Related papers: Remark on Cavalieri's Quadrature Formula

Let $a, b, c,$ and $n$ be integers, with $a$ nonzero and $n$ at least two. Necessary and sufficient conditions on these parameters are derived which guarantee that all solutions of the congruence \[ ax^2+bx+c \equiv 0\ \textrm{mod}\ n \]…

We demonstrate the fact that linearity is a meaningful symmetry in the sense of Lie and Noether. The role played by that `linearity symmetry' in the quadrature of linear ordinary second-order differential equations is reviewed, by the use…

The classical quadratic formula and some of its lesser known variants for solving the quadratic equation are reviewed. Then, a new formula for the roots of a quadratic polynomial is presented.

In this paper, we first derive some explicit formulas for the computation of the n-th order divergence operator in Malliavin calculus in the one-dimensional case. We then extend these results to the case of isonormal Gaussian space. Our…

The big $-1$ Jacobi polynomials $(Q_n^{(0)}(x;\alpha,\beta,c))_n$ have been classically defined for $\alpha,\beta\in(-1,\infty)$, $c\in(-1,1)$. We extend this family so that wider sets of parameters are allowed, i.e., they are non-standard.…

We formulate a solution to the Algebraic version of the Inverse Jacobi problem. Using this solution we produce explicit addition laws on any algebraic curve generalizing the law suggested by Leykin [2] in the case of (n, s) curves. This…

We show that the bicovariant first order differential calculi on a factorisable semisimple quantum group are in 1-1 correspondence with irreducible representations $V$ of the quantum group enveloping algebra. The corresponding calculus is…

The standard design principle for quadrature formulas is that they should be exact for integrands of a given class, such as polynomials of a fixed degree. We show how this principle fails to predict the actual behavior in four cases:…

The numerical analysis of variational integrators relies on variational error analysis, which relates the order of accuracy of a variational integrator with the order of approximation of the exact discrete Lagrangian by a computable…

Let $a_k(n)$ denotes the number of representations of a non-negative integer $n$ as sum of $k$ quadratic forms of the type $x^2+xy+y^2$ and $a_{\lambda_1,\lambda_2,\lambda_3\dots\lambda_k}(n)$ denotes the number of representations $n$ as a…

The fundamental problem of calculus of variations is considered when solutions are differentiable curves on locally convex spaces. Such problems admit an extension of the Euler-Lagrange equations [Orlov 2002] for continuously normally…

A class of numerical quadrature rules is derived, with equally-spaced nodes, and unit weights except at a few points at each end of the series, for which "corrections" (not using any further information about the integrand) are added to the…

The first objective of the paper is to estimate logarithmic partial derivative for meromorphic functions in several complex variables. Our estimations for logarithmic partial derivatives extend the results of Gundersen \cite{GG2} to the…

In this paper, we find a basis for the space of modular forms of weight $2$ on $\Gamma_1(48)$. We use this basis to find formulas for the number of representations of a positive integer $n$ by certain quaternary quadratic forms of the form…

Sequences of discrete Levy and adjoint Levy transformations for the multidimensional quadrilateral lattices are studied. After a suitable number of iterations we show how all the relevant geometrical features of the transformed…

In a recent paper by Ibragimov [N. H. Ibragimov, Invariant Lagrangians and a new method of integration of nonlinear equations, J. Math. Anal. Appl. 304 (2005) 212--235] a method was presented in order to find Lagrangians of certain…

In this largely expository article, we present a Kawamata-Viehweg type formulation of the (logarithmic) Akizuki-Nakano Vanishing Theorem. While the result is likely known to the experts, it does not seem to appear in the existing…

Let $f(X_1,\dots, X_n)$ be a nonzero multilinear noncommutative polynomial. If $A$ is a unital algebra with a surjective inner derivation, then every element in $A$ can be written as $f(a_1,\dots,a_n)$ for some $a_i\in A$.

A general Riccati equation is integrated in quadratures in case one of its coefficients is an arbitrary function and two others are expressed through it.

All squigonometric functions admit derivatives that can be expressed as polynomials of the squine and cosquine. We introduce a general framework that allows us to determine these polynomials recursively. We also provide an explicit formula…