EN ESTE ARTÍCULO DAMOS UN ENFOQUE MOTIVADO POR UN PAQUETE DE CLIFFORD A LA ECUACIÓN DE ONDA DE UN FÉRTIL DE $ 1/2 $ LIBRE EN EL COLECTOR DE SITTER, UNA BRANA CON TOPOLOGÍA $ M = \ MATHRM {S0} (4,1) / \ MATHRM { S0} (3,1) $ QUE VIVE EN EL ESPACIO-TIEMPO MAYORITARIO $ \ MATHBB {R} ^ {4,1} = (\ MATHRING {M} = \ MATHBB {R} ^ {5}, \ BOLDSYMBOL {\ MATHRING {G }} $ Y EQUIPADO CON UN CAMPO MÉTRICO $ \ BOLDSYMBOL {G: = - I} ^ {\ AST} \ BOLDSYMBOL {\ MATHRING {G}%} $ CON $ \ BOLDSYMBOL {I}: M \ RIGHTARROW \ MATHRING { M} $ SIENDO EL MAPA DE INCLUSIÓN. PARA OBTENER EL ANÁLOGO DE LA ECUACIÓN DE DIRAC EN EL ESPACIO-TIEMPO DE MINKOWSKI EN LA ESTRUCTURA $ \ MATHRING {M} $, FACTORIZAMOS ADECUADAMENTE LOS DOS INVARIANTES DE CASIMIR $ C_ {1} $ Y $ C_ {2} $ DEL ÁLGEBRA DE LIE DEL GRUPO DE SITTER USANDO LA RESTRICCIÓN DADA EN LA LINEALIZACIÓN DE $ C_ {2} $ COMO ENTRADA PARA LINEALIZAR $ C_ {1} $. DE ESTA MANERA OBTENEMOS UNA ECUACIÓN QUE LLAMAMOS \ TEXTBF {DHESS1, } QUE EN ESTUDIOS PREVIOS DE OTROS AUTORES SE POSTULÓ SIMPLEMENTE $. $ LUEGO DERIVAMOS UNA ECUACIÓN DE ONDA (LLAMADA \ TEXTBF {DHESS2}) PARA UN SPIN LIBRE $ 1/2 $ FERMION EN EL COLECTOR DE SITTER USANDO UN ARGUMENTO HEURÍSTICO QUE ES UN LA GENERALIZACIÓN OBVIA DE UN ARGUMENTO HEURÍSTICO (DESCRITO EN DETALLE EN EL APÉNDICE D) QUE PERMITE DERIVAR LA ECUACIÓN DE DIRAC EN EL ESPACIOTIEMPO DE MINKOWSKI Y QUE MUESTRA QUE UNA ECUACIÓN TAN FAMOSA NO EXPRESA NADA MÁS QUE EL HECHO DE QUE EL MOMENTO DE UNA PARTÍCULA LIBRE ES UN CAMPO VECTORIAL CONSTANTE TEMPORALES \ INTEGRAL DE UN CAMPO DE VELOCIDAD DADO. ES UN HECHO NOTABLE QUE \ TEXTBF {DHESS1} Y \ TEXTBF {DHESS2} \ COINCIDEN. UNO DE LOS PRINCIPALES INGREDIENTES EN NUESTRO TRABAJO ES EL USO DEL CONCEPTO DE HUERTOS DIRAC-HESTENES. LOS APÉNDICES B Y C RECUERDAN ESTE CONCEPTO Y SU RELACIÓN CON LOS CAMPOS COVARIANTES DE LA ESPINORA DE DIRAC USUALMENTE UTILIZADOS POR LOS FÍSICOS

THE DAGUM FAMILY OF ISOTROPIC COVARIANCE FUNCTIONS HAS TWO PARAMETERS THAT ALLOW FOR DECOUPLING OF THE FRACTAL DIMENSION AND THE HURST EFFECT FOR GAUSSIAN RANDOM FIELDS THAT ARE STATIONARY AND ISOTROPIC OVER EUCLIDEAN SPACES. SUFFICIENT CONDITIONS THAT ALLOW FOR POSITIVE DEFINITENESS IN RD OF THE DAGUM FAMILY HAVE BEEN PROPOSED ON THE BASIS OF THE FACT THAT THE DAGUM FAMILY ALLOWS FOR COMPLETE MONOTONICITY UNDER SOME PARAMETER RESTRICTIONS. THE SPECTRAL PROPERTIES OF THE DAGUM FAMILY HAVE BEEN INSPECTED TO A VERY LIMITED EXTENT ONLY, AND THIS PAPER GIVES INSIGHT INTO THIS DIRECTION. SPECIFICALLY, WE STUDY FINITE AND ASYMPTOTIC PROPERTIES OF THE ISOTROPIC SPECTRAL DENSITY (INTENDED AS THE HANKEL TRANSFORM) OF THE DAGUM MODEL. ALSO, WE ESTABLISH SOME CLOSED-FORM EXPRESSIONS FOR THE DAGUM SPECTRAL DENSITY IN TERMS OF THE FOX?WRIGHT FUNCTIONS. FINALLY, WE PROVIDE ASYMPTOTIC PROPERTIES FOR SUCH A CLASS OF SPECTRAL DENSITIES.

WE PROPOSE A NEW SET OF MASTER INTEGRALS WHICH CAN BE USED AS A BASIS FOR CERTAIN MULTILOOP CALCULATIONS IN MASSLESS GAUGE FIELD THEORIES. IN THESE THEORIES WE CONSIDER THREE-POINT FEYNMAN DIAGRAMS WITH ARBITRARY NUMBER OF LOOPS. THE CORRESPONDING MULTILOOP INTEGRALS MAY BE DECOMPOSED IN TERMS OF THIS SET OF THE MASTER INTEGRALS. WE CONSTRUCT A NEW REDUCTION PROCEDURE WHICH WE APPLY TO PERFORM THIS DECOMPOSITION.

ANALYTIC EXPRESSIONS FOR THE N-DIMENSIONAL DEBYE FUNCTION ARE OBTAINED BY THE METHOD OF BRACKETS. THE NEW EXPRESSIONS ARE SUITABLE FOR THE STUDY OF HEAT CAPACITY OF SOLIDS AND THE ANALYSIS OF THE ASYMPTOTIC BEHAVIOR OF THIS FUNCTION, BOTH IN THE HIGH AND LOW TEMPERATURE LIMITS.

WE CONSIDER A SIMPLE MODEL FOR QCD DYNAMICS IN WHICH DGLAP INTEGRO-DIFFERENTIAL EQUATION MAY BE SOLVED ANALYTICALLY. THIS IS A GAUGE MODEL WHICH POSSESSES DOMINANT EVOLUTION OF GAUGE BOSON (GLUON) DISTRIBUTION AND IN WHICH THE GAUGE COUPLING DOES NOT RUN. THIS MAY BE N = 4 SUPERSYMMETRIC GAUGE THEORY WITH SOFTLY BROKEN SUPERSYMMETRY, OTHER FINITE SUPERSYMMETRIC GAUGE THEORY WITH A LOWER LEVEL OF SUPERSYMMETRY, OR TOPOLOGICAL CHERN?SIMONS FIELD THEORIES. WE MAINTAIN ONLY ONE TERM IN THE SPLITTING FUNCTION OF UNINTEGRATED GLUON DISTRIBUTION AND SOLVE DGLAP ANALYTICALLY FOR THIS SIMPLIFIED SPLITTING FUNCTION. THE SOLUTION IS FOUND USING THE CAUCHY INTEGRAL FORMULA. THE SOLUTION RESTRICTS THE FORM OF THE UNINTEGRATED GLUON DISTRIBUTION AS A FUNCTION OF MOMENTUM TRANSFER AND OF BJORKEN X. THEN, WE CONSIDER AN ALMOST REALISTIC SPLITTING FUNCTION OF UNINTEGRATED GLUON DISTRIBUTION AS AN INPUT TO DGLAP EQUATION AND SOLVE IT BY THE SAME METHOD WHICH WE HAVE DEVELOPED TO SOLVE DGLAP EQUATION FOR THE TOY-MODEL. WE STUDY A RESULT OBTAINED FOR THE REALISTIC GLUON DISTRIBUTION AND FIND A SINGULAR BESSEL-LIKE BEHAVIOR IN THE VICINITY OF THE POINT X = 0 AND A SMOOTH BEHAVIOR IN THE VICINITY OF THE POINT X = 1.

A SIMPLE MODEL FOR QCD DYNAMICS IN WHICH THE DGLAP INTEGRO-DIFFERENTIAL EQUATION MAY BE SOLVED ANALYTICALLY HAS BEEN CONSIDERED IN OUR PREVIOUS PAPERS ARXIV:1611.08787 [HEP-PH] AND ARXIV:1906.07924 [HEP-PH]. WHEN SUCH A MODEL CONTAINS ONLY ONE TERM IN THE SPLITTING FUNCTION OF THE DOMINANT PARTON DISTRIBUTION, THEN BESSEL FUNCTION APPEARS TO BE THE SOLUTION TO THIS SIMPLIFIED DGLAP EQUATION. TO OUR KNOWLEDGE, THIS MODEL WITH ONLY ONE TERM IN THE SPLITTING FUNCTION FOR THE FIRST TIME HAS BEEN PROPOSED BY BLÜMLEIN IN ARXIV:HEP-PH/9506403. IN ARXIV:1906.07924 [HEP-PH] WE HAVE SHOWN THAT A DUAL INTEGRO-DIFFERENTIAL EQUATION OBTAINED FROM THE DGLAP EQUATION BY A COMPLEX MAP IN THE PLANE OF THE MELLIN MOMENT IN THIS MODEL MAY BE CONSIDERED AS THE BFKL EQUATION. THEN, IN ARXIV:1906.07924 WE HAVE APPLIED A COMPLEX DIFFEOMORPHISM TO OBTAIN A STANDARD INTEGRAL FROM GRADSHTEYN AND RYZHIK TABLES STARTING FROM THE CONTOUR INTEGRAL FOR PARTON DISTRIBUTION FUNCTIONS THAT IS USUALLY TAKEN BY CALCULUS OF RESIDUES. THIS STANDARD INTEGRAL FROM THESE TABLES APPEARS TO BE THE LAPLACE TRANSFORMATION OF JACOBIAN FOR THIS COMPLEX DIFFEOMORPHISM. HERE WE WRITE UP ALL THE FORMULAE BEHIND THIS TRICK IN DETAIL AND FIND OUT CERTAIN IMPORTANT POINTS FOR FURTHER DEVELOPMENT OF THIS STRATEGY. WE VERIFY THAT THE INVERSE LAPLACE TRANSFORMATION OF THE LAPLACE IMAGE OF THE BESSEL FUNCTION MAY BE REPRESENTED IN A FORM OF BARNES CONTOUR INTEGRAL.

BELOKUROV-USYUKINA LOOP REDUCTION METHOD HAS BEEN PROPOSED IN 1983 TO REDUCE A NUMBER OF RUNGS IN TRIANGLE LADDER-LIKE DIAGRAM BY ONE. THE DISADVANTAGE OF THE METHOD IS THAT IT WORKS IN D = 4 DIMENSIONS ONLY AND IT CANNOT BE USED FOR CALCULATION OF AMPLITUDES IN FIELD THEORY IN WHICH WE ARE REQUIRED TO PUT ALL THE INCOMING AND OUTGOING MOMENTA ON SHELL. WE GENERALIZE THE BELOKUROV-USYUKINA LOOP REDUCTION TECHNIQUE TO NON-INTEGER D = 4 ? 2? DIMENSIONS. IN THIS PAPER WE SHOW HOW A TWO-LOOP TRIANGLE DIAGRAM WITH PARTICULAR VALUES OF INDICES OF SCALAR PROPAGATORS IN THE POSITION SPACE CAN BE REDUCED TO A COMBINATION OF THREE ONE-LOOP SCALAR DIAGRAMS. IT IS KNOWN THAT ANY ONE-LOOP MASSLESS MOMENTUM INTEGRAL CAN BE PRESENTED IN TERMS OF APPELL?S FUNCTION F 4. THIS MEANS THAT PARTICULAR DIAGRAM CONSIDERED IN THE PRESENT PAPER CAN BE REPRESENTED IN TERMS OF APPELL?S FUNCTION F 4 TOO. SUCH A GENERALIZATION OF BELOKUROV-USYUKINA LOOP REDUCTION TECHNIQUE ALLOWS US TO CALCULATE THAT DIAGRAM BY THIS METHOD EXACTLY WITHOUT DECOMPOSITION IN TERMS OF THE PARAMETER ?.

IN THIS PAPER WE SHOW THAT SOME 3-DIMENSIONAL ISOMETRY ALGEBRAS, SPECIFICALLY THOSE OF TYPE I, II, III AND V (ACCORDING BIANCHI'S CLASSIFICATION), CAN BE OBTAINED AS EXPANSIONS OF THE ISOMETRIES IN 2 DIMENSIONS. IT IS SHOWN THAT IN GENERAL MORE THAN ONE SEMIGROUP WILL LEAD TO THE SAME RESULT. IT IS IMPOSSIBLE TO OBTAIN THE ALGEBRAS OF TYPE IV, VI-IX AS AN EXPANSION FROM THE ISOMETRY ALGEBRAS IN 2 DIMENSIONS. THIS MEANS THAT THE FIRST SET OF ALGEBRAS HAS PROPERTIES THAT CAN BE OBTAINED FROM ISOMETRIES IN 2 DIMENSIONS WHILE THE SECOND SET HAS PROPERTIES THAT ARE IN SOME SENSE INTRINSIC IN 3 DIMENSIONS. ALL THE RESULTS ARE CHECKED WITH COMPUTER PROGRAMS. THIS PROCEDURE CAN BE GENERALIZED TO HIGHER DIMENSIONS, WHICH COULD BE USEFUL FOR DIVERSE PHYSICAL APPLICATIONS.

WE CONSTRUCT A FAMILY OF TRIANGLE-LADDER DIAGRAMS THAT CAN BE CALCULATED USING THE BELOKUROV-USYUKINA LOOP REDUCTION TECHNIQUE IN D=4?2? DIMENSIONS. THE MAIN IDEA OF THE APPROACH WE PROPOSE IS TO GENERALIZE THIS LOOP REDUCTION TECHNIQUE EXISTING IN D=4 DIMENSIONS. WE DERIVE A RECURRENCE RELATION BETWEEN THE RESULT FOR AN L-LOOP TRIANGLE-LADDER DIAGRAM OF THIS FAMILY AND THE RESULT FOR AN (L-1)-LOOP TRIANGLELADDER DIAGRAM OF THE SAME FAMILY. BECAUSE THE PROPOSED METHOD COMBINES ANALYTIC AND DIMENSIONAL REGULARIZATIONS, WE MUST REMOVE THE ANALYTIC REGULARIZATION AT THE END OF THE CALCULATION BY TAKING THE DOUBLE UNIFORM LIMIT IN WHICH THE PARAMETERS OF THE ANALYTIC REGULARIZATION VANISH. IN THE POSITION SPACE, WE OBTAIN A DIAGRAM IN THE LEFT-HAND SIDE OF THE RECURRENCE RELATIONS IN WHICH THE RUNG INDICES ARE 1 AND ALL OTHER INDICES ARE 1 - ? IN THIS LIMIT. FOURIER TRANSFORMS OF DIAGRAMS OF THIS TYPE GIVE MOMENTUM SPACE DIAGRAMS WITH RUNG INDICES 1 - ? AND ALL OTHER INDICES 1. BY A CONFORMAL TRANSFORMATION OF THE DUAL SPACE IMAGE OF THIS MOMENTUM SPACE REPRESENTATION, WE RELATE SUCH A FAMILY OF TRIANGLE-LADDER MOMENTUM DIAGRAMS TO A FAMILY OF BOX-LADDER MOMENTUM DIAGRAMS WITH RUNG INDICES 1 - ? AND ALL OTHER INDICES 1. BECAUSE ANY DIAGRAM FROM THIS FAMILY IS REDUCIBLE TO A ONE-LOOP DIAGRAM, THE PROPOSED GENERALIZATION OF THE BELOKUROV-USYUKINA LOOP REDUCTION TECHNIQUE TO A NONINTEGER NUMBER OF DIMENSIONS ALLOWS CALCULATING THIS FAMILY OF BOX-LADDER DIAGRAMS IN THE MOMENTUM SPACE EXPLICITLY IN TERMS OF APPELL?S HYPERGEOMETRIC FUNCTION F4 WITHOUT EXPANDING IN POWERS OF THE PARAMETER ? IN AN ARBITRARY KINEMATIC REGION IN THE MOMENTUM SPACE.

IN THIS PAPER, WE PROCEED TO STUDY PROPERTIES OF MELLIN?BARNES (MB) TRANSFORMS OF USYUKINA?DAVYDYCHEV (UD) FUNCTIONS. IN OUR PREVIOUS PAPERS (ALLENDES ET AL., 2013 [13], KNIEHL ET AL., 2013 [14]), WE SHOWED THAT MULTI-FOLD MELLIN?BARNES (MB) TRANSFORMS OF USYUKINA?DAVYDYCHEV (UD) FUNCTIONS MAY BE REDUCED TO TWO-FOLD MB TRANSFORMS AND THAT THE HIGHER-ORDER UD FUNCTIONS MAY BE OBTAINED IN TERMS OF A DIFFERENTIAL OPERATOR BY APPLYING IT TO A SLIGHTLY MODIFIED FIRST UD FUNCTION. THE RESULT IS VALID IN DIMENSIONS, AND ITS ANALOG IN DIMENSIONS EXITS, TOO (GONZALEZ AND KONDRASHUK, 2013 [6]). IN ALLENDES ET AL. (2013) [13], THE CHAIN OF RECURRENCE RELATIONS FOR ANALYTICALLY REGULARIZED UD FUNCTIONS WAS OBTAINED IMPLICITLY BY COMPARING THE LEFT-HAND SIDE AND THE RIGHT-HAND SIDE OF THE DIAGRAMMATIC RELATIONS BETWEEN THE DIAGRAMS WITH DIFFERENT LOOP ORDERS. IN TURN, THESE DIAGRAMMATIC RELATIONS WERE OBTAINED USING THE METHOD OF LOOP REDUCTION FOR THE TRIANGLE LADDER DIAGRAMS PROPOSED IN 1983 BY BELOKUROV AND USYUKINA. HERE, WE REPRODUCE THESE RECURRENCE RELATIONS BY CALCULATING EXPLICITLY, VIA BARNES LEMMAS, THE CONTOUR INTEGRALS PRODUCED BY THE LEFT-HAND SIDES OF THE DIAGRAMMATIC RELATIONS. IN THIS A WAY, WE EXPLICITLY CALCULATE A FAMILY OF MULTI-FOLD CONTOUR INTEGRALS OF CERTAIN RATIOS OF EULER GAMMA FUNCTIONS. WE MAKE A CONJECTURE THAT SIMILAR RESULTS FOR THE CONTOUR INTEGRALS ARE VALID FOR A WIDER FAMILY OF SM

WE START WITH THE EXPLICIT SOLUTION, IN TERMS OF THE LAMBERT W FUNCTION, OF THE RENORMALIZATION GROUP EQUATION (RGE) FOR THE GAUGE COUPLING IN THE SUPERSYMMETRIC YANG-MILLS THEORY DESCRIBED BY THE WELL-KNOWN BETA FUNCTION OF NOVIKOV ET AL.(NSVZ). WE THEN CONSTRUCT A CLASS OF BETA FUNCTIONS FOR WHICH THE RGE CAN BE SOLVED IN TERMS OF THE LAMBERT W FUNCTION. THESE BETA FUNCTIONS ARE EXPRESSED IN TERMS OF A FUNCTION WHICH IS A TRUNCATED LAURENT SERIES IN THE INVERSE OF THE GAUGE COUPLING. THE PARAMETERS IN THE LAURENT SERIES CAN BE ADJUSTED SO THAT THE FIRST COEFFICIENTS OF THE TAYLOR EXPANSION OF THE BETA FUNCTION IN THE GAUGE COUPLING REPRODUCE THE FOUR-LOOP OR FIVE-LOOP QCD (OR SQCD) BETA FUNCTION.

WE ARGUE IN FAVOR OF THE INDEPENDENCE ON ANY SCALE, ULTRAVIOLET OR INFRARED, IN KERNELS OF THE EFFECTIVE ACTION EXPRESSED IN TERMS OF DRESSED ?=1 SUPERFIELDS FOR THE CASE OF ?=4 SUPER-YANG?MILLS THEORY. UNDER ?SCALE INDEPENDENCE? OF THE EFFECTIVE ACTION OF DRESSED MEAN SUPERFIELDS, WE MEAN ITS ?FINITENESS IN THE OFF-SHELL LIMIT OF REMOVING ALL THE REGULARIZATIONS?. THIS OFF-SHELL LIMIT IS SCALE INDEPENDENT BECAUSE NO SCALE REMAINS INSIDE THESE KERNELS AFTER REMOVING THE REGULARIZATIONS. WE USE TWO TYPES OF REGULARIZATION: REGULARIZATION BY DIMENSIONAL REDUCTION AND REGULARIZATION BY HIGHER DERIVATIVES IN ITS SUPERSYMMETRIC FORM. BASED ON THE SLAVNOV?TAYLOR IDENTITY, WE SHOW THAT DRESSED FIELDS OF MATTER AND OF VECTOR MULTIPLETS CAN BE INTRODUCED TO EXPRESS THE EFFECTIVE ACTION IN TERMS OF THEM. KERNELS OF THE EFFECTIVE ACTION EXPRESSED IN TERMS OF SUCH DRESSED EFFECTIVE FIELDS DO NOT DEPEND ON THE ULTRAVIOLET SCALE. IN THE CASE OF DIMENSIONAL REDUCTION, BY USING THE DEVELOPED TECHNIQUE, WE SHOW HOW THE PROBLEM OF INCONSISTENCY OF THE DIMENSIONAL REDUCTION CAN BE SOLVED. USING PIGUET AND SIBOLD FORMALISM, WE INDICATE THAT THE DEPENDENCE ON THE INFRARED SCALE DISAPPEARS OFF SHELL IN BOTH THE REGULARIZATIONS.

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.

THE METHOD OF BRACKETS IS A METHOD FOR THE EVALUATION OF DEFINITE INTEGRALS BASED ON A SMALL NUMBER OF RULES. THIS IS EMPLOYED HERE FOR THE EVALUATION OF MELLIN?BARNES INTEGRAL. THE FUNDAMENTAL IDEA IS TO TRANSFORM THESE INTEGRAL REPRESENTATIONS INTO A BRACKET SERIES TO OBTAIN THEIR VALUES. THE EXPANSION OF THE GAMMA FUNCTION IN SUCH A SERIES CONSTITUTE THE MAIN PART OF THIS NEW APPLICATION. THE POWER AND FLEXIBILITY OF THIS PROCEDURE IS ILLUSTRATED WITH A VARIETY OF EXAMPLES.

WE OBSERVE A PROPERTY OF ORTHOGONALITY OF THE MELLIN?BARNES TRANSFORMATION OF TRIANGLE ONE-LOOP DIAGRAMS, WHICH FOLLOWS FROM OUR PREVIOUS PAPERS (KONDRASHUK AND KOTIKOV IN JHEP 0808:106, 2008; KONDRASHUK AND VERGARA IN JHEP 1003:051, 2010; ALLENDES ET AL. IN J MATH PHYS 51:052304, 2010). IN THOSE PAPERS IT HAS BEEN ESTABLISHED THAT USYUKINA?DAVYDYCHEV FUNCTIONS ARE INVARIANT WITH RESPECT TO THE FOURIER TRANSFORMATION. THIS HAS BEEN PROVED AT THE LEVEL OF GRAPHS AND ALSO VIA THE MELLIN?BARNES TRANSFORMATION. WE PARTIALLY APPLY TO THE ONE-LOOP MASSLESS SCALAR DIAGRAM THE SAME TRICK IN WHICH THE MELLIN?BARNES TRANSFORMATION WAS INVOLVED AND OBTAIN THE PROPERTY OF ORTHOGONALITY OF THE CORRESPONDING MB TRANSFORMS UNDER INTEGRATION OVER CONTOURS IN TWO COMPLEX PLANES WITH CERTAIN WEIGHT. THIS PROPERTY IS VALID IN AN ARBITRARY NUMBER OF DIMENSIONS.

THE PRODUCT OF THE GLUON DRESSING FUNCTION AND THE SQUARE OF THE GHOST DRESSING FUNCTION IN THE LANDAU GAUGE CAN BE REGARDED TO REPRESENT, APART FROM THE INVERSE POWER CORRECTIONS 1/Q^{2N}, A NONPERTURBATIVE GENERALIZATION A(Q^2) OF THE PERTURBATIVE QCD RUNNING COUPLING A(Q^2)=ALPHA_S(Q^2)/PI. RECENT LARGE VOLUME LATTICE CALCULATIONS FOR THESE DRESSING FUNCTIONS STRONGLY INDICATE THAT SUCH A GENERALIZED COUPLING GOES TO ZERO AS A(Q^2)~Q^2 WHEN THE SQUARED MOMENTA Q^2 GO TO ZERO (Q^2

THIS PAPER IS DEVOTED TO THE CALCULATION BY MELLIN-BARNES TRANSFORM OF A ESPECIAL CLASS OF INTEGRALS. IT CONTAINS DOUBLE INTEGRALS IN THE POSITION SPACE IN D = 4-2E DIMENSIONS, WHERE E IS PARAMETER OF DIMENSIONAL REGULARIZATION. THESE INTEGRALS CONTRIBUTE TO THE EFFECTIVE ACTION OF THE N = 4 SUPERSYMMETRIC YANG-MILLS THEORY. THE INTEGRAND IS A FRACTION IN WHICH THE NUMERATOR IS A LOGARITHM OF RATIO OF SPACETIME INTERVALS, AND THE DENOMINATOR IS THE PRODUCT OF POWERS OF SPACETIME INTERVALS. ACCORDING TO THE METHOD DEVELOPED IN THE PREVIOUS PAPERS, IN ORDER TO MAKE USE OF THE UNIQUENESS TECHNIQUE FOR ONE OF TWO INTEGRATIONS, WE SHIFT EXPONENTS IN POWERS IN THE DENOMINATOR OF INTEGRANDS BY SOME MULTIPLES OF E. AS THE NEXT STEP, THE SECOND INTEGRATION IN THE POSITION SPACE IS DONE BY MELLIN-BARNES TRANSFORM. FOR NORMALIZING PROCEDURE, WE REPRODUCE FIRST THE KNOWN RESULT OBTAINED EARLIER BY GEGENBAUER POLYNOMIAL TECHNIQUE. THEN, WE MAKE ANOTHER SHIFT OF EXPONENTS IN POWERS IN THE DENOMINATOR TO CREATE THE LOGARITHM IN THE NUMERATOR AS THE DERIVATIVE WITH RESPECT TO THE SHIFT PARAMETER DELTA. WE SHOW THAT THE TECHNIQUE OF WORK WITH THE CONTOUR OF THE INTEGRAL MODIFIED IN THIS WAY BY USING MELLIN-BARNES TRANSFORM REPEATS THE TECHNIQUE OF WORK WITH THE CONTOUR OF THE INTEGRAL WITHOUT SUCH A MODIFICATION. IN PARTICULAR, ALL THE OPERATIONS WITH A SHIFT OF CONTOUR OF INTEGRATION OVER COMPLEX VARIABLES OF TWO-FOLD MELLIN-BARNES TRANSFORM ARE THE SAME AS BEFORE THE DELTA MODIFICATION OF INDICES, AND EVEN THE POLES OF RESIDUES COINCIDE. THIS CONFIRMS THE OBSERVATION MADE IN THE PREVIOUS PAPERS THAT IN THE POSITION SPACE ALL THE GREEN FUNCTION OF N = 4 SUPERSYMMETRIC YANG-MILLS THEORY CAN BE EXPRESSED IN TERMS OF UD FUNCTIONS. COMMENT: TALK AT EL CONGRESO DE MATEMATICA CAPRICORNIO, COMCA 2009, ANTOFAGASTA, CHILE AND AT DMFA SEMINAR, UCSC, CONCEPCION, CHILE, 24 PAGES; REVISED VERSION, INTRODUCTION IS MODIFIED, CONCLUSION IS ADDED, FIVE APPENDICES ARE ADDED, APPENDIX E IS NEW

THE S-EXPANSION METHOD IS A GENERALIZATION OF THE INÖNÜ-WIGNER (IW) CONTRACTION THAT ALLOWS TO STUDY NEW NON-TRIVIAL RELATIONS BETWEEN DIFFERENT LIE ALGEBRAS. BASICALLY, THIS METHOD COMBINES A LIE ALGEBRA G WITH A FINITE ABELIAN SEMIGROUP S IN SUCH A WAY THAT A NEW S-EXPANDED ALGEBRA GS CAN BE DEFINED. WHEN THE SEMIGROUP HAS A ZERO-ELEMENT AND/OR A SPECIFIC DECOMPOSITION, WHICH IS SAID TO BE RESONANT WITH THE SUBSPACE STRUCTURE OF THE ORIGINAL ALGEBRA, THEN IT IS POSSIBLE TO EXTRACT SMALLER ALGEBRAS FROM GS WHICH HAVE INTERESTING PROPERTIES. HERE WE GIVE A BRIEF DESCRIPTION OF THE S-EXPANSION, ITS APPLICATIONS AND THE MAIN MOTIVATIONS THAT LEAD US TO ELABORATE A JAVA LIBRARY, WHICH AUTOMATIZES THIS METHOD AND ALLOWS US TO REPRESENT AND TO CLASSIFY ALL POSSIBLE S-EXPANSIONS OF A GIVEN LIE ALGEBRA.