WE PROVIDE CANONICAL FORMS FOR THE HOMOGENEOUS POLYNOMIALS OF DEGREE FIVE. THEN WE CHARACTERIZE ALL THE PHASE PORTRAITS IN THE POINCARÉ DISK FOR ALL QUARTIC HOMOGENEOUS POLYNOMIAL DIFFERENTIAL SYSTEMS. MORE PRECISELY, THERE ARE EXACTLY 23 DIFFERENT TOPOLOGICAL PHASE PORTRAITS FOR THE QUARTIC HOMOGENEOUS POLYNOMIAL DIFFERENTIAL SYSTEMS.

WE DESCRIBE THE DYNAMICS OF THE 3-DIMENSIONAL COMPETITIVE LOTKA-VOLTERRA SYSTEMS (X) OVER DOT = X(A - X - Y - Z), (Y) OVER DOT = Y(B - X - Y - Z), (Z) = Z(C - X - Y - Z), PROVIDING THE PHASE PORTRAITS FOR ALL THE VALUES OF THE PARAMETERS A, B AND C WITH 0 < A < B < C IN THE POSITIVE OCTANT OF THE POINCARE BALL.

WE PROVIDE THE PHASE PORTRAITS OF THE 3-DIMENSIONAL LOTKA-VOLTERRA SYSTEMS X=X (Y+AZ), Y=(X+Z), Z=BZ(-AX+Y), FOR ALL THE VALUES OF THE PARAMETERS AND , IN THE FINITE REGION AND IN THE INFINITY REGION THROUGH THE POINCARÉ COMPACTIFICATION. WE ALSO STUDY THE INTEGRABILITY OF THE SYSTEM.

WE CHARACTERIZE WHEN THE UNIQUE EQUILIBRIUM POINT OF THE POSITIVE QUADRANT OF THE TWODIMENSIONAL LOTKA?VOLTERRA SYSTEM IS A GLOBAL ATTRACTOR IN THAT QUADRANT. ADDITIONALLY, WE CLASSIFY THE PHASE PORTRAITS OF THIS CLASS OF LOKTA?VOLTERRA SYSTEM.

IN THIS WORK WE CONSIDER THE KOLMOGOROV SYSTEM OF DEGREE 3 IN R2 AND R3 HAVING AN EQUILIBRIUM POINT IN THE POSITIVE QUADRANT AND OCTANT, RESPECTIVELY. WE PROVIDE SUFFICIENT CONDITIONS IN ORDER THAT THE EQUILIBRIUM POINT WILL BE A HOPF POINT FOR THE PLANAR CASE AND A ZERO-HOPF POINT FOR THE SPATIAL ONE. WE STUDY THE LIMIT CYCLES BIFURCATING FROM THESE EQUILIBRIA USING AVERAGING THEORY OF SECOND AND FIRST ORDER, RESPECTIVELY. WE NOTE THAT THE EQUILIBRIUM POINT IS LOCATED IN THE QUADRANT OR OCTANT WHERE THE KOLMOGOROV SYSTEMS HAVE BIOLOGICAL MEANING.

WE STUDY THE NUMBER OF LIMIT CYCLES BIFURCATING FROM THE ORIGIN OF A HAMILTONIAN SYSTEM OF DEGREE 4. WE PROVE, USING THE AVERAGING THEORY OF ORDER 7, THAT THERE ARE QUARTIC POLYNOMIAL SYSTEMS CLOSE THESE HAMILTONIAN SYSTEMS HAVING 3 LIMIT CYCLES.

BUILDING SIMULATION SOFTWARE TOOLS SUPPORT DESIGNERS TO ANALYSE AND IDENTIFY CERTAIN USERS' BEHAVIOURAL PATTERNS; BESIDES, THEY CAN PREDICT FUTURE TRENDS ABOUT THE ENERGY DEMAND AND CONSUMPTION IN BUILDINGS, AS WELL AS CO2 EMISSIONS, DESIGN ANALYSIS, ENERGY EFFICIENCY, OR LIGHTING. THESE TOOLS ALLOW TO COLLECT AND REPORT INFORMATION ABOUT SUCH PROCESSES. HOWEVER, UNDERSTANDING THE RESULTS FROM SIMULATIONS USUALLY IMPLIES INTERPRETING AN EXTREMELY LARGE AMOUNT OF DATA OR GRAPHS, WHICH CAN BE A COMPLEX TASK. THEREFORE, THERE IS A NEED OF ALTERNATIVES THAT EASE THIS INTERPRETATION OF RESULTS, HENCE COMPLEMENTING CLASSIC SIMULATION TOOLS. UNDER THE WIDESPREAD EN 15251 MODEL CRITERIA, THIS PAPER PRESENTS A NOVEL TECHNOLOGY TO IMPROVE REPORTING TOOLS OF BUILDING SIMULATION SOFTWARE BY USING LINGUISTIC DESCRIPTION OF DATA AND TIMESPAN COMPUTATIONAL PERCEPTIONS. A DATA-DRIVEN SOFTWARE ARCHITECTURE FOR AUTOMATICALLY GENERATING LINGUISTIC REPORTS IS HERE PROPOSED, WHICH PROVIDES DESIGNERS WITH A BETTER UNDERSTANDING OF THE DATA FROM BUILDING SIMULATION TOOLS. IN ORDER TO SHOW AND EXPLORE THE POSSIBILITIES OF THIS TECHNOLOGY, A SOFTWARE APPLICATION HAS BEEN DESIGNED, IMPLEMENTED AND EVALUATED BY EXPERTS. THE SURVEY SHOWED THAT USEFULNESS AND CLARIFICATION WERE BETTER EVALUATED THAN SIMPLICITY AND TIME-SAVING FOR THE THREE KINDS OF REPORT, THOUGH ALWAYS ABOVE 7 POINTS OUT OF 10, BEING MOST OF P-VALUES OF CONTINGENCY BELOW 0.05.

FINITE ENERGY QCD SUM RULES INVOLVING NUCLEON CURRENT CORRELATORS ARE USED TO DETERMINE SEVERAL QCD AND HADRONIC PARAMETERS IN THE PRESENCE OF AN EXTERNAL, UNIFORM, LARGE MAGNETIC FIELD. THE CONTINUUM HADRONIC THRESHOLD S0, NUCLEON MASS MN, CURRENT-NUCLEON COUPLINGN, TRANSVERSE VELOCITY V?, THE SPIN POLARIZATION CONDENSATE ?¯Q?12Q?, AND THE MAGNETIC SUSCEPTIBILITY OF THE QUARK CONDENSATE ?Q, ARE OBTAINED FOR THE CASE OF PROTONS AND NEUTRONS. DUE TO THE MAGNETIC FIELD, AND CHARGE ASYMMETRY OF LIGHT QUARKS UP AND DOWN, ALL THE OBTAINED QUANTITIES EVOLVE DIFFERENTLY WITH THE MAGNETIC FIELD, FOR EACH NUCLEON OR QUARK FLAVOR. WITH THIS APPROACH IT IS POSSIBLE TO OBTAIN THE EVOLUTION OF THE ABOVE PARAMETERS UP TO A MAGNETIC FIELD STRENGTH EB

WE STUDY A CHEMOSTAT SYSTEM OF THE FORM (X) OVER DOT = -QX + (R) OVER TILDE /K + Y XY, (Y) OVER DOT = ((C) OVER TILDE - Y)Q - (R) OVER TILDE/(A) OVER TILDE (K + Y) XY, WHERE Q > 0, (R) OVER TILDE > 0, K > 0, (C) OVER TILDE > 0 AND (A) OVER TILDE NOT EQUAL 0. THIS SYSTEM APPEARS IN COMPETITION MODELLING IN BIOLOGY. WE DESCRIBE ITS GLOBAL DYNAMICS ON THE POINCARE DISC AND STUDY ITS LIOUVILLIAN INTEGRABILITY. FOR THE FIRST TOPIC WE USE THE WELL-KNOWN POINCARE COMPACTIFICATION THEORY AND FOR THE SECOND ONE WE MAKE USE OF THE PUISEUX SERIES TO DERIVE THE STRUCTURE OF ALL THE IRREDUCIBLE INVARIANT ALGEBRAIC CURVES. (C) 2019 ELSEVIER LTD. ALL RIGHTS RESERVED.

IN THIS PAPER BY USING THE POINCARÉ COMPACTIFICATION OF ?3 WE MAKE A GLOBAL ANALYSIS OF THE MODEL X? = ?AX + Y + YZ, Y? = X ? AY + BXZ, Z? = CZ ? BXY. IN PARTICULAR WE GIVE THE COMPLETE DESCRIPTION OF ITS DYNAMICS ON THE INFINITY SPHERE. FOR A + C = 0 OR B = 1 THIS SYSTEM HAS INVARIANTS. FOR THESE VALUES OF THE PARAMETERS WE PROVIDE THE GLOBAL PHASE PORTRAIT OF THE SYSTEM IN THE POINCARÉ BALL. WE ALSO DESCRIBE THE ? AND ?-LIMIT SETS OF ITS ORBITS IN THE POINCARÉ BALL.

WE STUDY THE PHASE PORTRAITS ON THE POINCARE DISC FOR ALL THE LINEAR TYPE CENTERS OF POLYNOMIAL HAMILTONIAN SYSTEMS OF DEGREE 5 WITH HAMILTONIAN FUNCTION H(X,Y) = H-1(X) + H-2(Y), WHERE H-1(X) = 1/2X(2) + A(3)/3X(3) + A(4)/4X(4) + A(5)/5X(5) AND H-2(Y) = 1/2Y(2) + B(3)/3Y(3) + B(4)/4Y(4) + B(5)/5Y(5) AS FUNCTION OF THE SIX REAL PARAMETERS A(3), A(4), A(5), B(3), B(4) AND B(5) WITH A(5)B(5) NOT EQUAL 0. WE CHARACTERIZE THE TYPE AND MULTIPLICITY OF THE ROOTS OF THE POLYNOMIALS (P) OVER CAP (Y) = 1+ B(3)Y + B(4)Y(2) + B(5)Y(3) AND (Q) OVER CAP (X) = 1 + A(3)X A(4)X + A(5)X(3) AND WE PROVE THAT THE FINITE EQUILIBRIA ARE SADDLES, CENTERS, CUSPS OR THE UNION OF TWO HYPERBOLIC SECTORS. FOR THE INFINITE EQUILIBRIA WE FOUND THAT THERE ONLY EXIST TWO NODES ON THE POINCARE DISC WITH OPPOSITE STABILITY. WE ALSO CHARACTERIZE THE SEPARATRICES OF THE EQUILIBRIA AND ANALYZE THE POSSIBLE CONNECTIONS BETWEEN THEM. AS A COMPLEMENT WE USE THE ENERGY LEVEL TO COMPLETE THE GLOBAL PHASE PORTRAIT.