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Recupero Vincenzo - One of the best experts on this subject based on the ideXlab platform.

  • Slice regular semigroups
    'American Mathematical Society (AMS)', 2018
    Co-Authors: Ghiloni Riccardo, Recupero Vincenzo
    Abstract:

    In this paper we introduce the notion of slice regular right linear semigroup in a quaternionic Banach space. It is an operatorial function which is slice regular (a noncommutative counterpart of analyticity) and which satisfies a noncommutative semigroup law characterizing the exponential function in an infinite dimensional noncommutative setting. We prove that a right linear operator semigroup in a quaternionic Banach space is slice regular if and only if its generator is spherical sectorial. This result provides a connection between the slice regularity and the noncommutative semigroups theory, and characterizes those semigroups which can be represented by a noncommutative Cauchy Integral Formula. All our results are generalized to Banach two-sided modules having as a set of scalar any real associative *-algebra, Clifford algebras R_n included

  • Noncommutative Cauchy Integral Formula
    'Springer Science and Business Media LLC', 2017
    Co-Authors: Riccardo Ghiloni, Alessandro Perotti, Recupero Vincenzo
    Abstract:

    The aim of this paper is to provide and prove the most general Cauchy Integral Formula for slice regular functions and for continuously differentiable functions on a real alternative *-algebra. Slice regular functions represent a generalization of the classical concept of holomorphic function of a complex variable in the noncommutative and nonassociative settings. As an application, we obtain two kinds of local series expansion for slice regular functions

  • Semigroups over real alternative *-algebras: generation theorems and spherical sectorial operators
    American Mathematical Society ( AMS), 2016
    Co-Authors: Ghiloni Riccardo, Recupero Vincenzo
    Abstract:

    The aim of this paper is twofold. On one hand, generalizing some recent results obtained in the quaternionic setting, but using simpler techniques, we prove the generation theorems for semigroups in Banach spaces whose set of scalars belongs to the class of real alternative *-algebras, which includes, besides real and complex numbers, quaternions, octonions and Clifford algebras. On the other hand, in this new general framework, we introduce the notion of spherical sectorial operator and we prove that a spherical sectorial operator generates a semigroup that can be represented by a Cauchy Integral Formula. It follows that such a semigroup is analytic in tim

  • Noncommutative Cauchy Integral Formula
    2016
    Co-Authors: Ghiloni Riccardo, Perotti Alessandro, Recupero Vincenzo
    Abstract:

    The aim of this paper is to provide and prove the most general Cauchy Integral Formula for slice regular functions and for C^1 functions on a real alternative *-algebra. Slice regular functions represent a generalization of the classical concept of holomorphic function of a complex variable in the noncommutative and nonassociative settings. As an application, we obtain two kinds of local series expansion for slice regular functions.Comment: 13 pages, an example has been added at the end of Section 4. To appear in "Complex Analysis and Operator Theory

  • Slice regular semigroups
    2016
    Co-Authors: Ghiloni Riccardo, Recupero Vincenzo
    Abstract:

    In this paper we introduce the notion of slice regular right linear semigroup in a quaternionic Banach space. It is an operatorial function which is slice regular (a noncommutative counterpart of analyticity) and which satisfies a noncommutative semigroup law characterizing the exponential function in an infinite dimensional noncommutative setting. We prove that a right linear operator semigroup in a quaternionic Banach space is slice regular if and only if its generator is spherical sectorial. This result provides a connection between the slice regularity and the noncommutative semigroups theory, and characterizes those semigroups which can be represented by a noncommutative Cauchy Integral Formula. All our results are generalized to Banach two-sided modules having as a set of scalar any real associative *-algebra, Clifford R_n algebras included.Comment: A misprint in the second displayed Formula of Definition 6.11 (p. 28) has been correcte

Vladimír Souček - One of the best experts on this subject based on the ideXlab platform.

  • the Cauchy Integral Formula in hermitian quaternionic and mathfrak osp 4 2 osp 4 2 clifford analysis
    Computational Methods and Function Theory, 2020
    Co-Authors: F Brackx, H De Schepper, Roman Lávička, Vladimír Souček
    Abstract:

    As is the case for the theory of holomorphic functions in the complex plane, the Cauchy Integral Formula has proven to be a cornerstone of Clifford analysis, the monogenic function theory in higher dimensional euclidean space. In recent years, several new branches of Clifford analysis have emerged. Similarly as to how hermitian Clifford analysis in euclidean space $${\mathbb {R}}^{2n}$$ of even dimension emerged as a refinement of euclidean Clifford analysis by introducing a complex structure on $${\mathbb {R}}^{2n}$$, quaternionic Clifford analysis arose as a further refinement by introducing a so-called hypercomplex structure $${\mathbb {Q}}$$, i.e. three complex structures ($${\mathbb {I}}$$, $${\mathbb {J}}$$, $${\mathbb {K}}$$) which follow the quaternionic multiplication rules, on $${\mathbb {R}}^{4p}$$, the dimension now being a fourfold. Two, respectively four, differential operators lead to first order systems invariant under the action of the respective symmetry groups U(n) and Sp(p). Their simultaneous null solutions are called hermitian monogenic and quaternionic monogenic functions respectively. In this contribution we further elaborate on the Cauchy Integral Formula for hermitian and quaternionic monogenic functions. Moreover we establish Caychy Integral Formulae for $$\mathfrak {osp}(4|2)$$-monogenic functions, the newest branch of Clifford analysis refining quaternionic monogenicity by taking the underlying symplectic symmetry fully into account.

  • the Cauchy Integral Formula in hermitian quaternionic and osp 4 2 clifford analysis
    arXiv: Complex Variables, 2019
    Co-Authors: Fred Brackx, Hennie De Schepper, Roman Lávička, Vladimír Souček
    Abstract:

    As is the case for the theory of holomorphic functions in the complex plane, the Cauchy Integral Formula has proven to be a corner stone of Clifford analysis, the monogenic function theory in higher dimensional euclidean space. In recent years, several new branches of Clifford analysis have emerged. Similarly as hermitian Clifford analysis in euclidean space R^{2n} of even dimension emerged as a refinement of euclidean Clifford analysis by introducing a complex structure on R^{2n}, quaternionic Clifford analysis arose as a further refinement by introducing a so--called hypercomplex structure Q, i.e.\ three complex structures (I, J, K) which submit to the quaternionic multiplication rules, on R^{4p}, the dimension now being a fourfold. Two, respectively four, differential operators lead to first order systems invariant under the action of the respective symmetry groups U(n) and Sp(p). Their simultaneous null solutions are called hermitian monogenic and quaternionic monogenic functions respectively. In this contribution we further elaborate on the Caychy Integral Formula for hermitian and quaternionic monogenic functions. Moreover we establish Caychy Integral Formulae for osp(4|2)--monogenic functions, the newest branch of Clifford analysis refining quaternionic monogenicity by taking the underlying symplectic symmetry fully into account.

  • Cauchy’s Formula in Clifford Analysis: An Overview
    Trends in Mathematics, 2019
    Co-Authors: Fred Brackx, Hennie De Schepper, Roman Lávička, Vladimír Souček
    Abstract:

    The Clifford-Cauchy Integral Formula has proven to be a corner stone of the monogenic function theory, as is the case for the traditional Cauchy Formula in the theory of holomorphic functions in the complex plane. In the recent years, several new branches of Clifford analysis have emerged. Similarly as hermitian Clifford analysis was introduced in Euclidean space \(\mathbb {R}^{2n}\) of even dimension as a refinement of Euclidean Clifford analysis by the introduction of a complex structure on \(\mathbb {R}^{2n}\), quaternionic Clifford analysis arose as a further refinement by the introduction of a so-called hypercomplex structure \(\mathbb {Q}\), i.e. three complex structures (\(\mathbb {I}\), \(\mathbb {J}\), \(\mathbb {K}\)) which submit to the quaternionic multiplication rules, on Euclidean space \(\mathbb {R}^{4p}\), the dimension now being a fourfold. Two, respectively four differential operators are constructed, leading to invariant systems under the action of the respective symmetry groups U(n) and Sp(p). Their simultaneous null solutions are respectively called hermitian and quaternionic monogenic functions. The basics of hermitian monogenicity have been studied in e.g. Brackx et al. (Compl Anal Oper Theory 1(3):341–365, 2007; Complex Var Elliptic Equ 52(10–11):1063–1079, 2007; Appl Clifford Algebras 18(3–4):451–487, 2008). Quaternionic monogenicity has been developed in, amongst others, Pena-Pena (Complex Anal Oper Theory 1:97–113, 2007), Eelbode (Complex Var Elliptic Equ 53(10):975–987, 2008), Damiano et al. (Adv Geom 11:169–189, 2011), and Brackx et al. (Adv Appl Clifford Alg 24(4):955–980, 2014; Ann Glob Anal Geom 46:409–430, 2014). In this contribution, we give an overview of the ways in which a Cauchy Integral representation Formula has been established within each of these frameworks.

  • Cauchy s Formula in clifford analysis an overview
    Topics in Clifford analysis : special volume in honor of Wolfgang Sprößig, 2019
    Co-Authors: Fred Brackx, Hennie De Schepper, Roman Lávička, Vladimír Souček
    Abstract:

    The Clifford-Cauchy Integral Formula has proven to be a corner stone of the monogenic function theory, as is the case for the traditional Cauchy Formula in the theory of holomorphic functions in the complex plane. In the recent years, several new branches of Clifford analysis have emerged. Similarly as hermitian Clifford analysis was introduced in Euclidean space \(\mathbb {R}^{2n}\) of even dimension as a refinement of Euclidean Clifford analysis by the introduction of a complex structure on \(\mathbb {R}^{2n}\), quaternionic Clifford analysis arose as a further refinement by the introduction of a so-called hypercomplex structure \(\mathbb {Q}\), i.e. three complex structures (\(\mathbb {I}\), \(\mathbb {J}\), \(\mathbb {K}\)) which submit to the quaternionic multiplication rules, on Euclidean space \(\mathbb {R}^{4p}\), the dimension now being a fourfold. Two, respectively four differential operators are constructed, leading to invariant systems under the action of the respective symmetry groups U(n) and Sp(p). Their simultaneous null solutions are respectively called hermitian and quaternionic monogenic functions. The basics of hermitian monogenicity have been studied in e.g. Brackx et al. (Compl Anal Oper Theory 1(3):341–365, 2007; Complex Var Elliptic Equ 52(10–11):1063–1079, 2007; Appl Clifford Algebras 18(3–4):451–487, 2008). Quaternionic monogenicity has been developed in, amongst others, Pena-Pena (Complex Anal Oper Theory 1:97–113, 2007), Eelbode (Complex Var Elliptic Equ 53(10):975–987, 2008), Damiano et al. (Adv Geom 11:169–189, 2011), and Brackx et al. (Adv Appl Clifford Alg 24(4):955–980, 2014; Ann Glob Anal Geom 46:409–430, 2014). In this contribution, we give an overview of the ways in which a Cauchy Integral representation Formula has been established within each of these frameworks.

Ghiloni Riccardo - One of the best experts on this subject based on the ideXlab platform.

  • Slice regular semigroups
    'American Mathematical Society (AMS)', 2018
    Co-Authors: Ghiloni Riccardo, Recupero Vincenzo
    Abstract:

    In this paper we introduce the notion of slice regular right linear semigroup in a quaternionic Banach space. It is an operatorial function which is slice regular (a noncommutative counterpart of analyticity) and which satisfies a noncommutative semigroup law characterizing the exponential function in an infinite dimensional noncommutative setting. We prove that a right linear operator semigroup in a quaternionic Banach space is slice regular if and only if its generator is spherical sectorial. This result provides a connection between the slice regularity and the noncommutative semigroups theory, and characterizes those semigroups which can be represented by a noncommutative Cauchy Integral Formula. All our results are generalized to Banach two-sided modules having as a set of scalar any real associative *-algebra, Clifford algebras R_n included

  • Semigroups over real alternative *-algebras: generation theorems and spherical sectorial operators
    American Mathematical Society ( AMS), 2016
    Co-Authors: Ghiloni Riccardo, Recupero Vincenzo
    Abstract:

    The aim of this paper is twofold. On one hand, generalizing some recent results obtained in the quaternionic setting, but using simpler techniques, we prove the generation theorems for semigroups in Banach spaces whose set of scalars belongs to the class of real alternative *-algebras, which includes, besides real and complex numbers, quaternions, octonions and Clifford algebras. On the other hand, in this new general framework, we introduce the notion of spherical sectorial operator and we prove that a spherical sectorial operator generates a semigroup that can be represented by a Cauchy Integral Formula. It follows that such a semigroup is analytic in tim

  • Noncommutative Cauchy Integral Formula
    2016
    Co-Authors: Ghiloni Riccardo, Perotti Alessandro, Recupero Vincenzo
    Abstract:

    The aim of this paper is to provide and prove the most general Cauchy Integral Formula for slice regular functions and for C^1 functions on a real alternative *-algebra. Slice regular functions represent a generalization of the classical concept of holomorphic function of a complex variable in the noncommutative and nonassociative settings. As an application, we obtain two kinds of local series expansion for slice regular functions.Comment: 13 pages, an example has been added at the end of Section 4. To appear in "Complex Analysis and Operator Theory

  • Slice regular semigroups
    2016
    Co-Authors: Ghiloni Riccardo, Recupero Vincenzo
    Abstract:

    In this paper we introduce the notion of slice regular right linear semigroup in a quaternionic Banach space. It is an operatorial function which is slice regular (a noncommutative counterpart of analyticity) and which satisfies a noncommutative semigroup law characterizing the exponential function in an infinite dimensional noncommutative setting. We prove that a right linear operator semigroup in a quaternionic Banach space is slice regular if and only if its generator is spherical sectorial. This result provides a connection between the slice regularity and the noncommutative semigroups theory, and characterizes those semigroups which can be represented by a noncommutative Cauchy Integral Formula. All our results are generalized to Banach two-sided modules having as a set of scalar any real associative *-algebra, Clifford R_n algebras included.Comment: A misprint in the second displayed Formula of Definition 6.11 (p. 28) has been correcte

Vincenzo Recupero - One of the best experts on this subject based on the ideXlab platform.

  • Noncommutative Cauchy Integral Formula
    Complex Analysis and Operator Theory, 2017
    Co-Authors: Riccardo Ghiloni, Alessandro Perotti, Vincenzo Recupero
    Abstract:

    The aim of this paper is to provide and prove the most general Cauchy Integral Formula for slice regular functions and for $$C^1$$ C 1 functions on a real alternative *-algebra. Slice regular functions represent a generalization of the classical concept of holomorphic function of a complex variable in the noncommutative and nonassociative settings. As an application, we obtain two kinds of local series expansion for slice regular functions.

  • Semigroups over real alternative *-algebras: Generation theorems and spherical sectorial operators
    Transactions of the American Mathematical Society, 2015
    Co-Authors: Riccardo Ghiloni, Vincenzo Recupero
    Abstract:

    The aim of this paper is twofold. On one hand, generalizing some recent results obtained in the quaternionic setting, but using simpler techniques, we prove the generation theorems for semigroups in Banach spaces whose set of scalars belongs to the class of real alternative *-algebras, which includes, besides real and complex numbers, quaternions, octonions and Clifford algebras. On the other hand, in this new general framework, we introduce the notion of spherical sectorial operator and we prove that a spherical sectorial operator generates a semigroup that can be represented by a Cauchy Integral Formula. It follows that such a semigroup is analytic in time

Parra-ferrada Ivan - One of the best experts on this subject based on the ideXlab platform.

  • Analytical solution to DGLAP integro-differential equation in a simple toy-model with a fixed gauge coupling
    2016
    Co-Authors: Alvarez Gustavo, Cvetic Gorazd, Kniehl, Bernd A., Kondrashuk Igor, Parra-ferrada Ivan
    Abstract:

    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 ${\cal N} =4$ supersymmetric gauge theory with softly broken supersymmetry, other finite supersymmetric gauge theory with 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 by use of the Cauchy Integral Formula. The solution restricts form of the unintegrated gluon distribution as function of transfer momentum 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 behaviour in the vicinity of the point $x=0$ and a smooth behaviour in the vicinity of the point $x=1.

  • Analytical solution to DGLAP integro-differential equation in a simple toy-model with a fixed gauge coupling
    2016
    Co-Authors: Alvarez Gustavo, Cvetic Gorazd, Kniehl, Bernd A., Kondrashuk Igor, Parra-ferrada Ivan
    Abstract:

    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 ${\cal N} =4$ supersymmetric gauge theory with softly broken supersymmetry, other finite supersymmetric gauge theory with 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 by use of the Cauchy Integral Formula. The solution restricts form of the unintegrated gluon distribution as function of transfer momentum 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 behaviour in the vicinity of the point $x=0$ and a smooth behaviour in the vicinity of the point $x=1.$Comment: 25 page