Related papers: Varieties of Sums of Powers for binary forms
Symmetric tensor decomposition is an important problem with applications in several areas for example signal processing, statistics, data analysis and computational neuroscience. It is equivalent to Waring's problem for homogeneous…
We revisit recent work of Heath-Brown on the average order of the quantity r(L_1)r(L_2)r(L_3)r(L_4), for suitable binary linear forms L_1,..., L_4, for integers ranging over quite general regions. In addition to improving the error term in…
Several combinatorial identities are presented, involving Stirling functions of the second kind with a complex variable. The identities involve also Stirling numbers of the first kind, binomial coefficients and harmonic numbers.
Elimination theory has many applications, in particular, it describes explicitly an image of a complex line under rational transformation and determines the number of common zeroes of two polynomials in one variable. We generalize classical…
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Starting with a general discussion, a program is sketched for a quantization based on dilations. This resolving-power quantization is simplest for scalar field theories. The hope is to find a way to relax the requirement of locality so that…
We give a bijective parameter representation for a sum of squares of numbers being equal to another sum of squares of numbers.
Waring problem for forms is important and classical in mathematics. It has been widely investigated because of its wide applications in several areas. In this paper, we consider the Waring problem for binary forms with complex coefficients.…
Recently, the properties of a binomial sum related to the multi-link inverted pendulum enumeration problem have been studied. In this note, we establish bounds for this binomial sum.
Two types of finite series of products of harmonic numbers involving nonnegative integer powers are evaluated, also yielding two other important harmonic number identities. The recursion formulas for these sums are derived, which are easily…
In this article, we continue our recent investigations on bilinear sums and additive energies with modular square roots. Here we improve our recent results for the case when the ranges of variables are large. We use these results to make…
Shapes do not define a linear space. This paper explores the linear structure of deformations as a representation of shapes. This transforms shape optimization to a variant of optimal control. The numerical challenges of this point of view…
The irreducible decomposition of a unitary representation often contains continuous spectrum when restricted to a non-compact subgroup. The author singles out a nice class of branching problems where each irreducible summand occurs…
We consider the decomposition of bounded linear operators on Hilbert spaces in terms of functions forming frames. Similar to the singular-value decomposition, the resulting frame decompositions encode information on the structure and…
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Foundations of the formal series $*$ -- calculus in deformation quantisation are discussed. Several classes of continuous linear functionals over algebras applied in classical and quantum physics are introduced. The notion of nonnegativity…
We develop a theory of sesquilinear forms over finite fields, investigating their representations via polynomials and coefficient matrices, along with classification results for these forms. Through their connection to quadratic forms, we…
We study the quantitative relationship between the cones of nonnegative polynomials, cones of sums of squares and cones of sums of powers of linear forms. We derive bounds on the volumes (raised to the power reciprocal to the ambient…
We explicitly determine the Ap\'ery limits for the sums of powers of binomial coefficients. As an application, we prove a weak version of Franel's conjecture on the order of the recurrences for these sequences. Namely, we prove the…