The Experts below are selected from a list of 321 Experts worldwide ranked by ideXlab platform
Vladimir Petrov Kostov - One of the best experts on this subject based on the ideXlab platform.
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a property of a partial theta function
arXiv: Classical Analysis and ODEs, 2015Co-Authors: Vladimir Petrov KostovAbstract:The series $\theta (q,x):=\sum _{j=0}^{\infty}q^{j(j+1)/2}x^j$ converges for $|q|<1$ and defines a {\em partial theta function}. For any fixed $q\in (0,1)$ it has infinitely many negative zeros. It is known that for $q$ taking one of the {\em spectral} values $\tilde{q}_1$, $\tilde{q}_2$, $\ldots$ (where $0.3092493386\ldots =\tilde{q}_1<\tilde{q}_2<\cdots <1$, $\lim _{j\rightarrow \infty}\tilde{q}_j=1$) the function $\theta (q,.)$ has a double zero which is the rightmost of its Real zeros (the rest of them being simple). For $q\neq \tilde{q}_j$ the partial theta function has no multiple Real zeros. We prove that: 1) for $q\in (\tilde{q}_{j},\tilde{q}_{j+1}]$ the function $\theta$ is a product of a degree $2j$ Real Polynomial without Real roots and a function of the Laguerre-P\'olya class $\cal{LP-I}$; 2) for $q\in \mathbb{C}\backslash 0$, $|q|<1$, $\theta (q,x)=\prod _i(1+x/x_i)$, where $-x_i$ are the zeros of $\theta$; 3) for any fixed $q\in \mathbb{C}\backslash 0$, $|q|<1$, the function $\theta$ has at most finitely-many multiple zeros; 4) for any $q\in (-1,0)$ the function $\theta$ is a product of a Real Polynomial without Real zeros and a function of the Laguerre-P\'olya class $\cal{LP}$. 5) for any fixed $q\in \mathbb{C}\backslash 0$, $|q|<1$, and for $k$ sufficiently large, the function $\theta$ has a zero $\zeta _k$ close to $-q^{-k}$. These are all but finitely-many of the zeros of $\theta$.
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a refined Realization theorem in the context of the schur szegő composition
Bulletin Des Sciences Mathematiques, 2012Co-Authors: Vladimir Petrov KostovAbstract:Every Polynomial of the form $P = (x + 1)(x^{n−1} + c_1x^{n−2} + \cdots + c_{n−1})$ is representable as Schur-Szeg\H{o} composition of $n−1$ Polynomials of the form $(x +1)^{n−1}(x +a_i )$, where the numbers $a_i$ are unique up to permutation. We give necessary and sufficient conditions upon the possible values of the 8-vector whose components are the number of positive, zero, negative and complex roots of a Real Polynomial $P$ and the number of positive, zero, negative and complex among the quantities $a_i$ corresponding to $P$. A similar result is proved about entire functions of the form $e^xR$, where $R$ is a Polynomial.
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a Realization theorem in the context of the schur szegő composition
Functional Analysis and Its Applications, 2009Co-Authors: Vladimir Petrov KostovAbstract:Every Real Polynomial of degree n in one variable with root −1 can be represented as the Schur-Szegő composition of n − 1 Polynomials of the form (x + 1) n−1(x + a i ), where the numbers a i are uniquely determined up to permutation. Some a i are Real, and the others form complex conjugate pairs. In this note, we show that for each pair (ρ, r), where 0 ⩽ ρ, r ⩽ [n/2], there exists a Polynomial with exactly ρ pairs of complex conjugate roots and exactly r complex conjugate pairs in the corresponding set of numbers a i .
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on root arrangements for hyperbolic Polynomial like functions and their derivatives
Bulletin Des Sciences Mathematiques, 2007Co-Authors: Vladimir Petrov KostovAbstract:Abstract A Real Polynomial P of degree n in one Real variable is hyperbolic if its roots are all Real. A Real-valued function P is called a hyperbolic Polynomial-like function (HPLF) of degree n if it has n Real zeros and P ( n ) vanishes nowhere. Denote by x k ( i ) the roots of P ( i ) , k = 1 , … , n − i , i = 0 , … , n − 1 . Then in the absence of any equality of the form x i ( j ) = x k ( l ) ( ∗ ) one has ∀ i j , x k ( i ) x k ( j ) x k + j − i ( i ) ( ∗ ∗ ) (the Rolle theorem). For n ⩾ 4 (resp. for n ⩾ 5 ) not all arrangements without equalities ( ∗ ) of n ( n + 1 ) / 2 Real numbers x k ( i ) and compatible with ( ∗ ∗ ) (we call them admissible) are Realizable by the roots of hyperbolic Polynomials (resp. of HPLFs) of degree n and of their derivatives. For n = 5 we show that from 286 admissible arrangements, exactly 236 are Realizable by HPLFs; from these 236 arrangements, 116 are Realizable by hyperbolic Polynomials and 24 by perturbations of such.
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root arrangements of hyperbolic Polynomial like functions
Revista Matematica Complutense, 2006Co-Authors: Vladimir Petrov KostovAbstract:A Real Polynomial P of degree n in one Real variable is hyperbolic if its roots are all Real. A Real-valued function P is called a hyperbolic Polynomial-like function (HPLF) of degree n if it has n Real zeros and P (n) vanishes nowhere. Denote by x (i) k the roots of P
Alexander A Sherstov - One of the best experts on this subject based on the ideXlab platform.
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near optimal lower bounds on the threshold degree and sign rank of ac 0
Symposium on the Theory of Computing, 2019Co-Authors: Alexander A SherstovAbstract:The threshold degree of a Boolean function f∶{0,1}n→{0,1} is the minimum degree of a Real Polynomial p that represents f in sign: sgn p(x)=(−1)f(x). A related notion is sign-rank, defined for a Boolean matrix F=[Fij] as the minimum rank of a Real matrix M with sgn Mij=(−1)Fij. Determining the maximum threshold degree and sign-rank achievable by constant-depth circuits (AC0) is a well-known and extensively studied open problem, with complexity-theoretic and algorithmic applications. We give an essentially optimal solution to this problem. For any є>0, we construct an AC0 circuit in n variables that has threshold degree Ω(n1−є) and sign-rank exp(Ω(n1−є)), improving on the previous best lower bounds of Ω(√n) and exp(Ω(√n)), respectively. Our results subsume all previous lower bounds on the threshold degree and sign-rank of AC0 circuits of any given depth, with a strict improvement starting at depth 4. As a corollary, we also obtain near-optimal bounds on the discrepancy, threshold weight, and threshold density of AC0, strictly subsuming previous work on these quantities. Our work gives some of the strongest lower bounds to date on the communication complexity of AC0.
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near optimal lower bounds on the threshold degree and sign rank of ac 0
arXiv: Computational Complexity, 2019Co-Authors: Alexander A SherstovAbstract:The threshold degree of a Boolean function $f\colon\{0,1\}^n\to\{0,1\}$ is the minimum degree of a Real Polynomial $p$ that represents $f$ in sign: $\mathrm{sgn}\; p(x)=(-1)^{f(x)}.$ A related notion is sign-rank, defined for a Boolean matrix $F=[F_{ij}]$ as the minimum rank of a Real matrix $M$ with $\mathrm{sgn}\; M_{ij}=(-1)^{F_{ij}}$. Determining the maximum threshold degree and sign-rank achievable by constant-depth circuits ($\text{AC}^{0}$) is a well-known and extensively studied open problem, with complexity-theoretic and algorithmic applications. We give an essentially optimal solution to this problem. For any $\epsilon>0,$ we construct an $\text{AC}^{0}$ circuit in $n$ variables that has threshold degree $\Omega(n^{1-\epsilon})$ and sign-rank $\exp(\Omega(n^{1-\epsilon})),$ improving on the previous best lower bounds of $\Omega(\sqrt{n})$ and $\exp(\tilde{\Omega}(\sqrt{n}))$, respectively. Our results subsume all previous lower bounds on the threshold degree and sign-rank of $\text{AC}^{0}$ circuits of any given depth, with a strict improvement starting at depth $4$. As a corollary, we also obtain near-optimal bounds on the discrepancy, threshold weight, and threshold density of $\text{AC}^{0}$, strictly subsuming previous work on these quantities. Our work gives some of the strongest lower bounds to date on the communication complexity of $\text{AC}^{0}$.
Khazhgali Kozhasov - One of the best experts on this subject based on the ideXlab platform.
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The Real Polynomial Eigenvalue Problem is Well Conditioned on the Average
Foundations of Computational Mathematics, 2020Co-Authors: Carlos Beltrán, Khazhgali KozhasovAbstract:We study the average condition number for Polynomial eigenvalues of collections of matrices drawn from some random matrix ensembles. In particular, we prove that Polynomial eigenvalue problems defined by matrices with random Gaussian entries are very well conditioned on the average.
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the Real Polynomial eigenvalue problem is well conditioned on the average
arXiv: Numerical Analysis, 2018Co-Authors: Carlos Beltrán, Khazhgali KozhasovAbstract:We study the average condition number for Polynomial eigenvalues of collections of matrices drawn from various random matrix ensembles. In particular, we prove that Polynomial eigenvalue problems defined by matrices with Gaussian entries are very well-conditioned on the average.
J. Selva - One of the best experts on this subject based on the ideXlab platform.
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Computation of spectral and root MUSIC through Real Polynomial rooting
IEEE Transactions on Signal Processing, 2005Co-Authors: J. SelvaAbstract:We present a technique to compute the Spectral and (Unitary) Root MUSIC estimations from the roots of a Real Polynomial in the problem of estimating the angles of arrival to a Uniform Linear Array (ULA). The computed estimations are the actual MUSIC or (Unitary) Root MUSIC estimations, but the need for a one-dimensional (1-D) search in Spectral MUSIC is eliminated, and the number of Polynomial coefficients to compute in (Unitary) Root MUSIC is halved. Besides, the proposed technique makes it possible to perform the Polynomial rooting step in these methods through Real arithmetic.
Ivan Dokmanic - One of the best experts on this subject based on the ideXlab platform.
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Real Polynomial gram matrices without Real spectral factors
arXiv: Signal Processing, 2019Co-Authors: Puoya Tabaghi, Ivan DokmanicAbstract:It is well known that a non-negative definite Polynomial matrix (a Polynomial Gramian) $G(t)$ can be written as a product of its Polynomial spectral factors, $G(t) = X(t)^H X(t)$. In this paper, we give a new algebraic characterization of spectral factors when $G(t)$ is Real-valued. The key idea is to construct a representation set that is in bijection with the set of Real Polynomial Gramians. We use the derived characterization to identify the set of all complex Polynomial matrices that generate Real-valued Gramians, and we formulate a conjecture that typical rank-deficient Real Polynomial Gramians have Real spectral factors.
