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## Uma prova de incompletude da aritmética baseada no teorema das definições recursivas; A proof of incompleteness for arithmetic by means of the Theorem of the Definion by Recursion

Vicente, Luciano
Fonte: Biblioteca Digitais de Teses e Dissertações da USP Publicador: Biblioteca Digitais de Teses e Dissertações da USP
Tipo: Dissertação de Mestrado Formato: application/pdf
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Esta dissertação estabelece a incompletude de um sistema formal cujas únicas constantes não-lógicas são 0 e s (respectivamente, o número natural 0 e a função sucessor segundo a interpretação standard), fundamentando-se, para tanto, em um teorema cuja prova necessita essencialmente da maquinária lógica de segunda-ordem e que foi designado de Teorema das Definições Recursivas.; We establish here the incompleteness of the formal system S2 for arithmetic_a formal system whose signature is {0, s}_by means of the Theorem of the Definition by Recursion (TDR). However, unlike the standard proofs of incompleteness, the proof of TDR, by virtue of restricted signature, uses essentially the power of second-order logic.

## Ensino de fatos aritméticos para escolares com deficiência intelectual; Teaching arithmetic facts to students with intellectual disabilities

Cechin, Michelle Brugnera Cruz; Costa, Adriana Corrêa; Dorneles, Beatriz Vargas
Tipo: Artigo de Revista Científica Formato: application/pdf
POR
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Este estudo identificou os procedimentos de contagem usados por crianças com deficiência intelectual e verificou os efeitos de um programa de intervenção direcionado ao ensino de fatos aritméticos. Participaram três crianças, entre oito e 12 anos, de uma escola da rede municipal de ensino de Porto Alegre, Rio Grande do Sul, que atende fundamentalmente às classes socioeconômicas baixas. A partir da revisão de literatura sobre os processos cognitivos envolvidos na resolução de problemas aditivos e as implicações para o seu ensino, avaliou-se a eficácia de um modelo de intervenção pedagógica como um recurso para o avanço no uso dos procedimentos de contagem. Foi aplicado um programa em 10 encontros, realizados uma vez por semana, com duração de aproximadamente cinquenta minutos cada. A proposta caracterizou-se pelo ensino direto, explícito e sistemático, através de sequências de instrução, partindo dos procedimentos de contagem usados pelos estudantes, que foram avaliados em dois momentos (pré-teste e pós-teste). Verificou-se que houve um avanço nos procedimentos de contagem utilizados pelas crianças após a intervenção, revelando que o programa foi eficaz. Mesmo intervenções de curta duração, como é o caso desta...

## Traces, ideals, and arithmetic means

Kaftal, Victor; Weiss, Gary
Tipo: Artigo de Revista Científica
EN
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This article grew out of recent work of Dykema, Figiel, Weiss, and Wodzicki (Commutator structure of operator ideals) which inter alia characterizes commutator ideals in terms of arithmetic means. In this paper we study ideals that are arithmetically mean (am) stable, am-closed, am-open, soft-edged and soft-complemented. We show that many of the ideals in the literature possess such properties. We apply these notions to prove that for all the ideals considered, the linear codimension of their commutator space (the “number of traces on the ideal”) is either 0, 1, or ∞. We identify the largest ideal which supports a unique nonsingular trace as the intersection of certain Lorentz ideals. An application to elementary operators is given. We study properties of arithmetic mean operations on ideals, e.g., we prove that the am-closure of a sum of ideals is the sum of their am-closures. We obtain cancellation properties for arithmetic means: for principal ideals, a necessary and sufficient condition for first order cancellations is the regularity of the generator; for second order cancellations, sufficient conditions are that the generator satisfies the exponential Δ2-condition or is regular. We construct an example where second order cancellation fails...

## Spontaneous meta-arithmetic as the first step toward school algebra

Caspi, Shai; Sfard, Anna
Fonte: Grupo de Investigaci??n Did??ctica de la Matem??tica: Pensamiento Num??rico, (FQM-193), Publicador: Grupo de Investigaci??n Did??ctica de la Matem??tica: Pensamiento Num??rico, (FQM-193),
Tipo: Artigo de Revista Científica
ENG
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Taking as a point of departure the vision of school algebra as a formalized meta-discourse of arithmetic, we have been following six pairs of 7th-grade students (12-13 years old) as they gradually modify their spontaneous meta-arithmetic toward the ???official??? algebraic form of talk. In this paper we take a look at the very beginning of this process. Preliminary analyses of data have shown, unsurprisingly, that while reflecting on arithmetic processes and relations, the uninitiated 7th graders were employing colloquial means, which could not protect them against occasional ambiguities. More unexpectedly, this spontaneous meta-arithmetic, although not supported by any previous algebraic schooling, displayed some algebra-like features, not to be normally found in everyday discourses.; Tomando como punto de partida la visi??n del ??lgebra escolar como un meta-discurso formalizado de la aritm??tica, hemos estado siguiendo a seis pares de estudiantes de 7?? curso (12-13 a??os) cuando modifican gradualmente su meta-aritm??tica espont??nea hacia la forma algebraica ???oficial??? de hablar. En este art??culo miramos el principio de este proceso. Los an??lisis preliminares de los datos han mostrado, como era de esperar, que mientras reflexionaban sobre los procesos y relaciones aritm??ticas...

## Second order arithmetic means in operator ideals

Kaftal, Victor; Weiss, Gary
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
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Equality of the second order arithmetic means of two principal ideals does not imply equality of their first order arithmetic means (second order equality cancellation). We provide fairly broad sufficient conditions on one of the principal ideals for this implication to hold true. We present also sufficient conditions for second order inclusion cancellations. These conditions are formulated in terms of the growth properties of the ratio of regularity sequence associated to the sequence of s-number of a generator of the principal ideal. These results are then extended to general ideals.; Comment: 19 pages. To appear in Operators and Matrices

## Traces on operator ideals and arithmetic means

Kaftal, Victor; Weiss, Gary
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
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This article - a part of a multipaper project investigating arithmetic mean ideals - investigates the codimension of commutator spaces [I, B(H)] of operator ideals on a separable Hilbert space, i.e., How many traces can an ideal support?" We conjecture that the codimension can be only zero, one, or infinity. Using the arithmetic mean (am) operations on ideals introduced by Dykema, Figiel, Weiss, and Wodzicki, and the analogous am operations at infinity that we develop in this article, the conjecture is proven for all ideals not contained in the largest am-infinity stable ideal and not containing the smallest am-stable ideal. It is also proven for all soft-edged ideals (i.e., I= IK(H)) and all soft-complemented ideals (i.e., I= I/K(H)), which include many classical operator ideals. In the process, we prove that an ideal of trace class operators supports a unique trace (up to scalar multiples) if and only if it is am-infinity stable and that, for a principal ideal, am-infinity stability is equivalent to regularity at infinity of the sequence of s-numbers of the generator. Furthermore, we apply trace extension methods to two problems on elementary operators studied by V. Shulman and to Fuglede-Putnam type problems of the second author.; Comment: 41 pages...

## A survey on the interplay between arithmetic mean ideals, traces, lattices of operator ideals, and an infinite Schur-Horn majorization theorem

Kaftal, Victor; Weiss, Gary
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
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The work of Dykema, Figiel, Weiss, and Wodzicki on the structure of commutators showed that arithmetic means play an important role in the study of operator ideals, and we explored their role in a multipaper project which we survey in this article. We start by presenting the notions of arithmetic mean ideals and arithmetic mean at infinity ideals. Then we explore their connections with commutator spaces, traces, elementary operators, lattice and sublattice structure of ideals, arithmetic mean ideal cancellation properties of first and second order, and softness properties - a term that we introduced but a notion ubiquitous in the literature on operator ideals. Arithmetic mean closure of ideals leads us to investigate majorization for infinite sequences and this in turn leads us to an infinite Schur-Horn majorization theorem which extends theorems by A. Neumann, by Arveson and Kadison, and by Antezana, Massey, Ruiz and Stojanoff. We also list ten open questions that we encountered in the development of this material.; Comment: 33 pages

## Weighted Quasi-Arithmetic Means and Invariance of Types 1, 2, and 3

Horwitz, Alan
Tipo: Artigo de Revista Científica
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Let m_{n} and m_{n-1} be an n mean and an n-1 mean, respectively, n\geq3. If x=(x_{1},...,x_{n}), let {\pi}_{\neqj}x=(x_{1},...,x_{j-1},x_{j+1},...,x_{n}). m_{n-1} and m_{n} are said to form a type 1 invariant pair if m_{n}(m_{n-1}({\pi}_{\neq1}x),m_{n-1}({\pi}_{\neq2}x),...,m_{n-1}({\pi}_{\neqn}x))=m_{n}(x) for all x\inR^{n}. m_{n-1} and m_{n} are said to form a type 2 invariant pair if m_{n}(x,m_{n-1}(x))=m_{n-1}(x) for all x\inR_{+}^{n-1}. If x=(x_{1},...,x_{n-1}), let {\pi}_{=j}x=(x_{1},...,x_{j-1},x_{j},x_{j},x_{j+1},...,x_{n-1})\inR_{+}^{n}. m_{n-1} and m_{n} are said to form a type 3 invariant pair if m_{n-1}(m_{n}({\pi}=_{1}x),...,m_{n}({\pi}_{=n-1}x))=m_{n-1}(x) for all x\inR_{+}^{n-1}. Let m_{h,w,n}(a_{1},...,a_{n})=h^{-1}(((\sum_{k=1}^{n}w(a_{k})h(a_{k}))/(\sum_{k=1}^{n}w(a_{k})))), where h(x) is continuous and monotone, and w(x) is continuous and positive, on (0,\infty) denote the family of weighted quasi--arithmetic means in n variables. We prove that if m_{h,w,n} and m_{h,w,n-1} form a type 1 or type 3 invariant pair, then m_{h,w,n} and m_{h,w,n-1} are quasi--arithmetic means. The method of proof involves deriving equations for certain partial derivatives of order 3 of m_{h,w,n} on the diagonal of R_{+}^{n}. The proof also requires an equation relating certain partial derivatives of order 3 for type 1 or type 3 invariant pairs of means. We also show that any pair of weighted quasi--arithmetic means m_{h...

## A new estimate of the difference among quasi-arithmetic means

Pasteczka, Paweł
Tipo: Artigo de Revista Científica
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In the 1960s Cargo and Shisha proved some majorizations for the distance among quasi-arithmetic means (defined as f^{-1}(\sum_{i=1}^{n} w_i f(a_i) for any continuous, strictly monotone function f:I->R, where I is an interval, and a=(a_1,...,a_n) is a vector with entries in I, w=(w_1,...,w_n) is a sequence of corresponding weights w_i>0, w_1+...+w_n=1). Nearly thirty years later, in 1991, P\ales presented an iff condition for a sequence of quasi-arithmetic means to converge to another QA mean. It was closely related with the three parameters' operator (f(x)-f(z))/(f(x)-f(y)). The author presented recently an estimate for the distance among such quasi-arithmetic means whose underlying functions satisfy some smoothness conditions. Used was the operator f -> f''/f' introduced in the 1940s by Mikusi\'nski and \L{}ojasiewicz. It is natural to look for similar estimate(s) in the case of the underlying functions not being smooth. For instance, by the way of using P\ales' operator. This is done in the present note. Moreover, the result strengthens author's earlier estimates.; Comment: 7 pages

## Scales of quasi-arithmetic means determined by invariance property

Pasteczka, Paweł
Tipo: Artigo de Revista Científica
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It is well known that if $\mathcal{P}_t$ denotes a set of power means then the mapping $\mathbb{R} \ni t \mapsto \mathcal{P}_t(v) \in (\min v, \max v)$ is both 1-1 and onto for any non-constant sequence $v = (v_1,\dots,\,v_n)$ of positive numbers. Shortly: the family of power means is a scale. If $I$ is an interval and $f \colon I \rightarrow \mathbb{R}$ is a continuous, strictly monotone function then $f^{-1}(\tfrac{1}{n} \sum f(v_i))$ is a natural generalization of power means, so called quasi-arithmetic mean generated by $f$. A famous folk theorem says that the only homogeneous, quasi-a\-rith\-me\-tic means are power means. We prove that, upon replacing the homogeneity requirement by an invariant-type axiom, one gets a family of quasi-arithmetic means building up a scale, too.; Comment: 11 pages

## On the Variability Estimation of Lognormal Distribution Based on Sample Harmonic and Arithmetic Means

Ji, Edward Y.; Ji, Brian L.
Tipo: Artigo de Revista Científica
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For the lognormal distribution, an unbiased estimator of the squared coefficient of variation is derived from the relative ratio of sample arithmetic to harmonic means. Analytical proofs and simulation results are presented.; Comment: 7 pages, 2 figures

## Some Best Possible Inequalities Concerning Certain Bivariate Means

Zhao, Tie-Hong; Chu, Yu-Ming; Liu, Bao-Yu
Tipo: Artigo de Revista Científica
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In this paper, some inequalities of bounds for the Neuman-S\'{a}ndor mean in terms of weighted arithmetic means of two bivariate means are established. Bounds involving weighted arithmetic means are sharp.; Comment: 8 pages

## Limit properties in a family of quasi-arithmetic means

Pasteczka, Paweł
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
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It is known that the family of power means tends to maximum pointwise if we pass argument to infinity. We will give some necessary and sufficient condition for the family of quasi-arithmetic means generated by a functions satisfying certain smoothness conditions to have analogous property.; Comment: 10 pages

## The precision of the arithmetic mean, geometric mean and percentiles for citation data: An experimental simulation modelling approach

Thelwall, Mike
Tipo: Artigo de Revista Científica
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When comparing the citation impact of nations, departments or other groups of researchers within individual fields, three approaches have been proposed: arithmetic means, geometric means, and percentage in the top X%. This article compares the precision of these statistics using 97 trillion experimentally simulated citation counts from 6875 sets of different parameters (although all having the same scale parameter) based upon the discretised lognormal distribution with limits from 1000 repetitions for each parameter set. The results show that the geometric mean is the most precise, closely followed by the percentage of a country's articles in the top 50% most cited articles for a field, year and document type. Thus the geometric mean citation count is recommended for future citation-based comparisons between nations. The percentage of a country's articles in the top 1% most cited is a particularly imprecise indicator and is not recommended for international comparisons based on individual fields. Moreover, whereas standard confidence interval formulae for the geometric mean appear to be accurate, confidence interval formulae are less accurate and consistent for percentile indicators. These recommendations assume that the scale parameters of the samples are the same but the choice of indicator is complex and partly conceptual if they are not.; Comment: Thelwall...

## The best bounds for Toader mean in terms of the centroidal and arithmetic means

Hua, Yun; Qi, Feng
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
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In the paper, the authors discover the best constants $\alpha_{1}$, $\alpha_{2}$, $\beta_{1}$, and $\beta_{2}$ for the double inequalities $$\alpha_{1}\bar{C}(a,b)+(1-\alpha_{1}) A(a,b)< T(a,b) <\beta_{1} \bar{C}(a,b)+(1-\beta_{1})A(a,b)$$ and $$\frac{\alpha_{2}}{A(a,b)}+\frac{1-\alpha_{2}}{\bar{C}(a,b)}<\frac1{T(a,b)} <\frac{\beta_{2}}{A(a,b)}+\frac{1-\beta_{2}}{\bar{C}(a,b)}$$ to be valid for all $a,b>0$ with $a\ne b$, where $$\bar{C}(a,b)=\frac{2(a^{2}+ab+b^{2})}{3(a+b)},\quad A(a,b)=\frac{a+b}2,$$ and $$T(a,b)=\frac{2}{\pi}\int_{0}^{{\pi}/{2}}\sqrt{a^2{\cos^2{\theta}}+b^2{\sin^2{\theta}}}\,\td\theta$$ are respectively the centroidal, arithmetic, and Toader means of two positive numbers $a$ and $b$. As an application of the above inequalities, the authors also find some new bounds for the complete elliptic integral of the second kind.; Comment: 7 pages

## On almost everywhere convergence of strong arithmetic means of Fourier series

Wilson, Bobby
Tipo: Artigo de Revista Científica
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This article establishes a real-variable argument for Zygmund's theorem on almost everywhere convergence of strong arithmetic means of partial sums of Fourier series on $\mathbb{T}$, up to passing to a subsequence. Our approach extends to, among other cases, functions that are defined on $\mathbb{T}^d$, which allows us to establish an analogue of Zygmund's theorem in higher dimensions.

## Sharp two parameter bounds for logarithmic and arithmetic-geometric means

Chu, Yu-Ming; Qiu, Ye-Fang; Wang, Miao-Kun; Ma, Xiao-Yan
Tipo: Artigo de Revista Científica
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For fixed $s\geq 1$ and $t_{1},t_{2}\in(0,1/2)$ we prove that the inequalities $G^{s}(t_{1}a+(1-t_{1})b,t_{1}b+(1-t_{1})a)A^{1-s}(a,b)>AG(a,b)$ and $G^{s}(t_{2}a+(1-t_{2})b,t_{2}b+(1-t_{2})a)A^{1-s}(a,b)>L(a,b)$ hold for all $a,b>0$ with $a\neq b$ if and only if $t_{1}\geq 1/2-\sqrt{2s}/(4s)$ and $t_{2}\geq 1/2-\sqrt{6s}/(6s)$. Here $G(a,b)$, $L(a,b)$, $AG(a,b)$ and $A(a,b)$ are the geometric, logarithmic, arithmetic-geometric and arithmetic means of $a$ and $b$, respectively.; Comment: 8 pages

## A Note on the Weighted Harmonic-Geometric-Arithmetic Means Inequalities

Maze, Gerard; Wagner, Urs
Tipo: Artigo de Revista Científica
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In this note, we derive non trivial sharp bounds related to the weighted harmonic-geometric-arithmetic means inequalities, when two out of the three terms are known. As application, we give an explicit bound for the trace of the inverse of a symmetric positive definite matrix and an inequality related to the coefficients of polynomials with positive roots.; Comment: Reviewed article, to appear in Math. Ineq. App

## Characterization of generalized quasi-arithmetic means

Matkowski, Janusz; Páles, Zsolt
Tipo: Artigo de Revista Científica
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In this paper we characterize generalized quasi-arithmetic means, that is means of the form $M(x_1,...,x_n):=(f_1+...+f_n)^{-1}(f_1(x_1)+...+f_n(x_n))$, where $f_1,...,f_n:I\to\mathbb{R}$ are strictly increasing and continuous functions. Our characterization involves the Gauss composition of the cyclic mean-type mapping induced by $M$ and a generalized bisymmetry equation.

## Quasi-arithmetic means of covariance functions with potential applications to space-time data

Porcu, E.; Mateu, J.; Christakos, G.