In my lecture at the 2011 Congress on Logic, Methodology, and the Philosophy of Science in Nancy, France, I spoke on an additional axiom of set theory — the Proper Forcing Axiom — which has proved very successful in settling combinatorial problems concerning uncountable sets. Since I have already written a exposition on this subject [43], I have decided to address a broader question in this art...

Forcing is a technique that was discovered by Cohen in the mid 20th century, and it is particularly useful in proving independence results. The process involves adding new, generic elements to your universe and using them to prove a result. In many applications this is encapsulated by the addition of suitable axioms, often these are Forcing Axioms. One of the first such principles is known as M...

We characterize the degrees of freedom ηX in a system with two multiple antenna transmitters and two multiple antenna receivers. With M antennas at all nodes we find ⌊ 43M⌋ ≤ ηX ≤ 4 3M . The MMK scheme proposed in [1] is seen to achieve the maximum multiplexing gain rounded down to the nearest integer and is exactly optimal when M is a multiple of 3. Compared to zero forcing that allows only M ...

Multiple-input multiple-output (MIMO) are used in wireless communications to achieve high data rates within the limited frequency spectrum. MIMO uses multiple transmitting and receiving antennas so as to utilize the advantages of spatial diversity. The performance analysis of MIMO system over AWGN and Rician fading channels using OSTBC4 as space time coding with Zero Forcing (ZF) receiver is pr...

The notion of realizability algebra, which was introduced in [17, 18], is a tool to study the proof-program correspondence and to build new models of set theory, which we call realizability models of ZF. It is a variant of the well known notion of combinatory algebra, with a new instruction cc, and a new type for the environments. The sets of forcing conditions, in common use in set theory, are...

Hindman and Leader introduced the set $$0^{+}$$ of ultrafilters on $$\left( 0,1\right) $$ , characterized smallest ideal 0^{+},+\right) proved Central Set Theorem near 0. Recently Polynomial has been by Bergelson, Johnson Jr. Moreira. In this note, we will