Solvability conditions and monotone iterative scheme for boundary-value problems related to nonlinear monotone potential operators

This article deals with boundary-value problems (BVPs) for the second-order nonlinear differential equations with monotone potential operators of type Au := ??(k(|?u|²)?u(x)) + q(u ²)u(x), x ∈ Ω ? R ⁿ . An analysis of nonlinear problems shows that the potential of the operator A as well as the potential of related BVP plays an important role not only for solvability of these problems and linearization of the nonlinear operator, but also for the strong convergence of solutions of corresponding linearized problems. A monotone iterative scheme for the considered BVP is proposed.     

The Weyl bundle

Let F be a symplectic vector bundle over a space X. We construct a bundle of elementary C¹-algebras over X, and prove that the Dixmier-Douady invariant of this bundle is zero. The underlying Hilbert bundles, with their associated module structures, determine a characteristic class: we prove that this class is the second Stiefel-Whitney class of F.     

Double unconditional bases associated with a quadratic characteristic parameter problem

A detailed discussion is given of the quadratic characteristic value problem, $\text{x = λPx + (}\text{1}\text{λ}\text{)Qx}$, where P and Q are compact selfadjoint operators in Hilbert space. Theorems are proven showing that unconditional bases of various types may be constructed from the root vectors of $\text{A(λ) = λP + (}\text{1}\text{λ}\text{)Q}$. These theorems are of interest in the stability theory of linear evolution equations, particularly for application to certain problems in the linearized theory of hydrodynamic stability.     

Classification and unfoldings of degenerate Hopf bifurcations

This paper initiates the classification, up to symmetry-covariant contact equivalence, of perturbations of local Hopf bifurcation problems which do not satisfy the classical non-degeneracy conditions. The only remaining hypothesis is that ±i should be simple eigenvalues of the linearized right-hand side at criticality. Then the Lyapunov-Schmidt method allows a reduction to a scalar equation G(x, λ) = 0, where G(?x, λ) = ?G(x, λ). A definition is given of the codimension of G, and a complete classification is obtained for all problems with codimension ?3, together with the corresponding universal unfoldings. The perturbed bifurcation diagrams are given for the cases with codimension ?2, and for one case with codimension 3; for this last case one of the unfolding parameters is a “modal” parameter, such that the topological codimension equals in fact 2. Formulas are given for the calculation of the Taylor coefficients needed for the application of the results, and finally the results are applied to two simple problems: a model of glycolytic oscillations and the Fitzhugh nerve equations.     

Multipoint Boundary-Value Solution of Two-Point Boundary-Value Problems

An iterative scheme, in which two-point boundary-value problems (TPBVP) are solved as multipoint boundary-value problems (MPBVP), which are independent TPBVPs in each iteration and on each subdomain, is derived for second-order ordinary differential equations. Several equations are solved for illustration. In particular, the algorithm is described in detail for the first boundary-value problem (FBVP) and second boundary-value problem (SBVP). A possible extension to higher-order BVPs is discussed briefly. The procedure may be used when the original TPBVP cannot be solved (does not converge) in a single long domain. It is suitable for implementation on computers with parallel processing. However, that issue is beyond the scope of this paper. The long domain is cut into a large number of subdomains and, based on assumed boundary conditions at the interface points, the resulting local BVPs are solved by any convenient conventional method. The local solutions are then patched by using simple matching formulas, which are derived below, rather than solving large systems of algebraic equations, as it is done in similar existing methods. Assuming that the local solutions are obtained by the most efficient methods, the overall convergence speed depends on the speed of matching. The proposed matching algorithm is based on a fixed-point iteration and has only a linear convergence rate. The rate can be made quadratic by applying standard accelerating schemes, which is beyond the scope of this article.     

Isospectral flow method for nonnegative inverse eigenvalue problem with prescribed structure

The nonnegative inverse eigenvalue problem is that given a family of complex numbers λ={λ₁,…,λ_n}, find a nonnegative matrix of order n with spectrum λ. This problem is difficult and remains unsolved partially. In this paper, we focus on its generalization that the reconstructed nonnegative matrices should have some prescribed entries. It is easy to see that this new problem will come back to the common nonnegative inverse eigenvalue problem if there is no constraint of the locations of entries. A numerical isospectral flow method which is developed by hybridizing the optimization theory and steepest descent method is used to study the reconstruction. Moreover, an error estimate of the numerical iteration for ordinary differential equations on the matrix manifold is presented. After that, a numerical method for the nonnegative symmetric inverse eigenvalue problem with prescribed entries and its error estimate are considered. Finally, the approaches are verified by the numerical test results.     

Linear and nonlinear nonlocal boundary value problems for differential-operator equations

This study focuses on nonlocal boundary value problems (BVPs) for linear and nonlinear elliptic differential-operator equations (DOEs) that are defined in Banach-valued function spaces. The considered domain is a region with varying bound and depends on a certain parameter. Some conditions that guarantee the maximal L_p -regularity and Fredholmness of linear BVPs, uniformly with respect to this parameter, are presented. This fact implies that the appropriate differential operator is a generator of an analytic semigroup. Then, by using these results, the existence, uniqueness and maximal smoothness of solutions of nonlocal BVPs for nonlinear DOEs are shown. These results are applied to nonlocal BVPs for regular elliptic partial differential equations, finite and infinite systems of differential equations on cylindrical domains, in order to obtain the algebraic conditions that guarantee the same properties.     

A variational approach to define robustness for parametric multiobjective optimization problems

In contrast to classical optimization problems, in multiobjective optimization several objective functions are considered at the same time. For these problems, the solution is not a single optimum but a set of optimal compromises, the so-called Pareto set. In this work, we consider multiobjective optimization problems that additionally depend on an external parameter ${\lambda \in \mathbb{R}}$ , so-called parametric multiobjective optimization problems. The solution of such a problem is given by the λ-dependent Pareto set. In this work we give a new definition that allows to characterize λ-robust Pareto points, meaning points which hardly vary under the variation of the parameter λ. To describe this task mathematically, we make use of the classical calculus of variations. A system of differential algebraic equations will turn out to describe λ-robust solutions. For the numerical solution of these equations concepts of the discrete calculus of variations are used. The new robustness concept is illustrated by numerical examples.     

The determinant bundle on the moduli space of stable triples over a curve

We construct a holomorphic Hermitian line bundle over the moduli space of stable triples of the form (E₁, E₂,?), where E₁ and E₂ are holomorphic vector bundles over a fixed compact Riemann surfaceX, and?: E₂ → E₁ is a holomorphic vector bundle homomorphism. The curvature of the Chern connection of this holomorphic Hermitian line bundle is computed. The curvature is shown to coincide with a constant scalar multiple of the natural Kähler form on the moduli space. The construction is based on a result of Quillen on the determinant line bundle over the space of Dolbeault operators on a fixed C^∞ Hermitian vector bundle over a compact Riemann surface.     

Semistability vs. nefness for (Higgs) vector bundles

Generalizing a result of Miyaoka, we prove that the semistability of a vector bundle E on a smooth projective curve over a field of characteristic zero is equivalent to the nefness of any of certain divisorial classes θs, λs in the Grassmannians Grs(E) of locally-free quotients of E and in the projective bundles PQs, respectively (here 0<s<rkE and Qs is the universal quotient bundle on Grs(E)). The result is extended to Higgs bundles. In that case a necessary and sufficient condition for semistability is that all classes λs are nef. We also extend this result to higher-dimensional complex projective varieties by showing that the nefness of the classes λs is equivalent to the semistability of the bundle E together with the vanishing of the characteristic class .     

On the Picard bundle

Fix a holomorphic line bundle ξ over a compact connected Riemann surface X of genus g, with g?2, and also fix an integer r such that degree(ξ)>r(2g−1). Let Mξ(r) denote the moduli space of stable vector bundles over X of rank r and determinant ξ. The Fourier-Mukai transform, with respect to a Poincaré line bundle on X×J(X), of any F∈Mξ(r) is a stable vector bundle on J(X). This gives an injective map of Mξ(r) in a moduli space associated to J(X). If g=2, then Mξ(r) becomes a Lagrangian subscheme.     

Weakly complex Einstein-Finsler vector bundle

Let (E, F) be a complex Finsler vector bundle over a compact Kähler manifold (M, g) with Kähler form Φ. We prove that if (E, F) is a weakly complex Einstein-Finsler vector bundle in the sense of Aikou (1997), then it is modeled on a complex Minkowski space. Consequently, a complex Einstein-Finsler vector bundle (E, F) over a compact Kähler manifold (M, g) is necessarily Φ-semistable and (E, F) = (E₁, F₁) ? · · · ? (Ek; Fk); where F _j := F |E _j , and each (E _j , F _j ) is modeled on a complex Minkowski space whose associated Hermitian vector bundle is a Φ-stable Einstein-Hermitian vector bundle with the same factor c as (E, F).     

Existence of solutions to derivative-dependent,nonlinear singular boundary value problems

This article examines two-point boundary value problems (BVPs) for second-order, singular ordinary differential equations where the right-hand-side of the differential equation may depend on the derivative of the solution. We introduce a method to obtain a priori bounds on all potential solutions, including their “derivatives”, to the singular BVP under consideration. The approach is based on the application of differential inequalities of singular type. The ideas are then applied to yield new existence results for solutions.     

Determinant line bundle on moduli space of parabolic bundles

In Biswas and Raghavendra (Proc Indian Acad Sci (Math Sci) 103:41–71, 1993; Asian J Math 2:303–324, 1998), a parabolic determinant line bundle on a moduli space of stable parabolic bundles was constructed, along with a Hermitian structure on it. The construction of the Hermitian structure was indirect: The parabolic determinant line bundle was identified with the pullback of the determinant line bundle on a moduli space of usual vector bundles over a covering curve. The Hermitian structure on the parabolic determinant bundle was taken to be the pullback of the Quillen metric on the determinant line bundle on the moduli space of usual vector bundles. Here a direct construction of the Hermitian structure is given. For that we need to establish a version of the correspondence between the stable parabolic bundles and the Hermitian–Einstein connections in the context of conical metrics. Also, a recently obtained parabolic analog of Faltings’ criterion of semistability plays a crucial role.     

Exotic characteristic classes of quaternionic bundles

For aC ^∞ quaternionic vector bundle, the odd-dimensional real Chern classes vanish, and this allows for a construction of secondary (exotic) characteristic classes associated with a pair of quaternionic structures of a given complex vector bundle. This construction is then applied to obtain exotic characteristic classes associated with an automorphismβ of the holomorphic tangent bundle of a Kähler manifold. These results are the complex analoga of those given for the higher order Maslov classes in [V2].     

A difference-analytical method for the computation of eigenvalues of the fourth-order equations with separated boundary conditions

We propose a method for the computation of eigenvalues with odd multiplicities for spectral boundary-value problems of the 4th order with separated boundary conditions. In this method approximate eigenvalues appear as zeros of a certain function f(λ) which admits an explicit representation.     

The FMM accelerated PIES with the modified binary tree in solving potential problems for the domains with curvilinear boundaries

The paper presents an accelerating of solving potential boundary value problems (BVPs) with curvilinear boundaries by modified parametric integral equations system (PIES). The fast multipole method (FMM) known from the literature was included into modified PIES. To consider complex curvilinear shapes of a boundary, the modification of a binary tree used by the FMM is proposed. The FMM combined with the PIES, called the fast PIES, also allows a significant reduction of random access memory (RAM) utilization. Therefore, it is possible to solve complex engineering problems on a standard personal computer (PC). The proposed algorithm is based on the modified PIES and allows for obtaining accurate solutions of complex BVPs described by the curvilinear boundary at a reasonable time on the PC.

     

Nonlinear eigenvalue problems and spherical fibrations of Banach spaces

Eigenvalue problems of the form g′(v) = λh′(v) are considered, with the normalizations g(v) = r or h(v) = r, where g and h are real-valued C¹ functions on a real Banach space which are invariant under a periodic linear isometry. Theorems are proved on the existence of solutions λ(r), v(r), and on their dependence upon the normalization constant r > 0. In particular, the relation, as r → 0, of λ(r), v(r) to solutions of the linearized problem g″(0)v = λh″(0)v is discussed. The theorems are applied to elliptic problems for Euler-Lagrange operators corresponding to multiple integral functionals on closed subspaces of Sobolev spaces.     

Mutually-adjoint boundary-value problems with deviation from the characteristic for a multidimensional Gellerstedt equation

In this paper, we study mutually-adjoint boundary-value problems with a deviation from the characteristic for multidimensional Gellerstedt equation. In [3, 4], for the equation of the vibration of a string, the boundary-value problem with a deviation from the characteristic was studied, where the main attention was paid to the study of such problems for hyperbolic equations. For hyperbolic equations on the plane, this problem was studied in [5, 9].     

The Distributions of the Entries of Young Tableaux

Let T be a standard Young tableau of shape λ⊢k. We show that the probability that a randomly chosen Young tableau of n cells contains T as a subtableau is, in the limit n→∞, equal to f^λ/k!, where f^λ is the number of all tableaux of shape λ. In other words, the probability that a large tableau contains T is equal to the number of tableaux whose shape is that of T, divided by k!. We give several applications, to the probabilities that a set of prescribed entries will appear in a set of prescribed cells of a tableau, and to the probabilities that subtableaux of given shapes will occur. Our argument rests on a notion of quasirandomness of families of permutations, and we give sufficient conditions for this to hold.