We present a short elementary proof of the following Twelve Points Theorem: Let M be a convex polygon with vertices at the lattice points, containing a single lattice point in its interior. Denote by m (resp. m) the number of lattice points in the boundary of M (resp. in the boundary of the dual polygon). Then

We present a short elementary proof of the following Twelve Points Theorem: Let M be a convex polygon with vertices at the lattice points, containing a single lattice point in its interior. Denote by m (resp. m) the number of lattice points in the boundary of M (resp. in the boundary of the dual polygon). Then

Let (m, b) be a pair of natural numbers. For m even (resp. m odd and b ≥ 2) we show that if there is an m-dimensional non-formal compact oriented manifold with first Betti number b1 = b, there is also a symplectic (resp. contact) manifold with these properties.

We consider a Stein manifold M of dimension ≥ 2 and a compact subset K ⊂ M such that M ′ = M\K is connected. Let S be a compact differential manifold, and let MS , resp. M ′ S stand for the complex manifold of maps S → M , resp. S → M , of some specified regularity that are homotopic to constant. We prove that any holomorphic function on M ′ S continues analytically to MS (perhaps as a multival...

We establish a stronger symmetry between the numbers of northeast and southeast chains in the context of 01-fillings of moon polyominoes. Let M be a moon polyomino with n rows and m columns. Consider all the 01-fillings of M in which every row has at most one 1. We introduce four mixed statistics with respect to a bipartition of rows or columns of M. More precisely, let S ⊆ {1, 2, . . . , n} an...

The aim of this paper is to introduce $(L,M)$-fuzzy closurestructure where $L$ and $M$ are strictly two-sided, commutativequantales. Firstly, we define $(L,M)$-fuzzy closure spaces and getsome relations between $(L,M)$-double fuzzy topological spaces and$(L,M)$-fuzzy closure spaces. Then, we introduce initial$(L,M)$-fuzzy closure structures and we prove that the category$(L,M)$-{bf FC} of $(L,M...

The aim of this paper is to extend the truth value table oflattice-valued convergence spaces to a more general case andthen to use it to introduce and study the quantale-valued fuzzy Scotttopology in fuzzy domain theory. Let $(L,*,varepsilon)$ be acommutative unital quantale and let $otimes$ be a binary operationon $L$ which is distributive over nonempty subsets. The quadruple$(L,*,otimes,varep...

and to be power bounded (notation: T ∈ (PW)) if (1) holds for every polynomial of the special form p(ζ) = ζ where n is a positive integer. If T ∈ (PB) [resp., T ∈ (PW)], then there is a smallest number M ≥ 1 which satisfies (1) [resp., (1) restricted]. This number will be called the polynomial bound [resp., the power bound] of T and will be denoted by Mpb(T ) [resp., Mpw(T )]. (If T / ∈ (PB) [r...

Let R be a commutative Noetherian ring, I an ideal of R and M a non-zero R-module. In this paper we calculate the extension of annihilator of local cohomology modules H^t_I(M), t≥0, under the ring extension R⊂R[X] (resp. R⊂R[[X]]). By using this extension we will present some of the faithfulness conditions of local cohomology modules, and show that if the Lynch's conjecture, i...

According to the statistical random sampling method, estimation has been taken confidence interval into account in statistics due to the unknown probability for the deviations of point estimate. In terms of fuzziness or vagueness, confidence interval is defined as a range of numerical values, not a single value, so we have to discover an estimate for that interval at first. Based on the above-m...