The Major Subdegree Problem of A. H. Lachlan (first posed in 1967) has become the longest-standing open question concerning the structure of the computably enumerable (c.e.) degrees. A c.e. degree a is a major subdegree of a c.e. degree b > a if for any c.e. degree x, 0′ = b ∨ x if and only if 0′ = a ∨ x. In this paper, we show that every c.e. degree b 6= 0 or 0′ has a major subdegree, answerin...

Two Riemannian manifolds are said to have C-conjugate geodesic flows if there exist an C diffeomorphism between their unit tangent bundles which intertwines the geodesic flows. We obtain a number of rigidity results for the geodesic flows on compact 2-step Riemannian nilmanifolds: For generic 2-step nilmanifolds the geodesic flow is C rigid. For special classes of 2-step nilmanifolds, we show t...

A new class of directed trees is introduced. formula for the group inverse matrices associated with any tree belonging to this obtained. This answers affirmatively, a conjecture Catral et al., class.

For a polarized abelian variety, Z. Jiang and G. Pareschi introduce an invariant show that the polarization is basepoint free or projectively normal if small. Their result generalized to higher syzygies by F. Caucci, is, satisfies property $(N_p)$ In this paper, we study relation between degrees of subvarieties with respect polarization. threefolds, give upper bound using subvarieties. particul...

Let K− 4 denote the graph obtained from K4 by removing one edge. Let k be an integer with k ≥ 2. Kawarabayashi conjectured that if G is a graph of order n ≥ 4k + 1 with σ2(G) ≥ n + k, then G has k vertex-disjoint copies of K− 4 . In this paper, we settle this conjecture affirmatively.

A conjecture of Mirkin, Muchnik, and Smith is answered affirmatively which connects the inconsistency function, a biologically meaningful similarity/dissimilarity measure for a gene tree and a species tree, to the mutation cost function, a combinatorial measure based on the mapping of trees. A linear-time algorithm for computing the mutation cost function is also derived from the conjecture.

We derive an explicit formula for the well-known Chern-Moser-Weyl tensor nondegenerate real hypersurfaces in complex space terms of their defining functions. The is considerably simplified when applying to "pluriharmonic perturbations" sphere or a Fefferman approximate solution Monge-Amp\`ere equation. As application, we show that CR invariant one-form $X_{\alpha}$ constructed recently by Case ...

Abstract We affirmatively solve the analogue of Lord Rayleigh’s conjecture on Riemannian manifolds with positive Ricci curvature for any clamped plates in 2 and 3 dimensions, sufficiently large dimensions beyond 3. These results complement those from flat (Ashbaugh Benguria Duke Math J 78(1):1–17, 1995; Nadirashvili Arch Ration Mech Anal 129(1):1–10, 1995) negatively curved (Kristály Adv 367:10...