On some combinatorial identities and harmonic sums

For any $m,n\in\mathbb{N}$ we first give new proofs for the following well known combinatorial identities \begin{equation*} S_n(m)=\sum\limits_{k=1}^n\binom{n}{k}\frac{(-1)^{k-1}}{k^m}=\sum\limits_{n\geq r_1\geq r_2\geq...\geq r_m\geq 1}\frac{1}{r_1r_2\cdots r_m} \end{equation*} and $$\sum\limits_{k=1}^n(-1)^{n-k}\binom{n}{k}k^n = n!,$$ and then we produce the generating function and an integral representation for $S_n(m)$. Using them we evaluate many interesting finite and infinite harmonic sums in closed form. For example, we show that $$\zeta(3)=\frac{1}{9}\sum\limits_{n=1}^\infty\frac{H_n^3+3H_nH_n^{(2)}+2H_n^{(3)}}{2^n},$$ and $$\zeta(5)=\frac{2}{45}\sum\limits_{n=1}^{\infty}\frac{H_n^4+6H_n^2H_n^{(2)}+8H_nH_n^{(3)}+3\left(H_n^{(2)}\right)^2+6H_n^{(4)}}{n2^n},$$ where $H_n^{(i)}$ are generalized harmonic numbers defined below.Comment: to appear in Int. J. Number Theory, 201

CASE OF ITALY-ALBANIA RELATIONS: POLICY CULTURAL INTERACTION (1878-1918)

After the Treaty of Berlin, there was a considerable change in the political status ofthe Balkans. Especially in Albania, the political and cultural studies of neighboring countries started to operate. Italy, with the aim of expanding its colonies throughthe weakening Ottoman Empire and by depending on its old heritage, started torealize certain plans in Albania. In our study, Italy's enhancement of cultural oppression on Albania between the years 1878 and 1918 by cultural means will be revealed by probing the newspapers and archival documents of that period.Attention will be drawn on the significant impact of educational and cultural interaction on political relations such as the demand to adopt the Latin alphabet in Albania and increase of schooling, by emphasizing oppression of cultural transformation besides political interaction while creating this effect

Five-year follow-up of bilateral stimulation of the subthalamic nucleus in advanced Parkinson's disease

Background: Although the short-term benefits of bilateral stimulation of the subthalamic nucleus in patients with advanced Parkinson's disease have been well documented, the long-term outcomes of the procedure are unknown. Methods: We conducted a five-year prospective study of the first 49 consecutive patients whom we treated with bilateral stimulation of the subthalamic nucleus. Patients were assessed at one, three, and five years with levodopa (on medication) and without levodopa (off medication), with use of the Unified Parkinson's Disease Rating Scale. Seven patients did not complete the study: three died, and four were lost to follow-up. Results: As compared with base line, the patients' scores at five years for motor function while off medication improved by 54 percent (P<0.001) and those for activities of daily living improved by 49 percent (P<0.001). Speech was the only motor function for which off-medication scores did not improve. The scores for motor function on medication did not improve one year after surgery, except for the dyskinesia scores. On-medication akinesia, speech, postural stability, and freezing of gait worsened between year 1 and year 5 (P<0.001 for all comparisons). At five years, the dose of dopaminergic treatment and the duration and severity of levodopa-induced dyskinesia were reduced, as compared with base line (P<0.001 for each comparison). The average scores for cognitive performance remained unchanged, but dementia developed in three patients after three years. Mean depression scores remained unchanged. Severe adverse events included a large intracerebral hemorrhage in one patient. One patient committed suicide. Conclusions: Patients with advanced Parkinson's disease who were treated with bilateral stimulation of the subthalamic nucleus had marked improvements over five years in motor function while off medication and in dyskinesia while on medication. There was no control group, but worsening of akinesia, speech, postural stability, freezing of gait, and cognitive function between the first and the fifth year is consistent with the natural history of Parkinson's disease

Sharp bounds for harmonic numbers

In the paper, we first survey some results on inequalities for bounding harmonic numbers or Euler-Mascheroni constant, and then we establish a new sharp double inequality for bounding harmonic numbers as follows: For $n\in\mathbb{N}$, the double inequality -\frac{1}{12n^2+{2(7-12\gamma)}/{(2\gamma-1)}}\le H(n)-\ln n-\frac1{2n}-\gamma<-\frac{1}{12n^2+6/5} is valid, with equality in the left-hand side only when $n=1$, where the scalars $\frac{2(7-12\gamma)}{2\gamma-1}$ and $\frac65$ are the best possible.Comment: 7 page

Logarithmic integrals with applications to BBP and Euler-type sums

For real numbers $p,q>1$ we consider the following family of integrals: \begin{equation*} \int_{0}^{1}\frac{(x^{q-2}+1)\log\left(x^{mq}+1\right)}{x^q+1}{\rm d}x \quad \mbox{and}\quad \int_{0}^{1}\frac{(x^{pt-2}+1)\log\left(x^t+1\right)}{x^{pt}+1}{\rm d}x. \end{equation*} We evaluate these integrals for all $m\in\mathbb{N}$, $q=2,3,4$ and $p=2,3$ explicitly. They recover some previously known integrals. We also compute many integrals over the infinite interval $[0,\infty)$. Applying these results we offer many new Euler- BBP- type sums.Comment: Accepted for publication in the Bulletin of Malaysian Mathematical Sciences Societ