By employing the univariate series expansion of classical hypergeometric series formulae, Shen [L.-C. Shen, Remarks on some integrals and series involving the Stirling numbers and ζ(n), Trans. Amer. Math. Soc. 347 (1995) 1391-1399] and Choi and Srivastava [J. Choi, H.M. Srivastava, Certain classes of infinite series, Monatsh. Math. 127 (1999) 15-25; J. Choi, H.M. Srivastava, Explicit evaluation of Euler and related sums, Ramanujan J. 10 (2005) 51-70] investigated the evaluation of infinite series related to generalized harmonic numbers. More summation formulae have systematically been derived by Chu [W. Chu, Hypergeometric series and the Riemann Zeta function, Acta Arith. 82 (1997) 103-118], who developed fully this approach to the multivariate case. The present paper will explore the hypergeometric series method further and establish numerous summation formulae expressing infinite series related to generalized harmonic numbers in terms of the Riemann Zeta function ζ(m) with m = 5, 6, 7, including several known ones as examples.
The Cauchy problem for a simplified shallow elastic fluids model, one 3 × 3 system of Temple's type, is studied and a global weak solution is obtained by using the compensated compactness theorem coupled with the total variation estimates on the first and third Riemann invariants, where the second Riemann invariant is singular near the zero layer depth ( = 0). This work extends in some sense the previous works, (Serre, 1987) and (Leveque and Temple, 1985), which provided the global existence of weak solutions for 2 × 2 strictly hyperbolic system and (Heibig, 1994) for × strictly hyperbolic system with smooth Riemann invariants.
By means of two hypergeometric summation formulae, we establish two large classes of infinite series identities with harmonic numbers and central binomial coefficients. Up to now, these numerous formulae have hidden behind very few known identities of Apéry-like series for Riemann-zeta function, discovered mainly by Lehmer [14] and Elsner [12] as well as Borwein et al. [4, 5, 7]. All the computation and verification are carried out by an appropriately-devised Mathematica package.
Trigonometric sums over the angles equally distributed on the upper half plane are investigated systematically. Their generating functions and explicit formulae are established through the combination of the formal power series method and partial fraction decompositions.For a natural number n and trigonometric function T (x) (for example, sec-and csc-functions), consider the following finite sums with a free parameter y:When y = 0, the corresponding sums have extensively been studied by Chu and Marini [5,6] through partial fraction decompositions. Berndt [2] has employed the Cauchy residue theorem to treat the trigonometric reciprocity. Similar trigonometric sums have important applications in classical analysis, such as integer-valued problems by Byrne and Smith [3], Dedekind sums by Gessel [8] and Zagier [11], the matrix spectrum by Calogero [4, §2.4.5.3] as well as trigonometric approximation and interpolation in Kress [9, §8.2]. For the parametric trigonometric sums, refer to the most recent works due to Chu [7] and Mohlenkamp and Monzón [10].
Obesity is a primary risk factor for many diseases such as certain cancers. In this study, we have developed three algorithms including a random-walk based method OBNet, a shortest-path based method OBsp and a direct-overlap method OBoverlap, to reveal obesity-disease connections at protein-interaction subnetworks corresponding to thousands of biological functions and pathways. Through literature mining, we also curated an obesity-associated disease list, by which we compared the methods. As a result, OBNet outperforms other two methods. OBNet can predict whether a disease is obesity-related based on its associated genes. Meanwhile, OBNet identifies extensive connections between obesity genes and genes associated with a few diseases at various functional modules and pathways. Using breast cancer and Type 2 diabetes as two examples, OBNet identifies meaningful genes that may play key roles in connecting obesity and the two diseases. For example, TGFB1 and VEGFA are inferred to be the top two key genes mediating obesity-breast cancer connection in modules associated with brain development. Finally, the top modules identified by OBNet in breast cancer significantly overlap with modules identified from TCGA breast cancer gene expression study, revealing the power of OBNet in identifying biological processes involved in the disease.
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