Eigenvalue problem for an irregular λ-matrix

The solution of the eigenvalue problem is examined for the polyomial matrixD()=A_o^s+A₁^s–1+...+A_s when the matricesA ₀ andA ₂ (or one of them) are singular. A normalized process is used for solving the problem, permitting the determination of linearly independent eigenvectors corresponding to the zero eigenvalue of matrixD() and to the zero eigenvalue of matrixA ₀. The computation of the other eigenvalues ofD() is reduced to the same problem for a constant matrix of lower dimension. An ALGOL program and test examples are presented.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 58, pp. 80–92, 1976.     

On an inverse problem to Frobenius’ theorem

Let G be a finite group and e a positive integer dividing |G|, the order of G. Denoting \({L_e(G)=\{x\in G|\,x^e=1\}}\) , Frobenius proved that |L _e (G)| = ke for a positive integer k ≥ 1. In this paper, we give a complete classification of finite groups G with |L _e (G)| ≤ 2e for every e dividing |G|.     

The Navier–Stokes problem modified by an absorption term

It is well known that for the classical Navier–Stokes problem the best one can obtain is some decays in time of power type. With this in mind, we consider in this work, the classical Navier–Stokes problem modified by introducing, in the momentum equation, the absorption term |u|^σ?2 u, where σ > 1. For the obtained problem, we prove the existence of weak solutions for any dimension N ≥ 2 and its uniqueness for N = 2. Then we prove that, for zero body forces, the weak solutions extinct in a finite time if 1 < σ < 2 and exponentially decay in time if σ = 2. In the special case of a suitable force field which vanishes at some instant, we prove that the weak solutions extinct at the same instant provided 1 < σ < 2. We also prove that for non-zero body forces decaying at a power-time rate, the solutions decay at analogous power-time rates if σ > 2. Finally, we prove that for a general non-zero body force, the weak solutions exponentially decay in time for any σ > 1.     

An open problem on the optimality of an asymptotic solution to Duffing’s nonlinear oscillation problem

Based on the renormalization group method, Kirkinis (2012) [8] obtained an asymptotic solution to Duffing’s nonlinear oscillation problem. Kirkinis then asked if the asymptotic solution is optimal. In this paper, an affirmative answer to the open problem is given by means of the homotopy analysis method.     

Stability in an overdetermined problem for the Green’s function

In the plane, we consider the problem of reconstructing a domain from the normal derivative of its Green’s function (with fixed pole) relative to the Dirichlet problem for the Laplace operator. By means of the theory of conformal mappings, we derive stability estimates of Hölder type.     

Uniqueness and stability in an inverse problem for a Poisson's equation

Consider the Poisson's equation(?)″(x)=-e~(v-(?)) e~((?)-v)-N(x)with the Diriehlet boundary data,and we mainly investigate the inverse problem of determining the unknown function N(x)from a parameter function family.Some uniqueness and stability results in the inverse problem are obtained.     

Application of Maslov’s formula to finding an asymptotic solution to an elastic deformation problem

We propose a scheme of bifurcation analysis of equilibrium configurations of a weakly inhomogeneous elastic beam on an elastic base under the assumption of two-mode degeneracy; this scheme generalizes the Darinskii-Sapronov scheme developed earlier for the case of a homogeneous beam. The consideration of an inhomogeneous beam requires replacing the condition that the pair of eigenvectors of the operator from the linear part of the equation (at zero) is constant by the condition of the existence of a pair of vectors smoothly depending on the parameters whose linear hull is invariant with respect to . It is shown that such a pair is sufficient for the construction of the principal part of the key function and for analyzing the branching of the equilibrium configurations of the beam. The construction of the required pair of vectors is based on a formula for the orthogonal projection onto the root subspace of (from the theory of perturbations of self-adjoint operators in the sense of Maslov). The effect of the type of inhomogeneity of the beam on the formof its deflection is studied.     

On an affirmative answer to S. Lin?s problem

In this paper, we give an affirmative answer to the problem posed by S. Lin (2002, 2007) in [7] and [8], and give another answer to the question posed by Y. Ikeda, C. Liu and Y. Tanaka (2002) in [5].     

A failing eigenfunction expansion associated with an indefinite Sturm–Liouville problem

We consider the indefinite Sturm–Liouville problem $?f^{″}=\lambda rf$, $f^{\prime }(?1)=f^{\prime }(1)=0$ where $r\in L^1[?1,1]$ satisfies $xr(x)>0$. Conditions are presented such that the (normed) eigenfunctions $f_n$ form a Riesz basis of the Hilbert space $L_{|r|}^2[?1,1]$ (using known results for a modified problem). The main focus is on the non-Riesz basis case: We construct a function $f\in L_{|r|}^2[?1,1]$ having no eigenfunction expansion $f=\sum {\beta }_n{f}_n$. Furthermore, a sequence $({\alpha }_n)\in l^2$ is constructed such that the “Fourier series” $\sum {\alpha }_n{f}_n$ does not converge in $L_{|r|}^2[?1,1]$. These problems are closely related to the regularity property of the closed non-semibounded symmetric sesquilinear form $\mathfrak{t}[u,v]=\int u^{\prime }{\overline{v}}^{\prime }pdx$ with Dirichlet boundary conditions in $L^2[?1,1]$ where $p=1/r$. For the associated operator $T_{\mathfrak{t}}$ we construct elements in the difference between $\mathrm{dom}\mathfrak{t}$ and the domain of the associated regular closed form, i.e. $\mathrm{dom}{|T_{\mathfrak{t}}|}^{1/2}$.     

Energetic balance for the Rayleigh–Stokes problem of an Oldroyd-B fluid

Dissipation, the power due to the shear stress at the wall, the change of kinetic energy with time as well as the boundary layer thickness corresponding to the Rayleigh–Stokes problem for an Oldroyd-B fluid are established. The corresponding expressions of Maxwell, second grade and Newtonian fluids, performing the same motions, are obtained as the limiting cases of our general results. Specific features of the four models are emphasized by means of the asymptotic approximations and graphical representations. It is worth mentioning that in comparison with the Newtonian model, the power of the shear stress at the wall and the dissipation for Oldroyd-B fluids increase while the boundary layer thickness decreases.     

Lipschitz stability in an inverse problem for the Kuramoto–Sivashinsky equation

In this article, we present an inverse problem for the nonlinear 1D Kuramoto–Sivashinsky (KS) equation. More precisely, we study the nonlinear inverse problem of retrieving the anti-diffusion coefficient from the measurements of the solution on a part of the boundary and also at some positive time in the whole space domain. The Lipschitz stability for this inverse problem is our main result and it relies on the Bukhge?m–Klibanov method. The proof is indeed based on a global Carleman estimate for the linearized KS equation.     

A symmetric BEM–FEM method for an axisymmetric eddy current problem

This paper deals with the numerical solution of the time-harmonic eddy current model in an axisymmetric unbounded domain. To this end, a new symmetric BEM–FEM formulation is derived and also analyzed. Moreover, error estimates for the corresponding discretization are proven.     

Implementing an efficient fptas for the 0–1 multi-objective knapsack problem

In the present work, we are interested in the practical behavior of a new fully polynomial time approximation schemes (fptas) to solve the approximation version of the 0–1 multi-objective knapsack problem. The proposed methodology makes use of very general techniques (such as dominance relations in dynamic programming) and thus may be applicable in the implementation of fptas for other problems as well.     

Can an a priori error estimate for an approximate solution of an ill-posed problem be comparable with the error in data?

For a linear operator equation of the first kind with perturbed data, it is shown that the global (on typical sets) a priori error estimate for its approximate solution can have the same order as that for the approximate data only if the operator of the problem is normally solvable. If the operator of the problem is given exactly, this is possible only if the problem is well-posed (stable).     

The Rayleigh–Stokes problem for an edge in a generalized Oldroyd-B fluid

The velocity field corresponding to the Rayleigh–Stokes problem for an edge, in an incompressible generalized Oldroyd-B fluid has been established by means of the double Fourier sine and Laplace transforms. The fractional calculus approach is used in the constitutive relationship of the fluid model. The obtained solution, written in terms of the generalized G-functions, is presented as a sum of the Newtonian solution and the corresponding non-Newtonian contribution. The solution for generalized Maxwell fluids, as well as those for ordinary Maxwell and Oldroyd-B fluids, performing the same motion, is obtained as a limiting case of the present solution. This solution can be also specialized to give the similar solution for generalized second grade fluids. However, for simplicity, a new and simpler exact solution is established for these fluids. For β → 1, this last solution reduces to a previous solution obtained by a different technique.      

Reanalysis of an open problem associated with the fractional Schrödinger equation

It was recently shown that there are some difficulties in the solution method proposed by Laskin for obtaining the eigenvalues and eigenfunctions of the one-dimensional time-independent fractional Schrödinger equation with an infinite potential well encountered in quantum mechanics. In fact, this problem is still open. We propose a new fractional approach that allows overcoming the limitations of some previously introduced strategies. In deriving the solution, we use a method based on the eigenfunction of the Weyl fractional derivative. We obtain a solution suitable for computations in a closed form in terms of Mittag–Leffler functions and fractional trigonometric functions. It is a simple extension of the results previously obtained by Laskin et al.