Search results for "Polynomial"
showing 10 items of 566 documents
Finite Energy Sum Rules with Legendre Polynomial Kernels
2016
Abstract In this note we report about a method to deal with finite energy sum rules. With a reasonable knowledge of the main resonances of the spectrum, the method guarantees that we can find a nice duality matching between the low energy hadronic data and asymptotic QCD at high energies.
Up and down quark masses from Finite Energy QCD sum rules to five loops
2008
The up and down quark masses are determined from an optimized QCD Finite Energy Sum Rule (FESR) involving the correlator of axial-vector divergences, to five loop order in Perturbative QCD (PQCD), and including leading non-perturbative QCD and higher order quark mass corrections. This FESR is designed to reduce considerably the systematic uncertainties arising from the (unmeasured) hadronic resonance sector, which in this framework contributes less than 3-4% to the quark mass. This is achieved by introducing an integration kernel in the form of a second degree polynomial, restricted to vanish at the peak of the two lowest lying resonances. The driving hadronic contribution is then the pion …
Strange quark condensate from QCD sum rules to five loops
2007
It is argued that it is valid to use QCD sum rules to determine the scalar and pseudoscalar two-point functions at zero momentum, which in turn determine the ratio of the strange to non-strange quark condensates $R_{su} = \frac{}{}$ with ($q=u,d$). This is done in the framework of a new set of QCD Finite Energy Sum Rules (FESR) that involve as integration kernel a second degree polynomial, tuned to reduce considerably the systematic uncertainties in the hadronic spectral functions. As a result, the parameters limiting the precision of this determination are $\Lambda_{QCD}$, and to a major extent the strange quark mass. From the positivity of $R_{su}$ there follows an upper bound on the latt…
Data analysis procedures for pulse ELDOR measurements of broad distance distributions
2004
The reliability of procedures for extracting the distance distribution between spins from the dipolar evolution function is studied with particular emphasis on broad distributions. A new numerically stable procedure for fitting distance distributions with polynomial interpolation between sampling points is introduced and compared to Tikhonov regularization in the dipolar frequency and distance domains and to approximate Pake transformation. Distance distributions with only narrow peaks are most reliably extracted by distance-domain Tikhonov regularization, while frequency-domain Tikhonov regularization is favorable for distributions with only broad peaks. For the quantification of distribut…
Topological protection of highly entangled non-Gaussian two-photon states
2021
Abstract We study theoretically the evolution of entangled non-Gaussian two-photon states in disordered topological lattices. Specifically, we consider spatially entangled two-photon states, modulated by Laguerre polynomials up to the 3rd order, which feature ring-shaped spatial and spectral correlation patterns. Such states are discrete analogs of photon-subtracted squeezed states, which are ubiquitous in optical quantum information processing or sensing applications. We find that, in general, a higher degree of entanglement coincides with a loss of topological protection against disorder, this is in line with previous results for Gaussian two-photon states. However, we identify a particul…
Minimal Absent Words in Rooted and Unrooted Trees
2019
We extend the theory of minimal absent words to (rooted and unrooted) trees, having edges labeled by letters from an alphabet \(\varSigma \) of cardinality \(\sigma \). We show that the set \(\text {MAW}(T)\) of minimal absent words of a rooted (resp. unrooted) tree T with n nodes has cardinality \(O(n\sigma )\) (resp. \(O(n^{2}\sigma )\)), and we show that these bounds are realized. Then, we exhibit algorithms to compute all minimal absent words in a rooted (resp. unrooted) tree in output-sensitive time \(O(n+|\text {MAW}(T)|)\) (resp. \(O(n^{2}+|\text {MAW}(T)|)\) assuming an integer alphabet of size polynomial in n.
On Codimensions of Algebras with Involution
2020
Let A be an associative algebra with involution ∗ over a field F of characteristic zero. One associates to A, in a natural way, a numerical sequence \(c^{\ast }_n(A),\)n = 1, 2, …, called the sequence of ∗-codimensions of A which is the main tool for the quantitative investigation of the polynomial identities satisfied by A. In this paper we focus our attention on \(c^{\ast }_n(A),\)n = 1, 2, …, by presenting some recent results about it.
Defining relations of the noncommutative trace algebra of two 3×3 matrices
2006
The noncommutative (or mixed) trace algebra $T_{nd}$ is generated by $d$ generic $n\times n$ matrices and by the algebra $C_{nd}$ generated by all traces of products of generic matrices, $n,d\geq 2$. It is known that over a field of characteristic 0 this algebra is a finitely generated free module over a polynomial subalgebra $S$ of the center $C_{nd}$. For $n=3$ and $d=2$ we have found explicitly such a subalgebra $S$ and a set of free generators of the $S$-module $T_{32}$. We give also a set of defining relations of $T_{32}$ as an algebra and a Groebner basis of the corresponding ideal. The proofs are based on easy computer calculations with standard functions of Maple, the explicit prese…
The Lie algebra of polynomial vector fields and the Jacobian conjecture
1998
The Jacobian conjecture for polynomial maps ϕ:Kn→Kn is shown to be equivalent to a certain Lie algebra theoretic property of the Lie algebra\(\mathbb{D}\) of formal vector fields inn variables. To be precise, let\(\mathbb{D}_0 \) be the unique subalgebra of codimensionn (consisting of the singular vector fields),H a Cartan subalgebra of\(\mathbb{D}_0 \),Hλ the root spaces corresponding to linear forms λ onH and\(A = \oplus _{\lambda \in {\rm H}^ * } H_\lambda \). Then every polynomial map ϕ:Kn→Kn with invertible Jacobian matrix is an automorphism if and only if every automorphism Φ of\(\mathbb{D}\) with Φ(A)\( \subseteq A\) satisfies Φ(A)=A.
Homomorphisms on spaces of weakly continuous holomorphic functions
1999
Let X be a Banach space and let $X^{\ast }$ be its topological dual space. We study the algebra ${\cal H}_{w^\ast}(X^{\ast})$ of entire functions on $X^{\ast }$ that are weak-star continuous on bounded sets. We prove that every m-homogeneous polynomial of finite type P on $X^*$ that is weak-star continuous on bounded sets can be written in the form $P=\textstyle\sum\limits _{j=1}^q x_{1j}\cdots x_{mj}$ where $x_{ij} \in X$ , for all i,j. As an application, we characterize convolution homomorphisms on ${\cal H}_{w^\ast}(X^{\ast})$ and on the space ${\cal H}_{wu}(X)$ of entire functions on X which are weakly uniformly continuous on bounded subsets of X, assuming that X * has the approximation…