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The cone of nonnegative polynomials with nonnegative coefficients and linear operators preserving this cone

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Katkova O., Vishnyakova A.

Open Mathematics

2014

tom 12

nr 5

752-760

Let be the cone of real univariate polynomials of degree ≤ 2n which are nonnegative on the real axis and have nonnegative coefficients. We describe the extremal rays of this convex cone and the class of linear operators, acting diagonally in the standard monomial basis, preserving this cone.

Evaluation of the convolution sum involving the sum of divisors function for 22, 44 and 52

68%

Ntienjem E.

Open Mathematics

2017

tom 15

nr 1

446-458

The convolution sum, [...] ∑(l,m)∈N02αl+βm=nσ(l)σ(m), $ \begin{array}{} \sum\limits_{{(l\, ,m)\in \mathbb{N}_{0}^{2}}\atop{\alpha \,l+\beta\, m=n}} \sigma(l)\sigma(m), \end{array} $ where αβ = 22, 44, 52, is evaluated for all natural numbers n. Modular forms are used to achieve these evaluations. Since the modular space of level 22 is contained in that of level 44, we almost completely use the basis elements of the modular space of level 44 to carry out the evaluation of the convolution sums for αβ = 22. We then use these convolution sums to determine formulae for the number of representations of a positive integer by the octonary quadratic forms [...] a(x12+x22+x32+x42)+b(x52+x62+x72+x82), $a\,(x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2})+b\,(x_{5}^{2}+x_{6}^{2}+x_{7}^{2}+x_{8}^{2}),$ where (a, b) = (1, 11), (1, 13).

Ograniczanie wyników

2 Open Mathematics

1 Katkova O.

1 Ntienjem E.

1 Vishnyakova A.

1 2017

1 2014

Wyniki wyszukiwania

The cone of nonnegative polynomials with nonnegative coefficients and linear operators preserving this cone

Evaluation of the convolution sum involving the sum of divisors function for 22, 44 and 52