WE CONSIDER GALERKIN APPROXIMATIONS FOR THE EQUATIONS MODELING THE MOTION OF AN INCOMPRESSIBLE MAGNETO-MICROPOLAR FLUID IN A BOUNDED DOMAIN. WE DERIVE AN OPTIMAL UNIFORM IN TIME ERROR BOUND IN THE H1 AND L2 -NORMS FOR THE VELOCITY. THIS IS DONE WITHOUT EXPLICIT ASSUMPTION OF EXPONENTIAL STABILITY FOR A CLASS OF SOLUTIONS CORRESPONDING TO DECAYING EXTERNAL FORCE FIELDS. OUR STUDY IS DONE FOR NO-SLIP BOUNDARY CONDITIONS, BUT THE RESULTS OBTAINED ARE EASILY EXTENDED TO THE CASE OF PERIODIC BOUNDARY CONDITIONS.

CONVEXITY AND GENERALIZED CONVEXITY PLAY A CENTRAL ROLE IN MATHEMATICAL ECONOMICS AND OPTIMIZATION THEORY. SO, THE RESEARCH ON CRITERIA FOR CONVEXITY OR GENERALIZED CONVEXITY IS ONE OF THE MOST IMPORTANT ASPECTS IN MATHEMATICAL PROGRAMMING, IN ORDER TO CHARACTERIZE THE SOLUTIONS SET. MANY EFFORTS HAVE BEEN MADE IN THE FEW LAST YEARS TO WEAKEN THE CONVEXITY NOTIONS. IN THIS ARTICLE, TAKING IN MIND CRAVENS NOTION OF K-INVEXITY FUNCTION (WHEN K IS A CONE IN ? N ) AND MARTINS NOTION OF KARUSH?KUHN?TUCKER INVEXITY (HEREAFTER KKT-INVEXITY), WE DEFINE A NEW NOTION OF GENERALIZED CONVEXITY THAT IS BOTH NECESSARY AND SUFFICIENT TO ENSURE EVERY KKT POINT IS A GLOBAL OPTIMUM FOR PROGRAMMING PROBLEMS WITH CONIC CONSTRAINTS. THIS NEW DEFINITION IS A GENERALIZATION OF KKT-INVEXITY CONCEPT GIVEN BY MARTIN AND K-INVEXITY FUNCTION GIVEN BY CRAVEN. MOREOVER, IT IS THE WEAKEST TO CHARACTERIZE THE SET OF OPTIMAL SOLUTIONS. THE NOTIONS AND RESULTS THAT EXIST IN THE LITERATURE UP TO NOW ARE PARTICULAR INSTANCES OF THE ONES PRESENTED HERE.

NECESSARY CONDITIONS OF OPTIMALITY ARE PRESENTED FOR WEAKLY EFFICIENT SOLUTIONS TO MULTIOBJECTIVE MINIMIZATION PROBLEMS WITH INEQUALITY-TYPE CONSTRAINTS. THESE CONDITIONS ARE APPLIED WHEN THE CONSTRAINTS DO NOT NECESSARILY SATISFY ANY REGULARITY ASSUMPTIONS AND THEY ARE BASED ON THE CONCEPT OF 2-REGULARITY INTRODUCED BY IZMAILOV. IN GENERAL, THE OPTIMALITY CONDITIONS DO NOT PROVIDE THE COMPLETE WEAK PARETO OPTIMAL SET, SO 2-KKT-PSEUDOINVEX PROBLEMS ARE DEFINED. THIS NEW CONCEPT OF GENERALIZED CONVEXITY IS BOTH NECESSARY AND SUFFICIENT TO GUARANTEE THE CHARACTERIZATION OF ALL WEAKLY EFFICIENT SOLUTIONS BASED ON THE OPTIMALITY CONDITIONS AND IT IS THE WEAKEST ONE.

CONVEXITY AND GENERALIZED CONVEXITY PLAY A CENTRAL ROLE IN MATHEMATICAL PROGRAMMING FOR DUALITY RESULTS AND IN ORDER TO CHARACTERIZE THE SOLUTIONS SET. IN THIS PAPER, TAKING IN MIND CRAVEN'S NOTION OF K-INVEXITY FUNCTION (WHEN K IS A CONE IN R-N) AND MARTIN'S NOTION OF KARUSH-KUHN-TUCKER INVEXITY (HEREAFTER KKT-INVEXITY), WE DEFINE NEW NOTIONS OF GENERALIZED CONVEXITY FOR A MULTIOBJECTIVE PROBLEM WITH CONIC CONSTRAINTS. THESE NEW NOTIONS ARE BOTH NECESSARY AND SUFFICIENT TO ENSURE EVERY KARUSH-KUHN-TUCKER POINT IS A SOLUTION. THE STUDY OF THE SOLUTIONS IS ALSO DONE THROUGH THE SOLUTIONS OF AN ASSOCIATED SCALAR PROBLEM. A MOND-WEIR TYPE DUAL PROBLEM IS FORMULATED AND WEAK AND STRONG DUALITY RESULTS ARE PROVIDED. THE NOTIONS AND RESULTS THAT EXIST IN THE LITERATURE UP TO NOW ARE PARTICULAR INSTANCES OF THE ONES PRESENTED HERE.

AN OPTIMAL CONTROL PROBLEM WITH A CONVEX COST FUNCTIONAL SUBJECT TO A (LINEAR) NON-WELL-POSED PROBLEM (DIRICHLET HEAT EQUATION WITH A GIVEN FINAL DATA) IS CONSIDERED. THE CONTROL IS DISTRIBUTED AND A CONVEX CONSTRAINT ON THE CONTROL IS IMPOSED. FOR A GLOBALLY DISTRIBUTED CONTROL AND A CONVEX CONSTRAINT ON THE CONTROL WITH NON-EMPTY INTERIOR, WE DEDUCE FIRST-ORDER NECESSARY (AND SUFFICIENT) OPTIMALITY CONDITIONS USING THE SO-CALLED DUBOVITSKII?MILYUTIN FORMALISM, OBTAINING, IN PARTICULAR, THE EXISTENCE OF THE CORRESPONDING ADJOINT PROBLEM (WHICH IS AGAIN A NON-WELL-POSED PROBLEM). IN OTHER CASES (EITHER EMPTY INTERIOR CONVEX CONSTRAINT ON THE CONTROL OR PARTIALLY DISTRIBUTED CONTROL), WE ARRIVE AT THE OPTIMALITY CONDITIONS BUT ADMITTING THE EXISTENCE OF THE ADJOINT PROBLEM. FINALLY, NUMERICAL RESULTS ARE ALSO PRESENTED APPROXIMATING THE OPTIMALITY CONDITIONS FOR 1D DOMAINS BY FINITE DIFFERENCES IN TIME AND SPACE.

IN THIS WORK WE STUDY THE NUMERICAL RESOLUTION OF SYSTEMS OF FUZZY NONLINEAR EQUATIONS. MORE PRECISELY, WE DESCRIBE, ANALYZE AND SIMULATE NUMERICAL METHODS, SUCH AS NEWTON METHOD, IN ORDER TO APPROXIMATE EFFICIENTLY THE SOLUTIONS TO SUCH PROBLEMS. ONE OF THE MAIN ISSUES OF THIS TYPE OF PROBLEMS IS THAT THE STANDARD ANALYTICAL TECHNIQUES FOR FINDING SOLUTIONS, ARE NOT APPROPRIATE TO RESOLVE THEM. FOR THIS REASON, IN THIS PAPER WE FOCUS IN THE STUDY OF KNOWN RESULTS FOR THE CLASSICAL METHODS AND THEIR ADAPTATION TO THE RESOLUTION OF FUZZY PROBLEMS.

WE CONSIDER THE NONLINEAR MODEL DESCRIBING MICROPOLAR FLUID IN A BOUNDED SMOOTH REGION OF ? N (N=2,3) WITH DISTRIBUTED CONTROLS SUPPORTED IN SMALL SUBSET OF THIS DOMAIN. UNDER SUITABLE ASSUMPTIONS ON THE GALERKIN BASIS, WE INTRODUCE GALERKIN?S APPROXIMATIONS FOR THE CONTROLLABLE MICROPOLAR FLUID SYSTEM. BY USING THE HILBERT UNIQUENESS METHOD IN COMBINATION WITH A FIXED POINT ARGUMENT, WE PROVE THE EXACT CONTROLLABILITY RESULT FOR THIS FINITE-DIMENSIONAL SYSTEM.

WE ANALYZE A SYSTEM OF NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS MODELING THE STATIONARY FLOW INDUCED BY THE UPWARD SWIMMING OF CERTAIN MICROORGANISMS IN A FLUID. WE CONSIDER THE REALISTIC CASE IN WHICH THE EFFECTIVE VISCOSITY OF THE FLUID DEPENDS ON THE CONCENTRATION OF SUCH MICROORGANISMS. UNDER CERTAIN CONDITIONS, WE PROVE THE EXISTENCE AND UNIQUENESS OF SOLUTIONS FOR SUCH GENERALIZED BIOCONVECTIVE FLOW EQUATIONS

WE CONSIDER THE INITIAL BOUNDARY VALUE PROBLEM FOR THE SYSTEM OF EQUATIONS DESCRIBING THE NONSTATIONARY FLOW OF AN INCOMPRESSIBLE MICROPOLAR FLUID IN A DOMAIN OMEGA OF R3. UNDER HYPOTHESES THAT ARE SIMILAR TO THE NAVIER-STOKES EQUATIONS, BY USING AN ITERATIVE SCHEME, WE PROVE THE EXISTENCE AND UNIQUENESS OF STRONG SOLUTION IN L P (OMEGA), FOR P > 3.

WE EXTEND THE CONCEPT OF EXPANSIVE MEASURE [2] FROM HOMEOMORPHISM TO FLOWS. WE PROVE FOR CONTINUOUS FLOWS ON COMPACT SPACES THAT EVERY EXPANSIVE MEASURE HAS NO SINGULARITIES IN THE SUPPORT, IS APERIODIC, IS EXPANSIVE WITH RESPECT TO TIME-T MAPS (BUT NOT CONVERSELY), REMAINS EXPANSIVE UNDER TOPOLOGICAL EQUIVALENCE, VANISHES ALONG THE ORBITS AND IS NATURAL UNDER SUSPENSIONS. WE APPLY THESE PROPERTIES TO PROVE THAT THERE ARE NO EXPANSIVE FLOWS (IN THE SENSE OF [26]) OF ANY CLOSED SURFACE.

IN NON-REGULAR PROBLEMS THE CLASSICAL OPTIMALITY CONDITIONS ARE TOTALLY INAPPLICABLE. MEANINGFUL RESULTS WERE OBTAINED FOR PROBLEMS WITH CONIC CONSTRAINTS BY IZMAILOV AND SOLODOV (SIAM J CONTROL OPTIM 40(4):1280?1295, 2001). THEY ARE BASED ON THE SO-CALLED 2-REGULARITY CONDITION OF THE CONSTRAINTS AT A FEASIBLE POINT. IT IS WELL KNOWN THAT GENERALIZED CONVEXITY NOTIONS PLAY A VERY IMPORTANT ROLE IN OPTIMIZATION FOR ESTABLISHING OPTIMALITY CONDITIONS. IN THIS PAPER WE GIVE THE CONCEPT OF KARUSH?KUHN?TUCKER POINT TO REWRITE THE NECESSARY OPTIMALITY CONDITION GIVEN IN IZMAILOV AND SOLODOV (SIAM J CONTROL OPTIM 40(4):1280?1295, 2001) AND THE APPROPRIATE GENERALIZED CONVEXITY NOTIONS TO SHOW THAT THE OPTIMALITY CONDITION IS BOTH NECESSARY AND SUFFICIENT TO CHARACTERIZE OPTIMAL SOLUTIONS SET FOR NON-REGULAR PROBLEMS WITH CONIC CONSTRAINTS. THE RESULTS THAT EXIST IN THE LITERATURE UP TO NOW, EVEN FOR THE REGULAR CASE, ARE PARTICULAR INSTANCES OF THE ONES PRESENTED HERE.

WE STUDY A SYSTEM OF EQUATIONS MODELING THE STATIONARY MOTION OF INCOMPRESSIBLE ELECTRICAL CONDUCTING FLUID. BASED ON METHODS OF CLIFFORD ANALYSIS, WE REWRITE THE SYSTEM OF MAGNETOHYDRODYNAMICS FLUID IN THE HYPERCOMPLEX FORMULATION AND REPRESENT ITS SOLUTION IN CLIFFORD OPERATOR TERMS.

A STUDY OF THE CONVERGENCE OF WEAK SOLUTIONS OF THE NONSTATIONARY MICROPOLAR FLUIDS, IN BOUNDED DOMAINS OF ?N, WHEN THE VISCOSITIES TEND TO ZERO, IS ESTABLISHED. IN THE LIMIT, A FLUID GOVERNED BY AN EULER-LIKE SYSTEM IS FOUND.