For an essentially normal operator$T$, it is shown that there exists a unilateral shift of multiplicity$m$in$C$^{$*$}$(T)$if and only if γ($T$)≠0 and γ$(T)/m$. As application, we prove that the essential commutant of a unilateral shift and that of a bilateral shift are not isomorphic as$C$^{$*$}-algebras. Finally, we construct a natural$C$^{$*$}-algebra ε + ε_{*}on the Bergman space$L$_{$a$}^{$2$}($B$_{$n$}), and show that its essential commutant is generated by Toeplitz operators with symmetric continuous symbols and all compact operators.

The nonlinear asymptotic stability of Lane-Emden solutions is proved in this paper for spherically symmetric motions of viscous gaseous stars with the density dependent shear and bulk viscosities which vanish at the vacuum, when the adiabatic exponent γ lies in the stability regime (4/3, 2), by establishing the global-in-time regularity uniformly up to the vacuum boundary for the vacuum free boundary problem of the compressible Navier-Stokes-Poisson systems with spherical symmetry, which ensures the global existence of strong solutions capturing the precise physical behavior that the sound speed is C1/2-Hölder continuous across the vacuum boundary, the large time asymptotic uniform convergence of the evolving vacuum boundary, density and velocity to those of Lane-Emden solutions with detailed convergence rates, and the detailed large time behavior of solutions near the vacuum boundary. Those uniform convergence are of fundamental importance in the study of vacuum free boundary problems which are missing in the previous results for global weak solutions. Moreover, the results obtained in this paper apply to much broader cases of viscosities than those in Fang and Zhang (Arch Ration Mech Anal 191:195–243, 2009) for the theory of weak solutions when the adiabatic exponent γ lies in the most physically relevant range. Finally, this paper extends the previous local-in-time theory for strong solutions to a global-in-time one.

In this paper, we prove the mirror symmetry conjecture between the Saito–Givental theory of exceptional unimodular singularities on the Landau–Ginzburg B-side and the Fan–Jarvis– Ruan–Witten theory of their mirror partners on the Landau–Ginzburg A-side. On the B-side, we develop a perturbative method to compute the genus-0 correlation functions associated to the primitive forms. This is applied to the exceptional unimodular singularities, and we show that the numerical invariants match the orbifold-Grothendieck–Riemann–Roch and WDVV calculations in FJRW theory on the A-side. The coincidence of the full data at all genera is established by reconstruction techniques. Our result establishes the first examples of LG-LG mirror symmetry for weighted homogeneous polynomials of central charge greater than one (i.e. which contain negative degree deformation parameters).

We show that the classical fourth order accurate compact finite difference scheme with high order strong stability preserving time discretizations for convection diffusion problems satisfies a weak monotonicity property, which implies that a simple limiter can enforce the bound- preserving property without losing conservation and high order accuracy. Higher order accurate compact finite difference schemes satisfying the weak monotonicity will also be discussed.

Genome comparison is a vital research area of bioinformatics. For large-scale genome comparisons, the Multiple Sequence Alignment (MSA) methods have been impractical to use due to its algorithmic complexity. In this study, we propose a novel alignment-free method based on the one-to-one correspondence between a DNA sequence and its complete central moment vector of the cumulative Fourier power and phase spectra. In addition, the covariance between the four nucleotides in the power and phase spectra is included. We use the cumulative Fourier power and phase spectra to define a 28-dimensional vector for each DNA sequence. Euclidean distances between the vectors can measure the dissimilarity between DNA sequences. We perform testing with datasets of different sizes and types including simulated DNA sequences, exon-intron and complete genomes. The results show that our method is more accurate and efficient for performing hierarchical clustering than other alignment-free methods and MSA methods.