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Recently, many scholars have discovered that fractional calculus possess infinite memory and can better reflect the memory characteristics of neurons. Therefore, this paper studies the Hopf bifurcation of a fractional‐order network with short‐cut connections structure and self‐delay feedback. Firstly, we use the Laplace transform to obtain the characteristic equation of the model, which is the transcendental equation containing four times transcendental item. Secondly, by selecting the communication delay as the bifurcation parameter and the other delay as the constant in its stability interval, the conditions for the occurrence of Hopf bifurcation are established; the bifurcation diagrams are provided to ensure that the derived bifurcation findings are accurate. Thirdly, in the case of identical neurons, the crossing curves method is exploited to the fractional‐order functional function equation to extract the Hopf bifurcation curve. Finally, two numerical examples are employed to confirm the efficiency of the developed theoretical outcomes. [ABSTRACT FROM AUTHOR]

In this paper, I study optimal monetary policy in a simple New Keynesian model with loose commitment and stochastic credibility. The loose commitment framework breaks the commitment‐discretion dichotomy in optimal monetary policy problems and allows for intermediate cases between commitment and discretion. Under this framework, the central bank is imperfectly credible, meaning that it occasionally reneges on promised policy plans. I contribute to the literature by introducing time‐variation in the central bank's credibility. I model credibility as an exogenous two‐state Markov chain and use a recursive saddlepoint functional equation to solve the model. I find that greater persistence and frequency of credibility losses increase welfare losses. [ABSTRACT FROM AUTHOR]

This paper concerns the fractional derivative of the Lerch zeta function. The author already dealt with its functional equation. He reduced its computational cost and proved an approximate functional equation for this fractional derivative. Here, we study the mean square of this fractional derivative. Moreover, we deal with the distribution of zeros, showing some zero‐free regions. [ABSTRACT FROM AUTHOR]

We consider the magnetic Ginzburg–Landau equations on a closed manifold N$N$−ε2ΔAu=12(1−|u|2)u,ε2d∗dA=⟨∇Au,iu⟩,$$\begin{equation*} {\begin{cases} -\varepsilon ^2\Delta ^Au = \frac{1}{2}(1-|u|^{2})u,\\[1pt] \varepsilon ^2d^{*}dA=\langle \nabla ^Au,iu\rangle, \end{cases}} \end{equation*}$$formally corresponding to the Euler–Lagrange equations for the energy functional E(u,A)=12∫N|∇Au|2+ε2|dA|2+14ε2(1−|u|2)2,$$\begin{equation*} E(u,A)=\frac{1}{2}\int _{N}|\nabla ^Au|^{2}+\varepsilon ^2|dA|^{2}+\frac{1}{4\varepsilon ^2}{(1-|u|^{2})}^{2}, \end{equation*}$$where u:N→C$u\colon N\rightarrow \mathbb {C}$ and A$A$ is a 1‐form on N$N$. Given a codimension‐2 minimal submanifold M⊂N$M\subset N$, which is also oriented and non‐degenerate, we construct solutions (uε,Aε)$(u_\varepsilon,A_\varepsilon)$ such that uε$u_\varepsilon$ has a zero set consisting of a smooth surface close to M$M$. Away from M$M$, we have uε(x)→z|z|,Aε(x)→1|z|2(−z2dz1+z1dz2),x=expy(z1ν1(y)+z2ν2(y))$$\begin{align*} &u_\varepsilon (x)\rightarrow \frac{z}{|z|},\quad A_\varepsilon (x)\rightarrow \frac{1}{|z|^2}(-z_2dz_1+z_1dz_2),\\ & x=\exp _y(z_1\nu _1(y)+z_2\nu _2(y)) \end{align*}$$as ε→0$\varepsilon \rightarrow 0$, for all sufficiently small z≠0$z\ne 0$ and y∈M$y\in M$. Here, {ν1,ν2}$\lbrace \nu _1,\nu _2\rbrace$ is a normal frame for M$M$ in N$N$. This improves a recent result by De Philippis and Pigati (2022), who built a solution for which the concentration phenomenon holds in an energy, measure‐theoretical sense. [ABSTRACT FROM AUTHOR]

Nowadays there are many approximate methods for thermal conductivity calculation, that lead to satisfactory outcomes in engineering practice. With the new approach, the solution of the nonlinear thermal conductivity equation was investigated. In addition to this, we reviewed the approximate solution of the nonlinear equation of thermal conductivity with cubic nonlinearity. To solve the problems of mathematical analysis, differential and integral equations, and boundary value problems of mathematical physics, difference, and interpolation are applied. Thus, for thinking of the effectiveness and reasonableness of these approaches, it is crucial for their theoretical investigation. The solution to these questions was found for each class of equation and each of its methods in their way and was often represented as significant difficulty, and in many cases was an obstacle for the current time. A natural approach to solving this issue is the use of the ideas of functional analysis. The variational principle initially was considered as a variational approach for solving linear functional equations and finding eigenvalues of linear operators. As in any variational approach, the problem of solving an equation will be brought to finding the extremum of the certain function of a special type, given over the entire space. It was found that the approach is useful in a way of minimizing functions of the more general type. [ABSTRACT FROM AUTHOR]

In this work, we introduce a strongly continuous one‐parameter family of bounded linear operators that completely describes the well‐posedness of a second order abstract differential delay equation in the initial history space Lp([−r,0];X)$L^p([-r,0];X)$, r>0$r>0$. This family, which satisfies a specific functional equation is applied to characterize the mild solution of the considered second order delay equation. [ABSTRACT FROM AUTHOR]

In this study, a computational fluid dynamics (CFD) model was developed to simulate the aerodynamic performance of the National Renewable Energy Laboratory (NREL) offshore 5‐MW baseline wind turbine with single rotor and full wind turbine. Using statistical methods, the relation between pitch angle and performance when the speed is above the rated wind speed was analyzed; furthermore, other published data were compiled to establish the functional equations of power, thrust with various inflow wind speeds, and pitch angles. In addition, according to shape optimization based on parametric modeling, the two‐dimensional cross‐section of the wind turbine blade can be defined through a metasurface approach, and the three‐dimensional surface modeling of the wind turbine blade, nacelle, and tower is completed using the nonuniform rational B‐splines (NURBS) interpolator. In terms of aerodynamic simulation, the unstructured polygon mesh was used herein to discretize the space and simulate the highly curved and twisted surfaces of the blade. In this study, the compact and accurate mesh form obtained through a grid independence test will be used to analyze the distribution of the pressure coefficient, shear stress coefficient, and limited streamline on the blade surface at various inflow wind speeds and pitch angles to understand the differences between different turbulence models and the causes of power and thrust attenuation at high inflow wind speeds. In addition, the phenomena of blade‐tip vortices, dynamic stall, blade loading, and the interaction between nacelle and tower were collectively explored. [ABSTRACT FROM AUTHOR]

This paper investigates a class of higher order neutral functional dynamic equations with Laplacians. By using the generalized Riccati technique, integral averaging technique, and inequality technique, several new oscillation criteria are established. Our findings build on many previous results (Hassan and Kong, doi:10.1002/mma.4283; Wu and Sun, https://doi.org/10.1515/ms‐2015‐0166) and address cases that are not covered by existing criteria in the literature. Finally, we present some interesting examples to illustrate the versatility of new theorems. [ABSTRACT FROM AUTHOR]

In this paper, we prove three results on the density, respectively, local density and clustering of zeros of the Beurling zeta function ζ(s)$\zeta (s)$ close to the one‐line σ:=ℜs=1$\sigma :=\Re s=1$. The analysis here brings about some news, sometimes even for the classical case of the Riemann zeta function. As a complement to known results for the Selberg class, first we prove a Carlson type zero density estimate. Note that density results for the Selberg class rely on use of the functional equation of ζ, not available in the Beurling context. Our result sharpens results of Kahane, who proved an O(T)$O(T)$ estimate for zeros lying precisely just on a vertical line ℜs=a$\Re s=a$ in the critical strip. Next we deduce a variant of a well‐known theorem of Turán, extending its range of validity even for rectangles of height only h=2$h=2$. Finally, we extend a zero clustering result of Ramachandra from the Riemann zeta case. A weaker result — which, on the other hand, is a strong sharpening of the average result from the classic book of Montgomery — was worked out by Diamond, Montgomery and Vorhauer. On our way, we show that some obscure technicalities of the Ramachandra paper can be avoided. [ABSTRACT FROM AUTHOR]

Let Ωi, Ωo be bounded open connected subsets of ℝn that contain the origin. Let Ω(ϵ)≡Ωo\ϵΩi‾ for small ϵ > 0. Then, we consider a linear transmission problem for the Helmholtz equation in the pair of domains ϵΩi and Ω(ϵ) with Neumann boundary conditions on ∂Ωo. Under appropriate conditions on the wave numbers in ϵΩi and Ω(ϵ) and on the parameters involved in the transmission conditions on ϵ∂Ωi, the transmission problem has a unique solution (ui(ϵ, ·), uo(ϵ, ·)) for small values of ϵ > 0. Here, ui(ϵ, ·) and uo(ϵ, ·) solve the Helmholtz equation in ϵΩi and Ω(ϵ), respectively. Then, we prove that if x ∈ Ωo \ {0}, then uo(ϵ, x) can be expanded into a convergent power expansion of ϵ, κnϵlogϵ,δ2,nlog−1ϵ for ϵ small enough. Here, κn=1 if n is even and κn=0 if n is odd, and δ2, 2 ≡ 1 and δ2, n ≡ 0 if n ≥ 3. [ABSTRACT FROM AUTHOR]

Sobolev‐type fractional functional evolution equations have many applications in the modeling of many physical processes. Therefore, we investigate the existence of mild solutions for fractional‐order time‐delay evolution equation of Sobolev type with multiorders in a Banach space with the help of two various methods. First, we study abstract problem by assuming the existence and compactness of E−1 and introduce a closed‐form representation of a mild solution via a newly defined delayed Mittag–Leffler‐type function which is generated by nonpermutable and permutable linear‐bounded operators. Our second method is based on the theory of Sobolev‐type delayed resolvent families generated by the pair (A, E) and a subordination principle, and in this case, any restrictive assumptions are not required on E. Furthermore, we derive an exact analytical representation of solutions for multidimensional fractional functional dynamical systems with nonpermutable and permutable matrices. As an example, the Sobolev‐type delayed wave equation with a damping term is provided to illustrate the applications of the abstract results. [ABSTRACT FROM AUTHOR]

In this paper, a new and advanced quartic functional equation is introduced, and its classical stability results in various normed spaces are discussed. An appropriate illustration is also presented to invalidate the stability result for a very critical instance. At the end of this paper, we have concluded that the stability results are valid in various normed spaces. [ABSTRACT FROM AUTHOR]

The goal of this paper is to study the oscillation of the following second‐order nonlinear functional dynamic equation [r(t)(yΔ(t))α]Δ+Υ(t)yβ(δ(t))=0,t∈𝕋0=[t0,∞)∩𝕋.The main aim is to establish the necessary and sufficient conditions for the oscillation of the equation. The results are obtained when α > β and α < β. At the end, examples are provided to illustrate the analytical findings. We point out that the results are new even for the case 𝕋=ℝ and 𝕋=ℤ. [ABSTRACT FROM AUTHOR]

Many evolutionary processes from various fields of physical and engineering sciences undergo abrupt changes of state at certain moments of time between intervals of continuous evolution. These processes are more suitably modeled by impulsive differential equations. In this paper, we study the controllability of an impulsive fractional differential equation with infinite state‐dependent delay in an arbitrary Banach space. We apply semigroup theory and Schaefer fixed point theorem. As an application, we include an example to illustrate the theory. [ABSTRACT FROM AUTHOR]

Let f be a Bianchi modular form, that is, an automorphic form for GL(2) over an imaginary quadratic field F, and let p be a prime of F at which f is new. Let K be a quadratic extension of F, and L(f/K,s) the L‐function of the base‐change of f to K. Under certain hypotheses on f and K, the functional equation of L(f/K,s) ensures that it vanishes at the central point. The Bloch–Kato conjecture predicts that this should force the existence of non‐trivial classes in an appropriate global Selmer group attached to f and K. In this paper, we use the theory of double integrals developed by Salazar and the second author to construct certain p‐adic Abel–Jacobi maps, which we use to propose a construction of such classes via Stark–Heegner cycles. This builds on ideas of Darmon and in particular generalises an approach of Rotger and Seveso in the setting of classical modular forms. [ABSTRACT FROM AUTHOR]

A significant fraction of the experimental works on the rheology of polymer melts include an attempt to find the zero‐shear‐rate viscosity η0. This is done for good reasons, because η0 is a limiting property that depends only on thermodynamic variables and, importantly, the molecular and supermolecular structure of the melt. As with all limiting properties, η0 is impossible to measure directly. Fortunately with many melts, it can be estimated from viscosity measurements at very low shear rates or frequencies, but still remains one of those properties that becomes in the limit very prone to error. The common approach is to use a set of frequency‐ or shear‐rate‐dependent data and extrapolate to find η0. As with any extrapolation, the major question is the function used for the extrapolation. This question is addressed in some detail in this article. The question of which function to use was discarded in favor of using a large sample of 20 equations of many functional forms. This sample of randomly chosen equations was used to generate a set of η0 values, and the statistics of this distribution were examined, in the usual fashion, by description with an analytical probability density function that gives a high probability of being a likely generator of the data. In addition, a weighted average was proposed, where the weighting factor takes into account the quality of the fit. For testing these ideas, the room temperature melts of poly(vinyl isobutyl ether), poly(isobutylene), and poly(dimethyl siloxane) were used. The η0 of the latter was reachable; for the other resins, a falling ball technique was attempted. [ABSTRACT FROM AUTHOR]

We introduce motivic zeta functions for matroids. These zeta functions are defined as sums over the lattice points of Bergman fans, and in the realizable case, they coincide with the motivic Igusa zeta functions of hyperplane arrangements. We show that these motivic zeta functions satisfy a functional equation arising from matroid Poincaré duality in the sense of Adiprasito–Huh–Katz. In the process, we obtain a formula for the Hilbert series of the cohomology ring of a matroid, in the sense of Feichtner–Yuzvinsky. We then show that our motivic zeta functions specialize to the topological zeta functions for matroids introduced by van der Veer, and we compute the first two coefficients in the Taylor expansion of these topological zeta functions, providing affirmative answers to two questions posed by van der Veer. [ABSTRACT FROM AUTHOR]

Microreactor technology is increasingly applied in the chemical‐pharmaceutical industry for safer and more efficient drug production as compared to the traditional batch process. This technology is employed for the first time to study the production of 2‐[methyl(pyridin‐2‐yl)amino]ethanol, the first intermediate in rosiglitazone synthesis. Under the optimum operating conditions, a single microreactor chip at 160 °C allowed to produce the equivalent of more than five batch reactors at 120 °C. The kinetic study indicated that the reaction is second order. Thermodynamic parameters were determined by the Eyring equation and density functional theory (DFT) calculations with good agreement. DFT results suggested a concerted nucleophilic aromatic substitution reaction mechanism. [ABSTRACT FROM AUTHOR]

Riverine ecosystems provide important ecosystem services reflecting their unique forms and functions. While the effects of stressors such as land cover change, climate change and growing economies on riverine ecosystem services (RES) have been well researched, the effect of the structure of the river network itself is less understood. This paper compares the capacity of different river network topologies in the delivery of selected RES. For three contrasting synthetic river network topologies (Long Trellis Narrow; Coastal Dendritic; Inland Dendritic), we applied simple functional equations to model six RES: water supply, hydropower generation, sediment retention, nutrient uptake, flood attenuation and aquatic habitat provision. Results showed that the synthetic topologies deliver different levels of RES, driven by their differences in physical structure. For example, the Inland Dendritic network removed more nitrate and better attenuated flooding due to its relatively longer lower reaches but offered poorer prospects for water supply because the longer reaches were more susceptible to transmission losses (e.g., due to bed seepage). This study provides a valuable first step in understanding the effect of river network topology on RES delivery, and the relative strengths among network types. Understanding these effects can aid decision makers in the conservation and restoration of degraded river basins to preserve RES for future generations. [ABSTRACT FROM AUTHOR]

Let f be a holomorphic or Hecke–Maass cusp form for the full modular group SL(2,Z). In this paper we use the circle method to prove aWeyl-type subconvexity for GL(2) L-functions.We prove that ... for any ϵ > 0. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the existence of mild solution of functional integrodifferential equation with nonlocal conditions. To establish this results by using the resolvent operator theory and Sadovskii‐Krasnosel'skii type of fixed point theorem and to show the usefulness and the applicability of our results to a broad class of functional integrodifferential equations, an example is given to illustrate the theory. [ABSTRACT FROM AUTHOR]

In recent years, random functional or stochastic equations have been reported in a large class of problems. In many cases, an exact analytical solution of such equations is not available and, therefore, is of great importance to obtain their numerical approximation. This study presents a numerical technique based on Bernstein operational matrices for a family of stochastic fractional integro‐differential equations (SFIDE) by means of the trapezoidal rule. A relevant feature of this method is the conversion of the SFIDE into a linear system of algebraic equations that can be analyzed by numerical methods. An upper error bound, the convergence, and error analysis of the scheme are investigated. Three examples illustrate the accuracy and performance of the technique. [ABSTRACT FROM AUTHOR]

One of the most well known models of non‐expected utility is Gul's (1991) model of disappointment aversion. This model, however, is defined implicitly, as the solution to a functional equation; its explicit utility representation is unknown, which may limit its applicability. We show that an explicit representation can be easily constructed, using solely the components of the implicit representation. We also provide a more general result: an explicit representation for preferences in the betweenness class that also satisfy negative certainty independence (Dillenberger 2010) or its counterpart. We show how our approach gives a simple way to identify the parameters of the representation behaviorally and to study the consequences of disappointment aversion in a variety of applications. [ABSTRACT FROM AUTHOR]

We study the well‐posedness of the fractional degenerate integro‐differential equations (Pα):Dα(Mu)(t)=Au(t)+∫−∞ta(t−s)Au(s)ds+∫−∞tb(t−s)Bu(s)ds+f(t),(t∈T:=[0,2π]),in Lebesgue–Bochner spaces Lp(T;X) and Besov spaces Bp,qs(T;X), where A, B and M are closed linear operators on a Banach space X satisfying D(A)∩D(B)⊂D(M), D(A)∩D(B)≠{0}, α>0 and a,b∈L1(R+). We completely characterize the well‐posedness of (Pα) in the above vector‐valued function spaces on T by using operator‐valued Fourier multiplier. We also give an example that our abstract results may be applied. [ABSTRACT FROM AUTHOR]

Summary: In this article, a starting control problem with two intermediate moments of observation in the mixed linear problem for the second‐order hyperbolic equation with quadratic functional is studied. The necessary and sufficient optimality conditions are obtained in the form of variational inequality. [ABSTRACT FROM AUTHOR]

The conventional allometric method entails fitting a straight line to logarithmic transformations of the original bivariate data and then back‐transforming the resulting equation (at least implicitly) to form a two‐parameter power function, Y = a × Xb, on the arithmetic scale. Although the protocol is widely used in contemporary research, it commonly performs poorly and thereby leads investigators to form inaccurate impressions of the dominant pattern in their data. Here I re‐examine the metabolic allometry for six species of carabid beetle to illustrate the problem that arises when pattern in the original data can be described by two (or more) statistically equivalent equations with different functional form. Whereas conventional analyses of data for the beetles yielded only a single descriptive model for each dataset (i.e., the two‐parameter power equation), the more versatile protocol used here fitted two to four statistically equivalent equations (including the two‐parameter power function) to the same sets of observations. Conclusions based on just the power equation estimated by conventional allometry would be misleading because the equation does not afford a unique description for pattern in the data. Research highlights: Conventional allometry fits a two‐parameter power equation to untransformed data, but the two‐parameter equation may not be the only equation—or even the best—for describing pattern in the data. Conclusions based on conventional models consequently may be based on incomplete and misleading evidence. [ABSTRACT FROM AUTHOR]

Functional response models describe the relationship between prey density and per capita prey consumption rate by a predator. Type II functional responses, in which density‐dependent predation occurs via a decelerating feeding rate, seem to prevail in nature and are commonly described by Holling's disk equation. In the derivation of the disk equation, Holling did not include digestion time. Although some authors have later extended the interpretation of handling time by also including digestion time, this violates the key assumption of the disk equation that the processes of searching for and handling prey are mutually exclusive. The steady‐state satiation (SSS) equation is a functional response model that discriminates between handling and digestion time. The application of the SSS equation is underutilized so far in the ecological literature, probably due to its complexity. In this study, we first tested the viability of the SSS equation. Second, we investigated the mechanistic basis of the SSS equation, comparing the model's predictions with directly observed data. For this purpose, we used predator–prey systems of different taxa, that is, the ladybird beetle Hippodamia variegata preying on the aphid Aphis fabae, the lacewing Chrysoperla agilis preying on the aphid Myzus persicae, and the predatory mite Iphiseius degenerans preying on the thrips Frankliniella occidentalis. Our results show that the SSS equation is viable and can realistically describe type II functional response. In all predator–prey systems we tested, the model fitted the data reasonably well and provided realistic estimation of its parameters. [ABSTRACT FROM AUTHOR]

By using Aoki‐Rolewicz Theorem on p‐normalizing a quasi‐normed space, we prove stability results for Euler‐Lagrange quadratic functional equations in quasi‐Banach spaces. These results improve stability results and give the answer to Kim‐Rassias's question. [ABSTRACT FROM AUTHOR]

A class of one‐dimensional time‐fractional parabolic differential equations with delay effects of functional type in the time component is numerically investigated in this work. To that end, a compact difference scheme is constructed for the numerical solution of those equations based on the idea of separating the current state and the prehistory function. In these terms, the prehistory function is approximated by means of an appropriate interpolation–extrapolation operator. A discrete form of the fractional Gronwall inequality is employed to provide an optimal error estimate. The existence and uniqueness of the numerical solutions, the order of approximation error for the constructed scheme, the stability and the order of convergence are mathematically investigated in this work. [ABSTRACT FROM AUTHOR]

The aim of this paper is to introduce and investigate some of the primary generalizations and unifications of the Peters polynomials and numbers by means of convenient generating functions and p‐adic integrals method. Various fundamental properties of these polynomials and numbers involving some explicit series and integral representations in terms of the generalized Stirling numbers, generalized harmonic sums, and some well‐known special numbers and polynomials are presented. By using p‐adic integrals, we construct generating functions for Peters type polynomials and numbers (Apostol‐type Peters numbers and polynomials). By using these functions with their partial derivative eqautions and functional equations, we derive many properties, relations, explicit formulas, and identities including the Apostol‐Bernoulli polynomials, the Apostol‐Euler polynomials, the Boole polynomials, the Bernoulli polynomials, and numbers of the second kind, generalized harmonic sums. A brief revealing and historical information for the Peters type polynomials are given. Some of the formulas given in this article are given critiques and comments between previously well‐known formulas. Finally, two open problems for interpolation functions for Apostol‐type Peters numbers and polynomials are revealed. [ABSTRACT FROM AUTHOR]

We obtain a recursive formulation for a general class of optimization problems with forward‐looking constraints which often arise in economic dynamic models, for example, in contracting problems with incentive constraints or in models of optimal policy. In this case, the solution does not satisfy the Bellman equation. Our approach consists of studying a recursive Lagrangian. Under standard general conditions, there is a recursive saddle‐point functional equation (analogous to a Bellman equation) that characterizes a recursive solution to the planner's problem. The recursive formulation is obtained after adding a co‐state variable μt summarizing previous commitments reflected in past Lagrange multipliers. The continuation problem is obtained with μt playing the role of weights in the objective function. Our approach is applicable to characterizing and computing solutions to a large class of dynamic contracting problems. [ABSTRACT FROM AUTHOR]

This study aims at investigating the reachable set estimation for delayed memristive neural networks (MNNs) with bounded disturbances. Under the framework of Filippov's solution and differential inclusion theory, a reachable set estimation criterion is established by means of Lyapunov functional. The obtained condition guarantees that all the states of MNNs are bounded. Moreover, a delay‐dependent feedback controller design problem for MNNs is also considered. The exponential stability and stabilisability conditions are obtained. Two numerical examples are presented to demonstrate the usefulness of the proposed results. [ABSTRACT FROM AUTHOR]

We study the well‐posedness of the fractional differential equations with infinite delay (Pα):Dαu(t)=Au(t)+∫−∞ta(t−s)Au(s)ds+∫−∞tb(t−s)Bu(s)ds+f(t),(0≤t≤2π),on Lebesgue–Bochner spaces Lp(T;X) and Besov spaces Bp,qs(T;X), where A and B are closed linear operators on a Banach space X satisfying D(A)∩D(B)≠{0},  α>0 and a,b∈L1(R+). Under suitable assumptions on the kernels a and b, we completely characterize the well‐posedness of (Pα) in the above vector‐valued function spaces on T by using known operator‐valued Fourier multiplier theorems. We also give concrete examples where our abstract results may be applied. [ABSTRACT FROM AUTHOR]

This paper describes the classical investigation of various stability results of multiplicative inverse undecic and duodecic functional equations pertinent to Ulam stabiity theory in non‐Archimedean fields via fixed point technique and then presents two refutations that the stability results are invalid for control cases. [ABSTRACT FROM AUTHOR]

The problem of controlling two parallel interconnected asynchronous sequential machines (ASMs) by a single corrective controller is studied. The objective is to realise immediate recovery against transient faults that cause unauthorised state transitions to each ASM. Since only output feedback is available to the controller, the correction procedure must be able to accommodate state uncertainty underlying the composite machine. A computationally efficient scheme of controller synthesis is presented in the framework of corrective control theory. To validate the practicality of the scheme, the authors apply the proposed controller to a faulty asynchronous FPGA system in parallel connection. Experimental results confirm the instantaneous fault recovery of the closed-loop system. [ABSTRACT FROM AUTHOR]

We consider a time‐dependent model for the diffusion of a substance through an incompressible fluid in a perforated domain Ωϵ, Ωϵ⊂Ω⊂Rn with n = 3,4. The fluid flows in a domain containing a periodical set of "obstacles" (Ω\Ωϵ) placed along an inner (n − 1)‐dimensional manifold Σ⊂Ω. The size of the obstacles is much smaller than the size of the characteristic period ϵ. An advection term appears in the partial differential equation linking the fluid velocity with the concentration, while we assume a nonlinear adsorption law on the boundary of the obstacles. This law involves a monotone nonlinear function σ of the concentration and a large adsorption parameter. The "critical adsorption parameter" depends on the size of the obstacles , and, for different sizes, we derive the time‐dependent homogenized models. These models contain a "strange term" in the transmission conditions on Σ, which is a nonlinear function and inherits the properties of σ. The case in which the fluid velocity and the concentration do not interact is also considered for n ≥ 3. [ABSTRACT FROM AUTHOR]

In the present study, we have reparametrized the HISS exchange functional. The new "reHISS" exchange provides a balance between short‐ and mid‐range Hartree–Fock exchange (HFX) and a large total HFX coverage, with a fast convergence to zero HFX in the long range. The five parameters in this functional (according to equations 3 and 4 in the main text) are cSR = 0.15, cMR = 2.5279, cLR = 0, ωSR = 0.27, and ωLR = 0.2192. The combination of reHISS exchange with a reparametrized B97c‐type correlation functional (Chan et al., J. Comput. Chem. 2017, 38, 2307) and a D2 dispersion term (s6 = 0.6) gives the reHISSB‐D method. We find it to be more accurate than related screened‐exchange methods and, importantly, its accuracy is more uniform across different properties. Fundamentally, our analysis suggests that the good performance of the reHISS exchange is related to it capturing a near‐optimal proportion of HFX in the range of interelectronic distance that is important for many chemical properties, and we propose this range to be approximately 1–4Å. © 2018 Wiley Periodicals, Inc. Three‐range screened‐exchange density functional theory (SX‐DFT) protocol provides an opportunity for attaining superior accuracy when compared with two‐range SX‐DFT. The present study has established the optimal variations for mixing Hartree–Fock and DFT components. The authors have harnessed this knowledge to devise the reHISSB‐D method. It has improved accuracy when compared with related SX‐DFT methods, and provides a solid foundation for further refinement of SX‐DFT. [ABSTRACT FROM AUTHOR]

The aim of this paper is to construct a new method related to a family of operators to define generating functions for special numbers and polynomials. With the help of this method, we investigate various properties of these special numbers and polynomials with their generating functions, functional equations, and differential equations. We also give some algorithms to calculate the values of these numbers and polynomials. Moreover, we introduce some fundamental properties of this operator. By applying integral method to this operator, we obtain integral formulas together with combinatorial sums. Moreover, by applying this operator, the Lagrange inversion formula and convolution formula to these generating functions, we derive some novel identities and relations including combinatorial sums and combinatorial numbers including these new numbers and polynomials, ie, the Apostol‐Bernoulli numbers, the Apostol‐Euler numbers, the Stirling numbers, the central factorial numbers, and the other special numbers and polynomials. In addition, we give some examples derived from relations between composita and this operator on the set of formal power series. Finally, we give complex form of the Fourier series for these generating functions. By using these Fourier series, we not only derive some series relations and trigonometric sums including trigonometric function and hyperbolic functions but also give an example for a differential equation. [ABSTRACT FROM AUTHOR]

Let H be a fixed graph and [ABSTRACT FROM AUTHOR]

An equational condition is a set of equations in an algebraic language, and an algebraic structure satisfies such a condition if it possesses terms that meet the required equations. We find a single nontrivial equational condition which is implied by any nontrivial idempotent equational condition. [ABSTRACT FROM AUTHOR]

In this article, we use some greedy algorithms to avoid the ill-conditioning of the final linear system in unsymmetric Kansa collocation method. The greedy schemes have the same background, but we use them in different settings. In the first algorithm, the optimal trial points for interpolation obtained among a huge set of initial points are used for numerical solution of partial differential equations (PDEs). In the second algorithm, based on the Kansa's method, the PDE is discretized to a finite number of test functional equations, and a greedy sparse discretization is applied for approximating the linear functionals. Each functional is stably approximated by some few trial points with an acceptable accuracy. The third greedy algorithm is used to generate the test points. This paper shows that the greedily selection of nodes yields a better conditioning in contrast with usual full meshless methods. Some well-known PDE examples are solved and compared with the full unsymmetric Kansa's technique. [ABSTRACT FROM AUTHOR]

This study focuses on the exponential stabilisation problem for a general class of memristive neural networks subjected to both stochastic disturbance and time‐varying delays under periodically intermittent adaptive control. The stochastic disturbances are described as Brownian motions in the considered networks. An adaptive updated rule and a periodically intermittent adaptive control strategy are designed for the exponential stabilisation of memristive neural networks subjected to both stochastic disturbance and time‐varying delays. Then, by adopting the adaptive control technique, differential inclusion theory and set‐valued maps, and by building a new Lyapunov–Krasovskii functional, many novel sufficient conditions are proposed to guarantee exponential stabilisation for stochastic memristive neural networks. Different from existing results on stabilisation of stochastic memristive neural networks, the obtained criteria in this study are directly derived according to the parameters of networks. Finally, an example is carried out to demonstrate the validity of the theoretic results. [ABSTRACT FROM AUTHOR]

In this paper, a computational technique based on the pseudo-spectral method is presented for the solution of the optimal control problem constrained with elliptic variational inequality. In fact, our aim in this paper is to present a direct approach for this class of optimal control problems. By using the pseudo-spectral method, the infinite dimensional mathematical programming with equilibrium constraint, which can be an equivalent form of the considered problem, is converted to a finite dimensional mathematical programming with complementarity constraint. Then, the finite dimensional problem can be solved by the well-developed methods. Finally, numerical examples are presented to show the validity and efficiency of the technique. Copyright © 2017 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

We explore the set of preferences defined over temporal lotteries in an infinite horizon setting. We provide utility representations for all preferences that are both recursive and monotone. Our results indicate that the class of monotone recursive preferences includes Uzawa-Epstein preferences and risk-sensitive preferences, but leaves aside several of the recursive models suggested by Epstein and Zin (1989) and Weil (1990). Our representation result is derived in great generality using Lundberg's (1982, 1985) work on functional equations. [ABSTRACT FROM AUTHOR]

In this paper, we are introducing pertinent Euler-Lagrange-Jensen type k-quintic functional equations and investigate the 'Ulam stability' of these new k-quintic functional mappings f: X→ Y, where X is a real normed linear space and Y a real complete normed linear space. We also solve the Ulam stability problem for Euler-Lagrange-Jensen alternative k-quintic mappings. Copyright © 2016 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

The aim of this article is to define some new families of the special numbers. These numbers provide some further motivation for computation of combinatorial sums involving binomial coefficients and the Euler kind numbers of negative order. We can show that these numbers are related to the well-known numbers and polynomials such as the Stirling numbers of the second kind and the central factorial numbers, the array polynomials, the rook numbers and polynomials, the Bernstein basis functions and others. In order to derive our new identities and relations for these numbers, we use a technique including the generating functions and functional equations. Finally, we give not only a computational algorithm for these numbers but also some numerical values of these numbers and the Euler numbers of negative order with tables. We also give some combinatorial interpretations of our new numbers. Copyright © 2016 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

This study is concerned with the dwell‐time stability for impulsive systems under periodic or aperiodic impulses. A Lyapunov‐like functional approach is established to study the stability for impulsive systems. The Lyapunov‐like functional is time‐varying and decreasing but is not imposed definite positive nor continuous. A specific Lyapunov‐like functional is constructed by introducing the integral of state together with the cross‐terms of the integral and impulsive states. A tight bounding is obtained for the derivative of this functional with the help of the improved Jensen inequality reported recently and the integral equation of the impulsive system. On the basis of the Lyapunov‐like functional approach, new dwell‐time‐dependent stability results with ranged dwell‐time, maximal dwell‐time and minimal dwell‐time are derived for periodic or aperiodic impulsive systems. The stability results have less conservatism than some existing ones, which are illustrated by numerical examples. [ABSTRACT FROM AUTHOR]

Construction laborer assignment is the assignment of laborers in a team to the tasks for a daily work in a construction project. This study proposes a mathematical programming approach to examine the optimal construction laborer assignment problem with the objectives of productivity and occupational health and safety while considering equity between laborers. Several linearization techniques are presented to transform the mathematical programming model into a mixed-integer linear program, which can be solved by off-the-shelf mixed-integer linear solvers. The proposed model is applied to extensive numerical experiments and the results show that the mathematical programming approach outperforms a conventional heuristic by over 10% in terms of job completion time. This implies considerable overhead, equipment, and manpower savings for the construction industry. [ABSTRACT FROM AUTHOR]

Topological conjugacy for continuous skew product flows on topological group bundles and semidirect products is studied resulting in a splitting theorem. The results are used to discuss fiber entropy. [ABSTRACT FROM AUTHOR]

A robust iterative learning control (ILC) method is proposed for industrial batch processes with input delay subject to time‐varying uncertainties, based on a two‐dimensional (2D) system description of batch process operation. To compensate the input delay, a 2D state predictor is established to predict the augmented system states, such that a 2D ILC design is developed for the 'delay‐free' 2D system based on using only the measured output errors of current and previous cycles. Delay‐dependent stability conditions for the resulting 2D system are established in terms of matrix inequalities by defining a comprehensive 2D Lyapunov–Krasovskii functional candidate along with free‐weighting matrices. By solving these matrix inequalities using a cone complementarity linearisation method, the ILC controller is explicitly derived together with an adjustable H infinity performance index. An important merit is that perfect tracking can be realised for a batch process with arbitrarily long input delay if the delay‐free part of the 2D system can be stabilised, in no presence of time‐varying uncertainties. Moreover, the time integral of tracking error can be added as an extended 2D system state for ILC design to eliminate steady‐state output error for all batches. An illustrative example of injection moulding process is given to demonstrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]