If a Lie group acts on a manifold freely and properly, pulling back by the quotient map gives an isomorphism between the differential forms on the quotient manifold and the basic differential forms upstairs. We show that this result remains true for actions that are not necessarily free nor proper, as long as the identity component acts properly, where on the quotient space we take differential...

We introduce a new knowledge operator called ``knowing whether.'' Knowing whether an event occurred means either knowing that it occurred or knowing that it did not occur. We demonstrate the following difference between ``knowing whether'' and ``knowing that.'' In a multiple agent model, a sequence of events generated by successively applying ``knowing that'' operators, or their negations, may ...

4 Sobolev-Spaces 23 4.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . 23 4.2 Poisson’s Problem . . . . . . . . . . . . . . . . . . . . . . . . . 27 4.3 Abstract Eigenvalue Problem . . . . . . . . . . . . . . . . . . 30 4.4 Sobolev Spaces for Periodic Functions . . . . . . . . . . . . . . 32 4.5 Fractional Sobolev Spaces for Periodic Functions . . . . . . . . 32 4.6 Trace T...

in this paper, we introduce the concepts of norm and inner prod- uct on fuzzy linear spaces over fuzzy elds and discuss some fundamental properties.

This paper presents a generalization for systems of partial differential equations of Gronwall's classical integral inequality for ordinary differential equations. The proof is by reducing the vector integral inequality to a vector partial differential inequality and then using a vector generalization of Riemann's method to obtain the final inequality. The final inequality involves a matrix fun...

in this paper, we introduce intuitionistic fuzzy contraction mappingand prove a fixed point theorem in intuitionistic fuzzy metric spaces.

‎In this paper‎, ‎we consider a generalization of the real normed spaces and give some examples‎.  

in this paper, we propose a new definition of intuitionistic fuzzyquasi-metric and pseudo-metric spaces based on intuitionistic fuzzy points. weprove some properties of intuitionistic fuzzy quasi- metric and pseudo-metricspaces, and show that every intuitionistic fuzzy pseudo-metric space is intuitionisticfuzzy regular and intuitionistic fuzzy completely normal and henceintuitionistic fuzzy nor...

andApplied AnalysisHindawi Publishing Corporationhttp://www.hindawi.comVolume 2013 ISRNAppliedMathematicsHindawi Publishing Corporationhttp://www.hindawi.comVolume 2013Hindawi Publishing Corporationhttp://www.hindawi.comVolume 2013International Journal ofCombinatorics Hindawi Publishing Corporationhttp://www.hindawi.comVolume 2013<...