IN THIS PAPER WE CONSIDER TWO FAMILIES OF FOUR OR FIVE-DIMENSIONAL SOLVABLE LIE GROUPS WHICH ARE EXTENSIONS OF THE HEISENBERG GROUP. WE FIRST CALCULATE THE ENERGY OF AN ARBITRARY LEFT-INVARIANT VECTOR FIELDX ON THESE SPACES. THEN WE OBTAIN THE EXACT FORM OF ALL HARMONIC MAPS ON THESE SPACES AND PROVE THAT IF THESE SPACES ADMIT A NON-TRIVIAL HARMONIC MAP, THEN ANY HARMONIC MAP IS A CRITICAL POINT FOR THE ENERGY FUNCTIONAL RESTRICTED TO VECTOR FIELDS OF THE SAME LENGTH. LATER ON, WE INVESTIGATE THE EXISTENCE OF LEFT-INVARIANT YAMABE SOLITONS AND EXPLICITLY DESCRIBE ALL GEODESIC VECTORS ON THESE SPACES.

LET G BE A FINITE SOLVABLE GROUP. IN THIS PAPER WE CONSIDER THE CHARACTER GRAPH OF GAND STUDY SOME PARAMETERS OF THIS GRAPH. AT FIRST, WE ANSWER THIS QUESTION THAT WHEN IS THIS GRAPH HAMILTONIAN? THEN WE OBTAIN CONDITIONS WHICH IT IS A COMPLETE GRAPH. FINALLY, WE STUDY THE COLORING OF THIS GRAPH.

IN THIS TALK, WE GIVE A SURVEY OF SOME RECENT ADVANCES ON THE PROBLEM OF STUDYING SEMI-RATIONAL FINITE GROUPS...

IN THIS PAPER, THE CONCEPT OF COMMUTATORS IN BCH-ALGEBRAS IS INTRODUCED. SOME RELATED THEOREMS ON THESE NOTIONS ARE STATED AND PROVED, THE NOTIONS OF DERIVED SUBALGEBRAS AND SOLVABLE BCH-ALGEBRAS IN BCH-ALGEBRAS ARE INTRODUCED, AND SOME BASIC PROPERTIES IN SOLVABLE BCH-ALGEBRAS ARE GIVEN. IT IS OBTAINED THAT THE SUB-ALGEBRAS, ISOMORPHIC IMAGE AND INVERSE IMAGE OF A SOLVABLE BCH-ALGEBRAS ARE STILL SOLVABLE BCH-ALGEBRAS.