Probability theory has long provided a rigorous framework for quantifying uncertainty, yet its extension to infinite sets introduces profound conceptual challenges and opportunities. Contemporary ...

In this paper, we consider the function f p ( t )= 2p X 2 ( 2p t+p;p ) , where χ²(x; n) defined by X 2 ( x;p )= 2 −p/2 Γ( p/2 ) e −x/2 x p/2−1 , is the density function of a χ²-distribution with n ...

Set theory, the mathematical study of collections of objects, forms a foundation for much of modern mathematics, while cardinal functions provide a means to quantify the sizes of these sets, ...