The Experts below are selected from a list of 93552 Experts worldwide ranked by ideXlab platform
Vladimir Petrov Kostov - One of the best experts on this subject based on the ideXlab platform.
-
on a partial theta function and its spectrum
Proceedings of The Royal Society A: Mathematical Physical and Engineering Sciences, 2016Co-Authors: Vladimir Petrov KostovAbstract:The bivariate series defines a partial theta function . For fixed q (∣ q ∣ θ ( q , ·) is an entire function. For q ∈ (–1, 0) the function θ ( q , ·) has infinitely many negative and infinitely many positive Real Zeros. There exists a sequence of values of q tending to –1 + such that has a double Real Zero (the rest of its Real Zeros being simple). For k odd (respectively, k even) has a local minimum (respectively, maximum) at , and is the rightmost of the Real negative Zeros of (respectively, for k sufficiently large is the second from the left of the Real negative Zeros of ). For k sufficiently large one has . One has and .
-
on a partial theta function and its spectrum
arXiv: Classical Analysis and ODEs, 2015Co-Authors: Vladimir Petrov KostovAbstract:The bivariate series $\theta (q,x):=\sum _{j=0}^{\infty}q^{j(j+1)/2}x^j$ %(where $(q,x)\in {\bf C}^2$, $|q|<1$) defines a {\em partial theta function}. For fixed $q$ ($|q|<1$), $\theta (q,.)$ is an entire function. For $q\in (-1,0)$ the function $\theta (q,.)$ has infinitely many negative and infinitely many positive Real Zeros. There exists a sequence $\{ \bar{q}_j\}$ of values of $q$ tending to $-1^+$ such that $\theta (\bar{q}_k,.)$ has a double Real Zero $\bar{y}_k$ (the rest of its Real Zeros being simple). For $k$ odd (resp. for $k$ even) $\theta (\bar{q}_k,.)$ has a local minimum at $\bar{y}_k$ and $\bar{y}_k$ is the rightmost of the Real negative Zeros of $\theta (\bar{q}_k,.)$ (resp. $\theta (\bar{q}_k,.)$ has a local maximum at $\bar{y}_k$ and for $k$ sufficiently large $\bar{y}_k$ is the second from the left of the Real negative Zeros of $\theta (\bar{q}_k,.)$). For $k$ sufficiently large one has $-1<\bar{q}_{k+1}<\bar{q}_k<0$. One has $\bar{q}_k=1-(\pi /8k)+o(1/k)$ and $|\bar{y}_k|\rightarrow e^{\pi /2}=4.810477382\ldots$.
Petter Branden - One of the best experts on this subject based on the ideXlab platform.
-
obstructions to determinantal representability
Advances in Mathematics, 2011Co-Authors: Petter BrandenAbstract:There has recently been ample interest in the question of which sets can be represented by linear matrix inequalities (LMIs). A necessary condition is that the set is rigidly convex, and it has been conjectured that rigid convexity is also sufficient. To this end Helton and Vinnikov conjectured that any Real Zero polynomial admits a determinantal representation with symmetric matrices. We disprove this conjecture. By relating the question of finding LMI representations to the problem of determining whether a polymatroid is representable over the complex numbers, we find a Real Zero polynomial such that no power of it admits a determinantal representation. The proof uses recent results of Wagner and Wei on matroids with the half-plane property, and the polymatroids associated to hyperbolic polynomials introduced by Gurvits.
Junyi Wen - One of the best experts on this subject based on the ideXlab platform.
-
certified numerical Real root isolation for bivariate polynomial systems
International Symposium on Symbolic and Algebraic Computation, 2019Co-Authors: Jinsan Cheng, Junyi WenAbstract:In this paper, we present a new method for isolating Real roots of a bivariate polynomial system. Our method is a subdivision method which is based on Real root isolation of univariate polynomials and analyzing the local geometrical properties of the given system. We propose the concept of the orthogonal monotone system in a box and use it to determine the uniqueness and the existence of a simple Real Zero of the system in the box. We implement our method to isolate the Real Zeros of a given bivariate polynomial system. The experiments show the effectivity and efficiency of our method, especially for systems with high degrees and sparse terms. Our method also works for non-polynomial systems.
Xiaoshen Wang - One of the best experts on this subject based on the ideXlab platform.
-
counting Real connected components of trinomial curve intersections and m nomial hypersurfaces
Discrete and Computational Geometry, 2003Co-Authors: Maurice J Rojas, Xiaoshen WangAbstract:We prove that any pair of bivariate trinomials has at most five isolated roots in the positive quadrant. The best previous upper bounds independent of the polynomial degrees were much larger, e.g., 248832 (for just the non-degenerate roots) via a famous general result of Khovanski. Our bound is sharp, allows Real exponents, allows degeneracies, and extends to certain systems of n-variate fewnomials, giving improvements over earlier bounds by a factor exponential in the number of monomials. We also derive analogous sharpened bounds on the number of connected components of the Real Zero set of a single n-variate m-nomial.
-
counting Real connected components of trinomial curve intersections and m nomial hypersurfaces
arXiv: Algebraic Geometry, 2002Co-Authors: Maurice J Rojas, Xiaoshen WangAbstract:We prove that any pair of bivariate trinomials has at most 5 isolated roots in the positive quadrant. The best previous upper bounds independent of the polynomial degrees were much larger, e.g., 248832 (for just the non-degenerate roots) via a famous general result of Khovanski. Our bound is sharp, allows Real exponents, allows degeneracies, and extends to certain systems of n-variate fewnomials, giving improvements over earlier bounds by a factor exponential in the number of monomials. We also derive analogous sharpened bounds on the number of connected components of the Real Zero set of a single n-variate m-nomial.
Jinsan Cheng - One of the best experts on this subject based on the ideXlab platform.
-
certified numerical Real root isolation for bivariate polynomial systems
International Symposium on Symbolic and Algebraic Computation, 2019Co-Authors: Jinsan Cheng, Junyi WenAbstract:In this paper, we present a new method for isolating Real roots of a bivariate polynomial system. Our method is a subdivision method which is based on Real root isolation of univariate polynomials and analyzing the local geometrical properties of the given system. We propose the concept of the orthogonal monotone system in a box and use it to determine the uniqueness and the existence of a simple Real Zero of the system in the box. We implement our method to isolate the Real Zeros of a given bivariate polynomial system. The experiments show the effectivity and efficiency of our method, especially for systems with high degrees and sparse terms. Our method also works for non-polynomial systems.
