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449 Mathematical sciences
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Topological transcendental fields

- Chalebgwa, Taboka, Morris, Sidney

**Authors:**Chalebgwa, Taboka , Morris, Sidney**Date:**2022**Type:**Text , Journal article**Relation:**Axioms Vol. 11, no. 3 (2022), p.**Full Text:****Reviewed:****Description:**This article initiates the study of topological transcendental fields F which are subfields of the topological field C of all complex numbers such that F only consists of rational numbers and a nonempty set of transcendental numbers. F, with the topology it inherits as a subspace of C, is a topological field. Each topological transcendental field is a separable metrizable zero-dimensional space and algebraically is Q(T), the extension of the field of rational numbers by a set T of transcendental numbers. It is proven that there exist precisely 222ℵ0 of topological transcendental fields of the form ℚ(𝑇) with T a set of Liouville numbers, no two of which are homeomorphic.**Description:**This article initiates the study of topological transcendental fields F which are subfields of the topological field C of all complex numbers such that F only consists of rational numbers and a nonempty set of transcendental numbers. F, with the topology it inherits as a subspace of C, is a topological field. Each topological transcendental field is a separable metrizable zero-dimensional space and algebraically is Q(T), the extension of the field of rational numbers by a set T of transcendental numbers. It is proven that there exist precisely 2

**Authors:**Chalebgwa, Taboka , Morris, Sidney**Date:**2022**Type:**Text , Journal article**Relation:**Axioms Vol. 11, no. 3 (2022), p.**Full Text:****Reviewed:****Description:**This article initiates the study of topological transcendental fields F which are subfields of the topological field C of all complex numbers such that F only consists of rational numbers and a nonempty set of transcendental numbers. F, with the topology it inherits as a subspace of C, is a topological field. Each topological transcendental field is a separable metrizable zero-dimensional space and algebraically is Q(T), the extension of the field of rational numbers by a set T of transcendental numbers. It is proven that there exist precisely 222ℵ0 of topological transcendental fields of the form ℚ(𝑇) with T a set of Liouville numbers, no two of which are homeomorphic.**Description:**This article initiates the study of topological transcendental fields F which are subfields of the topological field C of all complex numbers such that F only consists of rational numbers and a nonempty set of transcendental numbers. F, with the topology it inherits as a subspace of C, is a topological field. Each topological transcendental field is a separable metrizable zero-dimensional space and algebraically is Q(T), the extension of the field of rational numbers by a set T of transcendental numbers. It is proven that there exist precisely 2

A continuous homomorphism of a thin set onto a fat set

- Chalebgwa, Taboka, Morris, Sidney

**Authors:**Chalebgwa, Taboka , Morris, Sidney**Date:**2022**Type:**Text , Journal article**Relation:**Bulletin of the Australian Mathematical Society Vol. 106, no. 3 (2022), p. 500-503**Full Text:**false**Reviewed:****Description:**A thin set is defined to be an uncountable dense zero-dimensional subset of measure zero and Hausdorff measure zero of an Euclidean space. A fat set is defined to be an uncountable dense path-connected subset of an Euclidean space which has full measure, that is, its complement has measure zero. While there are well-known pathological maps of a set of measure zero, such as the Cantor set, onto an interval, we show that the standard addition on

Sin, cos, exp and log of Liouville numbers

- Chalebgwa, Taboka, Morris, Sidney

**Authors:**Chalebgwa, Taboka , Morris, Sidney**Date:**2023**Type:**Text , Journal article**Relation:**Bulletin of the Australian Mathematical Society Vol. 108, no. 1 (2023), p. 81-85**Full Text:****Reviewed:****Description:**For any Liouville number, all of the following are transcendental numbers:, and the inverse functions evaluated at of the listed trigonometric and hyperbolic functions, noting that wherever multiple values are involved, every such value is transcendental. This remains true if 'Liouville number' is replaced by 'U-number', where U is one of Mahler's classes of transcendental numbers. © The Author(s), 2022. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

**Authors:**Chalebgwa, Taboka , Morris, Sidney**Date:**2023**Type:**Text , Journal article**Relation:**Bulletin of the Australian Mathematical Society Vol. 108, no. 1 (2023), p. 81-85**Full Text:****Reviewed:****Description:**For any Liouville number, all of the following are transcendental numbers:, and the inverse functions evaluated at of the listed trigonometric and hyperbolic functions, noting that wherever multiple values are involved, every such value is transcendental. This remains true if 'Liouville number' is replaced by 'U-number', where U is one of Mahler's classes of transcendental numbers. © The Author(s), 2022. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

Erds-liouville sets

- Chalebgwa, Taboka, Morris, Sidney

**Authors:**Chalebgwa, Taboka , Morris, Sidney**Date:**2023**Type:**Text , Journal article**Relation:**Bulletin of the Australian Mathematical Society Vol. 107, no. 2 (2023), p. 284-289**Full Text:****Reviewed:****Description:**In 1844, Joseph Liouville proved the existence of transcendental numbers. He introduced the set L of numbers, now known as Liouville numbers, and showed that they are all transcendental. It is known that L has cardinality c, the cardinality of the continuum, and is a dense G

**Authors:**Chalebgwa, Taboka , Morris, Sidney**Date:**2023**Type:**Text , Journal article**Relation:**Bulletin of the Australian Mathematical Society Vol. 107, no. 2 (2023), p. 284-289**Full Text:****Reviewed:****Description:**In 1844, Joseph Liouville proved the existence of transcendental numbers. He introduced the set L of numbers, now known as Liouville numbers, and showed that they are all transcendental. It is known that L has cardinality c, the cardinality of the continuum, and is a dense G

Erdos properties of subsets of the Mahler set S

- Chalebgwa, Taboka, Morris, Sidney

**Authors:**Chalebgwa, Taboka , Morris, Sidney**Date:**2023**Type:**Text , Journal article**Relation:**Bulletin of the Australian Mathematical Society Vol. 108, no. 3 (2023), p. 504-510**Full Text:****Reviewed:****Description:**Erd.os proved that every real number is the sum of two Liouville numbers. A set W of complex numbers is said to have the Erd.os property if every real number is the sum of two members of W. Mahler divided the set of all transcendental numbers into three disjoint classes S, T and U such that, in particular, any two complex numbers which are algebraically dependent lie in the same class. The set of Liouville numbers is a proper subset of the set U and has Lebesgue measure zero. It is proved here, using a theorem of Weil on locally compact groups, that if m ∈ [0,∞), then there exist 2c dense subsets W of S each of Lebesgue measure m such that W has the Erd.os property and no two of these W are homeomorphic. It is also proved that there are 2c dense subsets W of S each of full Lebesgue measure, which have the Erd.os property. Finally, it is proved that there are 2c dense subsets W of S such that every complex number is the sum of two members of W and such that no two of these W are homeomorphic. © 2023 The Author(s).**Description:**Erd.os proved that every real number is the sum of two Liouville numbers. A set W of complex numbers is said to have the Erd.os property if every real number is the sum of two members of W. Mahler divided the set of all transcendental numbers into three disjoint classes S, T and U such that, in particular, any two complex numbers which are algebraically dependent lie in the same class. The set of Liouville numbers is a proper subset of the set U and has Lebesgue measure zero. It is proved here, using a theorem of Weil on locally compact groups, that if m

**Authors:**Chalebgwa, Taboka , Morris, Sidney**Date:**2023**Type:**Text , Journal article**Relation:**Bulletin of the Australian Mathematical Society Vol. 108, no. 3 (2023), p. 504-510**Full Text:****Reviewed:****Description:**Erd.os proved that every real number is the sum of two Liouville numbers. A set W of complex numbers is said to have the Erd.os property if every real number is the sum of two members of W. Mahler divided the set of all transcendental numbers into three disjoint classes S, T and U such that, in particular, any two complex numbers which are algebraically dependent lie in the same class. The set of Liouville numbers is a proper subset of the set U and has Lebesgue measure zero. It is proved here, using a theorem of Weil on locally compact groups, that if m ∈ [0,∞), then there exist 2c dense subsets W of S each of Lebesgue measure m such that W has the Erd.os property and no two of these W are homeomorphic. It is also proved that there are 2c dense subsets W of S each of full Lebesgue measure, which have the Erd.os property. Finally, it is proved that there are 2c dense subsets W of S such that every complex number is the sum of two members of W and such that no two of these W are homeomorphic. © 2023 The Author(s).**Description:**Erd.os proved that every real number is the sum of two Liouville numbers. A set W of complex numbers is said to have the Erd.os property if every real number is the sum of two members of W. Mahler divided the set of all transcendental numbers into three disjoint classes S, T and U such that, in particular, any two complex numbers which are algebraically dependent lie in the same class. The set of Liouville numbers is a proper subset of the set U and has Lebesgue measure zero. It is proved here, using a theorem of Weil on locally compact groups, that if m

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