Matrix methods for perfect signal recovery underlying range space of operators

The purpose of this work is to investigate perfect reconstruction underlying range space of operators in finite dimensional Hilbert spaces by a new matrix method. To this end, first we obtain more structures of the canonical $K$-dual. % and survey optimal $K$-dual problem under probabilistic erasures. Then, we survey the problem of recovering and robustness of signals when the erasure set satisfies the minimal redundancy condition or the $K$-frame is maximal robust. Furthermore, we show that the error rate is reduced under erasures if the $K$-frame is of uniform excess. Toward the protection of encoding frame (K-dual) against erasures, we introduce a new concept so called $(r,k)$-matrix to recover lost data and solve the perfect recovery problem via matrix equations. Moreover, we discuss the existence of such matrices by using minimal redundancy condition on decoding frames for operators. We exhibit several examples that illustrate the advantage of using the new matrix method with respect to the previous approaches in existence construction. And finally, we provide the numerical results to confirm the main results in the case noise-free and test sensitivity of the method with respect to noise.

Investigation of the lightest hybrid meson candidate with a coupled-channel analysis of $${{\bar{p}}p}$$-, $$\pi ^- p$$- and $${\pi \pi }$$-Data

A sophisticated coupled-channel analysis is presented that combines different processes: the channels $${\pi ^0\pi ^0\eta }$$ π 0 π 0 η , $${\pi ^0\eta \eta }$$ π 0 η η and $${K^+K^-\pi ^0}$$ K + K - π 0 from $${{\bar{p}}p}$$ p ¯ p annihilations, the P- and D-wave amplitudes of the $$\pi \eta $$ π η and $$\pi \eta ^\prime $$ π η ′ systems produced in $$\pi ^-p$$ π - p scattering, and data from $${\pi \pi }$$ π π -scattering reactions. Hence our analysis combines the data sets used in two independent previous analyses published by the Crystal Barrel experiment and by the JPAC group. Based on the new insights from these studies, this paper aims at a better understanding of the spin-exotic $$\pi _1$$ π 1 resonances in the light-meson sector. By utilizing the K-matrix approach and realizing the analyticity via Chew-Mandelstam functions the amplitude of the spin-exotic wave can be well described by a single $$\pi _1$$ π 1 pole for both systems, $$\pi \eta $$ π η and $$\pi \eta ^\prime $$ π η ′ . The mass and the width of the $$\pi _1$$ π 1 -pole are measured to be $$(1623 \, \pm \, 47 \, ^{+24}_{-75})\, \mathrm {MeV/}c^2$$ ( 1623 ± 47 - 75 + 24 ) MeV / c 2 and $$(455 \, \pm 88 \, ^{+144}_{-175})\, \mathrm {MeV}$$ ( 455 ± 88 - 175 + 144 ) MeV .

Assessing Methods for Evaluating the Number of Components in Non-Negative Matrix Factorization

Non-negative matrix factorization is a relatively new method of matrix decomposition which factors an m × n data matrix X into an m × k matrix W and a k × n matrix H, so that X ≈ W × H. Importantly, all values in X, W, and H are constrained to be non-negative. NMF can be used for dimensionality reduction, since the k columns of W can be considered components into which X has been decomposed. The question arises: how does one choose k? In this paper, we first assess methods for estimating k in the context of NMF in synthetic data. Second, we examine the effect of normalization on this estimate’s accuracy in empirical data. In synthetic data with orthogonal underlying components, methods based on PCA and Brunet’s Cophenetic Correlation Coefficient achieved the highest accuracy. When evaluated on a well-known real dataset, normalization had an unpredictable effect on the estimate. For any given normalization method, the methods for estimating k gave widely varying results. We conclude that when estimating k, it is best not to apply normalization. If the underlying components are known to be orthogonal, then Velicer’s MAP or Minka’s Laplace-PCA method might be best. However, when the orthogonality of the underlying components is unknown, none of the methods seemed preferable.

Interactions of two and three mesons including higher partial waves from lattice QCD

We study two- and three-meson systems composed either of pions or kaons at maximal isospin using Monte Carlo simulations of lattice QCD. Utilizing the stochastic LapH method, we are able to determine hundreds of two- and three-particle energy levels, in nine different momentum frames, with high precision. We fit these levels using the relativistic finite-volume formalism based on a generic effective field theory in order to determine the parameters of the two- and three-particle K-matrices. We find that the statistical precision of our spectra is sufficient to probe not only the dominant s-wave interactions, but also those in d waves. In particular, we determine for the first time a term in the three-particle K-matrix that contains two-particle d waves. We use three Nf = 2 + 1 CLS ensembles with pion masses of 200, 280, and 340 MeV. This allows us to study the chiral dependence of the scattering observables, and compare to the expectations of chiral perturbation theory.

Lattice Clifford fractons and their Chern-Simons-like theory

We use Dirac matrix representations of the Clifford algebra to build fracton models on the lattice and their effective Chern-Simons-like theory. As an example, we build lattice fractons in odd D spatial dimensions and their (D+1) spacetime dimensional effective theory. The model possesses an anti-symmetric K matrix resembling that of hierarchical quantum Hall states. The gauge charges are conserved in sub-dimensional manifolds which ensures the fractonic behavior. The construction extends to any lattice fracton model built from commuting projectors and with tensor products of spin-1/2 degrees of freedom at the sites.

Unitarization Technics in Hadron Physics with Historical Remarks

We review a series of unitarization techniques that have been used during the last decades, many of them in connection with the advent and development of current algebra and later of Chiral Perturbation Theory. Several methods are discussed like the generalized effective-range expansion, K-matrix approach, Inverse Amplitude Method, Padé approximants and the N / D method. More details are given for the latter though. We also consider how to implement them in order to correct by final-state interactions. In connection with this some other methods are also introduced like the expansion of the inverse of the form factor, the Omnés solution, generalization to coupled channels and the Khuri-Treiman formalism, among others.