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By applying the MC algorithm and the Bauer-Muir transformation for continued fractions, in this paper we shall give six examples to show how to establish an infinite set of continued fraction formulas for certain Ramanujan-type series, such as Catalans constant, the exponential function, etc.
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We give an estimate of the general divided differences $[x_0,dots,x_m;f]$, where some of the $x_i$s are allowed to coalesce (in which case, $f$ is assumed to be sufficiently smooth). This estimate is then applied to significantly strengthen Whitney and Marchaud celebrated inequalities in relation to Hermite interpolation. For example, one of the numerous corollaries of this estimate is the fact that, given a function $fin C^{(r)}(I)$ and a set $Z={z_j}_{j=0}^mu$ such that $z_{j+1}-z_j geq lambda |I|$, for all $0le j le mu-1$, where $I:=[z_0, z_mu]$, $|I|$ is the length of $I$ and $lambda$ is some positive number, the Hermite polynomial ${mathcal L}(cdot;f;Z)$ of degree $le rmu+mu+r$ satisfying ${mathcal L}^{(j)}(z_ u; f;Z) = f^{(j)}(z_ u)$, for all $0le u le mu$ and $0le jle r$, approximates $f$ so that, for all $xin I$, [ big|f(x)- {mathcal L}(x;f;Z) big| le C left( mathop{rm dist} olimits(x, Z) right)^{r+1} int_{mathop{rm dist} olimits(x, Z)}^{2|I|}frac{omega_{m-r}(f^{(r)},t,I)}{t^2}dt , ] where $m :=(r+1)(mu+1)$, $C=C(m, lambda)$ and $mathop{rm dist} olimits(x, Z) := min_{0le j le mu} |x-z_j|$.
We establish partial semigroup property of Riemann-Liouville and Caputo fractional differential operators. Using this result we prove theorems on reduction of multi-term fractional differential systems to single-term and multi-order systems, and prove existence and uniqueness of solution to multi-term Caputo fractional differential systems
Using the Laplace derivative a Perron type integral, the Laplace integral, is defined. Moreover, it is shown that this integral includes Perron integral and to show that the inclusion is proper, an example of a function is constructed, which is Laplace integrable but not Perron integrable. Properties of integrals such as fundamental theorem of calculus, Hakes theorem, integration by parts, convergence theorems, mean value theorems, the integral remainder form of Taylors theorem with an estimation of the remainder, are established. It turns out that concerning the Alexiewiczs norm, the space of all Laplace integrable functions is incomplete and contains the set of all polynomials densely. Applications are shown to Poisson integral, a system of generalised ordinary differential equations and higher-order generalised ordinary differential equation.
In this paper we show a deterministic parallel all-pairs shortest paths algorithm for real-weighted directed graphs. The algorithm has $tilde{O}(nm+(n/d)^3)$ work and $tilde{O}(d)$ depth for any depth parameter $din [1,n]$. To the best of our knowledge, such a trade-off has only been previously described for the real-weighted single-source shortest paths problem using randomization [Bringmann et al., ICALP17]. Moreover, our result improves upon the parallelism of the state-of-the-art randomized parallel algorithm for computing transitive closure, which has $tilde{O}(nm+n^3/d^2)$ work and $tilde{O}(d)$ depth [Ullman and Yannakakis, SIAM J. Comput. 91]. Our APSP algorithm turns out to be a powerful tool for designing efficient planar graph algorithms in both parallel and sequential regimes. One notable ingredient of our parallel APSP algorithm is a simple deterministic $tilde{O}(nm)$-work $tilde{O}(d)$-depth procedure for computing $tilde{O}(n/d)$-size hitting sets of shortest $d$-hop paths between all pairs of vertices of a real-weighted digraph. Such hitting sets have also been called $d$-hub sets. Hub sets have previously proved especially useful in designing parallel or dynamic shortest paths algorithms and are typically obtained via random sampling. Our procedure implies, for example, an $tilde{O}(nm)$-time deterministic algorithm for finding a shortest negative cycle of a real-weighted digraph. Such a near-optimal bound for this problem has been so far only achieved using a randomized algorithm [Orlin et al., Discret. Appl. Math. 18].
A New Trinomial Recombination Tree Algorithm and Its Applications
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