Subspace arrangements as generalized star configurations

In this note, we show that any projective subspace arrangement can be realized as a generalized star configuration variety. This type of interpolation result may be useful in designing linear codes with prescribed codewords of minimum weight, as well as in answering a couple of questions about the number of equations needed to define a generalized star configuration variety.     

Locally symmetric left invariant Riemannian metrics on 3‐dimensional Lie groups

In this paper, we use the powerful tool Milnor bases to determine all the locally symmetric left invariant Riemannian metrics up to automorphism, on 3‐dimensional connected and simply connected Lie groups, by solving system of polynomial equations of constants structure of each Lie algebra . Moreover, we show that E ₀(2) is the only 3‐dimensional Lie group with locally symmetric left invariant Riemannian metrics which are not symmetric.     

Strict topologies on measure spaces

Let X be a measurable space, let 201600227:mana201600227-math-0001" class="section_image" src="/cms/asset/3a996abf-cbeb-4ab5-9a4b-59bda1f1c649/mana201600227-math-0001.png"> be a family of measurable subsets of it, and let 201600227:mana201600227-math-0002" class="section_image" src="/cms/asset/970be77c-80cf-4c42-86e3-49473a329277/mana201600227-math-0002.png"> be a subspace of complex measures on X that is also closed under restrictions of measures. In this paper we introduce the 201600227:mana201600227-math-0003" class="section_image" src="/cms/asset/fa9a9cf4-0546-4a69-9b46-623589583404/mana201600227-math-0003.png">‐convergence topology 201600227:mana201600227-math-0004" class="section_image" src="/cms/asset/992eee29-f447-403e-8088-93fe3e8b1483/mana201600227-math-0004.png"> and the 201600227:mana201600227-math-0005" class="section_image" src="/cms/asset/9e35217d-df15-4bc7-9183-f1af68a18d98/mana201600227-math-0005.png">‐strict topology 201600227:mana201600227-math-0006" class="section_image" src="/cms/asset/3154267c-04ba-4de0-86e5-0a700c789d75/mana201600227-math-0006.png"> on 201600227:mana201600227-math-0007" class="section_image" src="/cms/asset/fc601ef5-c7e5-4809-908a-04047915c119/mana201600227-math-0007.png">. Among other results, we find necessary and sufficient conditions for Hausdorff‐ness and coincide‐ness of these topologies. Applications to Lebesgue spaces, and also examples in Hausdorff topological spaces and locally compact groups are given.     

A classification of orbits admitting a unique invariant measure

We consider the space of countable structures with fixed underlying set in a given countable language. We show that the number of ergodic probability measures on this space that are $S_{\infty }$-invariant and concentrated on a single isomorphism class must be zero, or one, or continuum. Further, such an isomorphism class admits a unique $S_{\infty }$-invariant probability measure precisely when the structure is highly homogeneous; by a result of Peter J. Cameron, these are the structures that are interdefinable with one of the five reducts of the rational linear order $(\mathbb{Q},<)$.     

Existence and multiplicity of systems of Kirchhoff‐type equations with general potentials

This paper is concerned with the following systems of Kirchhoff‐type equations: Under more relaxed assumptions on V(x) and F(x,u,v), we first prove the existence of at least two nontrivial solutions for the aforementioned system by using Morse theory in combination with local linking arguments. Then by using the Clark theorem, the existence results of at least 2k distinct pairs of solutions are obtained. Some recent results from the literature are extended. Copyright © 2016 John Wiley & Sons, Ltd.     

Integral geometry of translation invariant functionals,II: The case of general convex bodies

In continuation of Part I, we study translative integral formulas for certain translation invariant functionals, which are defined on general convex bodies. Again, we consider local extensions and use these to show that the translative formulas extend to arbitrary continuous and translation invariant valuations. Then, we discuss applications to Poisson particle processes and Boolean models which contain, as a special case, some new results for flag measures.     

Mild solution of semilinear impulsive integro‐differential evolution equation in Banach spaces

Through solving the problem step by step and by applying the method of a C₀ semigroup of operators combined with the Banach contraction theorem, we investigate the existence and uniqueness of a mild solution of semilinear impulsive integro‐differential evolution equation in Banach spaces. In addition, an explicit iterative approximation sequence of the mild solution is derived. The assumed conditions in the present theorems are weaker and more general, and the results obtained are the generalizations and improvements of some known results. Examples are also given to illustrate our main results. Copyright © 2017 John Wiley & Sons, Ltd.