Recent characterizations of various types of differentiable generalized monotone maps by Karamardian– Schaible–Crouzeix and their strengthened versions by Crouzeix–Ferland are extended to the nonsmooth case. For nondifferentiable locally Lipschitz maps necessary and/or sufficient conditions for quasimonotonicity, pseudomonotonicity and strict/ strong pseudomonotonicity are derived. To accomplis...

We consider certain properties of maps of class C from R to Rd−1 that are strictly related to Sard’s theorem, and show that some of them can be extended to Lipschitz maps, while others still require some additional regularity. We also give counterexamples showing that, in term of regularity, our results are optimal.

We consider certain properties of maps of class C from R to Rd−1 that are strictly related to Sard’s theorem, and show that some of them can be extended to Lipschitz maps, while others still require some additional regularity. We also give examples showing that, in term of regularity, our results are optimal.

Recently a correspondence between non-propagating degrees of freedom in maximal supergravity and the very extended algebra E11 has been found. We perform a similar analysis for a supergravity theory with eight supercharges and very extended G2. In particular, in the context of d = 5 minimal supergravity, we study whether supersymmetry can be realised on higher-rank tensors with no propagating d...

Recently a correspondence between non-propagating degrees of freedom in maximal supergravity and the very extended algebra E11 has been found. We perform a similar analysis for a supergravity theory with eight supercharges and very extended G2. In particular, in the context of d = 5 minimal supergravity, we study whether supersymmetry can be realised on higher-rank tensors with no propagating d...

Let $R$ be a 2-torsion free semiprime ring with extended centroid $C$, $U$ the Utumi quotient ring of $R$ and $m,n>0$ are fixed integers. We show that if $R$ admits derivation $d$ such that $b[[d(x), x]_n,[y,d(y)]_m]=0$ for all $x,yin R$ where $0neq bin R$, then there exists a central idempotent element $e$ of $U$ such that $eU$ is commutative ring and $d$ induce a zero derivation on $(1-e)U$. ...

On a bounded strongly pseudo-convex domain X in C with a Lipschitz boundary, we prove that the ∂̄−Neumann operator N can be extended as a bounded operator from Sobolev (−1/2)−spaces to the Sobolev (1/2)−spaces. In particular, N is compact operator on Sobolev (−1/2)−spaces.

The contraction inequality for Rademacher averages is extended to Lipschitz functions with vector-valued domains, and it is also shown that in the bounding expression the Rademacher variables can be replaced by arbitrary iid symmetric and sub-gaussian variables. Example applications are given for multi-category learning, K-means clustering and learning-to-learn.