Relative Compactness and Product Stable Quotient Maps

relative compactness and product stable quotient maps

Sum-Product-Quotient Networks

We present a novel tractable generative model that extends Sum-Product Networks (SPNs) and significantly boosts their power. We call it Sum-Product-Quotient Networks (SPQNs), whose core concept is to incorporate conditional distributions into the model by direct computation using quotient nodes, e.g. P (A|B)= (A,B) P (B) . We provide sufficient conditions for the tractability of SPQNs that gene...

Stable Maps of 3-manifolds into the Plane and Their Quotient Spaces

We study stable maps / : M —*U of closed orientable 3-manifolds M into the plane using their quotient spaces, which are defined to be the spaces of the connected components of/-fibres and which are known to be 2-dimensional polyhedra. We show that every 3-manifold admits a stable map whose quotient space is homeomorphic to that of a stable map of 5 \ We also deduce a Morse-type inequality for s...

On strongly Jordan zero-product preserving maps

In this paper, we give a characterization of strongly Jordan zero-product preserving maps on normed algebras as a generalization of  Jordan zero-product preserving maps. In this direction, we give some illustrative examples to show that the notions of strongly zero-product preserving maps and strongly Jordan zero-product preserving maps are completely different. Also, we prove that the direct p...

Stable Exponentially Harmonic Maps Between Finsler Manifolds

Generalized tri-quotient maps and Čech-completeness

For a topological space X let K(X) be the set of all compact subsets of X. The purpose of this paper is to characterize Lindelöf Čech-complete spaces X by means of the sets K(X). Similar characterizations also hold for Lindelöf locally compact X, as well as for countably K-determined spaces X. Our results extend a classical result of J. Christensen.