We propose a definition of compact quantum groupoids in the setting of C -algebras, associate to such a quantum groupoid a regular C -pseudo-multiplicative unitary, and use this unitary to construct a dual Hopf C -bimodule and to pass to a measurable quantum groupoid in the sense of Enock and Lesieur. Moreover, we discuss examples related to compact and to étale groupoids and study principal co...

Conjecture [Schauder] For every non-empty convex subset C of a topological vector space E, a compact continuous mapping f : C → C has a fixed point, i.e., a point x∗ ∈ C such that f(x∗) = x∗. (See [16], problem 54). We recall that a mapping f : C → C is said to be compact if f(C) is contained in a compact subset of C. Schauder proved in 1930 that his conjecture holds for normed vector spaces an...

in this paper‎, ‎we introduce the notion of $(m,n)$-‎algebr‎aically compact modules as an analogue of algebraically‎ ‎compact modules and then we show that $(m,n)$-algebraically‎ ‎compactness‎ ‎and $(m,n)$-pure injectivity for modules coincide‎. ‎moreover‎, ‎further characterizations of a‎ ‎$(m,n)$-pure injective module over a commutative ring are given‎.

the polarimetric entropy and the alpha angle are two important parameters for analyzing polarimetric synthetic aperture radar data. they demonstrate the degree of randomness in the polarization of the received signal and scattering mechanism of scatterers respectively. recently, there has been an increasing interest in compact polarimetry due to its advantages in comparison with full polarimetr...

We give a short proof that the space of continuous functions from [0, 1] to [0, 1] is not compact in the compact-open topology. Suppose X and Y are compact topological spaces. Let C(X, Y ) be the space of continuous functions from X to Y , and give this space the compact-open topology. An interesting problem from topology is to prove or disprove that C(X, Y ) is compact. http://dx.doi.org/10.41...

3 is the boundary of two spherical caps of constant mean curvature H for any positive number H, which is at most the radius of C. It is natural to ask whether spherical caps are the only possible examples. Some examples of constant mean curvature immersed tori by Wente [7] indicate that there are compact genus-one immersed constant mean curvature surfaces with boundary C that are approximated b...