On Kernel Formulas and Dispersionless Hirota Equations

We rederive dispersionless Hirota equations of the dispersionless Toda hierarchy from the method of kernel formula provided by Carroll and Kodama. We then apply the method to derive dispersionless Hirota equations of the extended dispersionless BKP(EdBKP) hierarchy proposed by Takasaki. Moreover, we verify associativity equations (WDVV equations) in the EdBKP hierarchy from dispersionless Hirota equations and give a realization of associative algebra with structure constants expressed in terms of residue formula.Comment: 30 pages, minor corrections, references adde

Determining the strange and antistrange quark distributions of the nucleon

The difference between the strange and antistrange quark distributions, \delta s(x)=s(x)-\sbar(x), and the combination of light quark sea and strange quark sea, \Delta (x)=\dbar(x)+\ubar(x)-s(x)-\sbar(x), are originated from non-perturbative processes, and can be calculated using non-perturbative models of the nucleon. We report calculations of $\delta s(x)$ and $\Delta(x)$ using the meson cloud model. Combining our calculations of $\Delta(x)$ with relatively well known light antiquark distributions obtained from global analysis of available experimental data, we estimate the total strange sea distributions of the nucleon.Comment: 4 pages, 3 figures; talk given by F.-G. at QNP0

Optimal Topological Test for Degeneracies of Real Hamiltonians

We consider adiabatic transport of eigenstates of real Hamiltonians around loops in parameter space. It is demonstrated that loops that map to nontrivial loops in the space of eigenbases must encircle degeneracies. Examples from Jahn-Teller theory are presented to illustrate the test. We show furthermore that the proposed test is optimal.Comment: Minor corrections, accepted in Phys. Rev. Let

Uncertainties of predictions from parton distribution functions II: the Hessian method

We develop a general method to quantify the uncertainties of parton distribution functions and their physical predictions, with emphasis on incorporating all relevant experimental constraints. The method uses the Hessian formalism to study an effective chi-squared function that quantifies the fit between theory and experiment. Key ingredients are a recently developed iterative procedure to calculate the Hessian matrix in the difficult global analysis environment, and the use of parameters defined as components along appropriately normalized eigenvectors. The result is a set of 2d Eigenvector Basis parton distributions (where d=16 is the number of parton parameters) from which the uncertainty on any physical quantity due to the uncertainty in parton distributions can be calculated. We illustrate the method by applying it to calculate uncertainties of gluon and quark distribution functions, W boson rapidity distributions, and the correlation between W and Z production cross sections.Comment: 30 pages, Latex. Reference added. Normalization of Hessian matrix changed to HEP standar

Higher rank numerical ranges of normal matrices

The higher rank numerical range is closely connected to the construction of quantum error correction code for a noisy quantum channel. It is known that if a normal matrix $A \in M_n$ has eigenvalues $a_1, \..., a_n$, then its higher rank numerical range $\Lambda_k(A)$ is the intersection of convex polygons with vertices $a_{j_1}, \..., a_{j_{n-k+1}}$, where $1 \le j_1 < \... < j_{n-k+1} \le n$. In this paper, it is shown that the higher rank numerical range of a normal matrix with $m$ distinct eigenvalues can be written as the intersection of no more than $\max\{m,4\}$ closed half planes. In addition, given a convex polygon ${\mathcal P}$ a construction is given for a normal matrix $A \in M_n$ with minimum $n$ such that $\Lambda_k(A) = {\mathcal P}$. In particular, if ${\mathcal P}$ has $p$ vertices, with $p \ge 3$, there is a normal matrix $A \in M_n$ with $n \le \max\left\{p+k-1, 2k+2 \right\}$ such that $\Lambda_k(A) = {\mathcal P}$.Comment: 12 pages, 9 figures, to appear in SIAM Journal on Matrix Analysis and Application