Introductory Section

**Chapter 1**

*Francisco Bulnes*

**1. Introduction**

group theory [2].

homographic transformation.

the complex vector spaces Vec*C:*

finite dimension is computed as1

1 Here ⌊, ⌋, is a floor function.

**3**

Introductory Chapter: Frontiers

and Future Developments of the

The complex analysis development can establish three fundamental ways of modern research with the goal to solve certain conjectures and complete certain theories as the Lie groups representation theory, obtaining of a general cohomology for certain integral operators in the differential equations solution and extend these

The three ways are defined for the relations between the complex analysis, with

Likewise, some objects as modular forms can be the key for the multidisciplinary investigations in complex functions without the condition that *f z*ð Þ, be holomorphic in the upper half-plane. Then the modular forms are meromorphic functions [3]. The before, can be reflected in the satisfying of the certain functional equation Tfð Þ¼ *z ξ*, in which the conditions of the complex function *f z*ð Þ, respect to the transformation T, can be relaxed and the corresponding modular groups can be smaller groups. Such is the case, for example for the functional equation of the

The transcendence goes more beyond of the analyticity problems in complex analysis. The study of the complex Riemannian manifolds results relevant in algebraic geometry and strings theories, where complex Riemann surfaces as complex submanifolds can be useful in the determination of integration invariants. Also in the Fröbenius distribution for the determination of integral submanifolds as cycles, whose values are co-cycles in a spectrum of the category Spec*k*, of the category of

The theory of Riemann surfaces can be applied to the space *G* \ *H* ∗ , to obtain further information about modular forms and functions. Likewise, the modular and cusp forms spaces *Mk*ð Þ *G* , and *Sk*ð Þ *G* , for a group *G*, are of finite dimension, and [4] their dimensions can be computed thanks to the Riemann-Roch theorem in terms of the geometry of the G�action on *H:* For example to the Lie group *SL*ð Þ 2, , this

*k*

8 >>><

>>>:

*k*

12 � �, *k* � 2 mod12 ð Þ

(1)

12 � � þ 1, else

:

dim*CMk*ð Þ¼ SL 2ð Þ *; Z*

the cohomology with coefficients in holomorphic vector bundles, the hypercomplex analysis in a quaternionic algebra and the unitary representations in the Lie

Complex Analysis

solutions to the meromorphic context [1].

#### **Chapter 1**

## Introductory Chapter: Frontiers and Future Developments of the Complex Analysis

*Francisco Bulnes*

#### **1. Introduction**

The complex analysis development can establish three fundamental ways of modern research with the goal to solve certain conjectures and complete certain theories as the Lie groups representation theory, obtaining of a general cohomology for certain integral operators in the differential equations solution and extend these solutions to the meromorphic context [1].

The three ways are defined for the relations between the complex analysis, with the cohomology with coefficients in holomorphic vector bundles, the hypercomplex analysis in a quaternionic algebra and the unitary representations in the Lie group theory [2].

Likewise, some objects as modular forms can be the key for the multidisciplinary investigations in complex functions without the condition that *f z*ð Þ, be holomorphic in the upper half-plane. Then the modular forms are meromorphic functions [3]. The before, can be reflected in the satisfying of the certain functional equation Tfð Þ¼ *z ξ*, in which the conditions of the complex function *f z*ð Þ, respect to the transformation T, can be relaxed and the corresponding modular groups can be smaller groups. Such is the case, for example for the functional equation of the homographic transformation.

The transcendence goes more beyond of the analyticity problems in complex analysis. The study of the complex Riemannian manifolds results relevant in algebraic geometry and strings theories, where complex Riemann surfaces as complex submanifolds can be useful in the determination of integration invariants. Also in the Fröbenius distribution for the determination of integral submanifolds as cycles, whose values are co-cycles in a spectrum of the category Spec*k*, of the category of the complex vector spaces Vec*C:*

The theory of Riemann surfaces can be applied to the space *G* \ *H* ∗ , to obtain further information about modular forms and functions. Likewise, the modular and cusp forms spaces *Mk*ð Þ *G* , and *Sk*ð Þ *G* , for a group *G*, are of finite dimension, and [4] their dimensions can be computed thanks to the Riemann-Roch theorem in terms of the geometry of the G�action on *H:* For example to the Lie group *SL*ð Þ 2, , this finite dimension is computed as1 :

$$\dim\_{\mathbb{C}} \mathcal{M}\_k(\text{SL}(2, \mathbb{Z})) = \begin{cases} \left\lfloor \frac{k}{12} \right\rfloor, & k \equiv 2(\text{mod12})\\ \left\lfloor \frac{k}{12} \right\rfloor + 1, & \text{else} \end{cases} \tag{1}$$

1 Here ⌊, ⌋, is a floor function.

In projective spaces, the functions are homogeneous polynomials. These algebro-geometric objects are the sections of a sheaf (one could also say a line bundle in this case). Precisely here arises the re-interpretation of the differential operators through the homogeneous polynomials of the corresponding lines bundles sections of their respective connections [5] (**Figure 1**).

holomorphic manifold with *G*� orbits (parabolic orbits) of *G=L*, which have an induced Haar measure in each closed submanifold given for the flags of the corresponding holomorphic vector *G*� bundle. Then their orbital integrals are calculated on said orbits, obtaining invariants applicable to whole the homogeneous space in question, *G=L:* This process called of orbitalization of the homogeneous space *G=L*, is based in major part in the inclusion of homogeneous spaces obtained

*Introductory Chapter: Frontiers and Future Developments of the Complex Analysis*

The spaces or cohomology groups of the complex corresponding of sheaves

ð Þ *G=H*, *νn* , result be fine representations of *G=L*, through of the flags bundle corresponding to the complex holomorphic *G*� bundle in *G=L:* The realizations of these unitary representations are the orbital integrals on the flags that are the *K*� orbits of the corresponding vector *G*� bundle. This is precisely used in representations of *G*� (a complex Lie group) constructed on spaces of holomorphic sections of vector bundles and generalizations as the studied for the modular forms. The problem that here arise (in the unitary representations) is the verifying of the Hermitian forms calculated through the diverse integral realizations with the corresponding representations choosing the smooth and minimal globalizing, adequate. For it is used in a general version of the complex cohomology known as

However, this is not reduced to the obtaining of representations of complex groups. Also the obtaining of an analytic function *f z*ð Þ, of a space to other is realized accord to the integral cohomology ð Þ� *n* � 1 � *q ∂*� cohomology *Hn*�1,*q n* ð Þ *=D*,*V* ,

which represents a transformation in cycles (lines) of a flag manifold ¼ ð*L*∣*L*⊂ 4Þ. Likewise complex functions in certain complex manifolds can be obtained through complex or hypercomplex functions defined in complex subspaces (or submanifolds) which could be varieties in an algebraic context, for certain conditions of its cohomology. For example in a complex sheaves cohomology, results interesting for problems of differential equations whose operators are

Other study is the singularities study through varieties considering newly the complex varieties, in a pure algebraic treatment, or using the analyticity as a concept required for contour integrals to define a singularity as a pole of an analytic function. Likewise in concrete, through an analytic functions study, can be extended and generalized the Cauchy integral as through integrals of contour can be generalized functional in a cohomology of contours (cohomological functional). The Morera's and Cauchy-Goursat's theorems can be applied in a sense geometrical more general as cohomological functional of contours whose residue value can be modulo

*l S*1 � � ¼ 2*π:* For example, two singularities or poles. This could represent the surface

z�z1 � 1 *z*�*z*2

less than 2 and thus lie inside one contour. Likewise, the contour integral can be split into two smaller integrals using the Cauchy-Goursat theorem having finally the

This is a good example of traditional cohomological functional element of

, the integral operator that obtains a harmonic function in 4, through

*f w*½ � ð Þþ þ *x* ð Þ *y* þ iz *ζ*,ð Þþ *y* � iz ð Þ *w* � *x ζ*, *ζ dζ*, (2)

, in 2

*:* Likewise, for example, if we consider

, comes given for [8, 9]:

*z*2þ2*z*þ2 *:* The moduli space of these points are

� �dz ¼ 0 � 2*π*i � 2*πi* ¼ �4*πi:*

for reduction of *G=L*, through its holonomy [7].

*DOI: http://dx.doi.org/10.5772/intechopen.93934*

*∂*� cohomology called Serre generalization [7].

where *D*, is a linearly concave domain on 2

calculated functions of *C*∞ 3 � �, on lines *S*1

ð *S*1

of the real part of the function *g z*ð Þ¼ *z*2

contour integral [10] ∮ *C g z*ð Þdz ¼ ∮ *C* 1 � 1

*φ*ð Þ¼ *w*, *x*, *y*, *z*

germs of these sheaves.

H*f Π* � *ℓ*0

**5**

, *Ωr* ð Þ¼ *:*

*Hs*

the space 2

However, there is other conjecture established years ago, which establish the sentence.

**Conjecture 1.1.** All complex function (analytic function) *f z*ð Þ, can be determined for hypercomplex functions [6].

This could be obtained in a clear way, without pass for the holomorphicity.

This represents all a joint research program between the hypercomplex analysis and the integral operators cohomology, and for it is established determine an integral operators theory that establish equivalences between cohomology classes on different complex base spaces and the cycles and co-cycles of these under the corresponding integral operators.

Likewise, can be obtained integral representations that are realizations of certain unitary representations of Lie groups such as *U n*ð Þ, SU 2ð Þ, SUð Þ *n* , SU 2, 2 ð Þ, SUð Þ *p*, *q* , and *U p*ð Þ , *q* , for the obtaining of the general solution of a partial differential equations in contexts of the quaternion algebra. And not only that, also are contemplated the equivalences of the corresponding cohomology classes obtained through the restricted objects to a space of certain dimension, for example, the hypercomplex analysis in 4 ffi 2 , and in 4, is a quaternion analysis. Other example we can find in the integrals of the integral geometry of *G*2,4ð Þ , through the space 3 ð Þ , which establishes a twistor geometry [2, 7].

However, the isomorphism relations in homogeneous vector bundles are equivalent to the followed in cohomology developed in vector tomography. Likewise, the intertwining integral operators between cohomological classes of both contexts (respective cycles) result be cohomological classes in contexts of lines bundles (or equivalent co-cycles). In this sense a conjecture that can be proved is: "the Penrose transform is a vector Radon transform on homogeneous vector bundles sections in a Riemannian manifold" And considering the re-construction of a Riemannian manifold through cycles, we can consider the following conjecture also.

**Conjecture 1.2.** [2] The twistor transform is the generalization of the Radon transform on lines bundles and the Penrose transform is a specialization of the twistor transform in *S*4*:*

Of fact, the twistor transform can be determined for the Penrose transforms pair on *G*�orbits of projective spaces �, and þ*:* Then here arises the mentioned before in modular forms, the use of the lines bundles restricted to the polynomial rings corresponding to the subjacent operators on the complex vector spaces. In a more general sense, the vector bundles which refer us, are the seated in the homogeneous spaces *G=L*, with homogeneous subspaces *G=Q*, where is subjacent a

#### **Figure 1.**

*The* j�*invariant on the complex interval* ½ �þ �2, 2 ½ � 0, 1 *i: It supports working with modular transformations and evaluating theta sums. With this construction of blocks, one can easily implement other modular forms such as Eisenstein series. (*http://fredrikj.net/blog/2014/10/modular-forms-in-arb/*).*

*Introductory Chapter: Frontiers and Future Developments of the Complex Analysis DOI: http://dx.doi.org/10.5772/intechopen.93934*

holomorphic manifold with *G*� orbits (parabolic orbits) of *G=L*, which have an induced Haar measure in each closed submanifold given for the flags of the corresponding holomorphic vector *G*� bundle. Then their orbital integrals are calculated on said orbits, obtaining invariants applicable to whole the homogeneous space in question, *G=L:* This process called of orbitalization of the homogeneous space *G=L*, is based in major part in the inclusion of homogeneous spaces obtained for reduction of *G=L*, through its holonomy [7].

The spaces or cohomology groups of the complex corresponding of sheaves *Hs* ð Þ *G=H*, *νn* , result be fine representations of *G=L*, through of the flags bundle corresponding to the complex holomorphic *G*� bundle in *G=L:* The realizations of these unitary representations are the orbital integrals on the flags that are the *K*� orbits of the corresponding vector *G*� bundle. This is precisely used in representations of *G*� (a complex Lie group) constructed on spaces of holomorphic sections of vector bundles and generalizations as the studied for the modular forms. The problem that here arise (in the unitary representations) is the verifying of the Hermitian forms calculated through the diverse integral realizations with the corresponding representations choosing the smooth and minimal globalizing, adequate. For it is used in a general version of the complex cohomology known as *∂*� cohomology called Serre generalization [7].

However, this is not reduced to the obtaining of representations of complex groups. Also the obtaining of an analytic function *f z*ð Þ, of a space to other is realized accord to the integral cohomology ð Þ� *n* � 1 � *q ∂*� cohomology *Hn*�1,*q n* ð Þ *=D*,*V* , where *D*, is a linearly concave domain on 2 *:* Likewise, for example, if we consider the space 2 , the integral operator that obtains a harmonic function in 4, through calculated functions of *C*∞ 3 � �, on lines *S*1 , in 2 , comes given for [8, 9]:

$$\rho(w,\mathbf{x},\mathbf{y},\mathbf{z}) = \int\_{\mathcal{S}^d} f[(w+\mathbf{x}) + (\mathbf{y} + \mathbf{i}\mathbf{z})\zeta, (\mathbf{y} - \mathbf{i}\mathbf{z}) + (w - \mathbf{x})\zeta, \zeta] d\zeta,\tag{2}$$

which represents a transformation in cycles (lines) of a flag manifold ¼ ð*L*∣*L*⊂ 4Þ. Likewise complex functions in certain complex manifolds can be obtained through complex or hypercomplex functions defined in complex subspaces (or submanifolds) which could be varieties in an algebraic context, for certain conditions of its cohomology. For example in a complex sheaves cohomology, results interesting for problems of differential equations whose operators are germs of these sheaves.

Other study is the singularities study through varieties considering newly the complex varieties, in a pure algebraic treatment, or using the analyticity as a concept required for contour integrals to define a singularity as a pole of an analytic function.

Likewise in concrete, through an analytic functions study, can be extended and generalized the Cauchy integral as through integrals of contour can be generalized functional in a cohomology of contours (cohomological functional). The Morera's and Cauchy-Goursat's theorems can be applied in a sense geometrical more general as cohomological functional of contours whose residue value can be modulo *l S*1 � � ¼ 2*π:* For example, two singularities or poles. This could represent the surface of the real part of the function *g z*ð Þ¼ *z*2 *z*2þ2*z*þ2 *:* The moduli space of these points are less than 2 and thus lie inside one contour. Likewise, the contour integral can be split into two smaller integrals using the Cauchy-Goursat theorem having finally the contour integral [10] ∮ *C g z*ð Þdz ¼ ∮ *C* 1 � 1 z�z1 � 1 *z*�*z*2 � �dz ¼ 0 � 2*π*i � 2*πi* ¼ �4*πi:* This is a good example of traditional cohomological functional element of H*f Π* � *ℓ*0 , *Ωr* ð Þ¼ *:*

In projective spaces, the functions are homogeneous polynomials. These algebro-geometric objects are the sections of a sheaf (one could also say a line bundle in this case). Precisely here arises the re-interpretation of the differential operators through the homogeneous polynomials of the corresponding lines bundles

However, there is other conjecture established years ago, which establish the

This could be obtained in a clear way, without pass for the holomorphicity. This represents all a joint research program between the hypercomplex analysis and the integral operators cohomology, and for it is established determine an integral operators theory that establish equivalences between cohomology classes on different complex base spaces and the cycles and co-cycles of these under the

Likewise, can be obtained integral representations that are realizations of certain unitary representations of Lie groups such as *U n*ð Þ, SU 2ð Þ, SUð Þ *n* , SU 2, 2 ð Þ, SUð Þ *p*, *q* , and *U p*ð Þ , *q* , for the obtaining of the general solution of a partial differential equations in contexts of the quaternion algebra. And not only that, also are contemplated the equivalences of the corresponding cohomology classes obtained through the restricted objects to a space of certain dimension, for example, the hypercomplex

However, the isomorphism relations in homogeneous vector bundles are equivalent to the followed in cohomology developed in vector tomography. Likewise, the intertwining integral operators between cohomological classes of both contexts (respective cycles) result be cohomological classes in contexts of lines bundles (or equivalent co-cycles). In this sense a conjecture that can be proved is: "the Penrose transform is a vector Radon transform on homogeneous vector bundles sections in a Riemannian manifold" And considering the re-construction of a Riemannian man-

**Conjecture 1.2.** [2] The twistor transform is the generalization of the Radon transform on lines bundles and the Penrose transform is a specialization of the

*The* j�*invariant on the complex interval* ½ �þ �2, 2 ½ � 0, 1 *i: It supports working with modular transformations and evaluating theta sums. With this construction of blocks, one can easily implement other modular forms such*

*as Eisenstein series. (*http://fredrikj.net/blog/2014/10/modular-forms-in-arb/*).*

Of fact, the twistor transform can be determined for the Penrose transforms pair on *G*�orbits of projective spaces �, and þ*:* Then here arises the mentioned before in modular forms, the use of the lines bundles restricted to the polynomial rings corresponding to the subjacent operators on the complex vector spaces. In a more general sense, the vector bundles which refer us, are the seated in the homogeneous

, and in 4, is a quaternion analysis. Other example we can find

ð Þ , which

**Conjecture 1.1.** All complex function (analytic function) *f z*ð Þ, can be

in the integrals of the integral geometry of *G*2,4ð Þ , through the space 3

ifold through cycles, we can consider the following conjecture also.

spaces *G=L*, with homogeneous subspaces *G=Q*, where is subjacent a

sections of their respective connections [5] (**Figure 1**).

determined for hypercomplex functions [6].

*Advances in Complex Analysis and Applications*

corresponding integral operators.

establishes a twistor geometry [2, 7].

analysis in 4 ffi 2

twistor transform in *S*4*:*

**Figure 1.**

**4**

sentence.

As a way to extend the meromorphic functions, the Nevanlinna theory searches generalizations of extensions of analytic functions to algebroid functions, holomorphic curves [10], holomorphic maps between complex manifolds of arbitrary dimension, quasi-regular mappings and minimal surfaces [11]. This is being applicable in moduli problems and Shimura varieties.

**References**

Netherlands.

1994.

**7**

[1] F. Bulnes, Integral Geometry Methods in the Geometrical Langlands

*DOI: http://dx.doi.org/10.5772/intechopen.93934*

*Introductory Chapter: Frontiers and Future Developments of the Complex Analysis*

Special Issue: Integral Geometry Methods on Derived Categories in the Geometrical Langlands Program. Vol. 3, No. 6-2, 2014, pp. 30-37. doi: 10.11648/j.

[10] H. Weyl (1943). Meromorphic functions and analytic curves. Princeton University Press. New Jersey, USA.

[11] Drasin, D. (1976). "The inverse problem of the Nevanlinna theory". Acta Math. 138 (1): 83–151. doi: 10.1007/

[12] Shimura, Goro (1971), Introduction to the arithmetic theory of automorphic

[13] H. Iwaniec, Spectral Methods of Automorphic Forms, Second Edition, (2002) (Volume 53 in Graduate Studies

Mathematical Society, Providence, RI,

functions, Publications of the Mathematical Society of Japan, 11, Tokyo: Iwanami Shoten, Theorem 2.33,

in Mathematics), American

pamj.s.2014030602.16

BF02392314

Proposition 2.26.

USA.

[2] Bulnes, F. and Shapiro, M. (2007) Sobre una Teoría General de Operadores Integrales para Análisis y Geometría, Appliedmath 3 (Pure Mathematics Section) IPN, IM-UNAM, CINVESTAV, COFFA, ESIME-IMPN, ESFM-IPN Ciudad de México, Mexico.

[3] Kichoon Y. (1999) Meromorphic Functions and Projective Curves, Kluwer Academic Publishers,

[4] Shimura, Goro, An Introduction to the Arithmetic Theory of Automorphic Functions, Princeton University Press,

[5] Available from: http://fredrikj.net/ blog/2014/10/modular-forms-in-arb/

[6] Sorin D. Gal (2004) Introduction to the Geometric Function theory of Hypercomplex variables, Nova Science

[7] Francisco Bulnes, Integral Geometry and Cohomology in Field Theory on the Space-Time as Complex Riemannian Manifold, Book Chapter of Advances in Complex Analysis, Intech, July 2020. doi: 10.5772/intechopen.92969

[8] Francisco Bulnes. Mathematical Electrodynamics: Groups, Cohomology Classes, Unitary Representations, Orbits and Integral Transforms in Electro-Physics, American Journal of Electromagnetics and Applications. Vol.3, No. 6, 2015, pp. 43-52. doi: 10.11648/j.ajea.20150306.12

[9] Francisco Bulnes, Ronin Goborov, Integral Geometry and Complex Space-Time Cohomology in Field Theory, Pure and Applied Mathematics Journal.

Publishers. New York, USA.

Program, Scientific Research Publishing, Inc. USA, 8 jun. 2016.

#### **2. Future research**

Finally, the future researches in complex analysis will be centered on major research of meromorphicity, in the automorphicity [12, 13], bi-holomorphicity, Riemann-Zeta function, Hodge theory and the complex ideals for improve methods of homogeneous polynomials in the complex vector bundles studied in complex Riemaniann manifolds and their submanifolds, involving singularities as poles or insolated singularities of rational curves. The automorphic forms generalize the Cauchy and Morera's theorems in an arithmetic sense inside the analytic functions theory and their probable holomorphicity. The automorphic forms are the other basic operations that arise inside the mathematics in the modular forms.

#### **Author details**

Francisco Bulnes IINAMEI, Research Department in Mathematics and Engineering, TESCHA, Mexico

\*Address all correspondence to<EMAIL_ADDRESS>
© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*Introductory Chapter: Frontiers and Future Developments of the Complex Analysis DOI: http://dx.doi.org/10.5772/intechopen.93934*

### **References**

As a way to extend the meromorphic functions, the Nevanlinna theory searches

holomorphic curves [10], holomorphic maps between complex manifolds of arbitrary dimension, quasi-regular mappings and minimal surfaces [11]. This is being

Finally, the future researches in complex analysis will be centered on major research of meromorphicity, in the automorphicity [12, 13], bi-holomorphicity, Riemann-Zeta function, Hodge theory and the complex ideals for improve methods of homogeneous polynomials in the complex vector bundles studied in complex Riemaniann manifolds and their submanifolds, involving singularities as poles or insolated singularities of rational curves. The automorphic forms generalize the Cauchy and Morera's theorems in an arithmetic sense inside the analytic functions theory and their probable holomorphicity. The automorphic forms are the other

basic operations that arise inside the mathematics in the modular forms.

IINAMEI, Research Department in Mathematics and Engineering, TESCHA,

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

\*Address all correspondence to<EMAIL_ADDRESS>
provided the original work is properly cited.

generalizations of extensions of analytic functions to algebroid functions,

applicable in moduli problems and Shimura varieties.

*Advances in Complex Analysis and Applications*

**2. Future research**

**Author details**

Francisco Bulnes

Mexico

**6**

[1] F. Bulnes, Integral Geometry Methods in the Geometrical Langlands Program, Scientific Research Publishing, Inc. USA, 8 jun. 2016.

[2] Bulnes, F. and Shapiro, M. (2007) Sobre una Teoría General de Operadores Integrales para Análisis y Geometría, Appliedmath 3 (Pure Mathematics Section) IPN, IM-UNAM, CINVESTAV, COFFA, ESIME-IMPN, ESFM-IPN Ciudad de México, Mexico.

[3] Kichoon Y. (1999) Meromorphic Functions and Projective Curves, Kluwer Academic Publishers, Netherlands.

[4] Shimura, Goro, An Introduction to the Arithmetic Theory of Automorphic Functions, Princeton University Press, 1994.

[5] Available from: http://fredrikj.net/ blog/2014/10/modular-forms-in-arb/

[6] Sorin D. Gal (2004) Introduction to the Geometric Function theory of Hypercomplex variables, Nova Science Publishers. New York, USA.

[7] Francisco Bulnes, Integral Geometry and Cohomology in Field Theory on the Space-Time as Complex Riemannian Manifold, Book Chapter of Advances in Complex Analysis, Intech, July 2020. doi: 10.5772/intechopen.92969

[8] Francisco Bulnes. Mathematical Electrodynamics: Groups, Cohomology Classes, Unitary Representations, Orbits and Integral Transforms in Electro-Physics, American Journal of Electromagnetics and Applications. Vol.3, No. 6, 2015, pp. 43-52. doi: 10.11648/j.ajea.20150306.12

[9] Francisco Bulnes, Ronin Goborov, Integral Geometry and Complex Space-Time Cohomology in Field Theory, Pure and Applied Mathematics Journal.

Special Issue: Integral Geometry Methods on Derived Categories in the Geometrical Langlands Program. Vol. 3, No. 6-2, 2014, pp. 30-37. doi: 10.11648/j. pamj.s.2014030602.16

[10] H. Weyl (1943). Meromorphic functions and analytic curves. Princeton University Press. New Jersey, USA.

[11] Drasin, D. (1976). "The inverse problem of the Nevanlinna theory". Acta Math. 138 (1): 83–151. doi: 10.1007/ BF02392314

[12] Shimura, Goro (1971), Introduction to the arithmetic theory of automorphic functions, Publications of the Mathematical Society of Japan, 11, Tokyo: Iwanami Shoten, Theorem 2.33, Proposition 2.26.

[13] H. Iwaniec, Spectral Methods of Automorphic Forms, Second Edition, (2002) (Volume 53 in Graduate Studies in Mathematics), American Mathematical Society, Providence, RI, USA.

Section 2

Complex Functions and

Some of Their Functional

Transforms

**9**

### Section 2


*2.4. Enzymatic Hydrolysis*

Solids resulting from consecutive stages of autohydrolysis (AS) and delignification under selected conditions were subjected to hydrolysis using the enzymatic complex Cellic CTec2 (Novozymes, Denmark). The enzymatic activity of Cellic CTec2 was 137 FPU (Filter Paper Units)/g of enzyme [30]. Hydrolysis assays were performed in Erlenmeyer flasks kept at 50 ◦C in an orbital incubator (150 rpm). The pH of media was adjusted to 4.8 by adding 50 mM citrate buffer. Experiments were performed for 0–96 h at LSR = 15 g/g using an enzyme to solid ratio (ESR) of 15 FPU/g substrate. All assays

were performed by triplicate. Aliquots of samples were withdrawn at selected times, centrifuged, filtered, diluted and assayed for glucose by HPLC as described in Section 2.5. Cellulose conversion into glucose, and xylan conversion into xylose were calculated as follows:

$$\% \text{cellulos conversion into glucose} = 100 \times \frac{[Glu]}{[Glu\_{Pot}]} \tag{5}$$

where [*Glu*] is the glucose concentration and [*GluPot*] is the potential glucose concentration (calculated assuming total conversion of the cellulose contained in the substrate);

$$\% \text{ xylan conversion to xylose } = 100 \times \frac{[\text{Xyl}]}{[\text{Xyl}\_{Pot}]} \tag{6}$$

where [*Xyl*] is the xylose concentration and [*XylPot*] is the potential xylose concentration (calculated assuming total conversion of the xylan contained in the substrate).


*Review* **Permeable Pavements Life Cycle Assessment: A Literature Review**

### **Lucas Niehuns Antunes \*, Enedir Ghisi and Liseane Padilha Thives**

Department of Civil Engineering, Federal University of Santa Catarina, Laboratory of Energy Efficiency in Buildings, Florianópolis-SC 88040-900, Brazil<EMAIL_ADDRESS>(E.G<EMAIL_ADDRESS>(L.P.T.) **\***Correspondence:lucas\_niehuns@hotmail.com(L.N.A.);Tel.:+55-48-3721-2115;Fax:+55-48-3721-5191

Received: 28 September 2018; Accepted: 1 November 2018; Published: 3 November 2018

**Abstract:** The number of studies involving life cycle assessment has increased significantly in recent years. The life cycle assessment has been applied to assess the environmental performance of water infrastructures, including the environmental impacts associated with construction, maintenance and disposal, mainly evaluating the amount of greenhouse gas emissions, as well as the consumption of energy and natural resources. The objective of this paper is to present an overview of permeable pavements and show studies of life cycle assessment that compare the environmental performance of permeable pavements with traditional drainage systems. Although the studies found in the literature present an estimate of the sustainability of permeable pavements, the great heterogeneity in the evaluation methods and results is still notable. Therefore, it is necessary to homogenize the phases of goal and scope, inventory analysis, impact assessment and interpretation. It is also necessary to define the phases and processes of the evaluation, as well as the minimum amount of data to be considered in the modelling of life cycle assessment, in order to avoid heterogeneity in the functional units and other components. Thus, more consistent results will lead to a real evaluation of the environmental impacts caused by permeable pavements. Life cycle assessment studies are essential to guide planning and decision-making, leading to systems that consider increasing water resources and reducing natural disasters and environmental impacts.

**Keywords:** permeable pavements; life cycle assessment; stormwater management; sustainability