Norms Of Polynomials Research Articles

Burgers-like equation for spontaneous breakdown of the chiral symmetry in QCD

We link the spontaneous breakdown of chiral symmetry in Euclidean QCD to the collision of spectral shock waves in the vicinity of zero eigenvalue of the Dirac operator. The mechanism, originating from complex Burgerʼs-like equation for viscid, pressureless, one-dimensional flows of eigenvalues, is similar to the recently observed weak-strong coupling phase transition in large Nc Yang–Mills theory. The spectral viscosity is proportional to the inverse size of the random matrix that replaces the Dirac operator in the universal (ergodic) regime. We obtain the exact scaling function and critical exponents of the chiral phase transition for the averaged characteristic polynomial for Nc⩾3 QCD. We reinterpret our results in terms of known properties of chiral random matrix models and lattice data.

AsymptoticL4norm of polynomials derived from characters

Littlewood investigated polynomials with coefficients in $\{-1,1\}$ (Littlewood polynomials), to see how small their ratio of norms $||f||_4/||f||_2$ on the unit circle can become as $deg(f)\to\infty$. A small limit is equivalent to slow growth in the mean square autocorrelation of the associated binary sequences of coefficients of the polynomials. The autocorrelation problem for arrays and higher dimensional objects has also been studied; it is the natural generalization to multivariable polynomials. Here we find, for each $n > 1$, a family of $n$-variable Littlewood polynomials with lower asymptotic $||f||_4/||f||_2$ than any known hitherto. We discover these through a wide survey, infeasible with previous methods, of polynomials whose coefficients come from finite field characters. This is the first time that the lowest known asymptotic ratio of norms $||f||_4/||f||_2$ for multivariable polynomials $f(z_1,...,z_n)$ is strictly less than what could be obtained by using products $f_1(z_1)... f_n(z_n)$ of the best known univariate polynomials.

Extremal Mahler measures and $L_s$ norms of polynomials related to Barker sequences

Extremal Mahler measures and $L_s$ norms of polynomials related to Barker sequences

Warm Season Temperature-Mortality Relationships in Chisinau (Moldova)

Results of the epidemiological study of relationships between air temperature and daily mortality in Chisinau (Moldova) are presented. The research’s main task included description of mortality dependence on different temperature variables and identification of thermal optimum (minimal mortality temperature, MMT). Total daily deaths were used to characterize the mortality of urban and rural populations in April–September of 2000–2008, excluding the extremely warm season of 2007. The simple moving average procedure and 2nd-order polynomials were used for daily mean (Tmean), maximum (Tmax), and minimum (Tmin) temperatures and mortality approximation. Thermal optimum for mortality in Chisinau (15.2 deaths) was observed at Tmean, Tmax, and Tmin about 22°C, 27-28°C, and 17-18°C, respectively. Considering these values as certain cut-points, the correlations between temperature and mortality were estimated below and above MMTs. With air temperatures below its optimal value, each additional 1°C increase of Tmean (Tmax, Tmin) was accompanied by 1.40% (1.35%, 1.52%) decrease in daily mortality. The increase of Tmean and Tmax above optimal values was associated with ~2.8% and 3.5% increase of mortality; results for Tmin were not statistically significant. The dependency of mortality on apparent temperature was somewhat weaker below MMT; a significant relationship above MMT was not identified.

Self-similarity of higher-order moving averages

In this work, higher-order moving average polynomials are defined by straightforward generalization of the standard moving average. The self-similarity of the polynomials is analyzed for fractional Brownian series and quantified in terms of the Hurst exponent H by using the detrending moving average method. We prove that the exponent H of the fractional Brownian series and of the detrending moving average variance asymptotically agree for the first-order polynomial. Such asymptotic values are compared with the results obtained by the simulations. The higher-order polynomials correspond to trend estimates at shorter time scales as the degree of the polynomial increases. Importantly, the increase of polynomial degree does not require to change the moving average window. Thus trends at different time scales can be obtained on data sets with the same size. These polynomials could be interesting for those applications relying on trend estimates over different time horizons (financial markets) or on filtering at different frequencies (image analysis).

The norm of polynomials in large random and deterministic matrices

Let X_N= (X_1^(N), ..., X_p^(N)) be a family of N-by-N independent, normalized random matrices from the Gaussian Unitary Ensemble. We state sufficient conditions on matrices Y_N =(Y_1^(N), ..., Y_q^(N)), possibly random but independent of X_N, for which the operator norm of P(X_N, Y_N, Y_N^*) converges almost surely for all polynomials P. Limits are described by operator norms of objects from free probability theory. Taking advantage of the choice of the matrices Y_N and of the polynomials P we get for a large class of matrices the "no eigenvalues outside a neighborhood of the limiting spectrum" phenomena. We give examples of diagonal matrices Y_N for which the convergence holds. Convergence of the operator norm is shown to hold for block matrices, even with rectangular Gaussian blocks, a situation including non-white Wishart matrices and some matrices encountered in MIMO systems.

Average characteristic polynomials in the two-matrix model

The two-matrix model is defined on pairs of Hermitian matrices \documentclass[12pt]{minimal}\begin{document}$(M_1,M_2)$\end{document}(M1,M2) of size n × n by the probability measure \documentclass[12pt]{minimal}\begin{document}$\frac{1}{Z_n} \exp (\hbox{Tr} (-V(M_1)-W(M_2)+\tau M_1M_2))$\break $ dM_1\ dM_2,$\end{document}1Znexp(Tr(−V(M1)−W(M2)+τM1M2))dM1dM2, where V and W are given potential functions and \documentclass[12pt]{minimal}\begin{document}$\tau \in \mathbb {R}$\end{document}τ∈R. We study averages of products and ratios of characteristic polynomials in the two-matrix model, where both matrices \documentclass[12pt]{minimal}\begin{document}$M_1$\end{document}M1 and \documentclass[12pt]{minimal}\begin{document}$M_2$\end{document}M2 may appear in a combined way in both numerator and denominator. We obtain determinantal expressions for such averages. The determinants are constructed from several building blocks: the biorthogonal polynomials, \documentclass[12pt]{minimal}\begin{document}$p_n(x)$\end{document}pn(x) and \documentclass[12pt]{minimal}\begin{document}$q_n(y),$\end{document}qn(y), associated with the two-matrix model; certain transformed functions \documentclass[12pt]{minimal}\begin{document}$\widetilde{P}_n(w)$\end{document}P̃n(w) and \documentclass[12pt]{minimal}\begin{document}$\widetilde{Q}_n(v)$\end{document}Q̃n(v); and finally Cauchy-type transforms of the four Eynard–Mehta kernels \documentclass[12pt]{minimal}\begin{document}$K_{1,1}$\end{document}K1,1, \documentclass[12pt]{minimal}\begin{document}$K_{1,2}$\end{document}K1,2, \documentclass[12pt]{minimal}\begin{document}$K_{2,1}$\end{document}K2,1, and \documentclass[12pt]{minimal}\begin{document}$K_{2,2}$\end{document}K2,2. In this way, we generalize known results for the one-matrix model. Our results also imply a new proof of the Eynard–Mehta theorem for correlation functions in the two-matrix model, and they lead to a generating function for averages of products of traces.

Average characteristic polynomials for multiple orthogonal polynomial ensembles

Multiple orthogonal polynomials (MOP) are a non-definite version of matrix orthogonal polynomials. They are described by a Riemann–Hilbert matrix Y consisting of four blocks Y 1 , 1 , Y 1 , 2 , Y 2 , 1 and Y 2 , 2 . In this paper, we show that det Y 1 , 1 ( det Y 2 , 2 ) equals the average characteristic polynomial (average inverse characteristic polynomial, respectively) over the probabilistic ensemble that is associated to the MOP. In this way we generalize the classical results for orthogonal polynomials, and also some recent results for MOP of type I and type II. We then extend our results to arbitrary products and ratios of characteristic polynomials. In the latter case an important role is played by a matrix-valued version of the Christoffel–Darboux kernel. Our proofs use determinantal identities involving Schur complements, and adaptations of the classical results by Heine, Christoffel and Uvarov.

Inequalities for products of polynomials II

In this paper, we continue the study of inequalities connecting the product of uniform norms of polynomials with the norm of their product, begun in [28]. Asymptotically sharp constants are known for such inequalities over arbitrary compact sets in the complex plane. We show here that such constants can be improved under some natural additional assumptions. Thus we find the best constants for rotationally symmetric sets. In addition, we characterize all sets that allow an improvement in the constant when the number of factors is fixed, and find the improved value.

Inequalities for products of polynomials I

We study inequalities connecting the product of uniform norms of polynomials with the norm of their product. This circle of problems include the Gelfond-Mahler inequality for the unit disk and the Kneser-Borwein inequality for the segment $[-1,1]$. Furthermore, the asymptotically sharp constants are known for such inequalities over arbitrary compact sets in the complex plane. It is shown here that this best constant is smallest (namely: 2) for a disk. We also conjecture that it takes its largest value for a segment, among all compact connected sets in the plane.

Expected Norms of Zero-One Polynomials

AbstractLet = {a0 + a1z + · · · + an–1zn–1 : aj ∈ ﹛0, 1﹜¯﹜, whose elements are called zero- one polynomials and correspond naturally to the 2n subsets of [n] := ﹛0, 1, … , n – 1﹜. We also let = ﹛α(z) ∈ : α(1) = m﹜, whose elements correspond to the subsets of [n] of size m, and let , whose elements are the zero-one polynomials of degree exactly n.Many researchers have studied norms of polynomials with restricted coefficients. Using ‖α‖p to denote the usual Lp norm of α on the unit circle, one easily sees that α(z) = a0+a1z+· · ·+aNzN ∈ ℝ[z] satisfies and , where .If α(z) ∈ , say α(z) = zβ1 + · · · + zβm where β1 < · · · < βm, then ck is the number of times k appears as a difference βi – βj . The condition that α ∈ satisfies ck ∈ ﹛0, 1﹜ for 1 ≤ k ≤ n – 1 is thus equivalent to the condition that ﹛β1, … , βm﹜ is a Sidon set (meaning all differences of pairs of elements are distinct).In this paper, we find the average of over α ∈ , α ∈ , and α ∈ . We further show that our expression for the average of over yields a new proof of the known result: if m = o(n1/4) and B(n,m) denotes the number of Sidon sets of size m in [n], then almost all subsets of [n] of size m are Sidon, in the sense that .

Reverse triangle inequalities for potentials

We study the reverse triangle inequalities for suprema of logarithmic potentials on compact sets of the plane. This research is motivated by the inequalities for products of supremum norms of polynomials. We find sharp additive constants in the inequalities for potentials, and give applications of our results to the generalized polynomials. We also obtain sharp inequalities for products of norms of the weighted polynomials w n P n , deg ( P n ) ≤ n , and for sums of potentials with external fields. An important part of our work in the weighted case is a Riesz decomposition for the weighted farthest-point distance function.

Ratios of Norms for Polynomials and Connected n-width Problems

Let \(G \subset{\mathbb{C}}\) be a bounded simply connected domain with boundary Γ and let \(E \subset G\) be a regular compact set with connected complement. In this paper we investigate asymptotics of the extremal constants: $$\chi_{n}= \inf\limits_{p \in {\mathcal{P}}_{k_{n}}} \sup\limits_{q \in {\mathcal{P}}_{n}-{k_{n}}} \frac{||pq||_E} {||pq||_\Gamma}, \quad n= 1, 2, . . . ,$$ where \(|| \cdot || _{K}\) is the supremum norm on a compact set K, \({\mathcal{P}}_m\) is the set of all algebraic polynomials of degree at most m, and \(k_{n}/n \rightarrow \theta \in [0, 1]\) as \(n \rightarrow \infty\). Subsequently, we obtain asymptotic behavior of the Kolmogorov k-widths, \(k=k_{n}\), of the unit ball An∞ of \(H^{\infty} \cap {\mathcal{P}}_{n}\) restricted to E in C(E), where H∞ is the Hardy space of bounded analytic functions on G and C(E) is the space of continuous functions on E.

Equivalent norms for polynomials on the sphere

We find necessary and sufficient conditions for a sequence of sets E L ⊂ § d in order to obtain the inequality where 1 ⩽ p < +∞, Q L is any polynomial of degree smaller or equal than L, μ is a doubling measure, and the constant C p is independent of L. From this description, it follows an uncertainty principle for functions in L 2 (§ d ). We also consider weighted uniform versions of this result.

The average norm of polynomials of fixed height

Let n > 0 be any integer and let F n := {nΣi=n a i z i :a i , =0,±1} be the set of all polynomials of height 1 and degree n. Let β n (m):= 1 3n+1 ΣP∈Fn ∥P∥ m m . Here ∥P∥ m m is the mth power of the L m norm on the boundary of the unit disc. So β n (m) is the'average of the mth power of the L m norm over F n . In this paper we give exact formulae for β n (m) for various values of m. We also give a variety of related results for different classes of polynomials including polynomials of fixed height H, polynomials with coefficients ±1 and reciprocal polynomials. The results are surprisingly precise. Typical of the results we get is.the following. Theorem 0.1. For n > 0, we have β n (2)=2 3(n+1), β n (4)=8 9n 2 +14 9n+2 3 and β(6)=16 9n 3 +4n 2 +26 9n+2 3.

A REMEZ-TYPE THEOREM FOR HOMOGENEOUS POLYNOMIALS

Remez-type inequalities provide upper bounds for the uniform norms of polynomials p on given compact sets K, provided that |p(x)| ⩽ 1 for every x ∈ K\E, where E is a subset of K of small measure. In this paper we prove sharp Remez-type inequalities for homogeneous polynomials on star-like surfaces in Rd. In particular, this covers the case of spherical polynomials (when d = 2 we deduce a result of Erdélyi for univariate trigonometric polynomials).

Positive oriented periodic solutions of the first-order complex ODE with polynomial nonlinear part

We study nonlinear ODE problems in the complex Euclidean space, with the right-hand side being polynomial with nonconstant periodic coefficients. As the coefficients functions, we admit only functions with vanishing Fourier coefficients for negative indices. This leads to an existence theorem which relates the number of solutions with the number of zeros of the averaged right-hand side polynomial. A priori estimates of the norms of solutions are based on the Wirtinger-Poincare-type inequality. The proof of existence theorem is based on the continuation method of Krasnosielski et al., Mawhin et al., and the Leray-Schauder degree. We give a few applications on the complex Riccati equation and some others.

On discrete norms of polynomials

For a polynomial p of degree n < N we compare two norms: ∥ p ∥ ≔ sup { | p ( z ) | : z ∈ C ; | z | = 1 } and ∥ p ∥ N ≔ sup p z j : j = 0 , … , N - 1 ; z j = e 2 π i j N . We show that there exist universal constants C 1 and C 2 such that 1 + C 1 log N N - n ≤ sup ∥ p ∥ ∥ p ∥ N : p ∈ P n ≤ C 2 log N N - n + 1 .

On the resultant property of the Fisher information matrix of a vector ARMA process

A matrix is called a multiple resultant matrix associated to two matrix polynomials when it becomes singular if and only if the two matrix polynomials have at least one common eigenvalue. In this paper a new multiple resultant matrix is introduced. It concerns the Fisher information matrix (FIM) of a stationary vector autoregressive and moving average time series process (VARMA). The two matrix polynomials are the autoregressive and the moving average matrix polynomials of the VARMA process. In order to show that the FIM is a multiple resultant matrix two new representations of the FIM are derived. To construct such representations appropriate matrix differential rules are applied. The newly obtained representations are expressed in terms of the multiple Sylvester matrix and the tensor Sylvester matrix. The representation of the FIM expressed by the tensor Sylvester matrix is used to prove that the FIM becomes singular if and only if the autoregressive and moving average matrix polynomials have at least one common eigenvalue. It then follows that the FIM and the tensor Sylvester matrix have equivalent singularity conditions. In a simple numerical example it is shown however that the FIM fails to detect common eigenvalues due to some kind of numerical instability. Whereas the tensor Sylvester matrix reveals it clearly, proving the usefulness of the results derived in this paper.

Average independence polynomials

The independence polynomial of a graph G is the function i ( G , x ) = ∑ k ⩾ 0 i k x k , where i k is the number of independent sets of vertices in G of cardinality k. We investigate here the average independence polynomial, where the average is taken over all independence polynomials of (labeled) graphs of order n. We prove that while almost every independence polynomial has a nonreal root, the average independence polynomials always have all real, simple roots.