On the Lipschitz equivalence of self‐affine sets

Recently Lipschitz equivalence of self‐similar sets on has been studied extensively in the literature. However for self‐affine sets the problem is more awkward and there are very few results. In this paper, we introduce a w‐Lipschitz equivalence by repacing the Euclidean norm with a pseudo‐norm w. Under the open set condition, we prove that any two totally disconnected integral self‐affine sets with a common matrix are w‐Lipschitz equivalent if and only if their digit sets have equal cardinality. The main methods used are the technique of pseudo‐norm and Gromov hyperbolic graph theory on iterated function systems.     

Half Exact Functors Associated with Cotorsion Pairs on Exact Categories

In the previous article “Hearts of twin cotorsion pairs on exact categories. J. Algebra, 394, 245–284 (2013)”, we introduced the notion o     

Hermitian adjacency matrix of the second kind for mixed graphs

This contribution gives an extensive study on spectra of mixed graphs via its Hermitian adjacency matrix of the second kind (N-matrix for short) introduced by Mohar [25]. This matrix is indexed by the vertices of the mixed graph, and the entry corresponding to an arc from u to v is equal to the sixth root of unity $\omega =\frac{1+\mathbf{i}\sqrt{3}}{2}$ (and its symmetric entry is $\overline{\omega }=\frac{1?\mathbf{i}\sqrt{3}}{2}$); the entry corresponding to an undirected edge is equal to 1, and 0 otherwise. The main results of this paper include the following: equivalent conditions for a mixed graph that shares the same spectrum of its N-matrix with its underlying graph are given. A sharp upper bound on the spectral radius is established and the corresponding extremal mixed graphs are identified. Operations which are called two-way and three-way switchings are discussed–they give rise to some cospectral mixed graphs. We extract all the mixed graphs whose rank of its N-matrix is 2 (resp. 3). Furthermore, we show that if $M_G$ is a connected mixed graph with rank 2, then $M_G$ is switching equivalent to each connected mixed graph to which it is cospectral. However, this does not hold for some connected mixed graphs with rank 3. We identify all mixed graphs whose eigenvalues of its N-matrix lie in the range $(?\alpha ,\alpha )$ for $\alpha \in \{\sqrt{2},\sqrt{3},2\}$.     

Characterization of canonical classes for systems of linear ODEs

The equivalence group is determined for systems of linear ordinary differential equations in both the standard form and the normal form. It is then shown that the normal form of linear systems reducible by an invertible point transformation to the canonical form y ⁽ⁿ⁾=0 consists of copies of the same iterative scalar equation. It is also shown that contrary to the scalar case, an iterative vector equation need not be reducible to the canonical form by an invertible point transformation. Other properties of iterative linear systems are also derived, as well as a simple algebraic formula for their general solution. Copyright © 2017 John Wiley & Sons, Ltd.     

Anti‐inflammatory bioactive equivalence of combinatorial components β‐carboline alkaloids identified in Arenaria kansuensis by two‐dimensional chromatography and solid‐phase extraction coupled with liquid–liquid extraction enrichment technology

Bioactive equivalent combinatorial components play a critical role in herbal medicines. However, how to discover and enrich them efficiently is a question for herbal pharmaceuticals researchers. In our work, a novel two‐dimensional reversed‐phase/hydrophilic interaction high‐performance liquid chromatography method was established to perform real‐time components trapping and combining for preparation and isolation of coeluting components. Arenaria kansuensis was taken as an example, and solid‐phase extraction coupled with liquid–liquid extraction as a simple and efficient method for enriching trace components, reversed phase column coupled with hydrophilic interaction liquid chromatography XAmide column as two‐dimensional chromatography technology for isolation and preparation of coeluting constituents, enzyme‐linked immune‐sorbent assay as bio‐guided assay, and anti‐inflammatory bioactivity evaluation for bioactive constituents. A combination of 12 β‐carboline alkaloids was identified as anti‐inflammatory bioactive equivalent combinatorial components from A. kansuensis , which accounts for 1.9% w/w of original A. kansuensis . This work answers the key question of which are real anti‐inflammatory components from A. kansuensis and provides a fast and efficient approach for discovering and enriching trace β‐carboline alkaloids from herbal medicines for the first time. More importantly, the discovery of bioactive equivalent combinatorial components could improve the quality control of herbal products and inspire a herbal medicine based on combinatorial therapeutics.     

Group Classification of Equations of the Form y″ = f(x,y)

The problem of classification of ordinary differential equations of the form y = f(x,y) by admissible local Lie groups of transformations is solved. Standard equations are listed on the basis of the equivalence concept. The classes of equations admitting a oneparameter group and obtained from the standard equations by invariant extension are described.     

Unique Orbital Normal Form for Vector Fields of Hopf-Zero Singularity

Normal forms for vector fields of Hopf-zero singularity in R³ are studied. Multiple Lie bracket method is used to give unique normal forms under both conjugacy and orbital equivalence for such vector fields with a generic quadratic part.Dedicated to Professor Shui-Nee Chow on the occasion of his 60th birthday.     

A thermodynamic approach with internal variables using Lagrange formalism. Part II. Continuous symmetries in the case of the time–temperature equivalence

In the first part of this contribution, the Lie-symmetries of the principle of least action associated to the constitutive equations of the DNLR formalism of relaxation have been presented. We examine in this second part the continuous symmetries corresponding to the simple case of stress relaxation under isothermal conditions. The well-known principle of time/temperature equivalence is discussed in terms of variational symmetry for the Jacobi’s action functional, and connected to the Onsager’s relation near the thermodynamic equilibrium.