HL-differentiability is equivalent to MB#-differentiability

In 1972, it was announced by Averbukh and Smolyanov that HL-differentiability is equivalent to FB^#-differentiability. The proof has not been published till now. Here we prove a stronger result, namely, the one formulated in the title.     

CI-algebra is equivalent to dual Q-algebra

In this paper, we introduce the notion of dual Q-algebras and we show that the CI-algebras are equivalent to the dual Q-algebras.     

A homotopy equivalence that is not homotopic to a topological embedding

An open subset W of Sⁿ, n 6 or N = 4, and a homotopy equivalence ƒ: S² × S^{n − 4} → W are constructed having the property that ƒ is not homotopic to any topological embedding.     

Scaled Enflo type is equivalent to Rademacher type

We introduce the notion of the scaled Enflo type of a metricspace, and show that for Banach spaces, scaled Enflo type pis equivalent to Rademacher type p.     

Congruence join semidistributivity is equivalent to a congruence identity

We show that a locally finite variety is congruence join semidistributive if and only if it satisfies a congruence identity that is strong enough to force join semidistributivity in any lattice. Received February 9, 2000; accepted in final form November 23, 2000.     

The triangular principle is equivalent to the triangular scheme

The triangular scheme is a diagrammatic characterization of congruence join-semidistributivity. The triangular principle is a variant of this condition, where one of the congruences is replaced by a tolerance. This paper contains two proofs showing that the triangular principle and the triangular scheme are equivalent for varieties. The first one is a routine argument using tame congruence theory, and works only for locally finite varieties. The second proof is an elementary, but nontrivial calculation with terms, which works for arbitrary varieties. This yields a stronger Mal’tsev-condition characterizing congruence join-semidistributivity than the one obtained from the triangular scheme. Received April 19, 2005; accepted in final form June 15, 2005.     

Not every pseudoalgebra is equivalent to a strict one

We describe a finitary 2-monad on a locally finitely presentable 2-category for which not every pseudoalgebra is equivalent to a strict one. This shows that having rank is not a sufficient condition on a 2-monad for every pseudoalgebra to be strictifiable. Our counterexample comes from higher category theory: the strict algebras are strict 3-categories, and the pseudoalgebras are a type of semi-strict 3-category lying in between Gray-categories and tricategories. Thus, the result follows from the fact that not every Gray-category is equivalent to a strict 3-category, connecting 2-categorical and higher-categorical coherence theory. In particular, any nontrivially braided monoidal category gives an example of a pseudoalgebra that is not equivalent to a strict one.     

Convexity is equivalent to midpoint convexity combined with strict quasiconvextiy

It was recently shown by Nikodem that a function defined on an open convex subset of R ⁿ is convex if and only if it is midpoint convex and quasiconvex. It is shown that quasiconvexity can be replaced by strict quasiconvexity and that the openness condition can be removed altogether. The domain can then be taken from a general real linear space. There will also be given some related results of a “local” nature     

The Erdős-Mordell inequality is equivalent to non-positive curvature

We prove that, in plane absolute geometry, the Erdős-Mordell inequality is equivalent to the statement that the sum of the angles of a triangle is less than or equal to two right angles.      

The Riemann-Lebesgue property is equivalent to the complete continuity property

It is known that every space with the complete continuity propertyalso has the Riemann–Lebesgue property. It turns out thatthe converse implication is also valid.     

For cones in Hilbert spaces, regularity is equivalent to self-duality

The theorem stating that a cone in a Hilbert space is regular if and only if it is self-dual is proved and applied to obtain new proofs of earlier results.Translated fromMatematicheskie Zametki, Vol. 64, No. 4, pp. 616–621, October, 1998.     

On the mean number of encryptions for tree-based broadcast encryption schemes

The challenge of stateless-receiver broadcast encryption lies in minimizing storage and the number of encryptions while maintaining system security. Tree-based key distribution schemes offer the best known trade-off between the two parameters. Examples include the complete subtree scheme [D. Wallner, et al., Internet draft, http://www.ietf.org/ID.html [10]; C.K. Wong, et al., in: Proc. SIGCOMM, 1998, pp. 68–79 [11]], the subset difference scheme [D. Naor, et al., in: CRYPTO 2001, Lecture Notes in Comput. Sci., vol. 2139, 2001, pp. 41–62 [7]], and the layered subset difference scheme [D. Halevy, A. Shamir, in: CRYPTO 2002, Lecture Notes in Comput. Sci., vol. 2442, 2002, pp. 47–60 [5]]. We introduce generating functions for this family of schemes, which lead to analysis of the mean number of encryptions over all privileged sets of users. We also derive the mean number of encryptions when the number of privileged users is fixed. We expect that the techniques introduced as well as the results in this work will find applications in related areas.     

On when a graded ring is graded equivalent to a crossed product

Let be a ring graded by a group . We are concerned with describing those -graded rings that are graded equivalent to -crossed products. We give necessary and sufficient conditions for when a strongly graded ring is graded equivalent to a crossed product, provided that the 1-component is either Azumaya or semiperfect. Our result uses the torsion product theorem of Bass and Guralnick. We also construct various examples of such rings.

     

Every projective Schur algebra is Brauer equivalent to a radical abelian algebra

We prove that any projective Schur algebra over a field K isequivalent in Br(K) to a radical abelian algebra. This was conjecturedin 1995 by Sonn and the first author of this paper. As a consequence,we obtain a characterization of the projective Schur group bymeans of Galois cohomology. The conjecture was known for algebrasover fields of positive characteristic. In characteristic zerothe conjecture was known for algebras over fields with a Henselianvaluation over a local or global field of characteristic zero.     

The differentiable chain functor is not homotopy equivalent to the continuous chain functor

Let S_∗ and be the functors of continuous and differentiable singular chains on the category of differentiable manifolds. We prove that the natural transformation , which induces homology equivalences over each manifold, is not a natural homotopy equivalence.