We prove new estimates for the volume of a Lorentzian manifold and show especially that cosmological spacetimes with crushing singularities have finite volume. 0. Introduction Let N be a (n + 1)-dimensional Lorentzian manifold and suppose that N can be decomposed in the form (0.1) N = N0 ∪N− ∪N+, where N0 has finite volume and N− resp. N+ represent the critical past resp. future Cauchy developm...

where m is the normalized Lebesgue measure on T. By the Fatou Theorem, Kf can also be defined on T by its non-tangential boundary values. After such an extension one can view the Cauchy integral as a “transform”, i.e., an operator in L(m) which sends f into the boundary values of Kf . It is well known, and not difficult to see, that the Cauchy transform is unbounded for p = 1 or p = ∞. However,...

in this paper, the maximal dissipative extensions of a symmetric singular 1d discrete hamiltonian operator with maximal deficiency indices (2,2) (in limit-circle cases at ±∞) and acting in the hilbert space ℓ_{ω}²(z;c²) (z:={0,±1,±2,...}) are considered. we consider two classes dissipative operators with separated boundary conditions both at -∞ and ∞. for each of these cases we establish a self...

and Applied Analysis 3 In order to prove that the sequence {xn} is a Cauchy sequence with respect to norm ‖·‖C, we introduce an equivalent norm and show that {xn} is a Cauchy sequence with respect to the new one. Basing on the condition H2 , we see that there are two positive constantsM and m such that m ≤ y t ≤ M for all t ∈ R. Define the new norm ‖ · ‖1 by ‖u‖1 sup { 1 y t ‖u t ‖E : t ∈ R } ,...

1. Systems of 1st order ordinary differential equations 2 1.1. Existence of solutions 2 1.2. Uniqueness 4 1.3. Differentiability of solutions with respect to a parameter 6 2. Quasi-linear pdes 10 2.1. Step 1 10 2.2. Step 2 11 2.3. Cauchy problem 11 3. Degenerate case 12 4. Examples 14 4.1. Example 1 14 4.2. Example 2 14 4.3. Example 3 14 4.4. A maximum principle for a first order pde 15 5. Full...

We show that  higher derivations on a Hilbert$C^{*}-$module associated with the Cauchy functional equation satisfying generalized Hyers--Ulam stability.  

In this paper, Lorentz Transformation(LT) is derived by an alternate method, using photon clocks, placed at the locations of the concerned events, which are initially synchronised using a light signal(Einstein synchrony). Then, it is shown that the second term in the time dilation equation of the LT, is the term responsible for time travel and further it is shown that this term arises merely du...