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, 2003 In Chapter 11 we considered problems that can be cast in the language of graph theory: If we draw some special graphs in the plane, into how many parts do these graphs divide the plane? Indeed, we start with a set of lines; we consider the intersections of the given lines as nodes of the graph, and the segments arising on these lines as the edges of ...
László Lovász +2 more
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Corrected Euler-Maclaurin’s formulae
Rendiconti del Circolo Matematico di Palermo, 2005The aim of this paper is to derive corrected Euler-Maclaurin's formulae, i.e. open type quadrature formulae where the integral is approximated not only with the values of the function in points (5a+b)/6, (a+b)/2 and (a+5b)/6, but also with values of the first derivative in end points of the interval.
Iva Franjić, Josip Pečarić
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Three applications of Euler’s formula
1998A graph is planar if it can be drawn in the plane ℝ2 without crossing edges (or, equivalently, on the 2-dimensional sphere S 2). We talk of a plane graph if such a drawing is already given and fixed. Any such drawing decomposes the plane or sphere into a finite number of connected regions, including the outer (unbounded) region, which are referred to ...
Günter M. Ziegler, Martin Aigner
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The Mathematical Gazette, 1949
The most surprising thing about this formula is its use of the Bernoulli numbers, and it is natural to ask why they appear. The answer is that it is not the B’s which insist on entry, but the numbers Ar = Br/r!. We use the notation where all other B with odd subscripts vanish, and those with even subscripts are alternately positive and negative.
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The most surprising thing about this formula is its use of the Bernoulli numbers, and it is natural to ask why they appear. The answer is that it is not the B’s which insist on entry, but the numbers Ar = Br/r!. We use the notation where all other B with odd subscripts vanish, and those with even subscripts are alternately positive and negative.
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On the generalized Euler-Maclaurin formula [PDF]
ZAMM, 2005 The paper deals with a numerical quadrature formula. The method is based on the classical trapezoidal method with Gregory-type end correction. For this method, the author derives an error expansion in powers of the step size. The coefficients of the expansion are given in terms of higher order derivatives of the integrand, multiplied by certain numbers
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Panamerican mathematical journal, 2001
Modified versions of the Euler-Simpson formula, for Lipschitzian functions, functions of bounded variation and functions with derivatives in L_p-spaces, are given and applied to prove some inequalities and quadrature formulae.
Pečarić, Josip +2 more
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Modified versions of the Euler-Simpson formula, for Lipschitzian functions, functions of bounded variation and functions with derivatives in L_p-spaces, are given and applied to prove some inequalities and quadrature formulae.
Pečarić, Josip +2 more
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International Journal of Mathematical Education in Science and Technology, 2001
A formula for tilings of a rectangle, analogous to Euler's formula for polyhedra, is discussed, with particular reference to how it may be used in a classroom investigation.
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A formula for tilings of a rectangle, analogous to Euler's formula for polyhedra, is discussed, with particular reference to how it may be used in a classroom investigation.
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A Product Formula for Euler's Totient
Bulletin of the London Mathematical Society, 1985The author generalizes some results about the intersections (or projections) of given integral lattices with (or into) rational subspaces, and applies them to get a product formula for a numerical invariant associated to subspaces of \({\mathbb{R}}^ n\), which are orthogonal to \((1,1,...,,1)\in {\mathbb{R}}^ n\).
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A generalization of the Euler-Maclaurin formula
Mathematical Notes, 1997The author proves the following theorem: If \(f\) and \(f^{(r)}\) are of bounded variation on \([n,\infty)\), and \(\lim f^{(\nu)}(x)= 0\) for \(\nu\in [0,r]\) as \(x\to +\infty\), then for \(x\in [-\pi,\pi]\setminus \{0\}\) the following relation is valid: \[ \begin{multlined} \sum^\infty_{k= n} f(k)e^{ikx}= \int^\infty_n f(u) e^{iux}du+ {1\over 2} f ...
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The American Mathematical Monthly, 1936
(1936). An Euler Summation Formula. The American Mathematical Monthly: Vol. 43, No. 1, pp. 9-21.
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(1936). An Euler Summation Formula. The American Mathematical Monthly: Vol. 43, No. 1, pp. 9-21.
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