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Register a new userPacking theory derived from phyllotaxis and products of linear forms
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textit{Parastichies} are spiral patterns observed in plants and numerical patterns generated using golden angle method. We generalize this method for botanical pattern formation, by using Markoff theory and the theory of product of linear forms, to obtain a theory for (local) packing of any Riemannian manifolds of general dimensions $n$ with a locally diagonalizable metric, including the Euclidean spaces. Our method is based on the property of some special lattices that the density of the lattice packing maintains a large value for any scale transformations in the directions of the standard Euclidean axes, and utilizes maps that fulfill a system of partial differential equations. Using this method, we prove that it is possible to generate almost uniformly distributed point sets on any real analytic Riemann surfaces. The packing density is bounded below by approximately 0.7. A packing with logarithmic-spirals and a 3D analogue of the Vogel spiral are obtained as a result. We also provide a method to construct $(n+1)$-dimensional Riemannian manifolds with diagonal and constant-determinant metrics from $n$-dimensional manifolds with such a metric, which generally works for $n = 1, 2$. The obtained manifolds have the self-similarity of biological growth characterized by increasing size without changing shape.
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This paper is devoted to the factorization of multivariate polynomials into products of linear forms, a problem which has applications to differential algebra, to the resolution of systems of polynomial equations and to Waring decomposition (i.e., decomposition in sums of d-th powers of linear forms; this problem is also known as symmetric tensor decomposition). We provide three black box algorithms for this problem. Our main contribution is an algorithm motivated by the application to Waring decomposition. This algorithm reduces the corresponding factorization problem to simultaenous matrix diagonalization, a standard task in linear algebra. The algorithm relies on ideas from invariant theory, and more specifically on Lie algebras. Our second algorithm reconstructs a factorization from several bi-variate projections. Our third algorithm reconstructs it from the determination of the zero set of the input polynomial, which is a union of hyperplanes.
A generalized Riemann hypothesis states that all zeros of the completed Hecke $L$-function $L^*(f,s)$ of a normalized Hecke eigenform $f$ on the full modular group should lie on the vertical line $Re(s)=frac{k}{2}.$ It was shown by Kohnen that there exists a Hecke eigenform $f$ of weight $k$ such that $L^*(f,s) eq 0$ for sufficiently large $k$ and any point on the line segments $Im(s)=t_0, frac{k-1}{2} < Re(s) < frac{k}{2}-epsilon, frac{k }{2}+epsilon < Re(s) < frac{k+1}{2},$ for any given real number $t_0$ and a positive real number $epsilon.$ This paper concerns the non-vanishing of the product $L^*(f,s)L^*(f,w)$ $(s,win mathbb{C})$ on average.
We obtain new restrictions on the linear programming bound for sphere packing, by optimizing over spaces of modular forms to produce feasible points in the dual linear program. In contrast to the situation in dimensions 8 and 24, where the linear programming bound is sharp, we show that it comes nowhere near the best packing densities known in dimensions 12, 16, 20, 28, and 32. More generally, we provide a systematic technique for proving separations of this sort.
We study the convex relaxation of a polynomial optimization problem, maximizing a product of linear forms over the complex sphere. We show that this convex program is also a relaxation of the permanent of Hermitian positive semidefinite (HPSD) matrices. By analyzing a constructive randomized rounding algorithm, we obtain an improved multiplicative approximation factor to the permanent of HPSD matrices, as well as computationally efficient certificates for this approximation. We also propose an analog of van der Waerdens conjecture for HPSD matrices, where the polynomial optimization problem is interpreted as a relaxation of the permanent.
Let $lambda_{pi}(1,n)$ be the Fourier coefficients of the Hecke-Maass cusp form $pi$ for $SL(3,mathbb{Z})$. The aim of this article is to get a non trivial bound on the sum which is non-linear additive twist of the coefficients $lambda_{pi}(1,n)$. More precisely, for any $0 < beta < 1$ we have $$sum_{n=1}^{infty} lambda_{pi}(1,n) , eleft(alpha n^{beta}right) Vleft(frac{n}{X}right) ll_{pi, alpha,epsilon} X^{frac{3 }{4}+frac{3 beta}{10} + epsilon}$$ for any $epsilon>0$. Here $V(x)$ is a smooth function supported in $[1,2]$ and satisfies $V^{(j)}(x) ll_{j} 1$.
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