Metric Geometry in Normed Spaces

This chapter focuses on the metric geometry of Teichmüller space. It first explains how one can think of Teich(Sɡ) as the space of complex structures on Sɡ. To this end, the chapter defines quasiconformal maps between surfaces and presents a solution to the resulting Teichmüller's extremal problem. It also considers the correspondence between complex structures and hyperbolic structures, along with the Teichmüller mapping, Teichmüller metric, and the proof of Teichmüller's uniqueness and existence theorems. The fundamental connection between Teichmüller's theorems, holomorphic quadratic differentials, and measured foliations is discussed as well. Finally, the chapter describes the Grötzsch's problem, whose solution is tied to the proof of Teichmüller's uniqueness theorem.

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On the stability of a nonic functional equation in quasi-normed spaces

JOURNAL OF ADVANCES IN PHYSICS ◽

10.24297/jap.v12i3.47 ◽

2016 ◽

Vol 12 (3) ◽

pp. 4368-4374

Author(s):

Soo Hwan Kim

Keyword(s):

Functional Equation ◽

Ulam Stability ◽

Normed Spaces ◽

The Stability

In this paper, we extend normed spaces to quasi-normed spacesÂ and prove the generalized Hyers-Ulam stability of a nonic functional equation:$$\aligned&f(x+5y) - 9f(x+4y) + 36f(x+3y) - 84f(x+2y) + 126f(x+y) - 126f(x)\\&\qquad + 84f(x-y)-36f(x-2y)+9f(x-3y)-f(x-4y) = 9 ! f(y),\endaligned$$where $9 ! = 362880$ in quasi-normed spaces.

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Solutions and Stability of Generalized Mixed Type QCA-Functional Equations in Random Normed Spaces

Annals of the Alexandru Ioan Cuza University - Mathematics ◽

10.2478/v10157-012-0047-2 ◽

2013 ◽

Vol 59 (2) ◽

pp. 299-320

Author(s):

M. Eshaghi Gordji ◽

Y.J. Cho ◽

H. Khodaei ◽

M. Ghanifard

Keyword(s):

Functional Equation ◽

General Solution ◽

Functional Equations ◽

Mixed Type ◽

Additive Functional ◽

Normed Spaces ◽

Additive Functional Equation ◽

Probabilistic Normed Spaces ◽

Random Normed Spaces ◽

Generalized Stability

Abstract In this paper, we investigate the general solution and the generalized stability for the quartic, cubic and additive functional equation (briefly, QCA-functional equation) for any k∈ℤ-{0,±1} in Menger probabilistic normed spaces.

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ON LACUNARY STATISTICAL CONVERGENCE AND SOME PROPERTIES IN FUZZY N-NORMED SPACES

i-manager’s Journal on Mathematics ◽

10.26634/jmat.7.3.14868 ◽

2018 ◽

Vol 7 (3) ◽

pp. 1

Author(s):

RECAİ TÜRKMEN MUHAMMED ◽

Keyword(s):

Statistical Convergence ◽

Normed Spaces ◽

Lacunary Statistical Convergence

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Hyers-Ulam stability of quadratic forms in 2-normed spaces

Demonstratio Mathematica ◽

10.1515/dema-2019-0038 ◽

2019 ◽

Vol 52 (1) ◽

pp. 496-502

Author(s):

Won-Gil Park ◽

Jae-Hyeong Bae

Keyword(s):

Banach Spaces ◽

Quadratic Forms ◽

Functional Equations ◽

Ulam Stability ◽

Normed Spaces

AbstractIn this paper, we obtain Hyers-Ulam stability of the functional equationsf (x + y, z + w) + f (x − y, z − w) = 2f (x, z) + 2f (y, w),f (x + y, z − w) + f (x − y, z + w) = 2f (x, z) + 2f (y, w)andf (x + y, z − w) + f (x − y, z + w) = 2f (x, z) − 2f (y, w)in 2-Banach spaces. The quadratic forms ax2 + bxy + cy2, ax2 + by2 and axy are solutions of the above functional equations, respectively.

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