Cauchy–Schwarz functions and convex partitions in the ray space of a supertropical quadratic form

Zur Izhakian; Manfred Knebusch

doi:10.1080/03081087.2021.1919049

Cauchy–Schwarz functions and convex partitions in the ray space of a supertropical quadratic form

Zur Izhakian^* (Corresponding Author), Manfred Knebusch

^*Corresponding author for this work

Mathematical Science

University of Regensburg

Research output: Contribution to journal › Article › peer-review

1 Citation (Scopus)

Abstract

Rays are classes of an equivalence relation on a module V over a supertropical semiring. They provide a version of convex geometry, supported by a ‘supertropical trigonometry’ and compatible with quasilinearity, in which the CS-ratio takes the role of the Cauchy–Schwarz inequality. CS-functions that emerge from the CS-ratio are a useful tool that helps to understand the variety of quasilinear stars in the ray space Ray(V). In particular, these functions induce a partition of Ray(V) into convex sets, and thereby a finer convex analysis which includes the notions of median, minima, glens, and polars.

Original language	English
Pages (from-to)	5502-5546
Number of pages	45
Journal	Linear and Multilinear Algebra
Volume	70
Issue number	20
Early online date	6 May 2021
DOIs	https://doi.org/10.1080/03081087.2021.1919049
Publication status	Published - 1 Jan 2022

Bibliographical note

Acknowledgments
The authors thank the referee for the helpful suggestions and comments.

Keywords

Algebra and Number Theory
supertropical algebra
supertropical modules
bilinear forms
quadratic forms
quadratic pairs
ray spaces
convex sets
quasilinear sets
Cauchy-Schwarz ratio
Cauchy-Schwarz functions
QL-stars

Access to Document

10.1080/03081087.2021.1919049Licence: Unspecified

Cite this

@article{39dcfac74bfa40c4a68bfc318d2dcd12,

title = "Cauchy–Schwarz functions and convex partitions in the ray space of a supertropical quadratic form",

abstract = "Rays are classes of an equivalence relation on a module V over a supertropical semiring. They provide a version of convex geometry, supported by a {\textquoteleft}supertropical trigonometry{\textquoteright} and compatible with quasilinearity, in which the CS-ratio takes the role of the Cauchy–Schwarz inequality. CS-functions that emerge from the CS-ratio are a useful tool that helps to understand the variety of quasilinear stars in the ray space Ray(V). In particular, these functions induce a partition of Ray(V) into convex sets, and thereby a finer convex analysis which includes the notions of median, minima, glens, and polars.",

keywords = "Algebra and Number Theory, supertropical algebra, supertropical modules, bilinear forms, quadratic forms, quadratic pairs, ray spaces, convex sets, quasilinear sets, Cauchy-Schwarz ratio, Cauchy-Schwarz functions, QL-stars",

author = "Zur Izhakian and Manfred Knebusch",

note = "Acknowledgments The authors thank the referee for the helpful suggestions and comments.",

year = "2022",

month = jan,

day = "1",

doi = "10.1080/03081087.2021.1919049",

language = "English",

volume = "70",

pages = "5502--5546",

journal = "Linear and Multilinear Algebra",

issn = "0308-1087",

publisher = "Taylor and Francis Ltd.",

number = "20",

}

TY - JOUR

T1 - Cauchy–Schwarz functions and convex partitions in the ray space of a supertropical quadratic form

AU - Izhakian, Zur

AU - Knebusch, Manfred

N1 - Acknowledgments The authors thank the referee for the helpful suggestions and comments.

PY - 2022/1/1

Y1 - 2022/1/1

N2 - Rays are classes of an equivalence relation on a module V over a supertropical semiring. They provide a version of convex geometry, supported by a ‘supertropical trigonometry’ and compatible with quasilinearity, in which the CS-ratio takes the role of the Cauchy–Schwarz inequality. CS-functions that emerge from the CS-ratio are a useful tool that helps to understand the variety of quasilinear stars in the ray space Ray(V). In particular, these functions induce a partition of Ray(V) into convex sets, and thereby a finer convex analysis which includes the notions of median, minima, glens, and polars.

AB - Rays are classes of an equivalence relation on a module V over a supertropical semiring. They provide a version of convex geometry, supported by a ‘supertropical trigonometry’ and compatible with quasilinearity, in which the CS-ratio takes the role of the Cauchy–Schwarz inequality. CS-functions that emerge from the CS-ratio are a useful tool that helps to understand the variety of quasilinear stars in the ray space Ray(V). In particular, these functions induce a partition of Ray(V) into convex sets, and thereby a finer convex analysis which includes the notions of median, minima, glens, and polars.

KW - Algebra and Number Theory

KW - supertropical algebra

KW - supertropical modules

KW - bilinear forms

KW - quadratic forms

KW - quadratic pairs

KW - ray spaces

KW - convex sets

KW - quasilinear sets

KW - Cauchy-Schwarz ratio

KW - Cauchy-Schwarz functions

KW - QL-stars

U2 - 10.1080/03081087.2021.1919049

DO - 10.1080/03081087.2021.1919049

M3 - Article

SN - 0308-1087

VL - 70

SP - 5502

EP - 5546

JO - Linear and Multilinear Algebra

JF - Linear and Multilinear Algebra

IS - 20

ER -

Cauchy–Schwarz functions and convex partitions in the ray space of a supertropical quadratic form

Abstract

Bibliographical note

Keywords

Access to Document

Fingerprint

Cite this