Looking backward: From Euler to Riemann

We survey the main ideas in the early history of the subjects on which Riemann worked and that led to some of his most important discoveries. The subjects discussed include the theory of functions of a complex variable, elliptic and Abelian integrals, the hypergeometric series, the zeta function, topology, differential geometry, integration, and the notion of space. We shall see that among Riemann's predecessors in all these fields, one name occupies a prominent place, this is Leonhard Euler. The final version of this paper will appear in the book \emph{From Riemann to differential geometry and relativity} (L. Ji, A. Papadopoulos and S. Yamada, ed.) Berlin: Springer, 2017

Symbolic Evaluations Inspired by Ramanujan's Series for 1/pi

We introduce classes of Ramanujan-like series for $\frac{1}{\pi}$, by devising methods for evaluating harmonic sums involving squared central binomial coefficients, as in the family of Ramanujan-type series indicated below, letting $H_{n} = 1 + \frac{1}{2} + \cdots + \frac{1}{n}$ denote the $n^{\text{th}}$ harmonic number: \begin{align*} & \sum _{n=1}^{\infty } \frac{\binom{2 n}{n}^2 H_n}{16^n(2 n - 1)} = \frac{ 8 \ln (2) - 4 }{\pi}, \\ & \sum _{n=1}^{\infty } \frac{\binom{2 n}{n}^2 H_n}{16^n(2 n-3)} = \frac{120 \ln (2)-68 }{27 \pi}, \\ & \sum _{n = 1}^{\infty } \frac{ \binom{2 n}{n}^2 H_n}{ 16^{ n} (2 n - 5)} = \frac{10680 \ln (2) -6508}{3375 \pi }, \\ & \cdots \end{align*} In this direction, our main technique is based on the evaluation of a parameter derivative of a beta-type integral, but we also show how new integration results involving complete elliptic integrals may be used to evaluate Ramanujan-like series for $\frac{1}{\pi}$ containing harmonic numbers. We present a generalization of the recently discovered harmonic summation formula $$\sum_{n=1}^{\infty} \binom{2n}{n}^{2} \frac{H_{n}}{32^{n}} = \frac{\Gamma^{2} \left( \frac{1}{4} \right)}{4 \sqrt{\pi}} \left( 1 - \frac{4 \ln(2)}{\pi} \right)$$ through creative applications of an integration method that we had previously introduced. We provide explicit closed-form expressions for natural variants of the above series. At the time of our research being conducted, up-to-date versions of Computer Algebra Systems such as Mathematica and Maple could not evaluate our introduced series, such as $$\sum _{n=1}^{\infty } \frac{ \binom{2 n}{n}^2 H_n}{32^n (n + 1)} = 8-\frac{2 \Gamma^2 \left(\frac{1} {4}\right)}{\pi ^{3/2}}-\frac{4 \pi ^{3/2}+16 \sqrt{\pi } \ln (2)}{\Gamma^2 \left(\frac{1}{4}\right)}.$$ We also introduce a class of harmonic summations for Catalan's constant $G$ and $\frac{1}{\pi}$ such as the series $$\sum _{n=1}^{\infty } \frac{ \binom{2 n}{n}^2 H_n}{ 16^{n} (n+1)^2} = 16+\frac{32 G-64 \ln (2)}{\pi }-16 \ln (2),$$ which we prove through a variation of our previous integration method for constructing $\frac{1}{\pi}$ series. We also present a new integration method for evaluating infinite series involving alternating harmonic numbers, and we apply a Fourier--Legendre-based technique recently introduced by Campbell et al., to prove new rational double hypergeometric series formulas for expressions involving $\frac{1}{\pi^2}$, especially the constant $\frac{\zeta(3)}{\pi^2}$, which is of number-theoretic interest

Symbolic Evaluations Inspired by Ramanujan's Series for 1/pi

We introduce classes of Ramanujan-like series for $\frac{1}{\pi}$, by devising methods for evaluating harmonic sums involving squared central binomial coefficients, as in the family of Ramanujan-type series indicated below, letting $H_{n} = 1 + \frac{1}{2} + \cdots + \frac{1}{n}$ denote the $n^{\text{th}}$ harmonic number: \begin{align*} & \sum _{n=1}^{\infty } \frac{\binom{2 n}{n}^2 H_n}{16^n(2 n - 1)} = \frac{ 8 \ln (2) - 4 }{\pi}, \\ & \sum _{n=1}^{\infty } \frac{\binom{2 n}{n}^2 H_n}{16^n(2 n-3)} = \frac{120 \ln (2)-68 }{27 \pi}, \\ & \sum _{n = 1}^{\infty } \frac{ \binom{2 n}{n}^2 H_n}{ 16^{ n} (2 n - 5)} = \frac{10680 \ln (2) -6508}{3375 \pi }, \\ & \cdots \end{align*} In this direction, our main technique is based on the evaluation of a parameter derivative of a beta-type integral, but we also show how new integration results involving complete elliptic integrals may be used to evaluate Ramanujan-like series for $\frac{1}{\pi}$ containing harmonic numbers. We present a generalization of the recently discovered harmonic summation formula $$\sum_{n=1}^{\infty} \binom{2n}{n}^{2} \frac{H_{n}}{32^{n}} = \frac{\Gamma^{2} \left( \frac{1}{4} \right)}{4 \sqrt{\pi}} \left( 1 - \frac{4 \ln(2)}{\pi} \right)$$ through creative applications of an integration method that we had previously introduced. We provide explicit closed-form expressions for natural variants of the above series. At the time of our research being conducted, up-to-date versions of Computer Algebra Systems such as Mathematica and Maple could not evaluate our introduced series, such as $$\sum _{n=1}^{\infty } \frac{ \binom{2 n}{n}^2 H_n}{32^n (n + 1)} = 8-\frac{2 \Gamma^2 \left(\frac{1} {4}\right)}{\pi ^{3/2}}-\frac{4 \pi ^{3/2}+16 \sqrt{\pi } \ln (2)}{\Gamma^2 \left(\frac{1}{4}\right)}.$$ We also introduce a class of harmonic summations for Catalan's constant $G$ and $\frac{1}{\pi}$ such as the series $$\sum _{n=1}^{\infty } \frac{ \binom{2 n}{n}^2 H_n}{ 16^{n} (n+1)^2} = 16+\frac{32 G-64 \ln (2)}{\pi }-16 \ln (2),$$ which we prove through a variation of our previous integration method for constructing $\frac{1}{\pi}$ series. We also present a new integration method for evaluating infinite series involving alternating harmonic numbers, and we apply a Fourier--Legendre-based technique recently introduced by Campbell et al., to prove new rational double hypergeometric series formulas for expressions involving $\frac{1}{\pi^2}$, especially the constant $\frac{\zeta(3)}{\pi^2}$, which is of number-theoretic interest

2018 Furman University Faculty Scholarship Reception Program

On February 23, 2018, the Libraries and the Office of the Provost hosted Furman’s second Faculty Scholarship Reception to recognize and celebrate the scholarly publications and creative works of Furman faculty members. The reception, held in the Blackwell Atrium of the James B. Duke Library, showcased scholarship published by 70 faculty members during the 2017 calendar year. The following faculty provided four-minute presentations on their scholarly or creative works: Omar Camenates, Associate Professor, Music Kevin Hutson, Professor, and Liz Bouzarth, Associate Professor, Mathematics Amy Jonason, Assistant Professor, Sociology Bill Pierce, Professor, and Scott Murr, Assistant Professor, Health Sciences Greg Springsteen, Associate Professor, Chemistr

Analytic Aspects of the Riemann Zeta and Multiple Zeta Values

This manuscript contains two parts. The first part contains fast converging series representations involving $\zeta(2n)$ for Apery's constant $\zeta(3)$. These representations are obtained via Clausen acceleration formulae. Moreover, we also find evaluations for more general rational zeta series involving $\zeta(2n)$ and binomial coefficients. The second part will be devoted to the multiple zeta and special Hurwitz zeta values (multiple $t$-values). In this part, using a new approach involving integer powers of $\arcsin$ which come from particular values of the Gauss hypergeometric function, we are able to provide new proofs for the evaluations of $\zeta(2, 2, \ldots, 2)$, and $t(2, 2, \ldots, 2)$. Moreover, we are able to evaluate $\zeta(2, 2, \ldots, 2, 3)$, and $t(2, 2, \ldots, 2, 3)$ in terms of rational zeta series involving $\zeta(2n)$. On the other hand, using properties of the Clausen functions we can express these rational zeta series as a finite $\mathbb{Q}$-linear combinations of powers of $\pi$ and odd zeta values. In particular, we deduce the famous formula of Zagier for the Hoffman elements in a special case. Zagier's formula is a remarkable example of both strength and the limits of the motivic formalism used by Brown in proving Hoffman's conjecture where the motivic argument does not give us a precise value for the special multiple zeta values $\zeta(2, 2, \ldots, 2, 3, 2, 2,\ldots, 2)$ as rational linear combinations of products $\zeta(m)\pi^{2n}$ with $m$ odd. In \cite{Zagier1} the formula is proven indirectly by computing the generating functions of both sides in closed form and then showing that both are entire functions of exponential growth and that they agree at sufficiently many points to force their equality

Modulação da resposta imunológica através da manipulação de agregados proteicos

Protein aggregation is induced by a wide variety of cellular stresses, including amino acid starvation, virus infection, endoplasmic reticulum stress, lipopolysaccharide, and oxidative stress. It has been suggested that altered proteostasis impacts the inflammatory response, but the underlying mechanism between altered proteostasis and inflammation is still poorly understood. Here, we aim to analyse the impact of protein aggregation in the response of human Dendritic Cells, focusing on plasmacytoid dendritic cells (pDCs). The number of aggregates was manipulated in a pDC cell line, CAL-1, by using both autophagy and proteasome inhibitors. Proteasome inhibition in pDCs induced assembly of large p62-based aggregates, together with an increase on IL-1β secretion and cell death, in an irreversible and cell specific manner. The same effects were not observed upon autophagy inhibition nor on a monocytic cell line. To study the mechanism behind the increase in the inflammatory response upon proteasome inhibition, p62-knockout cells were generated using the CRISPR/Cas9 system. Our data suggests that neither p62 nor the NLRP3 inflammasome are required for induction of cell death upon proteasome inhibition in pDCs. Overall, we conclude that proteasome inhibition induces an inflammatory response specific to pDCs. We propose that this effect must be considered when using proteasome inhibitors as potential drugs for the treatment of pDCs derived immune-mediated disorder and thus, mores studies should be done to clarify the outcome of proteasome inhibition on pDCs.A agregação de proteínas é um processo que pode ser induzido por uma grande variedade de stresses celulares, incluindo privação de aminoácidos, infeção viral, stress do retículo endoplasmático, lipopolissacarídeos, e stress oxidativo. Alterações na proteostase parecem ter um impacto na resposta inflamatória, mas os mecanismos subjacentes a este impacto são ainda pouco conhecidos. Esta tese tem por objetivo analisar de que forma a agregação de proteínas influencia a resposta das células dendríticas humanas, em particular das células dendríticas plasmocitóides (pDCs). Foram utilizados inibidores de autofagia e do proteossoma para manipular o número de agregados na linha celular de pDCs, CAL-1. A inibição do proteossoma em pDCs induziu a formação de agregados constituídos por p62 de grandes dimensões, e levou à secreção de IL-1β e morte celular, de forma irreversível e específica destas células. Os mesmos efeitos não foram observados após inibição da autofagia nem se verificaram numa outra linha celular, monocítica. Para estudar o mecanismo por detrás do aumento da resposta inflamatória após a inibição do proteossoma, foram criadas células knockout para o p62 usando o sistema CRISPR/Cas9. Os nossos resultados sugerem que nem o p62 nem o inflamassoma de NLRP3 são necessários para a indução da morte celular por inibição do proteossoma em pDCs. De forma geral, concluímos que a inibição do proteossoma induz uma resposta inflamatória específica para pDCs. Propomos que esse efeito deve ser tido em consideração quando se utilizem inibidores de proteossoma como potenciais fármacos para o tratamento de distúrbios mediados por pDCs e, portanto, mais estudos devem ser feitos para esclarecer o efeito da inibição do proteossoma em pDCs.Mestrado em Biomedicina Molecula