Analysis of search-extension method for finding multiple solutions of nonlinear problem

For numerical computations of multiple solutions of the nonlinear elliptic problemΔu f(u)=0 inΩ, u=0 onΓ, a search-extension method (SEM) was proposed and systematically studied by the authors. This paper shall complete its theoretical analysis. It is assumed that the nonlinearity is non-convex and its solution is isolated, under some conditions the corresponding linearized problem has a unique solution. By use of the compactness of the solution family and the contradiction argument, in general conditions, the high order regularity of the solution u∈H~(1 α),α>0 is proved. Assume that some initial value searched by suitably many eigenbases is already fallen into the neighborhood of the isolated solution, then the optimal error estimates of its nonlinear finite element approximation are shown by the duality argument and continuation method.     

Stabilization of fractional bilinear systems with multiple inputs

In this paper, we study in a constructive way the stabilization problem of fractional bilinear systems with multiple inputs. Using the quadratic Lyapunov functions and some additional hypotheses on the unit sphere, we construct stabilizing feedback laws for the considered fractional bilinear system. A numerical example is given to illustrate the efficiency of the obtained result.     

Structure of multiple solutions for nonlinear differential equations

Based on the eigensystem {λj,φj}of -Δ, the multiple solutions for nonlinear problem Δu f(u) =0 in Ω, u=0 on Ω are approximated. A new search-extension method (SEM), which consists of three steps in three level subspaces, is proposed. Numerical simulations for several typical nonlinear cases, i.e. f(u) = u~3,u~2(u-p),u~2(u~2 -p),     

λ-Convergence of multiple Walsh–Paley series and sets of uniqueness

Mathematical Notes - λ-convergent multiple Walsh–Paley series on a multidimensional dyadic group are studied. It is proved that, for all λ > 1, any arbitrary finite union of...     

Convergence and localization of multiple Fourier series for classes of bounded Λ-variation

Previously obtained results for convergence and localization of multiple trigonometric Fourier series for functions from classes of bounded Λ-variation and embedding of these classes into each other are strengthened in the paper. The case when sequences Λ and M have a limit of the ratio Σ _n=1 ^N 1/λ _n /Σ _n=1 ^N 1/µ _n is considered. A more strict condition, the existence of a limit for the ratio λ _n /µ _n was considered before.     

Existence of multiple positive solutions for one-dimensional p-Laplacian operator

In this paper, we consider the multipoint boundary value problem for one-dimensional p-Laplacian $$(\phi_{p}(u'))'+f(t,u,u')=0,\quad t\in [0,1],$$ subject to the boundary value conditions: $$u'(0)=\sum_{i=1}^{n-2}\alpha_{i}u'(\xi_{i}),\qquad u(1)=\sum_{i=1}^{n-2}\beta_{i}u(\xi_{i}),$$ where φ _p (s)=|s| ^p?2?s,p>1;ξ _i ∈(0,1) with 0<ξ ₁<ξ ₂<???<ξ _n?2<1 and α _i ,β _i satisfy α _i ,β _i ∈[0,∞),0≤∑ _i=1 ^n?2 α _i <1 and 0≤∑ _i=1 ^n?2 β _i <1. Using a fixed point theorem for operators in a cone, we provide sufficient conditions for the existence of multiple positive solutions to the above boundary value problem.     

N-cyclic functions and multiple subharmonic solutions of Duffingʼs equation

We introduce, in the abstract framework of finite isometry groups on a Hilbert space, a generalization of antiperiodicity called N-cyclicity. The non-existence of N-cyclic solutions of a certain type for the autonomous ODE $x^{″}+g(x)=0$ implies the existence of N different subharmonic solutions for some forced equations of the type $x^{″}+g(x)+c{x}^{\prime }=\epsilon f(t)$ where c and ε are some positive constants and f is, for instance, a sinusoidal function.     

Numerical study of multiple periodic flow states in spherical Couette flow

The supercritical flow states of the spherical Couette flow between two concentric spheres with the inner sphere rotating are investigated via direct numerical simulation using a three-dimensional finite difference method. For comparison with experiments of Nakabayashi et al. and Wimmer, a narrow gap and a medium gap with clearance ratio β=0.06 and 0.18 respectively are considered for the Reynolds number range covering the     

On the Gauss-Weierstrass summability of multiple trigonometric series at µ-smoothness points

The notion of μ-smoothness points of periodic functions of several variables is introduced and the rate of convergence of the Gauss-Weierstrass means of relevant Fourier series at these points is investigated.     

Perturbation behavior of a multiple eigenvalue in generalized Hermitian eigenvalue problems

We show that a multiple eigenvalue has different sensitivities under perturbations in a generalized Hermitian eigenvalue problem. Our result provides a solution to a question raised by Stewart and Sun. We also show how this difference of sensitivities plays a role in the eigenvalue forward error analysis after the Rayleigh-Ritz process, for which we present an approach that provides tight bounds.     

Lebesgue constants of multiple de la Vallée-Poussin sums over polyhedra

One considers linear summation methods for the multiple Fourier series the multidimensional analogues of the de la Vallé-Poussin sums. The summation of the Fourier series is carried out over the homotheties of an m-dimensional starshaped polyhedron . It is shown that if has rational vertices, then the Lebesgue constants of the considered methods, with the accuracy of O((p+1)^–1. log^m–1 (n+2)) are equal to where is the Fourier transform of the function . The exact value of the principal term of the Lebesgue constant is computed in two particular cases: 1) is obtained from an m-dimensional cube by means of a linear nonsingular transformation; 2) =0. is an m-dimensional simplex.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 125, pp. 154–165, 1983.     

Generalized Jacobi–Trudi determinants and evaluations of Schur multiple zeta values

We present new determinant expressions for regularized Schur multiple zeta values. These generalize the known Jacobi–Trudi formulas and can be used to quickly evaluate certain types of Schur multiple zeta values. Using these formulas we prove that every Schur multiple zeta value with alternating entries in 1 and 3 can be written as a polynomial in Riemann zeta values. Furthermore, we give conditions on the shape, which determine when such Schur multiple zetas are polynomials purely in odd or in even Riemann zeta values.     

The existence of multiple positive solutions for a?class of?semipositone Dirichlet boundary value problems

In this paper, we consider the existence of multiple positive solutions for the following singular semipositone Dirichlet boundary value problem: $$\left\{\begin{array}{l}-x''(t)=p(t)f(t, x) +q(t),\quad t\in(0,1),\\[4pt]x(0) =0,\qquad x(1) = 0,\end{array}\right.$$ where p:(0,1)??[0,+??) and f:[0,1]×[0,+??)??[0,+??) are continuous, q:(0,1)??(???,+??) is Lebesgue integrable. Under certain local conditions and superlinear or sublinear conditions on f, by using the fixed point theorem, some sufficient conditions for the existence of multiple positive solutions are established for the case in which the nonlinearity is allowed to be sign-changing.     

Adaptive compensation of uncertain Euler–Lagrange systems with multiple time-varying actuator faults

This work proposes the command tracking problem for uncertain Euler–Lagrange (EL) systems with multiple partial loss of effectiveness (PLOE) actuator faults. Compared to existing fault-tolerant controllers for EL systems, the proposed adaptive controller accounts for parametric uncertainties in the system and multiple time-varying actuator fault parameters. The proposed method can also handle an infinite number of fault cases. The closed-loop fault-tolerant system is treated as a switched dynamical system, and a switched system stability is established using multiple Lyapunov functions. It is shown that all signals are bounded in each sub-interval and at the switching instances, and asymptotic tracking can be obtained only for a finite number of fault occurrences, whereas tracking error is bounded for the infinite case. Finally, a simulation example on a robotic manipulator is presented to show the effectiveness of the proposed method.     

Existence of multiple nontrivial solutions for a Schrödinger–Poisson system

We consider a Schrödinger–Poisson system in R³${\mathbb{R}}^3$ with potential indefinite in sign and a general nonlinearity. We use the direct variational method and Morse theory to obtain the existence of multiple nontrivial solutions for this system.     

Improving the accuracy of multiple integral evaluation by applying Romberg’s method

Romberg’s method, which is used to improve the accuracy of one-dimensional integral evaluation, is extended to multiple integrals if they are evaluated using the product of composite quadrature formulas. Under certain conditions, the coefficients of the Romberg formula are independent of the integral’s multiplicity, which makes it possible to use a simple evaluation algorithm developed for one-dimensional integrals. As examples, integrals of multiplicity two to six are evaluated by Romberg’s method and the results are compared with other methods.     

Pooling of spare parts between multiple users: How to share the benefits?

Companies that maintain capital goods (e.g., airplanes or power plants) often face high costs, both for holding spare parts and due to downtime of their technical systems. These costs can be reduced by pooling common spare parts between multiple companies in the same region, but managers may be unsure about how to share the resulting costs or benefits in a fair way that avoids free riders. To tackle this problem, we study several players, each facing a Poisson demand process for an expensive, low-usage item. They share a stock point that is controlled by a continuous-review base stock policy with full backordering under an optimal base stock level. Costs consist of penalty costs for backorders and holding costs for on-hand stock. We propose to allocate the total costs proportional to players’ demand rates. Our key result is that this cost allocation rule satisfies many appealing properties: it makes all separate participants and subgroups of participants better off, it stimulates growth of the pool, it can be easily implemented in practice, and it induces players to reveal their private information truthfully. To obtain these game theoretical results, we exploit novel structural properties of the cost function in our (S − 1, S) inventory model.     

Improved two-step Newton’s method for computing simple multiple zeros of polynomial systems

Numerical Algorithms - Given a polynomial system f that is associated with an isolated singular zero ξ whose Jacobian matrix is of corank one, and an approximate zero x that is close to...