Prediction and Analysis of Tides and Tidal Currents

An efficient algorithm of tidal harmonic analysis and prediction is presented in this paper. The analysis is strengthened by utilising known relationships between tidal constituents found at a neighbouring reference site. The system of linear equations of the least-squares solution is enhanced with included constraint equations. In the case of inadequate data, ill-conditioning in the system of equations that has appeared in other algorithms is conveniently avoided. In solving the resultant normal equations, Goertzel's recurrence formula is adopted so that the whole compuation time is dramatically reduced.Published versio

Розв’язання рівнянь з використанням загального ряду Фібоначчі

The effective method of solving equations using Fibonacci general series has been shown. The concept ofthe Fibonacci general series has been applied; the recurrence formula of the population volume change has beenobtained. We get the equation based on the recurrence formulas; the largest of this equation is the index of thepopulation volume change. There are some examples of the linear solving equations by means usage the Fibonaccigeneral series in the frame of Excel.Рассмотрен эффективный метод решения уравнения с использованием общих рядов Фибоначчи. Введенопонятие общего ряда Фибоначчи. Приведена рекуррентная формула изменения объема популяции. На основерекуррентных формул получено уравнение, наибольший корень которого является индексом изменения объемапопуляции. Примеры решения линейных уравнений сделаны в электронных таблицах Excel с использованиемобщих рядов Фибоначчи.Роглянуто ефективний метод розв’язання рівняння з використанням загальних рядів Фібоначчі. Введенопоняття загального ряду Фібоначчі. Наведено рекурентну формулу зміни обсягу популяції. На основірекурентних формул отримано рівняння, найбільший корінь якого є індексом зміни обсягу популяції. Велектронних таблицях Excel зробленo приклади розв’язання лінійних рівнянь із використанням загальних рядівФібоначчі

A toolbox to solve coupled systems of differential and difference equations

We present algorithms to solve coupled systems of linear differential equations, arising in the calculation of massive Feynman diagrams with local operator insertions at 3-loop order, which do {\it not} request special choices of bases. Here we assume that the desired solution has a power series representation and we seek for the coefficients in closed form. In particular, if the coefficients depend on a small parameter \ep (the dimensional parameter), we assume that the coefficients themselves can be expanded in formal Laurent series w.r.t.\ \ep and we try to compute the first terms in closed form. More precisely, we have a decision algorithm which solves the following problem: if the terms can be represented by an indefinite nested hypergeometric sum expression (covering as special cases the harmonic sums, cyclotomic sums, generalized harmonic sums or nested binomial sums), then we can calculate them. If the algorithm fails, we obtain a proof that the terms cannot be represented by the class of indefinite nested hypergeometric sum expressions. Internally, this problem is reduced by holonomic closure properties to solving a coupled system of linear difference equations. The underlying method in this setting relies on decoupling algorithms, difference ring algorithms and recurrence solving. We demonstrate by a concrete example how this algorithm can be applied with the new Mathematica package \texttt{SolveCoupledSystem} which is based on the packages \texttt{Sigma}, \texttt{HarmonicSums} and \texttt{OreSys}. In all applications the representation in $x$-space is obtained as an iterated integral representation over general alphabets, generalizing Poincar\'{e} iterated integrals