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In this paper, we find all Pell and Pell-Lucas numbers written in the form −2a − 3b + 5c, in nonnegative integers a, b, c, with 0 ⩽ max{a, b} ⩽ c. [ABSTRACT FROM AUTHOR] more...

For an integer k ≥ 2 , let { F n (k) } n ≥ 2 − k be the k –generalized Fibonacci sequence which starts with 0 , ... , 0 , 1 (a total of k terms) and for which each term afterwards is the sum of the k preceding terms. In this paper, for an integer d ≥ 2 which is square-free, 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 k –generalized Fibonacci number, with a couple of parametric exceptions which we completely characterize. This paper extends previous work from [18] for the case k = 2 and [17] for the case k = 3. [ABSTRACT FROM AUTHOR] more...

Let { F n } n ≥ 0 be the sequence of Fibonacci numbers defined by F 0 = 0 , F 1 = 1 and F n + 2 = F n + 1 + F n for all n ≥ 0 . In this paper, for an integer d ≥ 2 which is square-free, we show that there is at most one value of the positive integer x participating in the Pell equation x 2 - d y 2 = ± 4 which is a sum of two Fibonacci numbers, with a few exceptions that we completely characterize. [ABSTRACT FROM AUTHOR] more...

In this paper, we study the Kirchhoff equations with perturbation in R3. Applying the finite reduction method, we prove that the equation has one‐bump positive solutions under some suitable conditions that are given in Section 1. [ABSTRACT FROM AUTHOR] more...

Background: Proper reduction method for Lumbar degenerative spondylolisthesis (LDS) is still controversial. The aim of this study was to determine the safety and effectiveness of lever reduction combined with traditional elevating-pull reduction technique for the treatment of elderly patients with LDS.Methods: From May 2015 to December 2017, 142 elderly patients (≥65 years) diagnosed with LDS were enrolled in this study with a mean follow-up of 25.42 ± 8.31 months. All patients were operated using lever reduction combined with traditional elevating-pull reduction technique. Patient age, sex, body mass index, bone mineral density, preoperative comorbidities, surgical duration, blood loss, and surgical complications were collected form patient charts. Clinical data as visual analog scale (VAS), Oswestry Disability Index (ODI), and 36-Item Short Form Health Survey (SF-36) were collected preoperatively, 1 month postoperatively, and at the final follow-up. Radiographic evaluation included slip percentage, slip angle (SA), lumbar lordosis (LL), and fusion status.Results: The clinical parameters of VASback, VASleg, ODI, and SF-36 had significantly improved at both follow-ups after surgery. A significant improvement was indicated for slippage reduction at both follow-ups, showing no significant correction loss after surgery. SA significantly increased after surgery and was well maintained at the final follow-up. LL was not affected by the surgery. At the final follow-up, complete fusion was obtained in 121 patients (85.2%) and partial fusion in 21 (14.8%). Revision surgery was performed for one patient. Screw loosening was observed in 3 (2.11%) cases. No nerve root injury or adjacent segment disease was observed.Conclusions: This new lever reduction combined with traditional elevating-pull reduction technique for the surgical treatment of elderly patients with LDS is both safe and effective. Satisfactory correction and fusion rates were achieved with acceptable correction loss and reduction-related complications. [ABSTRACT FROM AUTHOR] more...

Damage detection of bridge structures during their operating lifetime is essential. In this paper, two approaches, All Degrees of Freedom and Reduction of the Degrees of Freedom methods, are used to detect the damages in structures. The first method considers All Degrees of Freedom of the structure and the second method, Reduction of the Degrees of Freedom. Since the sensors are installed only on a few degrees of freedom, the responses are available for some of them. The Degrees of Freedom must be reduced and System Equivalent Reduction Expansion Process method is one of the most efficient ways to solve the problem. This research aimed to identify the damage of structures using the Modal Strain Energy method by reducing the structural degree of freedom. Two standard examples are used and the results compared to different damage cases to examine the efficiency of the mentioned method. The results illustrated the proper performance of the Reduction of the Degrees of Freedom method to identify the damage in truss structures. By increasing the number of modes, Reduction of the Degrees of Freedom method detects considerably more accurate the damaged elements, especially when the noise is considered. Also, based on the outcomes to identify damaged elements, it is possible to consider more modes instead of more sensors. [ABSTRACT FROM AUTHOR] more...

Using the lower bound of linear forms in logarithms of Matveev and the theory of continued fractions by means of a variation of a result of Dujella and Petho, we find all k-Fibonacci and k-Lucas numbers which are Fermat numbers. Some more general results are given. [ABSTRACT FROM AUTHOR] more...

Copyright of Journal of Natural Sciences, Life & Applied Sciences is the property of Arab Journal of Sciences & Research Publishing (AJSRP) and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.) more...

We construct multiple sign-changing solutions for the nonhomogeneous nonlocal equation (-ΔΩ)su = |u|4/N-2s u + ε ƒ(x) in Ω, under zero Dirichlet boundary conditions in a bounded domain Ω in RN, N > 4s, s ∈ (0,1], with f ∈ L∞(Ω), ƒ ≥ 0 and ƒ ≠ 0. Here, ε > 0 is a small parameter, and (−∆Ω)s represents a type of nonlocal operator sometimes called the spectral fractional Laplacian. We show that the number of sign-changing solutions goes to infinity as ε → 0 when it is assumed that Ω and ƒ have certain smoothness and possess certain symmetries, and we are also able to establish accurately the contribution of the nonhomogeneous term in the found solutions. Our proof relies on the Lyapunov-Schmidt reduction method. [ABSTRACT FROM AUTHOR] more...

We consider the fractional nonlinear Schrödinger equation in this paper. Applying the finite reduction method, we prove that the equation has positive solutions with peaks on a Clifford torus under some suitable conditions. [ABSTRACT FROM AUTHOR] more...

Let (u n) n ≥ 0 be a non-degenerate binary recurrence sequence with positive discriminant and p be a fixed prime number. In this paper, we are interested in finding a finiteness result for the solutions of the Diophantine equation u n 1 + u n 2 + ⋯ + u n t = p z with n 1 > n 2 > ⋯ > n t ≥ 0 . Moreover, we explicitly find all the powers of three which are sums of three balancing numbers using lower bounds for linear forms in logarithms. Further, we use a variant of the Baker–Davenport reduction method in Diophantine approximation due to Dujella and Pethő. [ABSTRACT FROM AUTHOR] more...

In this paper, we find all the solutions of the title Diophantine equation in nonnegative integer variables (m; n; x), where Pk is the k th term of the Pell sequence. [ABSTRACT FROM AUTHOR]

Abstract The materials particle size reduction is an efficient, but often a time and energy intensive operation. Targeting the optimization of composite manufacturing process through low-cost materials selection, production efficiency, and minimization of energy consumption, this study investigated two dry particle size reduction methods –slow grinding and fast collision milling– as a potential one-step non-chemical filler/matrix compatibilization alternative, for a Marine/Polypropylene Bio-composite sheet material, for building applications. Two local coastal litter materials from Sydney, kelp brown algae (Ecklonia spp.) and bivalve mollusc seashell (Verenidae spp.) were selected as combined multi-functional bio-fillers. Four bio-composites prototypes were manufactured by isothermal hot-compression method, formulated to represent the most relevant permutations of the powders, with filler/matrix 60/40 wt%, respectively. Mixed wood waste sawdust was incorporated as secondary reinforcement to the blend, as a third of the total filler load. 20 specimens were prepared and tested. The effect of the size reduction methods on the particles properties were characterized by bulk, and surface sensitive qualitative and quantitative analyses. The surface roughness, wettability, fracture surface, and mechanical properties of the bio-composite prototypes are reported. The overall results indicated four distinctive filler morphologies were obtained: laminar, flakes, slabs, oblong particles, in which the physicochemical properties, topographical properties, and crystal structure greatly differed. The prototype panels manufactured with mechanically milled powders presented the peak mechanical properties for flexural strength, attributable to an enhanced materials interfacial behaviour. Additionally, as result of the different powder properties, competitive advantages were attained for moisture resistance and appearance grade, thus as potential candidates for architectural interior applications. Graphical abstract Image 104661 Highlights • Two dry particle size reduction methods are examined for marine bio-fillers. • Matrix/filler non-chemical enhanced compatibilization alternative technique. • Cost-effective clean manufacturing of particulate fines from waste materials. • Enhanced hydrophobic blends with high-recyclability potential were achieved. • Low-carbon Marine/Polypropylene Bio-composite sheet material. [ABSTRACT FROM AUTHOR] more...

In this paper, we find all the solutions of the title Diophantine equation in nonnegative integer variables (m; n; x), where Pk is the k th term of the Pell sequence. [ABSTRACT FROM AUTHOR] more...

The superposition method is used to develop an effective method for analytical solution of the problem of the harmonic bending vibrations of parallelogram-shaped bimorph piezoceramic plates. The reduction of infinite series allows deriving a finite-dimensional system of algebraic equations by satisfying given boundary conditions by minimizing the standard deviation and using the collocation method. [ABSTRACT FROM AUTHOR] more...

A set of m positive integers {a1, a2, . . . , am} is called a Diophantine m-tuple if aiaj + 1 is a perfect square for all 1 ≤ i < j ≤ m. Dujella proved that there is no Diophantine sextuple and that there are at most finitely many Diophantine quintuples. In particular, a folklore conjecture concerning Diophantine m-tuples states that no Diophantine quintuple exists at all. We prove this conjecture. [ABSTRACT FROM AUTHOR] more...

For any integer k≥2, the sequence of the k-generalized Fibonacci numbers (or k-bonacci numbers) is defined by the k initial values F−(k−2)(k)=⋯=F0(k)=0 and F1(k)=1 and such that each term afterwards is the sum of the k preceding ones. In this paper, we search for repdigits (i.e., a number whose decimal expansion is of the form aa…a, with a∈[1,9]) in the sequence (Fn(k)Fn(k+m))n, for m∈[1,9]. This result generalizes a recent work of Bednařík and Trojovská (the case in which (k,m)=(2,1)). Our main tools are the transcendental method (for Diophantine equations) together with the theory of continued fractions (reduction method). more...

Hot dipping Zn-Al-Mg coatings were prepared by rapid induction heating combined with gas protection. The influence of oxide scale on the structure and surface quality of a hot-dip Zn-6Al-3Mg alloy coating was studied in this paper. The results showed that the reaction of Fe-Al was suppressed by the scale on the surface of the steel plate. When the thickness of scale was 10 μm and the steel entry temperature was 900 °C, the surface quality of the coating was good. The Zn-Al-Mg coatings mainly consisted of the ternary eutectic structure of Zn/Al/MgZn2 and Fe4Al13 at the interface. When the scale thickness was 2–3 μm with the same steel entry temperature, the surface quality of the coating was poor, and serious stripe-like protrusion defects were formed on the surface of the coating, which was mainly caused by the Fe4Al13 phase separating from the substrate / coating interface into the overlay. more...

In this paper, we study the problem of the explicit intersection of two sequences. More specifically, we find all repdigits (i.e., numbers with only one repeated digit in its decimal expansion) which can be written as the product of a Fibonacci by a Tribonacci number (both with the same indexes). To work on this problem, our approach is to combine lower bounds from the Baker’s theory with reduction methods (based on the theory of continued fractions) due to Dujella and Pethö. more...

The sequence of the k-generalized Fibonacci numbers ( F n ( k ) ) n is defined by the recurrence F n ( k ) = ∑ j = 1 k F n − j ( k ) beginning with the k terms 0 , … , 0 , 1 . In this paper, we shall solve the Diophantine equation F n ( k ) = ( F m ( l ) ) 2 + 1 , in positive integers m , n , k and l. more...

We consider the problem Δ2u = V(x)up + ϵ in RN with u,Δu→0 as |x|→ + ∞, where p=N+4N−4, N ≥ 5, V is a positive continuous potential. Our aim is to construct high‐energy solutions for this equation by applying the finite‐dimensional reduction method and the penalization method. [ABSTRACT FROM AUTHOR] more...

This paper proposes a sensitivity-based border-search and jump reduction method for optimum design of spatial trusses. It is considered as a two-phase optimization approach, where at the first phase, the first local optimum is found by few analyses, after the whole searching space is limited employing an efficient random strategy, and the second phase involves finding a sequence of local optimum points using the variables sensitivity with respect to corresponding values of constraints violation. To reach the global solution at phase two, a sequence of two sensitivity-based operators of border-search operator and jump operator are introduced until convergence is occurred. Sensitivity analysis is performed using numerical finite difference method. To do structural analysis, a link between open source software of OpenSees and MATLAB was developed. Spatial truss problems were attempted for optimization in order to show the fastness and efficiency of proposed technique. Results were compared with those reported in the literature. It shows that the proposed method is competitive with the other optimization methods with a significant reduction in number of analyses carried. [ABSTRACT FROM AUTHOR] more...

In this paper, we find all solutions of the exponential Diophantine equation Bn+1x - Bnx = Bm in positive integer variables (m, n, x), where Bk is the k-th term of the Balancing sequence. [ABSTRACT FROM AUTHOR] more...

Abstract In this paper we prove a conjecture that a D (4) -quintuple does not exist using both classical and new methods. Also, we give a new version of the Rickert's theorem that can be applied on some D (4) -quadruples. [ABSTRACT FROM AUTHOR] more...

In this paper, we show that there is at most one value of the positive integer X participating in the Pell equation X² - dY² = k, where k ∈ {±1, ±4}, which is a Padovan number, with a few exceptions that we completely characterize. [ABSTRACT FROM AUTHOR] more...

Let (Pn)n≥0 be the Padovan sequence given by P0 = 0, P1 = P2 = 1 and the recurrence formula Pn+3 = Pn+1 + Pn for all n ≥ 0. In this note we study and completely solve the Diophantine equation Pn - 2m = Pn1 - 2m1 in non-negative integers (n, m, n1, m1). [ABSTRACT FROM AUTHOR] more...

For an integer k≥2, let {Fn(k)}n≥0 be the k-generalized Fibonacci sequence which starts with 0,…,0,1> (k terms) and each term afterwards is the sum of the k preceding terms. In this paper, we find all integers c having at least two representations as a difference between a k-generalized Fibonacci number and a power of 2 for any fixed k≥4. This paper extends previous work from Ddamulira et al. (Proc Math Sci 127(3): 411-421, 2017. 10.1007/s12044-017-0338-3) for the case k=2 and Bravo et al. (Bull Korean Math Soc 54(3): 069-1080, 2017. 10.4134/BKMS.b160486) for the case k=3. [ABSTRACT FROM AUTHOR] more...

Copyright of Journal of Engineering Geology / Gongcheng Dizhi Xuebao is the property of Journal of Engineering Geology and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.) more...

Stresses in gear contact simulations, performed using elastic multibody systems, are recovered. A single gear pair is used for stress investigations and an impact is chosen as simulation case representing an extremely dynamical situation. The gears are modeled as a reduced elastic multibody system allowing a fast computation of the dynamical problem. Depending on the projection matrix which is used for model order reduction, stresses can sometimes not be recovered accurately throughout the whole gear. Thus, the main focus in this paper lies on the selection of the functions which make up the projection matrix and, therefore, determine the elastic deformations and the quality of recovered stresses. However, the chosen set of modes does not only affect stress calculation, it also strongly affects the computation of the dynamics of the gear system and, thus, the computational effort, and may lead to serious drawbacks. This issue is discussed, too. Upon that, several different mode sets are analyzed trying to minimize the computational effort of the elastic multibody system for the given problem while still being able to recover accurate stress values on distinct geometric areas. The stress values are compared with a finite element reference computation. The novel contribution of this paper is the determination of a minimal set of modes including ones assigned to the nodes of the gear contact surface, which are able to accurately recover stresses but minimize the numerical drawbacks. [ABSTRACT FROM AUTHOR] more...

A reduction method was proposed to synthesize better dispersed molybdenum oxide quantum dots (MoO x QDs) with stable fluorescence properties by using molybdenum disulfide (MoS 2 ) and hydrogen peroxide (H 2 O 2 ) as precursor and oxidant, and using ethanol as reducing agent and surfactant of oleylamine as modified agent. The morphology of the as-prepared MoO x QDs with average size of 1.96 nm were observed, calculated and analyzed from TEM and AFM detections. Excitation-dependent photoluminescence emission of MoO x QDs could be obtained and MoO x QDs exhibited blue fluorescence under UV light irradiation. Factors that influenced the syntheses of MoO x QDs were discussed, and the highest value of fluorescence intensity was exhibited with the adding of 5 mL oleylamine and the reaction time of 24 h. The cell imaging behaviors, the physiological stabilities and in vitro cytotoxicity testing of MoO x QDs were further discussed with HeLa cells as model. And the results showed that it might be a promising biomedical material as high quality fluorescent probes for bio-imaging. [ABSTRACT FROM AUTHOR] more...

From minimal surfaces such as Simons' cone and catenoids, using refined Lyapunov–Schmidt reduction method, we construct new solutions for a free boundary problem whose free boundary has two components. In dimension 8, using variational arguments, we also obtain solutions which are global minimizers of the corresponding energy functional. This shows that the theorem of Valdinoci et al. [41,42] is optimal. [ABSTRACT FROM AUTHOR] more...

We investigate the design of a prediction-based controller for a linear system subject to a time-varying input delay, not necessarily First-In/First-Out (FIFO). This means that the input signals can be reordered. The feedback law uses the current delay value in the prediction. It does not exactly compensate for the delay in the closed-loop dynamics but does not require to predict future delay values, contrary to the standard prediction technique. Modeling the input delay as a transport Partial Differential Equation, we prove asymptotic stabilization of the system state, that is, robust delay compensation, providing that the average ℒ 2 -norm of the delay time-derivative over some time-window is sufficiently small and that the average time between two discontinuities (average dwell time) is sufficiently large. [ABSTRACT FROM AUTHOR] more...

The stability of a prediction-based controller for linear time-invariant (LTI) systems is studied in the presence of time-varying input and output delays. The uncertain delay case is treated as well as the partial state knowledge case. The reduction method is used in order to prove the convergence of the closed-loop system including the state observer, the predictor and the plant. Explicit conditions that guarantee the closed-loop stability are given, thanks to a Lyapunov-Razumikhin analysis. Simulations illustrate the theoretical results. [ABSTRACT FROM AUTHOR] more...

The general solution of the boundary problem of planar vibrations of parallelogram-shaped piezoceramic plates is constructed. The solution is presented as infinite series, with each term satisfying the motion equations of a piezoceramic plate element. The series coefficients are determined with functional equations, generated by the boundary conditions of the problem. These equations can be solved using the two approaches, based on minimization of the standard deviation and the collocation method. In the case of practical application of finite sums, both approaches lead to the search of solving the systems of linear algebraic equations. Quantitative estimates of the dynamic characteristics of piezoceramic plates are obtained, their analysis permits of evaluating the plate geometry effect. This method provides high accuracy of calculation results. [ABSTRACT FROM AUTHOR] more...

Copyright of Journal of Food Safety & Quality is the property of Journal of Food Safety & Quality Editorial Department and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.) more...