Inverse Jacobian multipliers and Hopf bifurcation on center manifolds

In this paper we consider a class of higher dimensional differential systems in Rⁿ${\mathbb{R}}^n$ which have a two dimensional center manifold at the origin with a pair of pure imaginary eigenvalues. First we characterize the existence of either analytic or C^∞$C^{\infty }$ inverse Jacobian multipliers of the systems around the origin, which is either a center or a focus on the center manifold. Later we study the cyclicity of the system at the origin through Hopf bifurcation by using the vanishing multiplicity of the inverse Jacobian multiplier.     

A family of quadratic forms associated to quadratic mappings of spheres

The general form of a real quadratic mapping of spheres can be determined by studying the diagonalization of each form in an associated family of quadratic forms. In particular, the eigenvalues provide a means for detecting maps which are of the Hopf type. When the eigenvalues are nonzero for every form in the family, the forms associated to $\text{?:}\text{S}{}^{\mathrm{n}}\text{→}\text{S}{}^{\mathrm{m}}$ give rise to a quadratic form on the tangent bundle of the unit sphere Sⁿ. If ? is of the Hopf type, nondegeneracy of each form occurs only when n=1,3,7,15.     

An iterative adaptive finite element method for elliptic eigenvalue problems

We consider the task of resolving accurately the n$n$th eigenpair of a generalized eigenproblem rooted in some elliptic partial differential equation (PDE), using an adaptive finite element method (FEM). Conventional adaptive FEM algorithms call a generalized eigensolver after each mesh refinement step. This is not practical in our situation since the generalized eigensolver needs to calculate n$n$ eigenpairs after each mesh refinement step, it can switch the order of eigenpairs, and for repeated eigenvalues it can return an arbitrary linear combination of eigenfunctions from the corresponding eigenspace. In order to circumvent these problems, we propose a novel adaptive algorithm that only calls a generalized eigensolver once at the beginning of the computation, and then employs an iterative method to pursue a selected eigenvalue–eigenfunction pair on a sequence of locally refined meshes. Both Picard’s and Newton’s variants of the iterative method are presented. The underlying partial differential equation (PDE) is discretized with higher-order finite elements (hp$hp$-FEM) but the algorithm also works for standard low-order FEM. The method is described and accompanied with theoretical analysis and numerical examples. Instructions on how to reproduce the results are provided.     

Analysis of symmetric matrix valued functions I

For any symmetric function f:Rⁿ?Rⁿ$f:{\mathbb{R}}^n?{\mathbb{R}}^n$, one can define a corresponding function on the space of n×n$n\times n$ real symmetric matrices by applying f$f$ to the eigenvalues of the spectral decomposition. We show that this matrix valued function inherits from f$f$ the properties of continuity, Lipschitz continuity, strict continuity, directional differentiability, Frechet differentiability, continuous differentiability.     

Bifurcation theory for finitely smooth planar autonomous differential systems

In this paper we establish bifurcation theory of limit cycles for planar $C^k$ smooth autonomous differential systems, with $k\in \mathbb{N}$. The key point is to study the smoothness of bifurcation functions which are basic and important tool on the study of Hopf bifurcation at a fine focus or a center, and of Poincaré bifurcation in a period annulus. We especially study the smoothness of the first order Melnikov function in degenerate Hopf bifurcation at an elementary center. As we know, the smoothness problem was solved for analytic and $C^{\infty }$ differential systems, but it was not tackled for finitely smooth differential systems. Here, we present their optimal regularity of these bifurcation functions and their asymptotic expressions in the finite smooth case.     

Reducibility of steady-state bifurcations in coupled cell systems

A general theory for coupled cell systems was formulated recently by I. Stewart, M. Golubitsky and their collaborators. In their theory, a coupled cell system is a network of interacting dynamical systems whose coupling architecture is expressed by a directed graph called a coupled cell network. An equivalence relation on cells in a regular network (a coupled cell network with identical nodes and identical edges) determines a new network called quotient network by identifying cells in the same equivalence class and determines a quotient system as well. In this paper we develop an idea of reducibility of bifurcations in coupled cell systems associated with regular networks. A bifurcation of equilibria from subspace where states of all cells are equal is called a synchrony-breaking bifurcation. We say that a synchrony-breaking steady-state bifurcation is reducible in a coupled cell system if any bifurcation branch for the system is lifted from those for some quotient system. First, we give the complete classification of codimension-one synchrony-breaking steady-state bifurcations in 1-input regular networks (where each cell receives only one edge). Second, we show that under a mild condition on the multiplicity of critical eigenvalues, codimension-one synchrony-breaking steady-state bifurcations in generic coupled cell systems associated with an n  -cell coupled cell network with _Dn$D_n$ symmetry, a regular network, is reducible for n>2$n>2$.     

On the distribution of Laplacian eigenvalues of trees

For a tree T$T$ with n$n$ vertices, we apply an algorithm due to Jacobs and Trevisan (2011) to study how the number of small Laplacian eigenvalues behaves when the tree is transformed by a transformation defined by Mohar (2007). This allows us to obtain a new bound for the number of eigenvalues that are smaller than 2. We also report our progress towards a conjecture on the number of eigenvalues that are smaller than the average degree.     

Quantizations of Kac–Moody algebras

We analyze the extent to which a quantum universal enveloping algebra of a Kac–Moody algebra g$\mathfrak{g}$ is defined by multidegrees of its defining relations. To this end, we consider a class of character Hopf algebras defined by the same number of defining relations of the same degrees as the Kac–Moody algebra g$\mathfrak{g}$. We demonstrate that if the generalized Cartan matrix A$A$ of g$\mathfrak{g}$ is connected then the algebraic structure, up to a finite number of exceptional cases, is defined by just one “continuous” parameter q$q$ related to a symmetrization of A$A$, and one “discrete” parameter m$\mathfrak{m}$ related to the modular symmetrizations of A$A$. The Hopf algebra structure is defined by n(n−1)/2$n(n-1)/2$ additional “continuous” parameters. We also consider the exceptional cases for Cartan matrices of finite or affine types in more detail, establishing the number of exceptional parameter values in terms of the Fibonacci sequence.     

Generalized matrix inversion is not harder than matrix multiplication

Starting from the Strassen method for rapid matrix multiplication and inversion as well as from the recursive Cholesky factorization algorithm, we introduced a completely block recursive algorithm for generalized Cholesky factorization of a given symmetric, positive semi-definite matrix A∈R^n×n$A\in {\mathbb{R}}^{n\times n}$. We used the Strassen method for matrix inversion together with the recursive generalized Cholesky factorization method, and established an algorithm for computing generalized {2,3}$\{2,3\}$ and {2,4}$\{2,4\}$ inverses. Introduced algorithms are not harder than the matrix–matrix multiplication.     

Hermitian matrices with a bounded number of eigenvalues

Conjugation covariants of matrices are applied to study the real algebraic variety consisting of complex Hermitian matrices with a bounded number of distinct eigenvalues. A minimal generating system of the vanishing ideal of degenerate three by three Hermitian matrices is given, and the structure of the corresponding coordinate ring as a module over the special unitary group is determined. The method applies also for degenerate real symmetric three by three matrices. For arbitrary n   partial information on the minimal degree component of the vanishing ideal of the variety of n×n$n\times n$ Hermitian matrices with a bounded number of eigenvalues is obtained, and some known results on sum of squares presentations of subdiscriminants of real symmetric matrices are extended to the case of complex Hermitian matrices.     

Weak Hopf lemma for the invariant Laplacian and related elliptic operators

We obtain a weak version of the Hopf lemma for the invariant Laplacian on the unit ball of the complex n$n$-space. We also show that our result is sharp in some sense. Motivated by this result, we also consider a class of degenerate elliptic operators with the degeneracy depending on the distance to the boundary of the domain. We study the dependence of the validity of Hopf lemma on the degree of degeneracy of the operator. We show that Hopf lemma holds if the degeneracy is small and fails in general if the degeneracy is large. What is more interesting is the critical case for which we show that certain weak version of Hopf lemma holds.     

On the adjacency matrix of a threshold graph

A threshold graph on n   vertices is coded by a binary string of length n−1$n-1$. We obtain a formula for the inertia of (the adjacency matrix of) a threshold graph in terms of the code of the graph. It is shown that the number of negative eigenvalues of the adjacency matrix of a threshold graph is the number of ones in the code, whereas the nullity is given by the number of zeros in the code that are preceded by either a zero or a blank. A formula for the determinant of the adjacency matrix of a generalized threshold graph and the inverse, when it exists, of the adjacency matrix of a threshold graph are obtained. Results for antiregular graphs follow as special cases.     

Stability and Hopf bifurcation for a three-component reaction–diffusion population model with delay effect

A delayed three-component reaction–diffusion population model with Dirichlet boundary condition is investigated. The existence and stability of the positive spatially nonhomogeneous steady state solution are obtained via the implicit function theorem. Moreover, taking delay τ$\tau$ as the bifurcation parameter, Hopf bifurcation near the steady state solution is proved to occur at the critical value _τ0${\tau }_0$. The direction of Hopf bifurcation is forward. In particular, by using the normal form theory and the center manifold reduction for partial functional differential equations, the stability of bifurcating periodic solutions occurring through Hopf bifurcations is investigated. It is demonstrated that the bifurcating periodic solution occurring at _τ0${\tau }_0$ is orbitally asymptotically stable. Finally, the general results are applied to four types of three species population models. Numerical simulations are presented to illustrate our theoretical results.     

Periodic travelling-wave solution of brusselator

The Hopf bifurcation Theorem requires that one at first knows all eigenvalues of the linearized system; and a difficulty in applying the Hopf Theorem is that in the general case we are unable to solve an algebraic equation of degreen (n3) with parameter coefficients.This paper firstly proves Theorem 1 which enables us to know the existence of the Hopf bifuecation by direct use of the coefficients of the characteristic equ ation of the linearized system for the case where the characteristic equation of the linearized system has degree 2, 3 or 4. Then by using Theorem 1, we gave out the condition for existing a periodic traveling-wave solution of Brusselator in diffusion by Hopf bifurcat on. Finally, we discuss the problem of bifurcation direction and the problem of stability of the bifurcation.     

Hopf subalgebras and tensor powers of generalized permutation modules

By means of a certain module V$V$ and its tensor powers in a finite tensor category, we study a question of whether the depth of a Hopf subalgebra R$R$ of a finite-dimensional Hopf algebra H$H$ is finite. The module V$V$ is the counit representation induced from R$R$ to H$H$, which is then a generalized permutation module, as well as a module coalgebra. We show that if in the subalgebra pair either Hopf algebra has finite representation type, or V$V$ is either semisimple with R^∗$R^{\ast }$ pointed, projective, or its tensor powers satisfy a Burnside ring formula over a finite set of Hopf subalgebras including R$R$, then the depth of R$R$ in H$H$ is finite. One assigns a nonnegative integer depth to V$V$, or any other H$H$-module, by comparing the truncated tensor algebras of V$V$ in a finite tensor category and so obtains upper and lower bounds for depth of a Hopf subalgebra. For example, a relative Hopf restricted module has depth 1, and a permutation module of a corefree subgroup has depth less than the number of values of its character.     

A new method for computing Moore–Penrose inverse matrices

The Moore–Penrose inverse of an arbitrary matrix (including singular and rectangular) has many applications in statistics, prediction theory, control system analysis, curve fitting and numerical analysis. In this paper, an algorithm based on the conjugate Gram–Schmidt process and the Moore–Penrose inverse of partitioned matrices is proposed for computing the pseudoinverse of an m×n$m\times n$ real matrix A$A$ with m≥n$m\ge n$ and rank r≤n$r\le n$. Numerical experiments show that the resulting pseudoinverse matrix is reasonably accurate and its computation time is significantly less than that of pseudoinverses obtained by the other methods for large sparse matrices.     

On the local representation of piecewise smooth equations as a Lipschitz manifold

We study systems of equations, F(x)=0$F(x)=0$, given by piecewise differentiable functions F:Rⁿ→R^k$F:{\mathbb{R}}^n\to {\mathbb{R}}^k$, k?n$k?n$. The focus is on the representability of the solution set locally as an (n−k)$(n-k)$-dimensional Lipschitz manifold. For that, nonsmooth versions of inverse function theorems are applied. It turns out that their applicability depends on the choice of a particular basis. To overcome this obstacle we introduce a strong full-rank assumption (SFRA) in terms of Clarke?s generalized Jacobians. The SFRA claims the existence of a basis in which Clarke?s inverse function theorem can be applied. Aiming at a characterization of SFRA, we consider also a full-rank assumption (FRA). The FRA insures the full rank of all matrices from the Clarke?s generalized Jacobian. The article is devoted to the conjectured equivalence of SFRA and FRA. For min-type functions, we give reformulations of SFRA and FRA using orthogonal projections, basis enlargements, cross products, dual variables, as well as via exponentially many convex cones. The equivalence of SFRA and FRA is shown to be true for min-type functions in the new case k=3$k=3$.