scholarly journals Simple connectedness of one space of complex-valued functions

2015 ◽  
Vol 23 ◽  
pp. 70
Author(s):  
A.M. Pas'ko
Keyword(s):  
Simple Connectedness ◽  
Simply Connected ◽  
Complex Valued

The spaces $\mathbb{C}{\Omega}_n$ have been defined. It has been established that the spaces $\mathbb{C}{\Omega}_n$, $n \geqslant 2$ are simply connected.

scholarly journals An extention of Nomizu’s Theorem –A user’s guide–

2016 ◽  
Vol 3 (1) ◽  
Author(s):  
Hisashi Kasuya
Keyword(s):  
Lie Group ◽  
Simple Proof ◽  
Graded Algebra ◽  
De Rham Cohomology ◽  
Differential Graded Algebra ◽  
Finite Dimensional ◽  
De Rham ◽  
Simply Connected ◽  
Complex Valued ◽  
Solvable Lie Group

AbstractFor a simply connected solvable Lie group G with a lattice Γ, the author constructed an explicit finite-dimensional differential graded algebra A*Γ which computes the complex valued de Rham cohomology H*(Γ\G, C) of the solvmanifold Γ\G. In this note, we give a quick introduction to the construction of such A*Γ including a simple proof of H*(A*Γ) ≅ H*(Γ\G, C).

scholarly journals Strongly simply connected Auslander algebras

1997 ◽  
Vol 39 (1) ◽  
pp. 21-27 ◽  
Author(s):  
Ibrahim Assem ◽  
Peter Brown
Keyword(s):  
Representation Theory ◽  
Closed Field ◽  
Interesting Problem ◽  
Indecomposable Modules ◽  
Finite Algebras ◽  
Isomorphism Classes ◽  
Finite Dimensional ◽  
Infinite Case ◽  
Simple Connectedness ◽  
Simply Connected

Letkbe an algebraically closed field. By an algebra is meant an associative finite dimensionalk-algebra A with an identity. We are interested in studying the representation theory of Λ, that is, in describing the category mod Λ of finitely generated right Λ-modules. Thus we may, without loss of generality, assume that Λ is basic and connected. For our purpose, one strategy consists in using covering techniques to reduce the problem to the case where the algebra is simply connected, then in solving the problem in this latter case. This strategy was proved efficient for representation-finite algebras (that is, algebras having only finitely many isomorphism classes of indecomposable modules) and representation-finite simply connected algebras are by now well-understood: see, for instance [5], [7],[8]. While little is known about covering techniques in the representation-infinite case, it is clearly an interesting problem to describe the representation-infinite simply connected algebras. The objective of this paper is to give a criterion for the simple connectedness of a class of (mostly representationinfinite) algebras.

On Simple Connectedness of Minimal Representation-Infinite Algebras

2015 ◽  
Vol 22 (04) ◽  
pp. 639-654
Author(s):  
Hailou Yao ◽  
Guoqiang Han
Keyword(s):  
Cohomology Group ◽  
Hochschild Cohomology ◽  
Closed Field ◽  
Minimal Representation ◽  
Algebraically Closed Field ◽  
Hochschild Cohomology Group ◽  
Simple Connectedness ◽  
Simply Connected ◽  
Equivalent Conditions ◽  
Infinite Algebras

Let A be a connected minimal representation-infinite algebra over an algebraically closed field k. In this paper, we investigate the simple connectedness and strong simple connectedness of A. We prove that A is simply connected if and only if its first Hochschild cohomology group H1(A) is trivial. We also give some equivalent conditions of strong simple connectedness of an algebra A.

Harmonic Mappings onto Convex Domains

1987 ◽  
Vol 39 (6) ◽  
pp. 1489-1530 ◽  
Author(s):  
Yusuf Abu-Muhanna ◽  
Glenn Schober
Keyword(s):  
Univalent Functions ◽  
Harmonic Mappings ◽  
Open Unit ◽  
Open Unit Disk ◽  
Convex Domains ◽  
Harmonic Solution ◽  
Simply Connected Domain ◽  
Image Position ◽  
Simply Connected ◽  
Complex Valued

Let D be a simply-connected domain and w0 a fixed point of D. Denote by SD the set of all complex-valued, harmonic, orientation-preserving, univalent functions f from the open unit disk U onto D with f(0) = w0. Unlike conformai mappings, harmonic mappings are not essentially determined by their image domains. So, it is natural to study the set SD.In Section 2, we give some mapping theorems. We prove the existence, when D is convex and unbounded, of a univalent, harmonic solution f of the differential equationwhere a is analytic and |a| < 1, such that f(U) ⊂ D and

scholarly journals On some properties of solutions of the p-harmonic equation

Filomat ◽  
2013 ◽  
Vol 27 (4) ◽  
pp. 577-591 ◽  
Author(s):  
Sh. Chen ◽  
S. Ponnusamy ◽  
X. Wang
Keyword(s):  
Unit Disk ◽  
Connected Domain ◽  
Harmonic Mappings ◽  
Simply Connected Domain ◽  
Harmonic Equation ◽  
Simply Connected ◽  
Complex Valued ◽  
Class Of Functions

A 2p-times continuously differentiable complex-valued function ? = u + iv in a simply connected domain ? ? C is p-harmonic if ? satisfies the p-harmonic equation ?p? = 0. In this paper, we investigate the properties of p-harmonic mappings in the unit disk |z| < 1. First, we discuss the convexity, the starlikeness and the region of variability of some classes of p-harmonic mappings. Then we prove the existence of Landau constant for the class of functions of the form D? = z?z - ??z, where f is p-harmonic in |z| < 1. Also, we discuss the region of variability for certain p-harmonic mappings. At the end, as a consequence of the earlier results of the authors, we present explicit upper estimates for Bloch norm for bi- and tri-harmonic mappings.

scholarly journals The Simple Connectedness of Tame Algebras with Separating Almost Cyclic Coherent Auslander–Reiten Components

2021 ◽  
Author(s):  
Piotr Malicki
Keyword(s):  
Hochschild Cohomology ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Tame Algebra ◽  
Finite Dimensional ◽  
Cohomology Space ◽  
Simple Connectedness ◽  
Tame Algebras ◽  
Simply Connected ◽  
Finite Dimensional Algebras

AbstractWe study the simple connectedness of the class of finite-dimensional algebras over an algebraically closed field for which the Auslander–Reiten quiver admits a separating family of almost cyclic coherent components. We show that a tame algebra in this class is simply connected if and only if its first Hochschild cohomology space vanishes.

A GENERALISATION OF THE CLUNIE–SHEIL-SMALL THEOREM II

2017 ◽  
Vol 95 (3) ◽  
pp. 457-466 ◽  
Author(s):  
MAŁGORZATA MICHALSKA ◽  
ANDRZEJ M. MICHALSKI
Keyword(s):  
Complex Plane ◽  
Univalent Functions ◽  
Horizontal Direction ◽  
Harmonic Mappings ◽  
Connected Sets ◽  
Simply Connected ◽  
Univalence Criteria ◽  
Complex Valued ◽  
Planar Harmonic Mappings

We study properties of the simply connected sets in the complex plane, which are finite unions of domains convex in the horizontal direction. These considerations allow us to state new univalence criteria for complex-valued local homeomorphisms. In particular, we apply our results to planar harmonic mappings obtaining generalisations of the shear construction theorem due to Clunie and Sheil-Small [‘Harmonic univalent functions’, Ann. Acad. Sci. Fenn. Ser. A. I. Math.9 (1984), 3–25].

Complex-valued Neural Networks

2011 ◽  
Vol 131 (1) ◽  
pp. 2-8
Author(s):  
Akira Hirose
Keyword(s):  
Neural Networks ◽  
Complex Valued
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