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We analyze the photometric data and spectroscopic data that collect on the $\delta$ Scuti star AE UMa. The fundamental and the first overtone frequencies are confirmed as $f_{0} = 11.62560$\,c\,d$^{-1}$ and $f_{1} = 15.03124$\,c\,d$^{-1}$, respectively, from the frequency content by analyzing of the 40 nights light curve spanning from 2009 to 2012. Additionally, another 37 frequencies are identified as either the harmonics or the linear combinations of the fundamental and the first overtone frequencies, among which 25 are newly detected. The rate of period change of the fundamental mode is determined as $(1/P_{0})(dP_{0}/dt) = 5.4(\pm 1.9) \times 10^{-9}$\,yr$^{-1}$ as revealed from the $O-C$ diagram based on the 84 newly determined times of maximum light combined with those derived from the literature. The spectroscopic data suggests that AE UMa is a population\,I $\delta$ Scuti star. With these physical properties, we perform theoretical explorations based on the stellar evolution code MESA on this target, considering that the variation of pulsation period is caused by secular evolutionary effects. We finally constraint the AE UMa with the physical parameters as: the mass of $1.805 \pm 0.055\ M_{\odot}$, the radius of $1.647 \pm 0.032 \times 10^{11}$\,cm, the luminosity of $1.381 \pm 0.048~(\log L/L_{\odot})$ and the age of $1.055 \pm 0.095 \times 10^{9}$\,yr. AE UMa can be the (Pop. I) $\delta$ Scuti star that locates just after the second turn-off of its evolutional track leaving the main sequence, a star in the phase of the Hertzsprung Gap with a helium core and a hydrogen-burning shell., Comment: 36 pages, 12 figures. arXiv admin note: text overlap with arXiv:1301.0319 by other authors

In this paper, we develop a theory of redshift distortion of the galaxy power spectrum in the discrete wavelet transform (DWT) representation. Because the DWT power spectrum is dependent of both the scale and shape (configuration) of the decomposition modes, it is sensitive to distortion of shape of the field. On the other hand, the redshift distortion causes a shape distortion of distributions in real space with respect to redshift space. Therefore, the shape-dependent DWT power spectrum is useful to detect the effect of redshift distortion. We first established the mapping between the DWT power spectra in redshift and real space. The mapping depends on the redshift distortion effects of (1) bulk velocity, (2) selection function and (3) pairwise peculiar velocity. We then proposed $\beta$-estimators using the DWT off-diagonal power spectra. These $\beta$-estimators are model-free even when the non-linear redshift distortion effect is not negligible. Moreover, these estimators do not rely on the assumption of whether the pairwise velocity dispersion being scale-dependent. The tests with N-body simulation samples show that the proposed $\beta$-estimators can yield reliable measurements of $\beta$ with about 20% uncertainty for all popular dark matter models. We also develop an algorithm for reconstruction of the power spectrum in real space from the redshift distorted power spectrum. The numerical test also shows that the real power spectrum can be well recovered from the redshift distorted power spectrum., Comment: AAS LaTeX file, 28 pages, 7 figures included, accepted for publication in ApJ

In this paper, we develop the method of analyzing the velocity field of cosmic matter with a multiresolution decomposition. This is necessary in calculating the redshift distortion of power spectrum in the discrete wavelet transform (DWT) representation. We show that, in the DWT analysis, the velocity field can be described by discrete variables, which are given by assignment of the number density and velocity into the DWT modes. These DWT variables are complete and not redundant. In this scheme, the peculiar velocity and pairwise velocity of galaxies or particles are given by field variables. As a consequence, the velocity dispersion (VD) and pairwise velocity dispersion (PVD) are no longer measured by number-counting or pair-counting statistic, but with the ensemble of the field variables, and therefore, they are free from the bias due to the number-counting and pair-counting. We analyzed the VD and PVD of the velocity fields given by the N-body simulation for models of the SCDM, $\tau$CDM and $\Lambda$CDM. The spectrum (scale-dependence) of the VD and PVD show that the length scale of the two-point correlation of the velocity field is as large as few tens h$^{-1}$ Mpc. Although the VD and PVD show similar behavior in some aspects, they are substantially different from each other. The VD-to-PVD ratio shows the difference between the scale-dependencies of the VD and PVD. More prominent difference between the VD and PVD is shown by probability distribution function. The one-point distribution of peculiar velocity is approximately exponential, while the pairwise velocity's is lognormal, i.e. of long tail. This difference indicates that the cosmic velocity field is typically intermittent., Comment: AAS LaTeX file, 46 pages, 15 figures included, accepted for publication in ApJ

The power spectrum estimator based on the discrete wavelet transform (DWT) for 3-dimensional samples has been studied. The DWT estimator for multi-dimensional samples provides two types of spectra with respect to diagonal and off-diagonal modes, which are very flexible to deal with configuration-related problems in the power spectrum detection. With simulation samples and mock catalogues of the Las Campanas redshift survey (LCRS), we show (1) the slice-like geometry of the LCRS doesn't affect the off-diagonal power spectrum with ``slice-like'' mode; (2) the Poisson sampling with the LCRS selection function doesn't cause more than 1-$\sigma$ error in the DWT power spectrum; and (3) the powers of peculiar velocity fluctuations, which cause the redshift distortion, are approximately scale-independent. These results insure that the uncertainties of the power spectrum measurement are under control. The scatter of the DWT power spectra of the six strips of the LCRS survey is found to be rather small. It is less than 1-$\sigma$ of the cosmic variance of mock samples in the wavenumber range $0.1 < k < 2$ h Mpc$^{-1}$. To fit the detected LCRS diagonal DWT power spectrum with CDM models, we find that the best-fitting redshift distortion parameter $\beta$ is about the same as that obtained from the Fourier power spectrum. The velocity dispersions $\sigma_v$ for SCDM and $\Lambda$CDM models are also consistent with other $\sigma_v$ detections with the LCRS. A systematic difference between the best-fitting parameters of diagonal and off-diagonal power spectra has been significantly measured. This indicates that the off-diagonal power spectra are capable of providing information about the power spectrum of galaxy velocity field., Comment: AAS LaTeX file, 41 pages, 10 figures included, accepted for publication in ApJ