecies interaction between the components, we aim to obtain some new technical estimates. more...

Let (M_k) be the sequence of Mulatu numbers defined by M_0=4, M_1=1, M_k=M_(k-1)+M_(k-2) and (F_k) be the Fibonacci sequence given by the recurrence F_k=F_(k-1)+F_(k-2) with the initial conditions F_0=0, F_1=1 for k≥2. In this paper, we showed that all Mulatu numbers, that are concatenations of two Fibonacci numbers are 11,28. That is, we solved the equation M_k=〖10〗^d F_m+F_n, where d indicates the number of digits of F_n. We found the solutions of this equation as (k,m,n,d)∈{(4,2,2,1),(6,3,6,1)}. Moreover the solutions of this equation displayed as M_4=(F_2 F_2 ) ̅=11 and M_6=(F_3 F_6 ) ̅=28. Here the main tools are linear forms in logarithms and Baker Davenport basis reduction method. more...

In this note we find all Fibonacci and Lucas numbers of the form $-2^a-3^b-5^c+7^d$ where $a, b, c, d$ are non-negative integers, with $0 \leq \max\{a, b, c\}\leq d$. This result gives an answer to a question posed by Qu, Zeng and Cao. more...

In this note we find all Fibonacci and Lucas numbers of the form -2^a-3^b-5^c+7^d where a, b, c, d are non-negative integers, with 0 ≤ max{a, b, c} ≤ d. This result gives an answer to a question posed by Qu, Zeng and Cao. more...

In this paper, we show that there is at most one value of the positive integer X participating in the Pell equation X 2 − d Y 2 = ± 1 , which is a difference of two Fibonacci numbers. [ABSTRACT FROM AUTHOR] more...

This paper is concerned with the existence of multiple non-radial positive and sign-changing solutions for the following Kirchhoff equation: − a + b ∫ ℝ 3 | ∇ u | 2 Δ u + (1 + λ Q (x)) u = | u | p − 2 u , in ℝ 3 , (0. 1) where a , b > 0 are constants, p ∈ (2 , 6) , λ is a parameter, and Q (x) is a potential function. Under the assumption on Q (x) with exponential decay at infinity, we construct multi-peak positive and sign-changing solutions for problem (0.1) as λ → ∞ (or 0), where the peaks concentrate at infinity. [ABSTRACT FROM AUTHOR] more...

Geometric programming is a well-known optimization tool for dealing with a wide range of nonlinear optimization and engineering problems. In general, it is assumed that the parameters of a geometric programming problem are deterministic and accurate. However, in the real-world geometric programming problem, the parameters are frequently inaccurate and ambiguous. To tackle the ambiguity, this paper investigates the geometric programming problem in an uncertain environment, with the coefficients as triangular and trapezoidal twofold uncertain variables. In this paper, we introduce uncertain measures in a generalized version and focus on more complicated twofold uncertainties to propose triangular and trapezoidal twofold uncertain variables within the context of uncertainty theory. We develop three reduction methods to convert triangular and trapezoidal twofold uncertain variables into singlefold uncertain variables using optimistic, pessimistic, and expected value criteria. Reduction methods are used to convert the geometric programming problem with twofold uncertainty into the geometric programming problem with singlefold uncertainty. Furthermore, the chance-constrained uncertain-based framework is used to solve the reduced singlefold uncertain geometric programming problem. Finally, a numerical example is provided to demonstrate the effectiveness of the procedures. [ABSTRACT FROM AUTHOR] more...

In this paper, we consider the multi-species nonlinear Schrödinger systems in R N : { − Δ u j + V j (x) u j = μ j u j 3 + ∑ i = 1 ; i ≠ j d β i , j u i 2 u j in R N , u j (x) > 0 in R N , u j (x) → 0 as | x | → + ∞ , j = 1 , 2 , ⋯ , d , where N = 2 , 3 , μ j > 0 are constants, β i , j = β j , i ≠ 0 are coupling parameters, d ≥ 2 and V j (x) are potentials. By Ljapunov-Schmidt reduction arguments, we construct infinitely many nonradial positive solutions of the above system under some mild assumptions on potentials V j (x) and coupling parameters { β i , j } , without any symmetric assumptions on the limit case of the above system. Our result, giving a positive answer to the conjecture in Pistoia and Viara [50] and extending the results in [50,52] , reveals new phenomenon in the case of N = 2 and d = 2 and is almost optimal for the coupling parameters { β i , j }. [ABSTRACT FROM AUTHOR] more...

In this paper, we consider the following logarithmic Schrödinger–Poisson system - Δ u + V (x) u + λ K (x) ϕ u = f (u) + u log u 2 , x ∈ R 3 , - Δ ϕ - ε 4 Δ 4 ϕ = λ K (x) u 2 , x ∈ R 3 , ef=""33_2023_2170_Article_Equ45.gif""> which has increasingly received interest due to the indefiniteness of the energy functional and fourth-order term in Poisson equation. By using variational method, we prove the existence and multiplicity of positive solutions. Finally, we obtain the asymptotic behavior of positive solutions as ε → 0 + and λ → 0 + , respectively. [ABSTRACT FROM AUTHOR] more...

A b-repdigit is a positive integer that has only one distinct digit in its base b expansion, i.e. a number of the form a(bm – 1)=(b – 1), for some positive integers m ≥ 1, b ≥ 2 and 1 ≤ a ≤ b – 1. Let r; s be non-zero integers with r ≥ 1 and s ∈ {±1}, let {Un}n ≥0 be the Lucas sequence given by Un+2 = rUn+1 + sUn, with U0 = 0 and U1 = 1: In this paper, we give effective bounds for the solutions of the Diophantine equation where a; b; n; k and m are positive integers such that 1 ≤ a ≤ b – 1; n; k ≥ 1 and 2 ≤ b ≤ 10. Then, we explicitly solve the above Diophantine equation for the Fibonacci, Pell and balancing sequences. [ABSTRACT FROM AUTHOR] more...

This article discusses the solvability of the quasilinear Maxwell-Stokes system with topological parameters using the reduction method. The system involves boundary value problems with either natural or Dirichlet boundary conditions. The introduction of topological parameters is necessary for the well-posedness of the problems and allows for a better understanding of the effect of domain topology on physical systems. The article also provides examples of nonlinear Maxwell equations and explains the transition from Maxwell system to Maxwell-Stokes system. The study aims to contribute to the understanding of the topological effect in three-dimensional systems. [Extracted from the article] more...

In this paper, we investigate the common terms of the Leonardo { L e n } n ≥ 0 and Jacobsthal { J m } m ≥ 0 sequences. To accomplish this, we use Baker's theory of linear forms in logarithms of algebraic numbers along with a variation of the Baker-Davenport reduction method to solve the Diophantine equation L e n = J m . [ABSTRACT FROM AUTHOR] more...

For k ≥ 2, let (Pn(k))n≥2-k be the k-generalized Pell sequence which starts with 0, ⋯, 0,1 (k terms) and each term afterwards is given by the linear recurrence Pn(k) = 2Pn-1(k) + Pn-2(k) + ⋯ + Pn-k(k), for all n ≥ 2. An integer n is said to be close to a positive integer m if n satisfies |n - m| < √m. In this paper, we solve the Diophantine inequality |pn(k) - 2m| < 2m/2 in positive unknowns k, n, and m. [ABSTRACT FROM AUTHOR] more...

A positive integer is called a repdigit if it has only one distinct digit in its decimal expansion. In this paper, we find all repdigits that are products of consecutive Pell or Pell--Lucas numbers. This paper continues previous work which dealt with finding occurrences of repdigits in the Pell and Pell--Lucas sequences. [ABSTRACT FROM AUTHOR] more...

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The ocean covers 70% of the earth’s area, and the seawater uranium is more than 1000 times that of terrestrial uranium ore. In order to increase the power generation and ensure the sustainable operation of nuclear power plants, the development of unconventional uranium resources is of great significance. As an effective means to solve shortage of terrestrial uranium resources, uranium extraction from seawater has received extensive attention in recent years. At present, many methods including membrane separation, chemical deposition, solvent extraction, ion exchange, reduction and adsorption have been applied to uranium extraction from seawater. This paper focuses on the recent research progress of adsorption method used inorganic materials, organic materials, carbon materials as adsorbent, and emerging electroreduction technologies, and further discusses the challenges and opportunities of these two methods. more...

In this paper, we show that there are only finitely many Narayana's numbers which can be written as a product of three repdigits in base g with g ≥ 2. Moreover, for 2 ≤ g ≤ 10, we determine all these numbers. [ABSTRACT FROM AUTHOR] more...

Let k ≥ 2. A generalization of the well-known Pell sequence is the k-Pell sequence whose first k terms are 0,..., 0, 1 and each term afterwards is given by the linear recurrence p n (k) = 2 P n − 1 (k) + P n − 2 (k) + ⋯ + P n − k (k) . The goal of this paper is to show that 11, 33, 55, 88 and 99 are all repdigits expressible as sum or difference of two k-Pell. The proof of our main theorem uses lower bounds for linear forms in logarithms of algebraic numbers and a modified version of Baker-Davenport reduction method (due to Dujella and Pethő). This extends a result of Bravo and Herrera [Repdigits in generalized Pell sequences, Arch. Math. (Brno) 56(4) (2020), 249–262]. [ABSTRACT FROM AUTHOR] more...

Recall that a repdigit in base g is a positive integer that has only one digit in its base g expansion; i.e., a number of the form a (g m - 1) / (g - 1) , for some positive integers m ≥ 1 , g ≥ 2 and 1 ≤ a ≤ g - 1 . In the present study, we investigate all Fibonacci or Lucas numbers which are expressed as products of three repdigits in base g. As illustration, we consider the case g = 10 where we show that the numbers 144 and 18 are the largest Fibonacci and Lucas numbers which can be expressible as products of three repdigits respectively. All this is done using linear forms in logarithms of algebraic numbers. [ABSTRACT FROM AUTHOR] more...

Monodispersed and spherical gold nanoparticles and over nanoparticles with average is about 50 nm and 45 nm respectively, were synthesized by chemical reduction method using gearing as the protective agents, chioroauric aid SOUSE and over nitrate as metal source, hydrant hydrate as riding agents. UV-vis spectrum (UV-vis), X-ray powder diffraction ( XRD), flourier transform mortared spectrometer (FT-IR)and scorning electron microscope (SEM)were used to check the as-prepared gold and over nanoparticles. Some influencing factors nan the process of preparation of gold and over nanoparticles are briefly discussed. The results show that the prepared gold and over nanopaerieles are small Nan is with narrow distriburion. The amounts of precursor soon, reducing agent and gelatin solution play important roles in the size and morphologies of gold and silver nanoparticles. As the amount of precursor solution, reducing agents and gelatin solution increased, the particles size showed a trend of first decreasing and then increasing. The optimized processing parameters nan this research for preparation of gold and silver nanoparticles can be obtained when the consumption of gelatin solution, precursor solution, and redoing agent is 5,2 and 2 mL,respectively. [ABSTRACT FROM AUTHOR] more...

The integer sequence defined by P n + 3 = P n + 1 + P n with initial conditions P 0 = P 1 = P 2 = 1 is known as the Padovan sequence (P n) n ∈ Z . The Perrin sequence (R m) m ∈ Z satisfies the same recurrence equation as the Padovan sequence but with starting values R 0 = 3 , R 1 = 0 , and R 2 = 2 . In this note, we solve the Diophantine equation P n = ± R m with (n , m) ∈ Z 2 . [ABSTRACT FROM AUTHOR] more...

Understanding the sources and controlling processes of various groundwater contaminants and their removal methods is extremely important, as groundwater contamination is intricately linked to human health. Chromium (Cr) is a common groundwater contaminant with both natural and anthropogenic origins. Dissolved Cr exists in hexavalent and trivalent forms – while the former is carcinogenic and more soluble – the latter is a micronutrient at low levels and is less soluble. Therefore, most chromium removal methods rely on reducing the hexavalent chromium to its trivalent state to decrease the Cr-toxicity. In recent years, several experimental methods have been attempted for hexavalent chromium removal from aqueous media/groundwater. This paper reviews the recent findings on Cr removal by important, effective, and widely used methods such as adsorption by nanoscale zero-valent Fe-based and conventional materials, electrocoagulation (EC), and bioremediation. The reaction pathways, mechanisms, and effectiveness of each method are also highlighted. The role of parameters such as solution pH and temperature, initial Cr(VI) concentration, contact time with the reducing agent, adsorbent dose, and the presence of competing ions on Cr removal was evaluated. Many of the methods exhibit high (>90%) Cr removal efficiency; the main challenge would be to apply these methods for large-scale water treatment. HIGHLIGHTS Adsorption, electrocoagulation, and biological methods for Cr removal are reviewed in this study.; Stabilizer-based nanoscale zero-valent iron has proven to be an emerging technique for chromium mitigation.; pH of the aqueous medium seems to be the most influential parameter in determining Cr removal.; Future research should focus more on scaling-up the existing methods.; more...