Let Ω be a finite set and T(Ω) the full transformation monoid on Ω. The rank of t∈T(Ω) is natural number |Ωt|. Given A⊆T(Ω), denote by 〈A〉 semigroup generated A. k fixed such that 2≤k≤|Ω|. In first part this paper we (almost) classify permutation groups G for all transformations t∈T(Ω), every element in St:=〈G,t〉 can written as product eg, where e2=e∈St g∈G. second prove, among other results, i...

The function k : Ω×Ω→ R is called a kernel on Ω if the matrix (k(xi, xj))1≤i,j≤n is positive semidefinite for all positive integers n and all x1, . . . , xn ∈ Ω. It is well-known that if k is a kernel on Ω, then there exists a Hilbert space H̃ and Φ̃ : Ω→ H̃ such that k(x, x′) = 〈Φ̃(x), Φ̃(x)〉H̃. While H̃ and Φ̃ are not uniquely determined by k, the Hilbert space of functionsHk = {〈v, Φ̃(·)〉H̃ : v ∈ H̃} i...

In this paper, we study the existence and nonexistence of multiple positive solutions for problem   ∆u+K(x)up = 0 in Ω. u > 0 in Ω, u ∈ H1 loc(Ω) ∩ C(Ω). u ∣∣ ∂Ω = 0, u → μ > 0 as |x| → ∞ where Ω = RN \ ω is an exterior domain in RN , ω ⊂ RN is a bounded domain with smooth boundary and N > 2. μ ≥ 0, p > 1 are some given constants. K(x) satisfies: K(x) ∈ Cα loc(Ω) and ∃C, ,M > 0 such t...

In this section I characterize the optimal choices of farmers. I discuss the optimal choices of farmers for a plot ω, conditional on allocating a positive fraction of plot ω to the production of crop k, (φi,k (ω) > 0), an event I denote ω ∈ Ωi,k. Then I move on to prove the three main Propositions in the paper. Using Assumptions #1 and #2, I first show how to calculate the fraction of land allo...

We introduce an abstract concept for decomposing spaces with respect to a substructuring of a bounded domain. In this setting we define weakly conforming finite element approximations of quadratic minimization problems. Within a saddle point approach the reduction to symmetric positive Schur complement systems on the skeleton is analyzed. Applications include weakly conforming variants of least...

For ψ ∈ W 1,p(Ω;Rm) and g ∈ W−1,p(Ω;Rd), 1 < p < +∞, we consider a sequence of integral functionals F k : W 1,p(Ω;Rm)× Lp(Ω;Rd×n)→ [0,+∞] of the form F k (u, v) =  ∫ Ω fk(x,∇u, v) dx if u− ψ ∈W 1,p 0 (Ω;Rm) and div v = g, +∞ otherwise, where the integrands fk satisfy growth conditions of order p, uniformly in k. We prove a Γ-compactness result for F k with respect to the weak topology of W 1,...

We present a simple proof of the interior approximate controllability for the following broad class of second-order equations in the Hilbert space L2 Ω : ÿ Ay 1ωu t , t ∈ 0, τ , y 0 y0, ẏ 0 y1, where Ω is a domain in N N ≥ 1 , y0, y1 ∈ L2 Ω , ω is an open nonempty subset of Ω, 1ω denotes the characteristic function of the set ω, the distributed control u belongs to L2 0, τ ;L2 Ω , and A : D A ⊂...

An epsilon number is a transfinite number which is a fixed point of an exponential map: ω ε = ε. The formalization of the concept is done with use of the tetration of ordinals (Knuth's arrow notation, ↑↑). Namely, the ordinal indexing of epsilon numbers is defined as follows: ε0 = ω ↑↑ ω, εα+1 = εα ↑↑ ω, and for limit ordinal λ: ε λ = lim α<λ εα = α<λ εα. Tetration stabilizes at ω: α ↑↑ β = α ↑...

In this paper we introduce and study a family of complexity functions of infinite words indexed by k ∈ Z+ ∪ {+∞}. Let k ∈ Z+ ∪ {+∞} and A be a finite non-empty set. Two finite words u and v in A are said to be k-Abelian equivalent if for all x ∈ A of length less than or equal to k, the number of occurrences of x in u is equal to the number of occurrences of x in v. This defines a family of equi...