Euler's 1760 paper on divergent series

AbstractThat Euler was quite aware of the subtleties of assigning a sum to a divergent series is amply demonstrated in his paper De seriebus divergentibus which appeared in Novi commentarii academiae scientiarum Petropolitanae 5 (1754/55), 205–237 (= Opera Omnia (1) 14, 585–617) in the year 1760. The first half of this paper contains a detailed exposition of Euler's views which should be more readily accessible to the mathematical community.The authors present here a translation from Latin of the summary and first twelve sections of Euler's paper with some explanatory comments. The remainder of the paper, treating Wallis' hypergeometric series and other technical matter, is described briefly. Appended is a short bibliography of works concerning Euler which are available to the English-speaking reader

Nested recursions with ceiling function solutions

Consider a nested, non-homogeneous recursion R(n) defined by R(n) = \sum_{i=1}^k R(n-s_i-\sum_{j=1}^{p_i} R(n-a_ij)) + nu, with c initial conditions R(1) = xi_1 > 0,R(2)=xi_2 > 0, ..., R(c)=xi_c > 0, where the parameters are integers satisfying k > 0, p_i > 0 and a_ij > 0. We develop an algorithm to answer the following question: for an arbitrary rational number r/q, is there any set of values for k, p_i, s_i, a_ij and nu such that the ceiling function ceiling{rn/q} is the unique solution generated by R(n) with appropriate initial conditions? We apply this algorithm to explore those ceiling functions that appear as solutions to R(n). The pattern that emerges from this empirical investigation leads us to the following general result: every ceiling function of the form ceiling{n/q}$ is the solution of infinitely many such recursions. Further, the empirical evidence suggests that the converse conjecture is true: if ceiling{rn/q} is the solution generated by any recursion R(n) of the form above, then r=1. We also use our ceiling function methodology to derive the first known connection between the recursion R(n) and a natural generalization of Conway's recursion.Comment: Published in Journal of Difference Equations and Applications, 2010. 11 pages, 1 tabl

Striatal mRNA expression patterns underlying peak dose L-DOPA-induced dyskinesia in the 6-OHDA hemiparkinsonian rat

L-DOPA is the primary pharmacological treatment for relief of the motor symptoms of Parkinson’s disease (PD). With prolonged treatment (⩾5 years) the majority of patients will develop abnormal involuntary movements as a result of L-DOPA treatment, known as L-DOPA-induced dyskinesia. Understanding the underlying mechanisms of dyskinesia is a crucial step toward developing treatments for this debilitating side effect. We used the 6-hydroxydopamine (6-OHDA) rat model of PD treated with a three-week dosing regimen of L-DOPA plus the dopa decarboxylase inhibitor benserazide (4 mg/kg and 7.5 mg/kg s.c., respectively) to induce dyskinesia in 50% of individuals. We then used RNA-seq to investigate the differences in mRNA expression in the striatum of dyskinetic animals, non-dyskinetic animals, and untreated parkinsonian controls at the peak of dyskinesia expression, 60 min after L-DOPA administration. Overall, 255 genes were differentially expressed; with significant differences in mRNA expression observed between all three groups. In dyskinetic animals 129 genes were more highly expressed and 14 less highly expressed when compared with non-dyskinetic and untreated parkinsonian controls. In L-DOPA treated animals 42 genes were more highly expressed and 95 less highly expressed when compared with untreated parkinsonian controls. Gene set cluster analysis revealed an increase in expression of genes associated with the cytoskeleton and phosphoproteins in dyskinetic animals compared with non-dyskinetic animals, which is consistent with recent studies documenting an increase in synapses in dyskinetic animals. These genes may be potential targets for drugs to ameliorate L-DOPA-induced dyskinesia or as an adjunct treatment to prevent their occurrence

The neural speed of familiar face recognition

International audienceRapidly recognizing familiar people from their faces appears critical for social interactions (e.g., to differentiate friend from foe). However, the actual speed at which the human brain can distinguish familiar from unknown faces still remains debated. In particular, it is not clear whether familiarity can be extracted from rapid face individualization or if it requires additional time consuming processing. We recorded scalp EEG activity in 28 subjects performing a go/no-go, famous/non-famous, unrepeated, face recognition task. Speed constraints were used to encourage subjects to use the earliest familiarity information available. Event related potential (ERP) analyses show that both the N170 and the N250 components were modulated by familiarity. The N170 modulation was related to behaviour: subjects presenting the strongest N170 modulation were also faster but less accurate than those who only showed weak N170 modulation. A complementary Multi-Variate Pattern Analysis (MVPA) confirmed ERP results and provided some more insights into the dynamics of face recognition as the N170 differential effect appeared to be related to a first transitory phase (transitory bump of decoding power) starting at around 140 ms, which returned to baseline afterwards. This bump of activity was henceforth followed by an increase of decoding power starting around 200 ms after stimulus onset. Overall, our results suggest that rather than a simple single-process, familiarity for faces may rely on a cascade of neural processes, including a coarse and fast stage starting at 140 ms and a more refined but slower stage occurring after 200 ms