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A Purely Algebraic Proof of the Fundamental Theorem of Algebra

100%

Błaszczyk P.

Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia

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2016

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tom 8

7-23

PL

Proofs of the fundamental theorem of algebra can be divided upinto three groups according to the techniques involved: proofs that rely onreal or complex analysis, algebraic proofs, and topological proofs. Algebraicproofs make use of the fact that odd-degree real polynomials have real roots.This assumption, however, requires analytic methods, namely, the intermediatevalue theorem for real continuous functions. In this paper, we developthe idea of algebraic proof further towards a purely algebraic proof of theintermediate value theorem for real polynomials. In our proof, we neither usethe notion of continuous function nor refer to any theorem of real and complexanalysis. Instead, we apply techniques of modern algebra: we extend thefield of real numbers to the non-Archimedean field of hyperreals via an ultraproductconstruction and explore some relationships between the subringof limited hyperreals, its maximal ideal of infinitesimals, and real numbers.

2

On a problem of Steve Kalikow

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Shelah S.

Fundamenta Mathematicae

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2000

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tom 166

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nr 1-2

137-151

EN

The Kalikow problem for a pair (λ,κ) of cardinal numbers,λ > κ (in particular κ = 2) is whether we can map the family of ω-sequences from λ to the family of ω-sequences from κ in a very continuous manner. Namely, we demand that for η,ν ∈ ω we have: η, ν are almost equal if and only if their images are. We show consistency of the negative answer, e.g., for $ℵ_ω$ but we prove it for smaller cardinals. We indicate a close connection with the free subset property and its variants.

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O ciałach uporządkowanych

100%

Błaszczyk P.

Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia

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2012

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tom 4

15-30

PL

In this paper, we present some basic facts concerning ordered fields. We review definitions of an ordered field, give an example of a field that admits many orderings, and present equivalent definitions of the axiom of Archimedes and the continuity axiom. We show how to extend an ordered field by means of an ultrapower construction and formal power series.

4

Embeddings of chains into chains

86%

Novák V., Novotný T.

Discussiones Mathematicae - General Algebra and Applications

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2005

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tom 25

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nr 1

5-21

EN

Continuity of isotone mappings and embeddings of a chain G into another chain are studied. Especially, conditions are found under which the set of points of discontinuity of such a mapping is dense in G.

5

Stability and Continuity of Functions of Least Gradient

72%

Hakkarainen H., Korte R., Lahti P., Shanmugalingam N.

Analysis and Geometry in Metric Spaces

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2015

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tom 3

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nr 1

EN

In this note we prove that on metric measure spaces, functions of least gradient, as well as local minimizers of the area functional (after modification on a set of measure zero) are continuous everywhere outside their jump sets. As a tool, we develop some stability properties of sequences of least gradient functions. We also apply these tools to prove a maximum principle for functions of least gradient that arise as solutions to a Dirichlet problem.

6

19th-century real analysis, forward and backward

72%

Bair J., Blaszczyk P., Heinig P., Kanovei V., Katz M.

Antiquitates Mathematicae

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2019

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tom 13

PL

Dziewiętnastowieczna rzeczywista analiza otrzymała duży impuls w pracach Cauchy'ego. Cauchy posługuje się pojęciem zmiennej, granicy i nieskończenie małej, ale znaczenie, które wiąże z tymi terminami jest inne niż u jemu współczesnych autorów. Niektórzy historycy Cauchy'ego utrzymują schemat pojęciowy zdominowany przez założenie teleologicznej natury ewolucji rzeczywistej analizy w kierunku z góry ustalonej konstrukcji. Zatem Gilain i Siegmund-Schultze zakładają, że odniesienia do granicy (limite) w pracy Cauchy'ego niekoniecznie oznaczają, że Cauchy posługiwał się kontinuum Archimedesa, podczas gdy nieskończenie małe były jedynie wygodną figurą retoryczną, dla której Cauchy miał na myśli pełne uzasadnienie w kategoriach granic Archimedesa. Istnieje jednak inna formalizacja procedur Cauchy'ego wykorzystujących jego granice(limite), bardziej spójna z wszechobecnym użyciem nieskończenie małych Cauchy'ego, w kategoriach standardowej części zasady nowoczesnej analizy nieskończenie małej. Podważamy błędne przekonanie, zgodnie z którym Cauchy został rzekomo zmuszony do nauczania nieskończenie małych w Ecole Polytechnique. Pokazujemy, że debata dotyczyła głównie kwestii ścisłości, oddzielnego od nieskończenie małych. Krytyka podejścia Cauchy'ego przez jego współczesnego de Prony rzuca światło na znaczenie ścisłości dla Cauchy'ego i jego współczesnych. Uważna lektura wyzwań Cauchy'ego spotkała się z poglądem na rolę Cauchy'ego w historii analizy i wskazuje, że był on pionierem nieskończenie małych technik, podobnie jak zapowiedzią Epsilontiki.

EN

19th-century real analysis received a major impetus from Cauchy's work. Cauchy mentions variable quantities, limits, and infinitesimals, but the meaning he attached to these terms is not identical to their modern meaning. Some Cauchy historians work in a conceptual scheme dominated by an assumption of a teleological nature of the evolution of real analysis toward a preordained outcome. Thus, Gilain and Siegmund-Schultze assume that references to limite in Cauchy's work necessarily imply that Cauchy was working with an Archimedean continuum, whereas infinitesimals were merely a convenient figure of speech, for which Cauchy had in mind a complete justification in terms of Archimedean limits. However, there is another formalization of Cauchy's procedures exploiting his limite, more consistent with Cauchy's ubiquitous use of infinitesimals, in terms of the standard part principle of modern infinitesimal analysis. We challenge a misconception according to which Cauchy was allegedly forced to teach infinitesimals at the Ecole Polytechnique. We show that the debate there concerned mainly the issue of rigor, a separate one from infinitesimals. A critique of Cauchy's approach by his contemporary de Prony sheds light on the meaning of rigor to Cauchy and his contemporaries. An attentive reading of Cauchy's work challenges received views on Cauchy's role in the history of analysis and indicates that he was a pioneer of infinitesimal techniques as much as a harbinger of the Epsilontik.

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Pseudotopologies with applications to one-parameter groups, von Neumann algebras, and Lie algebra representations

72%

Rusinek J.

Studia Mathematica

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1993

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tom 107

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nr 3

273-286

EN

For any pair E,F of pseudotopological vector spaces, we endow the space L(E,F) of all continuous linear operators from E into F with a pseudotopology such that, if G is a pseudotopological space, then the mapping L(E,F) × L(F,G) ∋ (f,g) → gf ∈ L(E,G) is continuous. We use this pseudotopology to establish a result about differentiability of certain operator-valued functions related with strongly continuous one-parameter semigroups in Banach spaces, to characterize von Neumann algebras, and to establish a result about integration of Lie algebra representations.

Ograniczanie wyników

2 Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia

1 Analysis and Geometry in Metric Spaces

1 Antiquitates Mathematicae

1 Discussiones Mathematicae - General Algebra and Applications

1 Fundamenta Mathematicae

1 Studia Mathematica

2 Błaszczyk P.

1 Bair J.

1 Blaszczyk P.

1 Hakkarainen H.

1 Heinig P.

1 Kanovei V.

1 Katz M.

1 Korte R.

1 Lahti P.

1 Novotný T.

1 Novák V.

1 Rusinek J.

1 Shanmugalingam N.

1 Shelah S.

1 2019

1 2016

1 2015

1 2012

1 2005

1 2000

1 1993

A Purely Algebraic Proof of the Fundamental Theorem of Algebra

On a problem of Steve Kalikow

O ciałach uporządkowanych

Embeddings of chains into chains

Stability and Continuity of Functions of Least Gradient

19th-century real analysis, forward and backward

Pseudotopologies with applications to one-parameter groups, von Neumann algebras, and Lie algebra representations