(1.1) |f(a)− f(b)| ≤ L |a− b| for every pair of points a, b ∈ A. We also say that a function is Lipschitz if it is L-Lipschitz for some L. The Lipschitz condition as given in (1.1) is a purely metric condition; it makes sense for functions from one metric space to another. In these lectures, we concentrate on the theory of Lipschitz functions in Euclidean spaces. In Section 2, we study extensio...

We construct an orientable ribbon surface F ⊂ B, which is universal in the following sense: any orientable 4-manifold M ∼= B ∪ 1-handles ∪ 2-handles can be represented as a cover of B branched over F .

We point out the similarities and differences between cylindrical and disk photonic shells separating regions of flat spacetime. PACS numbers: 04.20.-q , 04.20.Jb, 04.40.-b

The vegetation cover and rangeland production are the most important factors to determine the proper planning system for effective utilization and management of rangelands. In this research, the effect of climatic factors (temperature and rainfall) were studied on vegetation cover and forage production of four species (Festuca ovina, Bromus tomentellus, Dactylis glomerata and Agropyron trichoph...

Given a presentation of a finitely presented group, there is a natural way to represent the group as the fundamental group of a 2-complex. The first part of this paper demonstrates one possible way to represent a finitely presented algebra S in a similarly compact form. From a presentation of the algebra, we construct a quiver with relations whose path algebra is finite dimensional. When we adj...

Maier, in his seminal paper [1], presents that the problem of finding an optimal cover is NP-complete. Cotelea’s paper [2] proposes a problem decomposition method for finding an optimal cover. The basic idea of the method is: (1) a big and intractable problem (optimal cover problem) is broken down into some smaller problems; (2) particular solutions of these smaller problems are combined to con...

A space X is said to be strongly base-paracompact if there is a basis B for X with |B| = w(X) such that every open cover of X has a star-finite open refinement by members of B. Strongly paracompact spaces which are strongly base-paracompact are studied. Strongly base-paracompact spaces are shown have a family of functions F with cardinality equal to the weight such that every open cover has a l...

Consider a bipartite graph G = (V,E) with parts N and M as shown in Figure 1. The function f(A) = |NG(A)| for A ⊆ N , where NG(A) is the neighborhood of set A, is a submodular function. A practical example is adding sensors to observe an area. Adding a new sensor will either cover an entirely new area, or overlap with area that is already covered. For sensor subsets A and B with A ⊆ B, the new ...

Let G be a connected complex reductive group acting on a smooth algebraic variety X. Then the cotangent bundle T ∗ X on X carries a canonical symplectic structure, and the G-action induces a moment map Φ : T ∗ X → g . Consider the Hamiltonian vector fields attached to functions of the form f ◦ Φ with f ∈ C[g]. In this paper we study the asymptotic behavior of the associated flow (a so-called in...