We developed a Friedrichs-model-like scheme in studying the hadron resonance phenomenology and present that the hadron resonances might be regarded as the Gamow states produced by a Hamiltonian in which the bare discrete state is described by the result of usual quark potential model and the interaction part is described by the quark pair creation model. In a one-parameter calculation, the X(3862), X(3872), and X(3930) state could be simultaneously produced with a quite good accuracy by coupling the three P-wave states, χc2(2P ), χc1(2P ), χc0(2P ) predicted in the Godfrey-Isgur model to the DD, DD * , D * D * continuum states. At the same time, we predict that the hc(2P ) state is at about 3902 MeV with a pole width of about 54 MeV. In this calculation, the X(3872) state has a large compositeness. This scheme may shed more light on the long-standing problem about the general discrepancy between the prediction of the quark model and the observed values, and it may also provide reference for future search for the hadron resonance state.PACS numbers: 12.39. Jh, 13.25.Gv, 13.75.Lb, 11.55.Fv As regards the charmonium spectrum above the openflavor thresholds, general discrepancies between the predicted masses in the quark potential model and the observed values have been highlighted for several years. Typically, among the P-wave n 2s+1 L J = 2 3 P 2 , 2 3 P 1 , 2 3 P 0 , and 2 1 P 1 states, the X(3930), discovered by the Belle Collaboration [1], is now assigned to χ c2 (2P ) charmonium state though its mass is about 50 MeV lower than the prediction in the quark potential model [2][3][4]. The properties of the other P-wave states have not been firmly determined yet. The X(3872) was first observed in the B ± → K ± J/ψπ + π − by the Belle Collaboration in 2003 [5]. Although its quantum number is 1 ++ , the same as the χ c1 (2P ), the pure charmonium interpretation was soon given up for the difficulties in explaining its decays. The pure molecular state explanation of X(3872) also encounters difficulties in understanding its radiative decays. So its nature remains to be obscure up to now. As for the χ c0 (2P ) state, the X(3915) is assigned to it several years ago, but this assignment is questioned for the mass splitting between χ c2 (2P ) and χ c0 (2P ), and its dominant decay mode [6,7]. In Ref.[8], analyses of the angular distribution of X(3915) to the final leptonic and pionic states also support the possibility of being a 2 ++ state, which means that it might be the same tensor state as the X(3930). Very recently, the Belle Collaboration announced a new result about the signal of X(3862) which could be a candidate for the χ c0 (2P ) [9]. The 2 1 P 1 state has not been discovered yet. These puzzles have been discussed exhaustively in the literatures (see refs. [10][11][12] for example), but a consistent description is still missing.In this paper, we adopt the idea of Gamow states and the solvable extended Friedrichs model developed recently [13][14][15], usually discussed in the pure mathematical physics literatur...
The production of K * (892) 0 and φ(1020) in pp collisions at √ s = 8 TeV was measured by using Run 1 data collected by the ALICE collaboration at the CERN Large Hadron Collider (LHC). The p T-differential yields d 2 N/dyd p T in the range 0 < p T < 20 GeV/c for K * 0 and 0.4 < p T < 16 GeV/c for φ have been measured at midrapidity, |y| < 0.5. Moreover, improved measurements of the K * 0 (892) and φ(1020) at √ s = 7 TeV are presented. The collision energy dependence of p T distributions, p T-integrated yields, and particle ratios in inelastic pp collisions are examined. The results are also compared with different collision systems. The values of the particle ratios are found to be similar to those measured at other LHC energies. In pp collisions a hardening of the particle spectra is observed with increasing energy, but at the same time it is also observed that the relative particle abundances are independent of the collision energy. The p T-differential yields of K * 0 and φ in pp collisions at √ s = 8 TeV are compared with the expectations of different Monte Carlo event generators.
We study verifiable sufficient conditions and computable performance bounds for sparse recovery algorithms such as the Basis Pursuit, the Dantzig selector and the Lasso estimator, in terms of a newly defined family of quality measures for the measurement matrices. With high probability, the developed measures for subgaussian random matrices are bounded away from zero as long as the number of measurements is reasonably large. Comparing to the restricted isotropic constant based performance analysis, the arguments in this paper are much more concise and the obtained bounds are tighter.Numerical experiments are presented to illustrate our theoretical results. Index TermsCompressive sensing; q-ratio sparsity; q-ratio constrained minimal singular values; Convex-concave procedure. z =0, z 2 1 / z 2 2 ≤s Az 2 z 2 and obtained the error 2 bounds in terms of this quality measure of the measurement matrix. Similarly, in [16], the authors definedz ∞ with · ♦ denoting a general norm, and derived the performance bounds on the ∞ norm of the recovery error vector based on this quality measure. This kind of measures has also been used in establishing results for block sparsity recovery [17] and low-rank matrix recovery [18]. In this paper we generalize these two quantities to a more general quantity called q-ratio CMSV with 1 < q ≤ ∞, and establish the performance bounds for both q norm and 1 norm of the reconstruction error. A. ContributionsOur contribution mainly has four aspects. First, we proposed a sufficient condition based on a q-ratio sparsity level for the exact recovery using 1 minimization in the noise free case, and designed a convexconcave procedure to solve the corresponding non-convex problem, leading to an acceptable verification algorithm. Second, we introduced q-ratio CMSV and derived concise bounds on both q norm and 1 norm of the reconstruction error for the Basis Pursuit (BP) [19], the Dantzig selector (DS) [20], and the Lasso estimator [21] in terms of q-ratio CMSV. We established the corresponding stable and robust recovery results involving both sparsity defect and measurement error. Third, we demonstrated that for subgaussion random matrices, the q-ratio CMSVs are bounded away from zero with high probability, as long as the number of measurement is large enough. Finally, we presented algorithms to compute the q-ratio CMSV for an arbitrary measurement matrix, and studied the effects of different parameters on the proposed q-ratio CMSV. Moreover, we illustrated that q-ratio CMSV based bound is tighter than the RIC based one. B. Organization and NotationsThe paper is organized as follows. In Section II, we present the definitions of q-ratio sparsity and q-ratio CMSV, and give a sufficient condition for unique noiseless recovery based on the q-ratio sparsity and an inequality for the q-ratio CMSV. In Section III, we derive performance bounds on both q norm and 1 norm of the reconstruction errors for several convex recovery algorithms in terms of q-ratio CMSVs. In Section IV, we demonstrate that the subgaussi...
Explicitly using the block structure of the unknown signal can achieve better recovery performance in compressive censing. An unknown signal with block structure can be accurately recovered from underdetermined linear measurements provided that it is sufficiently block sparse. However, in practice, the block sparsity level is typically unknown. In this paper, we consider a soft measure of block sparsity, kα(x) = ( x 2,α/ x 2,1)and propose a procedure to estimate it by using multivariate isotropic symmetric α-stable random projections without sparsity or block sparsity assumptions. The limiting distribution of the estimator is given. Some simulations are conducted to illustrate our theoretical results.
In this paper, we propose a general scale invariant approach for sparse signal recovery via the minimization of the q-ratio sparsity. When 1 < q ≤ ∞, both the theoretical analysis based on q-ratio constrained minimal singular values (CMSV) and the practical algorithms via nonlinear fractional programming are presented. Numerical experiments are conducted to demonstrate the advantageous performance of the proposed approaches over the state-of-the-art sparse recovery methods.
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