A generalization of Obata’s theorem

In a complete Riemannian manifold (M, g) if the hessian of a real-valued function satisfies some suitable conditions, then it restricts the geometry of (M, g). In this paper we characterize all compact rank-one symmetric spaces as those Riemannian manifolds (M, g) admitting a real-valued functionu such that the hessian ofu has at most two eigenvalues ?u and $ - \frac{{u + 1}}{2}$ under some mild hypotheses on (M, g). This generalizes a well-known result of Obata which characterizes all round spheres.     

A generalization of Baer’s theorem

In this paper, we studied the supersolvability of the product of two subgroups and got a generalization of Baer’s theorem.     

A generalization of Forelli’s theorem

The purpose of this paper is to present a generalization of Forelli’s theorem. In particular, we prove an all dimensional version of the two-dimensional theorem of Chirka (Kompleks. Anal. i Prilozh, 232–240, 2006) of 2005.     

A generalization of Nekhoroshev’s theorem

Nekhoroshev discovered a beautiful theorem in Hamiltonian systems that includes as special cases not only the Poincaré theorem on periodic orbits but also the theorem of Liouville–Arnol’d on completely integrable systems [7]. Sadly, his early death precluded him publishing a full account of his proof. The aim of this paper is twofold: first, to provide a complete proof of his original theorem and second a generalization to the noncommuting case. Our generalization of Nekhoroshev’s theorem to the nonabelian case subsumes aspects of the theory of noncommutative complete integrability as found in Mishchenko and Fomenko [5] and is similar to what Nekhoroshev’s theorem does in the abelian case.     

A generalization of Jentzsch’s theorem

We study the asymptotic behavior of the roots of polynomials given by a linear summation method for partial sums of the Fourier series.     

A generalization of a converse to Schur’s theorem

Niroomand (Arch. Math. 94 (2010) 401–404) proved a converse to a theorem of Schur in the following sense. He proved that if G is a group such that [G, G] is finite and G/Z(G) is finitely generated, then G/Z(G) is finite, of order bounded above by [G, G] ^k where k is the minimal number of generators required for G/Z(G). Here, we give a completely elementary short proof of a further generalization.     

A generalization of Le Potier’s vanishing theorem

The main result is a general vanishing theorem for the cohomology of the ample vector bundles obtained as Schur functors of some vectors bundle which is not assumed to be ample itself. This is a generalization of Le Potiers vanishing theorem. It is also proven that for two partitions I and J such that IJ, the ampleness of S_IE implies that of S_JE.Mathematics Subject Classification (2000):14F17     

The Diffie–Hellman problem and generalization of Verheul’s theorem

Bilinear pairings on elliptic curves have been of much interest in cryptography recently. Most of the protocols involving pairings rely on the hardness of the bilinear Diffie–Hellman problem. In contrast to the discrete log (or Diffie–Hellman) problem in a finite field, the difficulty of this problem has not yet been much studied. In 2001, Verheul (Advances in Cryptology—EUROCRYPT 2001, LNCS 2045, pp. 195–210, 2001) proved that on a certain class of curves, the discrete log and Diffie–Hellman problems are unlikely to be provably equivalent to the same problems in a corresponding finite field unless both Diffie–Hellman problems are easy. In this paper we generalize Verheul’s theorem and discuss the implications on the security of pairing based systems.      

A generalization of a theorem by Bigard–Keimel and Conrad–Diem

Let G be an archimedean ℓ-group and \mathfrakP(G){\mathfrak{P}(G)} denote the set of all polar preserving bounded group endomorphisms of G. Bigard and Keimel in [Bull. Soc. Math. France 97 (1969), 381–398] and, independently, Conrad and Diem in [Illinois J. Math. 15 (1971), 222–240] proved that \mathfrakP(G){\mathfrak{P}(G)} is an archimedean ℓ-group with respect to the pointwise addition and ordering. This classical result is extended in this paper to certain sets of disjointness preserving bounded homomorphisms on archimedean ℓ-groups.     

A generalization of Bonnet’s theorem on Darboux surfaces

The well-known Bonnet theorem claims that, on a Darboux surface in three-dimensional Euclidean space, along each line of curvature, the corresponding principal curvature is proportional to the cube of another principal curvature. In the present paper, this theorem is generalized (with respect to dimension) to n-dimensional hypersurfaces of Euclidean spaces.     

Motzkin’s theorem of the alternative: a continuous-time generalization

We revisit Zalmai’s theorem, which is a partial generalization of Motzkin’s theorem of the alternative in the continuous-time setting. In particular, we provide two simple examples demonstrating that its existing proof is incorrect, and we demonstrate that a suitably modified variant of Zalmai’s theorem, concerned with the inconsistency of systems of convex inequalities and affine equalities, can be verified. We also derive two generalized variants of Motzkin’s theorem of the alternative in the continuous-time setting.     

A multidimensional generalization of Lagrange’s theorem on continued fractions

A multidimensional geometric analog of Lagrange’s theorem on continued fractions is proposed. The multidimensional generalization of the geometric interpretation of a continued fraction uses the notion of a Klein polyhedron, that is, the convex hull of the set of nonzero points in the lattice ? ⁿ contained inside some n-dimensional simplicial cone with vertex at the origin. A criterion for the semiperiodicity of the boundary of a Klein polyhedron is obtained, and a statement about the nonempty intersection of the boundaries of the Klein polyhedra corresponding to a given simplicial cone and to a certain modification of this cone is proved.     

A generalization of Sen–Brinon’s theory

Let K be a complete discrete valuation field of mixed characteristic and k be its residue field of prime characteristic p > 0. We assume that [k : k ^p ] = p ^h < ∞. Let G _K be the absolute Galois group of K and ${\mathcal{R}}$ be a Banach algebra over ${\mathbb{C}_p:=\widehat{\overline{K}}}$ with a continuous action of G _K . When k is perfect (i.e. h = 0), Sen studied the Galois cohomology ${{\rm H}^1(G_K, \mathcal{R}^\ast)}$ and Sen’s operator associated to each class (Sen Ann Math 127:647–661, 1988). In this paper we generalize Sen’s theory to the case h ≥ 0 by using Brinon’s theory (Brinon Math Ann 327:793–813, 2003). We also give another formulation of Brinon’s theorem (à la Colmez).