The Möbius inversion formula for Fourier series applied to Bernoulli and Euler polynomials

AbstractHurwitz found the Fourier expansion of the Bernoulli polynomials over a century ago. In general, Fourier analysis can be fruitfully employed to obtain properties of the Bernoulli polynomials and related functions in a simple manner. In addition, applying the technique of Möbius inversion from analytic number theory to Fourier expansions, we derive identities involving Bernoulli polynomials, Bernoulli numbers, and the Möbius function; this includes formulas for the Bernoulli polynomials at rational arguments. Finally, we show some asymptotic properties concerning the Bernoulli and Euler polynomials

Asymptotic estimates for Apostol-Bernoulli and Apostol-Euler polynomials

We analyze the asymptotic behavior of the Apostol-Bernoulli polynomials $\mathcal{B}_{n}(x;\lambda)$ in detail. The starting point is their Fourier series on $[0,1]$ which, it is shown, remains valid as an asymptotic expansion over compact subsets of the complex plane. This is used to determine explicit estimates on the constants in the approximation, and also to analyze oscillatory phenomena which arise in certain cases. These results are transferred to the Apostol-Euler polynomials $\mathcal{E}_{n}(x;\lambda)$ via a simple relation linking them to the Apostol-Bernoulli polynomials.Comment: 16 page

Acculturation process and life domains: Different perceptions of native and immigrant adults in italy

Background: Acculturation process has taken up a relevant place in cross-cultural psychology by demonstrating the strong relationships between cultural context and individual behavioral development. Aim: The purpose of this study is to analyse acculturation strategies and attitudes in different life domains of native and immigrant adults living in Italy, following the Relative Acculturation Extended Model (RAEM). Methods: The participants were 250 Italian native and 100 immigrant adults who completed a questionnaire with items to measure their acculturation strategies (real plane) and attitudes (ideal plane), in general and related to different life domains (peripheral and central). Results: Results revealed that the acculturation attitude of immigrants is integration, whereas Italians prefer their assimilation. Conclusion: However, when different life domains are taken into account, immigrants claim to put in practice and prefer integration in most of the domains, whereas Italians perceive immigrants are separated but they prefer their assimilation or integration, depending on the specific domain

Existence and reduction of generalized Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi polynomials

summary:One can find in the mathematical literature many recent papers studying the generalized Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi polynomials, defined by means of generating functions. In this article we clarify the range of parameters in which these definitions are valid and when they provide essentially different families of polynomials. In particular, we show that, up to multiplicative constants, it is enough to take as the “main family” those given by $$\Big ( \frac{2}{\lambda e^t+1} \Big )^\alpha e^{xt} = \sum _{n=0}^{\infty } \mathcal{E}^{(\alpha )}_{n}(x;\lambda ) \frac{t^n}{n!}\,, \qquad \lambda \in \mathbb{C}\setminus \lbrace -1\rbrace \,,$$ and as an “exceptional family” $$\Big ( \frac{t}{e^t-1} \Big )^\alpha e^{xt} = \sum _{n=0}^{\infty } \mathcal{B}^{(\alpha )}_{n}(x) \frac{t^n}{n!}\,,$$ both of these for $\alpha \in \mathbb{C}$

Some functional relations derived from the Lindelöf-Wirtinger expansion of the Lerch transcendent function

The Lindelöf-Wirtinger expansion of the Lerch transcendent function implies, as a limiting case, Hurwitz’s formula for the eponymous zeta function. A generalized form of M ¨obius inversion applies to the Lindelöf-Wirtinger expansion and also implies an inversion formula for the Hurwitz zeta function as a limiting case. The inverted formulas involve the dynamical system of rotations of the circle and yield an arithmetical functional equation

TRIZ and lean philosophies applied together in management activities

Purpose: In the current market, the Portuguese industry faces strong competition from countries with substantially lower operating costs. The agri-food industry is also subject to increasing competition, both nationally and internationally. The utilization of methodologies that, in addition to continuous improvement, provide the development of creative and innovative solutions may be relevant for highlighting and differentiating between organizations. In this work, Teoriya Resheniya Izobretatelskikh Zadach and Lean Philosophies have been implemented together. Methodologies as Matrix of Ideality Matrix of Contradiction, Single Minute Exchange of Die, 5S and the Substance-Field analyses were used as complementary tools to improve the production management activities of agri-food sector. Strongly focused on the issues of time wastage and the organization and management of the filling line, the implementation of these methodologies led to a reduction of setup times, of the operators’ movement and an improvement in the line’s management and organization, and improve the results of 5S audits, depending on the work station. The aims of this study is the implementation of diferent methodologies, philosophies and tools to improve management activities. Methodology/Approach: The methodology presented in this work involves using Lean and TRIZ methodologies together. Namely Matrix of Ideality, the Matrix of Contradictions, SMED, 5S and the Substance-Field Analysis, were used to optimize the management activities, in agro-food industrial sector. The methodology strongly focused on the issues of time wastage, organization and management of an filling line. Findings: The implementation of methodologies led to a reduction of setup times by 60% to 70%, a reduction of the operators’ movement during format changes by 26.2%, and an improvement in the line’s management and organization by 9% to 12%, depending on the work station. Research Limitation/implication: In the management activities, the results are not universal. Therefore, the same methodology applied in another industrial sector will present differents results. Nevertheless, the application of this methodology and the tools choosen only make sense if it facilitates the management of improvements. Originality/Value of paper: There is a need and demand for innovative solutions that lead to continuous improvement, production processes optimization and resource saving. The combination of different validated methologies, with new concrete studies, is always in scientific and technical interest and continuously sought-after (demanded, welcome) by academic and industrial sectors.publishersversionpublishe

Malignant pilomatricoma with multiple bone metastases in a dog: Histological and immunohistochemical study.

An eleven year-old mongrel dog was referred with a history of left forelimb lameness and an ulcerated mass on the neck. Histologically, the cutaneous neoplasm revealed cystic lobules composed of basaloid cells with abrupt transition to central keratotic material, containing pycnotic and shadow cells. Approximately 3 months after primary diagnosis, a lesion of the cortical bone on the left humerus was observed using X-ray. Samples obtained from the humerus were processed for histopathological examination and the neoplastic tissue was observed to be similar to the type identified in the neck. Based on these findings, the tumor was diagnosed as a malignant pilomatricoma (MP) with bone metastasis. MP is a rare skin tumor that originates from hair matrix cells. To date, only nine reports have been presented in dogs. In the present study, we discuss the cytological and histological patterns of MP, confirmed by immunohistochemistry using β catenin antibody