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Register a new userExact and approximate solutions to the minimum of $1+x+cdots+x^{2n}$
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The polynomial $f_{2n}(x)=1+x+cdots+x^{2n}$ and its minimizer on the real line $x_{2n}=operatorname{arg,inf} f_{2n}(x)$ for $ninBbb N$ are studied. Results show that $x_{2n}$ exists, is unique, corresponds to $partial_x f_{2n}(x)=0$, and resides on the interval $[-1,-1/2]$ for all $n$. It is further shown that $inf f_{2n}(x)=(1+2n)/(1+2n(1-x_{2n}))$ and $inf f_{2n}(x)in[1/2,3/4]$ for all $n$ with an exact solution for $x_{2n}$ given in the form of a finite sum of hypergeometric functions of unity argument. Perturbation theory is applied to generate rapidly converging and asymptotically exact approximations to $x_{2n}$. Numerical studies are carried out to show how many terms of the perturbation expansion for $x_{2n}$ are needed to obtain suitably accurate approximations to the exact value.
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This paper does exactly what the title says it does. It expands the given series to arrive at the familiar pentagonal number expansion, also known as the pentagonal number theorem, and recalls its application to partition numbers. The paper is translated from Eulers Latin original into German.
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We show that all meromorphic solutions of the stationary reduction of the real cubic Swift-Hohenberg equation are elliptic or degenerate elliptic. We then obtain them all explicitly by the subequation method, and one of them appears to be a new elliptic solution.
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In an earlier article together with Carlos DAndrea [BDKSV2017], we described explicit expressions for the coefficients of the order-$d$ polynomial subresultant of $(x-alpha)^m$ and $(x-beta)^n $ with respect to Bernsteins set of polynomials ${(x-alpha)^j(x-beta)^{d-j}, , 0le jle d}$, for $0le d<min{m, n}$. The current paper further develops the study of these structured polynomials and shows that the coefficients of the subresultants of $(x-alpha)^m$ and $(x-beta)^n$ with respect to the monomial basis can be computed in linear arithmetic complexity, which is faster than for arbitrary polynomials. The result is obtained as a consequence of the amazing though seemingly unnoticed fact that these subresultants are scalar multiples of Jacobi polynomials up to an affine change of variables.
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