Function Theory of One Complex Variable

Greene, Robert E.; Krantz, Steven G.

doi:10.1090/gsm/040

Cited by 84 publications

(95 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…Por fim, deve constitui parte das análise do ensino atual de AC, uma abordagem de autores de compêndios especializados (NEEDHAM, 2000;GREENE & KRANTZ, 2006;KRANTZ, 1990;SCHWERDTFEGER, 1979;SPIEGEL et all, 2009) …”

Section: Considerações Finaisunclassified

Engenharia Didática: Implicações Para a Pesquisa No Âmbito Do Ensino Em Análise Complexa - Ac

Alves

2016

CeN

View full text Add to dashboard Cite

Resumo O presente artigo aborda e descreve as duas fases iniciais previstas por um design de investigação nominado de Engenharia Abstract This article discusses and describes the two initial phases provided by a nominated research design of Didactical Engineering -DE. Thus, in view of an interest declared by the teachhing of Complex Analysis -CA, the work emphasizes the elements that hold the potential to constitute the two initial stages of an DE, nomitaded by preliminary and a priori analysis, with emphasis on description and conception of only two problems situations. So, in view of the long heritage of the French tradition in Didactics of

show abstract

Section: Considerações Finaisunclassified

Engenharia Didática: Implicações Para a Pesquisa No Âmbito Do Ensino Em Análise Complexa - Ac

Alves

2016

CeN

View full text Add to dashboard Cite

show abstract

“…The fact that the Stieltjes transform is well defined follows from the fact that for the domain the function 1 z−x is bounded. One can show that G(z) is analytical for all z ∈ {z : z ∈ C, (z) > 0} using Morera's theorem (see work by Greene and Krantz, 2006). Then, it is enough to show that the contour integral Γ G(z)dz = 0, for all closed contours Γ in z ∈ {z : z ∈ C, (z) > 0}.…”

Section: Stieltjes Transformmentioning

confidence: 99%

Contributions to High–Dimensional Analysis under Kolmogorov Condition

Pielaszkiewicz

2015

Linköping Studies in Science and Technology. Dissertations

View full text Add to dashboard Cite

This thesis is about high-dimensional problems considered under the so-called Kolmogorov condition. Hence, we consider research questions related to random matrices with p rows (corresponding to the parameters) and n columns (corresponding to the sample size), where p > n, assuming that the ratio p n converges when the number of parameters and the sample size increase.We focus on the eigenvalue distribution of the considered matrices, since it is a well-known information-carrying object. The spectral distribution with compact support is fully characterized by its moments, i.e., by the normalized expectation of the trace of powers of the matrices. Moreover, such an expectation can be seen as a free moment in the non-commutative space of random matrices of size p × p equipped with the functionalHere, the connections with free probability theory arise. In the relation to that field we investigate the closed form of the asymptotic spectral distribution for the sum of the quadratic forms. Moreover, we put a free cumulant-moment relation formula that is based on the summation over partitions of the number. This formula is an alternative to the free cumulant-moment relation given through non-crossing partitions of the set.Furthermore, we investigate the normalized E[Tr{W mi }] and derive, using the differentiation with respect to some symmetric matrix, a recursive formula for that expectation. That allows us to re-establish moments of the Marčenko-Pastur distribution, and hence the recursive relation for the Catalan numbers.In this thesis we also prove that theWe considerwhere mwhich is proven to be normally distributed. Moreover, we propose, based on these random variables, a test for the identity of the covariance matrix using a goodness-of-fit approach. The test performs very well regarding the power of the test compared to some presented alternatives for both the high-dimensional data (p > n) and the multivariate data (p ≤ n). Populärvetenskaplig sammanfattningI många tillämpningar förekommer slumpmatriser, det vill säga matriser vars element följer någon stokastisk fördelning. Detär ofta intressant att känna till hur egenvärdena för dessa slumpmässiga matriser uppför sig, det vill säga beräkna fördelningen för egenvärdena, den så kallade spektralfördelningen. Egenvärdenä ar informationsbärande objekt, då de ger information om till exempel stabilitet och inversen av den slumpmässiga matrisen genom det minsta egenvärdet.Telekommunikation och teoretisk fysikär två områden där detär intressant att studera slumpmatriser, matrisernas egenvärden och fördelningen för dessa egenvärden. Speciellt gäller det för stora slumpmatriser där man dåär intresserad av den asymptotiska spektralfördelningen. Ett exempel kan vara en kanalmatris X för ett flerdimensionellt kommunikationssystem, där fördelningen för egenvärdena för matrisen XX * bestämmer kanalkapaciteten och uppnåeligä overföringshastigheter.Spektralfördelningen har studerats ingående inom teorin för slumpmässiga matriser. I denna avhandling kombinerar vi resulta...

show abstract

“…Then F satisfies the condition for the Schwarz lemma and so we can get the following variation of the Schwarz lemma called the Schwarz-Pick lemma (see [6], [16]). …”

Section: Corollary 23 Let F Be a Holomorphic Function For |Z| < R Withmentioning

confidence: 99%