We consider the approximability of maximization and minimization variants of the Morse matching problem, posed as open problems by Joswig and Pfetsch [12]. We establish hardness results for MaxMorse matching and Min-Morse matching. In particular, we show that, for a simplicial complex with n simplices and dimension d ≥ 3, it is NP-hard to approximate Min-Morse matching within a factor of O(n), ...

The purpose of this article is to study the minimal free resolution of homogeneous coordinate rings of elliptic ruled surfaces. Let X be an irreducible projective variety and L a very ample line bundle on X , whose complete linear series defines the morphism φL : X −→ P(H (L)) Let S = ⊕∞ m=0 S H(X,L) and R(L) ⊕∞ m=0 H (X,L). Since R(L) is a finitely generated graded module over S, it has a mini...

For a convergent sequence {xi} generated by xi+l = ~(x~, xt_ l , . . . , Xi_d+l) , define the multiplicative efficiency measure E to be (log~p)/M, where p is the order of convergence and M is the number of multiplications or divisions needed to compute rp. Then, if 9 is any multivariate rational function, E < 1. Since E = 1 for the sequence {xi} generated by x~+l = x~ ~ + xi -~ with the limit -...

R = R(X,Z,L, σ) = H(X, R) = k ⊕H(X, R1)⊕H(X, R2)⊕ · · · By [RS2, Theorem 3.1], R is noetherian with qgr-R ' qgr-R. Proposition 1.2. [RS2, Proposition 3.20]. Keep the above assumptions and assume that L is also ample and generated by its global sections. Then there exists M ∈ N such that, for m ≥M : (1) In ⊗ L⊗m n is generated by its global sections for all n ≥ 1. (2) R(X,Z,L⊗m, σ) is generated ...

The development of multicomponent reactions (MCRs) designed to produce elaborate biologically active compounds has become an important area of research in organic, combinatorial, and medicinal chemistry. A comparative study of the synthesis of new bis-spiro-oxindolo(pyrrolizidines/pyrrolidines) ring systems by the cycloaddition of azomethine ylides generated by a decarboxylative route from sarc...

Let L be a lattice. If for each a < b ∈ L there is a lattice embedding of φ : L → [a, b] then L is called a semifractal. If, in addition, 0,1 ∈ L and φ can always be chosen such that φ(0) = a and φ(1) = b then L is said to be a 0–1-semifractal. Now let L be a bounded lattice. If for each a1 < b1 ∈ L and a2 < b2 ∈ L there is a lattice embedding ψ : [a1, b1] → [a2, b2] with ψ(a1) = a2 and ψ(b1) =...